TY - JOUR
AU - Opp, Christian, C
AB - Abstract I develop a model of venture capital (VC) intermediation that quantitatively explains central empirical facts about VC activity and can evaluate its macroeconomic relevance. The impact of VC-backed innovations is significantly larger than suggested by observed aggregate venture exit valuations, even after accounting for large exposures to systematic and uninsurable idiosyncratic risks. The risk properties of venture capital play a quantitatively important role in both explaining empirical regularities and shaping the value of ventures’ contributions to economic growth. The model is analytically tractable and yields exact solutions, despite the presence of matching frictions, imperfect risk sharing, and endogenous growth. Received January 16, 2018; editorial decision November 7, 2018 by Editor Stijn Van Nieuwerburgh. Over the past few decades, venture capital funds have emerged as prominent financial intermediaries facilitating the entry of young innovative start-ups, having backed over 60% of firms undertaking initial public offerings (IPOs) in the United States (Kaplan and Lerner 2010). Moreover, empirical studies find that venture capital has had significant influence on patented inventions.1 Notwithstanding this remarkable record, the VC industry also has gone through dramatic boom-bust cycles that appear anomalous to many observers. Gupta (2000) and Gompers and Lerner (2003) argue, for example, that extreme movements in VC activity are a symptom of irrational overreaction, resulting in periods in which too many firms are funded, followed by ones in which not enough firms have access to capital. Despite VC-backed companies’ salience in the economy, the literature lacks to date a framework that explains central empirical facts about the magnitude and cyclicality of VC activity and can quantitatively evaluate its macroeconomic impact. In this paper, I aim to take a step toward developing a unified macro-finance framework that fills this void. The proposed model features three building blocks that are essential to achieving this goal: (1) a macroeconomic environment in which innovations by VC-backed start-ups and regular firms can generate endogenous contributions to aggregate growth; (2) a micro-founded representation of the VC industry that captures central features of the industry’s decentralized market structure and economic function; and (3) an environment with empirically plausible asset pricing implications that allows appropriately interpreting information encoded in market prices and reflects agents’ attitudes toward growth and risk. The model not only accounts for the pricing of aggregate risk but also sheds light on the implications of extreme idiosyncratic income risk typical for venture capital. The framework draws on empirical data on VC intermediation, asset prices, and macroeconomic dynamics to inform the model’s parameters. The calibrated model successfully matches the magnitudes and salient dynamic properties of aggregate VC fund commitments, VC-backed IPOs and mergers and acquisitions (M&As), VC fund fee revenues, and payoffs to entrepreneurs, suggesting that these data are not necessarily evidence of investor irrationality. Both imperfect competition in product and venture capital markets and the pricing of macroeconomic risk exposures materially contribute toward quantitatively explaining the observed magnitudes and dynamics of VC intermediation. Yet the analysis also reveals that market prices of VC-backed IPOs and M&As miss a substantial component of the total value agents assign to VC-backed innovations, as firm valuations reflect only producers’ rents. In the presented model, innovations by new start-ups and regular firms can improve the qualities of goods produced in a cross-section of industries, creating an endogenous source of macroeconomic growth. The patent system allows firms to profit from their own innovations, but those profits are compromised when competitors develop better products. Ideas for such new products have heterogenous success prospects: whereas any firm can implement ideas of regular quality, costly VC fund intermediation is needed to reach and support entrepreneurs with superior ideas. In particular, VC funds have to recruit fund managers with scarce human capital to match with these entrepreneurs in a decentralized market and to support the implementation of their ideas. VC funds in the model thus can obtain access to “proprietary deal flow.”2 Under this market structure, VC funds can generate economic profits (“alpha”) that fund managers extract via fees. While VC intermediation leads to more potent innovative activity, the scale of funding varies in the cross-section and over time, depending on the private rents that are attainable. Intermediation may even break down in times when rents cannot cover the fees needed to compensate fund managers’ human capital. The rents from VC intermediation, in turn, depend on not only current innovative productivity in a given industry but also the joint dynamics of competition and macroeconomic conditions. In particular, competitive forces imply that persistent negative shocks to venture funding in aggregate downturns can be partially good news for ventures that obtained funding during a preceding boom and successfully entered the market. Although dividend growth suffers when aggregate growth decreases, these firms benefit significantly from persistent declines in the funding and entry of new competitors. Venture investments thus can become hedges against macroeconomic low-frequency risks associated with persistent slow-downs in technological progress, especially in those industries where entry is highly procyclical.3 The associated impact on risk premiums has first-order effects: low discount rates in booms help explain extreme increases in venture investment as observed during the internet boom of the late 1990s, as well as a negative empirical relation between VC fund inflows and future returns.4 Moreover, as the model captures both risk-based and non risk-based factors affecting average returns to VC investments, it provides a cautionary note to empirical studies relying on standard risk-based asset pricing models in the context of venture capital, adding to other well-known challenges with VC data. Within this framework, I evaluate VC-backed innovations’ macroeconomic impact. While these innovations contribute on average a few basis points to consumption growth, contributions can fluctuate dramatically over time, and have an economically significant effect on agents’ expected lifetime utility. Their value is equivalent to about 1.3% of lifetime consumption, a number more than twice as large as the average aggregate market value of venture exits (IPOs and acquisitions). While this value can be compared to exit valuations, it is a more comprehensive measure, as it encodes innovations’ full impact on agents’ lifetime consumption rather than merely the effect on producers’ rents. The analysis also reveals that almost half of this value is due to VC intermediation’s efficiency advantages relative to regular firms’ research and development (R&D) investment. The magnitudes of these results are large relative to existing estimates quantifying the importance of other macroeconomic phenomena (e.g., the costs of business cycles; see Lucas 2003), highlighting the relevance of policies affecting the VC industry.5 Moreover, accounting for VC investments’ risk properties and agents’ risk aversion is indeed of first-order importance for these quantitative evaluations. If venture investments created the same average growth contributions but without any aggregate fluctuations, their value would be approximately twice as large as the baseline estimate. In practice, uninsurable idiosyncratic income risk is an additional important source of risk affecting agents involved in the venture capital process. Although most entrepreneurs fail, a handful obtain billions of dollars in the case of a successful venture exit. Given this extremely dispersed empirical distribution, the literature has found that even mildly risk-averse entrepreneurs would assign a certainty equivalent value close to zero to signing a new venture capital contract (see Hall and Woodward 2010). To account for these uninsurable risks, I examine the model’s implications when entrepreneurs and VC fund managers have to stay exposed to the idiosyncratic risk associated with the performance-sensitive compensation they obtain. Whereas solving economies with heterogeneous agents and imperfect risk sharing generally requires approximation techniques and/or computationally intensive methods, the presented production economy provides a special environment that remains tractable and yields exact solutions, contributing to the literature studying labor income risk in incomplete markets models (Bewley 1986; Huggett 1993; Aiyagari 1994). This tractability follows from the fact that the model mirrors the above-discussed special feature of venture capital data; idiosyncratic venture risks are so extreme that mildly risk-averse agents assign effectively zero incremental certainty equivalent value to the highly uncertain payoffs they might receive from their own venture capital opportunities. Facing this type of risk, agents involved in venture capital optimally choose the same portfolio weights in tradable assets and the same consumption-to-tradable-wealth ratios as all other agents. Yet, even when their own compensation claims feature these extreme uninsurable risks, agents still assign significant value to VC-backed innovations’ aggregate impact, considering it as equivalent to 1.2% of their lifetime consumption. This percentage value does not much differ from the baseline estimate of 1.3%, because, in either case, only a minor fraction of VC-backed innovations’ aggregate impact is internalized through risky venture capital compensation. To my best knowledge, this is the first paper to develop a quantitative dynamic general equilibrium model that explains central empirical facts about the magnitude and cyclicality of venture capital activity and evaluates its impact on the macroeconomy. A large literature in macroeconomics and economic growth addresses technological change but abstracts from venture capital intermediation and the pricing of risk (see, e.g., Jovanovic and MacDonald 1994; Laitner and Stolyarov 2003, 2004). The most related papers in this literature are those by Greenwood and Jovanovic (1999) and Hobijn and Jovanovic (2001), who link expectations about future increases in the rate of creative destruction during technological revolutions to initial stock market declines. In my model, persistent variation in creative destruction also generates the corresponding reverse effect that expectations about future declines in entry amplify preceding booms. In particular, my model reveals that procyclical variation in entry materially reduces venture investments’ risk premiums, a result that would not be captured by a model that abstracts from the pricing of risk. For similar reasons, my paper deviates from the existing literature on finance and growth, such as King and Levine (1993b), who consider project selection in a quality-ladder model with constant aggregate growth (see also Greenwood and Jovanovic 1990; Bencivenga and Smith 1991; King and Levine 1993a; Levine and Zervos 1998; Rajan and Zingales 1998). This literature does not provide a model that quantitatively captures VC activity and does not address the key objects of interest of this study, such as cyclical variation in VC intermediation, the impact of aggregate risk and risk pricing, and the performance-sensitive compensation of entrepreneurs and VC managers. Finally, an influential literature analyzes the impact of (intermediary) leverage on macroeconomic activity and asset prices (see, e.g., Bernanke and Gertler 1989; Kiyotaki and Moore 1997; He and Krishnamurthy 2012, 2013; Brunnermeier and Sannikov 2014). This literature does not aim to address empirical facts about the venture capital industry and the mechanics of VC funds, which are unlevered financial institutions. With regards to its implications for risk premiums my paper is more closely related to the strand of literature that analyzes general equilibrium production-based models to examine the time-series and cross-sectional properties of returns (see, e.g., Gomes, Kogan, and Zhang 2003). Several papers in this literature address the asset pricing implications of technological innovation, although the papers in this literature do not consider venture capital intermediation and its effects on returns to investors and macroeconomic growth. In Garleanu, Kogan, and Panageas (2012) and Kogan, Papanikolaou, and Stoffman (2015) displacement risk arises as a priced risk factor in economies with imperfect risk sharing and can help explain empirical patterns in asset returns, such as the value premium. While my paper also analyzes the effects of imperfect risk sharing, displacement risk in my setting affects discount rates primarily through its impact on firms’ exposures to low-frequency risks associated with persistent changes in aggregate growth. Risk prices are, however, also affected by uninsurable idiosyncratic venture income risk, that is, the environment does not satisfy the irrelevance conditions laid out by Krueger and Lustig (2010) (in terms of both preferences and the joint distributional properties of idiosyncratic and aggregate shocks). Kung and Schmid (2015) consider an expanding varieties model similar to Comin and Gertler (2006) to analyze the asset pricing implications of persistent movements in productivity in the presence of recursive preferences. The authors provide empirical evidence for the existence of innovation-driven low-frequency movements in aggregate growth rates. Pastor and Veronesi (2009) show that the adoption of revolutionary technologies is initially associated with low discount rates, but that discount rates rise as the nature of risk changes from idiosyncratic to systematic, eventually leading to falling asset prices. In my model, low discount rates in booms also increase valuations and VC investment, consistent with empirical evidence that funding booms predict low returns (see footnote 4). Further, my paper speaks to the equilibrium dynamics of IPO volume and is thus related to theories of IPO waves (see Jovanovic and Rousseau 2001; Pastor and Veronesi 2005). Although the framework proposed in this paper captures a multitude of economic forces affecting venture capital activity and its linkages to the macroeconomy, it naturally also abstracts from a variety of channels that can create cyclicality, such as entrepreneurs’ self-fulfilling expectations about the implementation of innovations (Shleifer 1986), rational herd behavior (Scharfstein and Stein 1990), endogenous timing and learning (Gul and Lundholm 1995), firm-specific learning-by-doing (Stein 1997), high investment in new technologies due to incentives to learn about the curvature of the production function (Johnson 2007), low expected returns and high investment in risky technologies in the presence of relative wealth concerns (DeMarzo, Kaniel, and Kremer 2007), and complementarities in investors’ information production (Dow, Goldstein, and Guembel 2011). Finally, my paper abstracts from behavioral frictions, which might also contribute to the cyclical behavior of VC activity. 1. The Economy The economy consists of a cross-section of industries that produce intermediate goods. These goods are aggregated to a final consumption good. Innovations generate quality improvements in intermediate goods, creating an endogenous source of consumption growth. Four types of agents with distinct skills assist in the process of generating innovations and producing goods. These agents are referred to as entrepreneurs, network VCs, white-collar labor, and blue-collar labor. Entrepreneurs of heterogeneous productivity generate ideas for new innovations. Whereas any firm can implement ideas of regular quality, VC fund intermediation is needed to realize the superior ideas generated by the most productive entrepreneurs. Network VCs assist in connecting VC funds to the most productive entrepreneurs. White-collar labor is required by any firm implementing innovative ideas and by any VC fund aiming to reach and support the best entrepreneurs. Finally, blue-collar labor is used to produce intermediate goods based on existing innovations. 1.1 Agents and preferences Let the subscripts |$S \in \{E,V,W,B\}$| index entrepreneurs, network VCs, white-collar labor, and blue-collar labor, respectively. Each skill group |$S$| consists of a double-continuum of agents. As industries form a single-continuum, specifying double-continuums of agents ensures that each agent is atomistic relative to an industry. An agent with a skill |$S$| is uniquely identified by a double index |$(i_{S}', i_{S}'') \in [0,1]^2$|⁠. The set of all agents in the economy is given by |$\Phi \equiv \{E,V,W,B\} \times [0,1]^2$|⁠. At date |$0$|⁠, agents own identical shares of existing firms and are endowed with their skill. The presented setup can be thought of as the continuous-time counterpart of a discrete-time economy, where a period lasts for an interval of |$\Delta t$|⁠, and where I consider the limiting case |$\Delta t \rightarrow 0$|⁠. In this limit, agents’ preferences are described by stochastic differential utility (Duffie and Epstein 1992a,b), which is a continuous-time version of the recursive preferences of Epstein and Zin (1989) and Weil (1990). These preferences allow specifying risk aversion and elasticity of intertemporal substitution separately, which is key to matching asset pricing moments that are important for the analysis. Agents have homogenous preferences but will generally have differing consumption processes in equilibrium.6 The utility index over a consumption process |$\{C_{\tau}\}_{\tau=t}^{\infty}$| is defined as $ \begin{equation} J_{t}=\mathbb{E}_{t}\left[ \int_{t}^{\infty} m\left( C_{\tau} ,J_{\tau} \right) d\tau \right]\!. \end{equation} $ (1) Here, the function |$m\left( C,J\right)$| is a normalized aggregator of current consumption and continuation utility: $ \begin{equation} m\left( C,J\right) =\frac{\beta} {\rho} \left( \left( (1 - \gamma) J\right) ^{1- \frac{\rho} {1 - \gamma} }C^{\rho} -(1 - \gamma) J\right)\!, \end{equation} $ (2) with |$\rho \equiv 1-\frac{1}{\psi}$|⁠, where |$\beta >0$| is the rate of time preference, |$\gamma >0$| is the coefficient of relative risk aversion, and |$\psi >0$| is the elasticity of intertemporal substitution. 1.2 Labor markets Blue-collar labor and white-collar labor are supplied inelastically and obtain time-varying equilibrium wage rates denoted by |$w_{B}$| and |$w_{W}$|⁠, respectively. Each agent with these skills is endowed with one unit of labor, such that the aggregate supply of each type of labor is one. Section 1.4 will describe the market structure determining entrepreneurs’ and network VCs’ compensation. 1.3 Production and development of intermediate goods In this section, I describe the production of intermediate goods based on existing innovations and the heterogeneous success prospects of entrepreneurs’ ideas in generating new innovations. 1.3.1 Intermediate good production Intermediate good varieties, also referred to as industries or product lines, are indexed by |$v\in \Psi = \left[0,1\right]$|⁠. A firm that owns a patent for an intermediate good needs to employ one unit of blue-collar labor to manufacture one unit of this intermediate good. 1.3.2 Innovations improving product quality As typical for Schumpeterian growth models, innovations lead to quality improvements for intermediate goods (e.g., Grossman and Helpman 1991). Let |$\{M \left(v,t\right) \}_{0}^{t}$| be a Poisson process that keeps track of the number of innovations that occurred in industry |$v$| between time 0 and time |$t$|⁠. The intensity of this Poisson process is denoted by |$h(v,t)$| and is determined in equilibrium, as further described below. The quality of the best intermediate good available in industry |$v$| at time |$t$| is given by $ \begin{equation} q\left( v,t\right) =\kappa ^{M\left( v,t\right)} q\left( v,0\right)\!,~~\forall~~v\text{ and}~~t, \end{equation} $ (3) where |$\kappa >1$| and |$q\left( v,0\right) \in \mathbb{R}_{+}$|⁠. The “quality ladder” specified in Equation (3) implies that each innovation leads to a proportional quality increase by a factor |$\kappa$|⁠. 1.3.3 Generating and implementing innovative ideas At any point in time, an entrepreneur can generate either regular or superior innovative ideas in a given industry |$v$|⁠. Implementing superior ideas leads to a new innovation with a higher propensity than implementing regular ideas does. While the qualities of ideas are heterogenous, entrepreneurs can generate any quantity of ideas. At any point in time and in each industry |$v$|⁠, only one specific entrepreneur creates superior ideas, the most productive entrepreneur. The process determining these entrepreneurs is described below. Implementing a unit of regular ideas requires one unit of white-collar labor. Implementing a unit of superior ideas requires not only one unit of white-collar labor but also VC fund intermediation: a VC fund has to connect to the most productive entrepreneur and support the implementation of her ideas. Section 1.4 will describe this role of VC funds. Because the funding of regular ideas does not require active VC intermediation, it can be undertaken by regular firms and is not categorized as VC-backed innovative activity. 1.3.4 Process determining the most productive entrepreneurs The following stochastic process determines the most productive entrepreneur in each industry and at each point in time. Figure 1 graphically illustrates this process, while depicting a related process for network VCs’ connections, the latter of which will be described in Section 1.4. At the beginning of each period |$[t, t+ d t)$|⁠, a single-continuum slice of the double continuum of entrepreneurs obtains positive productivity shocks rendering them the most productive entrepreneurs in the single continuum of industries in the given period. Specifically, the entrepreneur selected to be the most productive entrepreneur in industry |$v$| has the index |$(i_{E}',i_{E}'')= (i_{E,t}^{*}, v )$|⁠, where |$i_{E,t}^{*}$| is an i.i.d. shock that follows a uniform distribution on the interval |$[0,1]$|⁠. Here, the random variable |$i_{E,t}^{*}$| determines which slice of the double continuum of entrepreneurs is selected in a given period, that is, it pins down the value of the first index element |$i_{E}'$| that is common to all entrepreneurs with elevated productivities. The specific industry in which each of these entrepreneurs is more productive is, in turn, determined by an entrepreneur’s second index |$i_{E}''$|⁠, which implies a constant association of each entrepreneur with a specific industry; for example, an entrepreneur with the second index element |$i_{E}''=0.3$| can only become the most productive entrepreneur in industry |$v=0.3$|⁠. Both indices, |$i_{E}''$| and |$v$|⁠, live on the unit interval |$[0,1]$|⁠, so exactly one entrepreneur is the most productive of all entrepreneurs in each industry |$v$|⁠. The described process implies that over any time interval |$\Delta t>0$|⁠, the expected number of positive shocks an entrepreneur |$(i_{E}',i_{E}'')$| receives in industry |$v=i_{E}''$| is simply |$\Delta t$|⁠. As each entrepreneur obtains elevated venture opportunities in at most a finite number of periods |$[t,t+dt)$| over any time interval |$\Delta t>0$|⁠, these opportunities are atomistic relative to the positive measure of opportunities materializing in an industry, a fact that will be relevant for privately optimal behavior. Moreover, this feature ensures that the setup remains tractable when risk sharing is imperfect, which is considered in Section 4. Figure 1 Open in new tabDownload slide Processes determining best entrepreneurs and connected network VCs The graph illustrates how specific entrepreneurs and network VCs are selected to obtain elevated venture opportunities in a given period. The two squares in the graph represent the double continuums of entrepreneurs indexed by |$(i_{E}',i_{E}'')$| and network VCs indexed by |$(i_{V}',i_{V}'')$|⁠. The vertical solid black line on the right represents the continuum of industries that are indexed by |$v$|⁠. In each period, the i.i.d. random variables |$(i_{E,t}^{*},i_{V,t}^{*}$|⁠) determine the single-continuum subsets of entrepreneurs and network VCs that obtain an elevated status. These single continuums are illustrated by vertical solid gray lines in the two squares. The horizontal black dotted arrows illustrate the constant mapping between the indices |$i_{E}''$| and |$i_{V}''$|⁠, on the one hand, and the industry indices |$v$|⁠, on the other hand. These constant mappings apply for all indices (⁠|$i_{E}'', i_{V}'', v$|⁠), such that in each period, each industry |$v$| has one best entrepreneur and one connected network VC. Figure 1 Open in new tabDownload slide Processes determining best entrepreneurs and connected network VCs The graph illustrates how specific entrepreneurs and network VCs are selected to obtain elevated venture opportunities in a given period. The two squares in the graph represent the double continuums of entrepreneurs indexed by |$(i_{E}',i_{E}'')$| and network VCs indexed by |$(i_{V}',i_{V}'')$|⁠. The vertical solid black line on the right represents the continuum of industries that are indexed by |$v$|⁠. In each period, the i.i.d. random variables |$(i_{E,t}^{*},i_{V,t}^{*}$|⁠) determine the single-continuum subsets of entrepreneurs and network VCs that obtain an elevated status. These single continuums are illustrated by vertical solid gray lines in the two squares. The horizontal black dotted arrows illustrate the constant mapping between the indices |$i_{E}''$| and |$i_{V}''$|⁠, on the one hand, and the industry indices |$v$|⁠, on the other hand. These constant mappings apply for all indices (⁠|$i_{E}'', i_{V}'', v$|⁠), such that in each period, each industry |$v$| has one best entrepreneur and one connected network VC. 1.3.5 Innovation arrivals at the industry level Before specifying the success prospects of individual ideas it is useful to define the endogenous Poisson arrival rate of innovations at the industry level. This endogenous arrival rate characterizes the intensity with which the quality index defined in Equation (3) jumps to the next-higher level. Suppose that a total measure |$n$| of efficiency-adjusted ideas are implemented in a given period in industry |$v$|⁠, where |$n$| will be defined explicitly below. The Poisson arrival rate of a new innovation in this industry is then given by $ \begin{equation} \label{eqn:JointArrivalRate} h(v,Z,n)=\theta \left(v,Z \right) n^{\eta}, \end{equation} $ (4) where |$0<\eta <1$|⁠, and where |$\theta(v,Z)$| is an exogenous industry-specific productivity process that is governed by the aggregate state variable |$Z$|⁠. The functional specification for |$h$| features diminishing returns to scale at the industry level, reflecting the notion that ideas implemented in the same industry are effectively competing for similar innovations. The state |$Z$| follows a time-homogeneous continuous time Markov chain taking values in the set |$\Omega =\left\{ 1,...,l\right\}$| with transition rates |$\lambda _{ZZ^{\prime} }$| that are collected in the generator matrix |$\Lambda$|⁠. I define counting processes |$N_{t}(Z,Z^{\prime})$| that keep track of the number of jumps between these Markov states. 1.3.6 Success prospects of superior and regular ideas Suppose that a measure |$n_{1}$| of the most productive entrepreneur’s superior ideas in industry |$v$| is implemented. The Poisson intensity with which one of these ideas succeeds is given by $ \begin{equation} \label{eq:h1} h_{1}(v,Z, n_{1})=\theta \left(v,Z\right) n_{1}^{\eta}. \end{equation} $ (5) In contrast, if a measure |$n_{2}$| of regular ideas—generated by any other entrepreneur—is implemented, one of these ventures succeeds with Poisson intensity: $ \begin{equation} \label{eq:hU} h_{2}(v,Z, n_{1},n_{2})=h(v,Z, n_{1}+\phi n_{2}) - h_{1}(v,Z, n_{1}), \end{equation} $ (6) where |$\phi \in (0,1)$| represents an efficiency discount. In case an innovation arrival generated by (6) occurs, ideas in the measure |$n_{2}$| have a uniform conditional probability of being the successful one. I define the total efficiency-adjusted quantity of ideas as |$n \equiv n_{1}+\phi n_{2}$|⁠. These success prospects for regular and superior ideas (Equations (5) and (6)) feature two key differences. First, whereas the success prospects of the most productive entrepreneur’s superior ideas, |$h_{1}$|⁠, depend only on the quantity of his own ideas that are implemented, |$n_{1}$|⁠, regular ideas’ success prospects, |$h_{2}$|⁠, depend on both |$n_{1}$| and |$n_{2}$|⁠. This difference can be interpreted as an intraperiod first-mover advantage for the most productive entrepreneur: because of decreasing marginal returns at the industry level, the most productive entrepreneur can “skim off” venture ideas with the highest marginal success rates. The second difference is the efficiency discount |$\phi$| that is applied to regular ideas. This discount implies that a given measure of superior ideas generated by the most productive entrepreneur is always more productive than an equally large measure of regular venture ideas is. 1.3.7 Patents If a firm innovates, it obtains a perpetual patent for the new intermediate good. Yet the patent system does not preclude other firms from developing new goods of even higher quality. Further, the economy features the standard restriction that firms owning patents of different qualities cannot contract to share the higher profits that could be earned through collusion.7 At any point in time, producers in any industry |$v$| compete as price-setting oligopolists with common marginal costs equal to the blue-collar wage rate |$w_{B}$|⁠. 1.4 Financing innovative activity In this section, I describe how regular and superior innovative ideas are financed. 1.4.1 Human capital needed for VC fund intermediation Implementing superior venture ideas requires that a VC fund matches with the most productive entrepreneur in a given industry and period and advises the start-up’s activities. To do so, a VC fund has to employ two types of human capital: (1) a network VC that currently has connections in the industry and (2) |$c_{\iota}$| units of white-collar labor that facilitate the process of reaching and advising the best entrepreneur. Fund managers with white collar human capital are “generalists” in the sense that their human capital also can, in principle, be employed by regular firms and in other industries. In contrast, a network VC’s connection is specific to a particular industry. Let |$\iota _{1}\left( v,t\right) \in \left\{ 0,1\right\}$| denote an indicator variable that takes the value 1 if, in equilibrium, a VC fund matches with the best entrepreneur in industry |$v$| at time |$t$| by successfully recruiting both the connected network VC and |$c_{\iota}$| units of white-collar labor. As agents require compensation commensurate with their outside opportunities, it is possible that, in equilibrium, no VC fund operates in a particular industry and period (⁠|$\iota_{1}(v,t)=0$|⁠), in which case superior ideas and network connections are not used. 1.4.2 Process determining network VCs with useful connections Network VCs’ connections to industries are governed by a stochastic process that is specified analogously to the one determining the most productive entrepreneurs. As a result, this process also has the stochastic properties outlined above and ensures a tractable analysis under imperfect risk sharing. Figure 1 graphically illustrates the processes for entrepreneurs and network VCs. At the beginning of each period |$[t, t+d t)$|⁠, an i.i.d. random variable |$i_{V,t}^{*}$| that is uniformly distributed on the interval |$[0,1]$| determines the single-continuum slice of the double continuum of network VCs that has useful connections in the single continuum of industries. The network VC that has useful connections in industry |$v$| in that period is then identified by the double index |$(i_{V}',i_{V}'')= (i_{V,t}^{*}, v)$|⁠. 1.4.3 VC fund objective and compensation of network VCs and entrepreneurs VC funds raise capital from agents to finance ventures of a particular industry and vintage, where a vintage refers to the set of ventures of a specific period. Ventures that succeed are sold by a VC fund through an IPO or an M&A transaction at an endogenously determined market price |$P$| that will depend on the industry and the state of the economy.8 Conditional on having successfully matched with the best entrepreneur in a given industry |$v$|⁠, a VC fund maximizes the market value of funded ideas, net of venture investment costs. The matched VC fund takes the market price of a successful venture, |$P$|⁠, and the white-collar compensation rate, |$w_{W}$|⁠, as given and chooses the quantity of ideas it funds, |$n_{1}$|⁠. The maximum surplus a matched VC fund can generate net of the compensation for white-collar fund managers that facilitated the matching is given by $ \begin{align} \label{eq:VC-Max-Surplus} \pi_{1} \equiv \max_{n_{1} \in \mathbb{R}^{+}_{0}} \left\{ h_{1}(v,Z, n_{1}) \cdot P - (n_{1} w_{W} + c_{\iota} w_{W} ) \right\} . \end{align} $ (7) Going forward, I refer to the market value of expected venture exits as VC-backed IPO volume, and define the corresponding variable |$ipo \equiv h_{1} P$|⁠. The VC fund uses capital commitments from investors to fund the implementation of venture ideas, |$n_{1} w_{W}$|⁠, and to compensate VC fund managers’ white-collar human capital, |$c_{\iota} w_{W}$|⁠. I introduce a variable representing these capital commitments, |$com\equiv(n_{1}w_{W}+c_{\iota}w_{W})$|⁠. The part of capital commitments that immediately flows to VC fund managers is akin to performance-insensitive “management fees” observed in practice. For states where |$com >0$|⁠, I define |$f^{man} \equiv \frac{c_{\iota} w_{W}}{com}$| as the fraction of capital commitments that is used for these types of fees. In return for providing capital, the fund provides investors with an ownership stake of size |$\frac{com}{ipo}$| in the fund’s investment portfolio (net of any fees), which implies that competitive fund investors break even. New VC funds are free to enter in each period. Competition between VC funds for talent implies that any positive surplus |$\pi_{1}>0$|⁠, if it exists, will be appropriated by the connected network VC and the most productive entrepreneur in a given industry, both of which are needed to implement the most productive entrepreneur’s ideas. Correspondingly, these two agents jointly obtain a venture ownership stake |$\frac{\pi_{1}}{ipo}$| in the funded ventures, whenever |$ipo>0$|⁠. The allocation of this ownership stake is determined by bilateral Nash (1950) bargaining between the connected network VC and the most productive entrepreneur.9 A network VCs’ bargaining power is given by |$\varrho \in [0,1]$|⁠. The stake a network VC obtains is akin to a performance-sensitive carry fee observed in practice, as it only yields a positive payoff after a successful exit, which is when fund investors also obtain a positive investment return. For states where capital commitments are positive, I define |$f^{car} \equiv \frac{\varrho \pi_{1}}{com}$| as the expected value of this performance-sensitive carry fee relative to commitments. In sum, the claims obtained by fund managers and investors in the model is closely related to what is observed in practice: investors become fund shareholders receiving the cash returned by venture investments, and fund managers retain a fixed management fee and a performance-sensitive carry fee (see, e.g., Hall and Woodward 2007, 2010). 1.4.4 Funding of regular innovative ideas All entities that fund regular innovative ideas (e.g., existing firms) have identical investment opportunities, act competitively, and fund ideas as long as they can make at least zero expected profits. Thus, these entities do not internalize the diminishing returns to scale specified in Equation (6). Yet, consistent with the interpretation of (5) and (6) as reflecting an intraperiod first-mover advantage for the most productive entrepreneur, entities funding regular ideas can condition their decisions on the period’s realization of |$n_{1}$|⁠.10 The measure of regular ideas implemented by new or existing firms, |$n_{2}$|⁠, is then determined by the free-entry condition:11 $ \begin{equation} \label{eq:free-entry} \max \left\{ n_{2} \in \mathbb{R}^{+}_{0} : \frac{h_{2}(v,Z, n_{1}, n_{2})}{n_{2}} P - w_{W}) \geq 0 \right\}. \end{equation} $ (8) This condition reflects that regular innovative ideas are funded up to the point at which firms break even in expectation. While the success rate per unit of regular ideas is |$\frac{h_{2}}{n_{2}}$| and the payoff in success is |$P$|⁠, implementing a unit of ideas requires compensating one unit of white-collar labor. Whereas these white-collar agents obtain compensation commensurate with their labor market outside options, entrepreneurs cannot extract additional compensation from generating regular ideas, as these ideas are not scarce. While VC funds are unlevered financial intermediaries, levered institutions, such as banks, are often a source of funding for regular firms’ R&D activity in practice, in particular in the case of small and medium-sized private enterprises. These intermediaries, in turn, might face financial frictions related to leverage or net worth constraints that do not feature explicitly in the presented setup. I will discuss the model’s components absorbing such channels when calibrating the model in Section 3.1.2. Contrary to the competitive entities funding regular innovative ideas, a VC fund that matched with the most productive entrepreneur in an industry has an exclusive relationship with that entrepreneur, and internalizes diminishing returns to scale among funded ideas. Ceteris paribus, this effect reduces the quantity of ideas a VC firm funds. On the other hand, a most productive entrepreneur’s venture ideas are more productive. To streamline the analysis, I will maintain the assumption that |$\phi < \eta$|⁠, which ensures that the productivity channel dominates, that is, at given market prices for innovations, a VC fund that matched with a best entrepreneur optimally funds strictly more efficiency-adjusted ideas than normal competitive firms with access to regular ideas do. 1.5 Final good production The final good is produced by competitive firms according to the Cobb-Douglas production function $ \begin{equation} Y_{t} =A_{t} \cdot \exp \left( \int_{\Psi}\log \left[ q\left( v,t\right) x\left( v,t|q\right) \right] dv\right)\!, \label{M_AssumptionY_Technology} \end{equation} $ (9) where |$x\left( v,t|q\right)$| is the quantity of the intermediate good in industry |$v$| of quality |$q$| used in the production process.12 As is standard in Schumpeterian growth models, there is no storage technology for intermediate and final goods. Produced intermediate goods are used in final good production in the same period, and the output flow of the final good |$Y_{t}$| equals the flow of aggregate consumption in equilibrium. The representation of the final good production function (9) already indicates that, in equilibrium, the best intermediate good in each industry, which is of quality |$q(v,t)$|⁠, will be used in the production of the final good. The factor |$A_{t}$| in Equation (9) follows the stochastic differential equation $ \begin{equation} \frac{d A_{t}} {A_{t}} =\delta \left( Z_{t} \right) dt+\sigma\left( Z_{t} \right) dB_{t} ,\text{ with}~~A \left( 0\right) >0, \label{M_AssumptionFactorSDE} \end{equation} $ (10) where |$B_{t}$| is a standard Brownian motion, and where the local drift |$\delta$| and the local risk exposure |$\sigma$| depend on the Markov state |$Z_{t}$|⁠. Introducing the process for |$A_{t}$| will allow the model to capture channels affecting aggregate growth other than endogenous innovative activity, which is essential for an empirically plausible model calibration. 1.6 Risk sharing I will initially consider the case in which all agents are free to trade financial claims in a frictionless Walrasian market where Arrow-Debreu securities are in zero net supply. In Section 4, I relax this assumption and analyze the model’s implications when risks associated with the performance-sensitive claims of venture capital compensation cannot be shared by agents via financial side contracts. 2. Decentralized Equilibrium In the following, I analyze Markov perfect equilibria of the decentralized economy. Appendix A provides standard definitions of an allocation and a decentralized equilibrium. The aggregate state variables in the economy are the Markov state |$Z$| and the level of aggregate output |$Y$|⁠. Because of the iso-elastic properties of the setup, the economy scales with |$Y$|⁠. Throughout, I will take advantage of this scaling property and directly consider scaled prices and compensation rates. I mark these scaled variables by a tilde. For example, the blue-collar and white-collar wage-to-output ratios will be denoted by |$\tilde{w}_{B}(Z)$| and |$\tilde{w}_{W}(Z)$|⁠, respectively. 2.1 Industry-level quantities and prices In this section, I analyze industry-level equilibrium conditions taking as given the dynamics of key aggregate variables, such as the drift and volatility of aggregate output, denoted by |$\mu(Z)$| and |$\sigma(Z)$| respectively, the stochastic discount factor, denoted by |$\xi _{t}$|⁠, and |$\tilde{w}_{B}(Z)$| and |$\tilde{w}_{W}(Z)$|⁠. In Section 2.2, I characterize the equilibrium conditions determining these endogenous aggregate variables. 2.1.1 Intermediate and final goods production Producers of the final good behave competitively and take the prices |$p_{x}(v,t|q)$| of intermediate goods of various available qualities in each industry as given. A final good producer’s first-order condition yields the unit elastic intermediate good demand: $ \begin{align} \label{eq:IntermediateDemand} x(v,t|q)= \frac{Y_{t}}{p_{x}(v,t|q)}. \end{align} $ (11) Given this intermediate good demand, the oligopoly equilibrium entails the best intermediate good producer (the incumbent) optimally using “limit pricing.” That is, the incumbent sets prices that, adjusted for quality, fall epsilon below the unit cost of production of the next-best intermediate good producer who could enter using its inferior patent. If the incumbent has a quality advantage of a factor |$\kappa$| relative to the next-best producer (which is the relevant case in equilibrium), then the profit-maximizing price is $ \begin{align} \label{eq:IntermediatePrice} p_{x}(v,t|q)= \kappa \cdot w_{B}(Y_{t},Z_{t}). \end{align} $ (12) The incumbent then earns the profits: $ \begin{align} \label{eq:IncumbentProfit} \pi_{x}(v,t)= (p_{x}(v,t|q) - w_{B}(Y_{t},Z_{t}) ) \cdot x(v,t|q) = \left(1- \frac{1}{\kappa} \right) Y_{t}. \end{align} $ (13) 2.1.2 Innovative activity Because of the above-mentioned scaling property of the economy, strategies determining the quantities of ideas receiving funding (⁠|$n_{1}$| and |$n_{2}$|⁠) do not depend on the current level of aggregate output |$Y$|⁠. In contrast, investment denominated in final good consumption units will scale with |$Y$|⁠. 2.1.2.1 Intensive margin of VC funding First, consider the maximization problem of a VC fund conditional on having matched with the best entrepreneur in industry |$v$|⁠, such that |$\iota_{1}(v,t) = 1$|⁠. The scaled maximum surplus this VC fund can attain net of the compensation paid to white-collar fund managers that facilitated the matching is $ \begin{align} \label{eq:OptimumConditionalVCSurplus} \tilde{\pi}_{1}(v,Z) = \max_{n_{1} \in \mathbb{R}^{+}_{0}} \left\{ h_{1}(v,Z, n_{1}) \tilde{P}(v,Z) -\left(n_{1}\tilde{w}_{W}(Z) +c_{\iota}\tilde{w}_{W}(Z) \right) \right\}. \end{align} $ (14) The first-order condition associated with (14) yields the optimal scale of VC funding conditional on a match: $ \begin{align} \label{eq:OptimumConditionalVCScreeningn1} n_{1}(v,Z) = \left( \eta \theta \left( v,Z\right)\frac{\tilde{P}\left( v,Z\right)} {\tilde{w} _{W}\left( Z\right)} \right)^{\frac{1}{1-\eta} }. \end{align} $ (15) In contrast, if no VC fund reaches and supports the best entrepreneur in a given industry (if |$\iota_{1}(v,t)=0$|⁠), then that entrepreneur’s superior ideas cannot be implemented, that is, |$n_{1}(v,t)=0$|⁠. 2.1.2.2 Extensive margin of VC funding A network VC that has a useful industry connection at time |$t$| chooses the probability with which she accepts the best available compensation offer from a VC fund and facilitates the fund’s matching with the best entrepreneur. I denote the corresponding Markov strategy by |$\bar{\iota}_{1}\left( v,Z_{t}\right)\equiv \Pr [ \iota _{1}(v,t)=1|Z_{t}]$|⁠. A connected network VC optimally facilitates the matching process only if a VC fund can generate a weakly positive surplus, |$\tilde{\pi}_{1} \geq 0$|⁠, and can thus offer the network VC a stake with weakly positive value, |$\varrho\tilde{\pi}_{1} \geq 0$|⁠. The matching probability |$\bar{\iota}_{1}(v,Z)$| thus satisfies the equilibrium conditions: $ \begin{equation} \label{eq:OptimumIota1} \begin{alignedat}{2} \bar{\iota}_{1}(v,Z)&=0 &\quad \text{ if}~~\tilde{\pi}_{1}(v,Z)<0,\\ \bar{\iota}_{1}(v,Z)&\in [0,1] &\!\!\text{if}~~\tilde{\pi}_{1}(v,Z)=0,\ \\ \bar{\iota}_{1}(v,Z)&= 1 &\quad \text{ if}~~\tilde{\pi}_{1}(v,Z)>0.\, \end{alignedat} \end{equation} $ (16) 2.1.2.3 Funding of regular innovative ideas Let |$n_{2}(v,Z,n_{1})$| denote the solution to the free-entry condition (8) pinning down the measure of regular ideas implemented by new or existing firms. Following the optimal strategy (15), a VC fund extends funding to the most productive entrepreneur until the marginal benefit of implementing superior ideas is equal to the implementation cost. Regular ideas, compared with the best entrepreneur’s marginal idea, have strictly lower success prospects, so plugging (15) into condition (8) yields the result that no additional regular ideas are funded (⁠|$n_{2}=0$|⁠) in periods when the most productive entrepreneur already obtained funding (when |$\iota_{1}=1$|⁠). Yet, when matching with the best entrepreneur fails (when |$\iota_{1}=0$|⁠), condition (8) implies that the following quantity of regular ideas is funded: $ \begin{align} \label{eq:n2opt} n_{2}(v,Z,0) = \frac{1}{\phi} \left( \frac{\phi \theta \left( v,Z_{t}\right) \tilde{P}\left( v,Z_{t}\right)} {\tilde{w}_{W}\left( Z_{t}\right)} \right) ^{\frac{1}{1-\eta} }. \end{align} $ (17) The above results lead to the following Proposition characterizing the equilibrium arrival rate of new innovations in each industry. Proposition 1 (Creative destruction). The expected arrival rate of new innovations in industry |$v$| is $ \begin{equation} \label{eq:hbarEquilibrium} \bar{h}\left( v,Z\right) = \bar{\iota} _{1}\left( v,Z\right) h(v,n_{1}\left( v,Z\right) )+\left( 1-\bar{\iota}_{1}\left( v,Z\right) \right) h(v,Z,\phi n_{2}\left(v,Z,0\right) ), \end{equation} $ (18) where the Markov strategies |$n_{1}\left( v,Z\right)$|⁠, |$\bar{\iota} _{1}\left( v,Z\right)$|⁠, and |$n_{2}\left(v,Z,0\right)$| are characterized in Equations (15), (16), and (17), respectively. As VC funds’ intermediation decisions affect both the intensive margin and the extensive margin of funding provided to the best entrepreneurs, they also affect the arrival rate of new innovations. This arrival rate |$\bar{h}(v,Z)$| is also referred to as the rate of creative destruction. It represents both the rate of entry by firms having developed new products in a given industry |$v$| and the rate of exit by firms that are disrupted by these new products. As entry poses a competitive threat, this rate has important implications for firms’ post-IPO prices and returns, as further described in the next subsection. 2.1.3 Market prices and expected returns Building on the results developed thus far, I now characterize the market prices of ventures that have succeeded in developing a new product. Proposition 2 (Market price). The market price of a firm that holds the patent to the highest-quality intermediate good in industry |$v$| is $ \begin{equation} \label{eq:IPOPrice} P\left( v,Y_{t},Z_{t}\right) =\mathbb{E}_{t}\left[ \int_{t}^{\tau ^{\ast} }\frac{ \xi _{\tau} }{\xi _{t}}\pi_{x}\left( v,\tau \right) d\tau \right] =\tilde{P}\left( v,Z_{t}\right) \cdot Y_{t}, \end{equation} $ (19) where |$\tau ^{\ast} \equiv \sup \left\{ \tau :M\left( v,\tau \right) =M\left( v,t\right) \right\}$|⁠, and where the profit flow rate |$\pi_{x}\left( v,t\right)$| is given in Equation (13). The Hamilton-Jacobi-Bellman (HJB) equation associated with (19) implies that the function |$\tilde{P}\left( v,Z\right)$| solves the following system of equations for all |$Z\in \Omega$|⁠: $ \begin{eqnarray} \label{eq:IPOpricelinEQ} 0 &=&1-\frac{1}{\kappa} -\left( r_{f}\left( Z\right) +rp^{D}\left( v,Z\right) +rp^{J}\left( v,Z\right) +\bar{h}\left( v,Z\right) -\mu \left( Z\right) \right) \tilde{P}\left( v,Z\right) \notag \\ &&+\sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }\left( \tilde{P}\left( v,Z^{\prime} \right) -\tilde{P}\left( v,Z\right) \right)\!. \end{eqnarray} $ (20) The expressions for the risk-free rate |$r_{f}$|⁠, the diffusion-risk premium |$rp^{D}$|⁠, and the jump-risk premium |$rp^{J}$| are provided in Appendices B and D. Proof. See Appendices B and D. ■ Proposition 2 highlights that the industry-specific rates of creative destruction, |$\bar{h}(v,Z)$|⁠, have important implications for ventures’ post-IPO market valuations. A firm has a higher propensity of being disrupted by yet another entrant when the rate of entry is expected to stay persistently high. The rates of creative destruction also vary with the aggregate state |$Z$|⁠, which implies that they further affect exposures to macroeconomic risks and the associated jump-risk premiums |$rp^{J}$|⁠. As the VC industry facilitates entry of new firms, its equilibrium behavior also affects these risk exposures. After an IPO, a firm’s expected return is given by the sum of the risk-free rate |$r_{f}$|⁠, the diffusion risk premium |$rp^{D}$|⁠, and the jump risk premium |$rp^{J}$|⁠. 2.1.3.1 Gross alpha versus net alpha Like in Berk and Green’s (2004) partial equilibrium model of the mutual fund industry, competition for VC fund managers’ talent ensures that fund investors make zero “net alpha” in equilibrium. This result is consistent with several empirical studies in the VC literature finding that returns to investors net of fees are approximately zero (see, e.g., Korteweg and Nagel 2014). In particular, after netting out fees to VC managers, VC fund investors in the model obtain a fraction |$\frac{com}{ipo}$| of the fund’s venture investment portfolio, which has a market value equal to the capital they committed to the fund, implying a zero net present value (NPV) investment, or, equivalently, zero net alpha. In contrast, a VC fund’s gross alpha is given by $ \begin{align} \alpha \equiv f^{man} + f^{car}, \end{align} $ (21) which defines alpha as the NPV generated by a VC fund, scaled by investors’ capital commitments |$com$|⁠.13 This ratio is a measure of abnormal expected returns at the venture investment stage in that it does not represent compensation for exposures to aggregate risk. 2.2 Aggregate quantities and prices The following proposition characterizes the dynamics of aggregate output |$Y$|⁠, which equals aggregate consumption in equilibrium. Proposition 3 (Aggregate output). Aggregate output follows the stochastic differential equation $ \begin{equation} \label{eq:ConsumptionDynamics} \frac{dY_{t}}{Y_{t}}=\mu \left( Z_{t}\right) dt+\sigma \left( Z_{t}\right) dB_{t}, \end{equation} $ (22) where the local drift |$\mu$| takes the following form: $ \begin{equation} \label{eq:ConsumptionDrift} \mu \left( Z_{t}\right) =\delta \left( Z_{t}\right) +\log \left[ \kappa \right] \int_{\Psi}\bar{h}\left( v,Z_{t}\right) dv. \end{equation} $ (23) Proof. See Appendix C. ■ Proposition 3 shows that expected aggregate growth |$\mu$| can be decomposed into the exogenous component |$\delta$| and an endogenous component that depends on the rates of creative destruction |$\bar{h}$| across industries. An endogenous fraction of creative destruction is, in turn, due to VC investments (see Proposition 1). Next, I characterize the dynamics of the stochastic discount factor under perfect risk sharing, where agents exhibit identical consumption growth in equilibrium. Section 4 will consider the case in which financial markets are incomplete and idiosyncratic venture income risk is uninsurable. Proposition 4 (Stochastic discount factor). The stochastic discount factor follows the Markov-modulated jump-diffusion process: $ \begin{align} \frac{d\xi_{t}}{\xi_{t}} & = -r_{f}\left( Z_{t}\right) dt-\gamma \sigma \left( Z_{t}\right) dB_{t}\nonumber\\ &\quad{} +\sum_{Z^{\prime} \neq Z_{t}}(e^{\zeta \left(Z_{t},Z^{\prime} \right)} -1)\left( dN_{t}\left( Z_{t}, Z^{\prime} \right) -\lambda _{Z_{t}Z^{\prime} }dt\right)\!, \end{align} $ (24) where the function |$\zeta (Z,Z^{\prime} )$| is determined by agents’ HJB equation provided in Appendix D. Proof. See Appendix D. ■ 2.2.1 Labor markets and wages Combining the optimal intermediate good demand (11) and intermediate goods prices (12) with the fact that one unit of blue-collar labor produces one unit of the final good leads to the following market-clearing condition for blue-collar labor: $ \begin{equation} \label{eq:Blue-Collar-Market-Clearing} \int_{\Psi}x\left( v,t|q\right) dv=\frac{1}{\kappa \cdot \tilde{w}_{B}\left( Z_{t}\right)} = 1. \end{equation} $ (25) It follows that the blue-collar wage-to-output ratio is constant and given by |$\tilde{w}_{B}= 1/ \kappa$|⁠. The white-collar wage-to-output ratio |$\tilde{w}_{W}\left( Z_{t}\right)$| solves the market-clearing condition: $ \begin{equation} \int_{\Psi} \left[ \bar{\iota}_{1}\left( v,Z\right) (n_{1}\left( v,Z\right) +c_{\iota})+ \left( 1-\bar{\iota}_{1}\left( v,Z\right) \right) n_{2}\left( v,Z,0\right) \right] dv = 1. \end{equation} $ (26) As characterized in Equations (15), (16), and (17), the Markov strategies for |$n_{1}$|⁠, |$\bar{\iota}_{1}$|⁠, and |$n_{2}$| depend on the white-collar wage rate |$\tilde{w}_{W}$| and market prices |$\tilde{P}$|⁠. The wage rate |$\tilde{w}_{W}$| varies across aggregate states |$Z$|⁠, as the demand for white-collar labor from VC funds and innovative firms responds to forward-looking information encoded in market prices |$\tilde{P}$|⁠. 3. Calibration and Evaluation In this section, I calibrate the model and evaluate its predictions. The calibration targets the dynamics of key variables relevant for macroeconomic dynamics, asset prices, and VC activity in U.S. data. 3.1 Choosing parameters Below, I first discuss preference parameters and dynamics of exogenous macroeconomic processes. Afterward, I address parameter choices relating to VC activity and endogenous growth. 3.1.1 Exogenous aggregate processes and preference parameters I parameterize the model based on a nine-state Markov chain for the aggregate state |$Z$|⁠. The exogenous drift component of aggregate output growth, |$\delta$|⁠, and the local volatility, |$\sigma$|⁠, are chosen so that the resultant equilibrium process for agents’ consumption growth matches the one considered in Chen (2010) state by state.14 In addition, I choose the values for preference parameters (⁠|$\beta$|⁠, |$\psi$|⁠, and |$\gamma$|⁠) used by Chen (2010) (see Table 1). Together, these choices discipline consumption dynamics and asset pricing moments of the economy and ensure reasonable aggregate risk premiums and risk-free rate dynamics.15 Throughout, I present state-dependent parameter values and equilibrium outcomes in three-by-three matrices that sort the nine Markov states along two dimensions: expected consumption growth, |$\mu$|⁠, and local volatility, |$\sigma$| (each categorized into Low, Med, and High). It is worth highlighting that every state |$Z$| is associated with a distinct drift |$\mu(Z)$|⁠; for example, drifts still vary within the category “High |$\mu$|⁠.” Table A1 in the appendix reports the values of |$\mu$| and |$\sigma$|⁠, and Table A2 lists the unconditional probabilities of the Markov states. Table A3 in the appendix provides the values of the exogenous components |$\delta$| of the aggregate growth drifts. Table 1 State-invariant parameters Parameter Variable Value Rate of time preference |$\beta$| |$0.015$| Elasticity of intertemporal substitution |$\psi$| |$1.500$| Coefficient of relative risk aversion |$\gamma$| |$7.500$| Efficiency discount of regular venture ideas |$\phi$| |$0.323$| Jump size of the quality index per innovation |$\kappa$| |$1.148$| Decreasing returns to scale parameter |$\eta$| |$0.585$| White-collar labor required for VC intermediation in industries |$\Psi'$| |$c_{\iota}$| |$0.027$| Network VCs’ bargaining power |$\varrho$| |$0.346$| Measure of industries |$\Psi'$| |$\int_{\Psi'}dv$| |$0.028$| Parameter Variable Value Rate of time preference |$\beta$| |$0.015$| Elasticity of intertemporal substitution |$\psi$| |$1.500$| Coefficient of relative risk aversion |$\gamma$| |$7.500$| Efficiency discount of regular venture ideas |$\phi$| |$0.323$| Jump size of the quality index per innovation |$\kappa$| |$1.148$| Decreasing returns to scale parameter |$\eta$| |$0.585$| White-collar labor required for VC intermediation in industries |$\Psi'$| |$c_{\iota}$| |$0.027$| Network VCs’ bargaining power |$\varrho$| |$0.346$| Measure of industries |$\Psi'$| |$\int_{\Psi'}dv$| |$0.028$| Open in new tab Table 1 State-invariant parameters Parameter Variable Value Rate of time preference |$\beta$| |$0.015$| Elasticity of intertemporal substitution |$\psi$| |$1.500$| Coefficient of relative risk aversion |$\gamma$| |$7.500$| Efficiency discount of regular venture ideas |$\phi$| |$0.323$| Jump size of the quality index per innovation |$\kappa$| |$1.148$| Decreasing returns to scale parameter |$\eta$| |$0.585$| White-collar labor required for VC intermediation in industries |$\Psi'$| |$c_{\iota}$| |$0.027$| Network VCs’ bargaining power |$\varrho$| |$0.346$| Measure of industries |$\Psi'$| |$\int_{\Psi'}dv$| |$0.028$| Parameter Variable Value Rate of time preference |$\beta$| |$0.015$| Elasticity of intertemporal substitution |$\psi$| |$1.500$| Coefficient of relative risk aversion |$\gamma$| |$7.500$| Efficiency discount of regular venture ideas |$\phi$| |$0.323$| Jump size of the quality index per innovation |$\kappa$| |$1.148$| Decreasing returns to scale parameter |$\eta$| |$0.585$| White-collar labor required for VC intermediation in industries |$\Psi'$| |$c_{\iota}$| |$0.027$| Network VCs’ bargaining power |$\varrho$| |$0.346$| Measure of industries |$\Psi'$| |$\int_{\Psi'}dv$| |$0.028$| Open in new tab 3.1.2 VC activity and endogenous growth The remaining parameters of the model relate to VC activity and the endogenous component of growth. VC activity is empirically concentrated in specific industries, such as IT and Pharma (see Kortum and Lerner 2000), and historically has been highly volatile and cyclical (see Gompers and Lerner 2004; Gompers et al. 2008). To capture these empirical regularities, I consider a parsimonious parameterization of the economy with two subsets of industries, denoted by |$\Psi'$| and |$\Psi''$|⁠. Across these two sets of industries, structural parameters and productivity dynamics differ; cross-sectional allocations of resources would be naturally symmetric across all industries if all industries faced the exact same technological conditions at any point in time. I designate the set of industries |$\Psi'$| as one in which VC intermediation may occur at times, and the set |$\Psi''$| as one in which VC intermediation does not obtain. The second set aims to represent industries that are atypical candidates for VC activity, as, for example, the steel industry. This specification with two sets of industries keeps the analysis with industry heterogeneity parsimonious, while still capturing the above-mentioned stylized facts. In particular, I assume that the parameter value of |$c_{\iota}$| is prohibitively high in industries |$\Psi''$|⁠, reflecting the notion that VC funds’ intermediation efforts provide insufficient net benefits here. In these industries, innovative activity is more profitable if undertaken by regular (existing) firms.16 Apart from the parameter values of |$c_{\iota}$|⁠, the two sets of industries also differ in terms of their productivity dynamics to capture the above-mentioned pronounced cyclicality of VC activity. In the following, I discuss how each of the remaining parameters is chosen to target moments in the data. To streamline this discussion, Appendix E provides additional details on the data and the construction of the empirical measures. Table 1 reports the values of all state-invariant model parameters. 3.1.2.1 Measure of industries |$\Psi'$| The measure of industries that potentially exhibit VC intermediation, |$\int_{\Psi'}dv$|⁠, is set such that the model matches the historical average ratio of capital commitments to the VC industry relative to aggregate consumption, which I estimate to be 0.3% based on data reported by the National Venture Capital Association (2012, 2017). 3.1.2.2 Innovation step size |$\kappa$| The parameter |$\kappa$| is chosen to match the innovation step size estimate in Acemoglu et al. (2013). Appendix E shows this parameter choice also implies that profits in the model are consistent with empirical evidence about IT sector markups. 3.1.2.3 Efficiency discount |$\phi$| I use Kortum and Lerner’s (2000) patent production function estimates to choose the parameter |$\phi$|⁠, which measures the relative efficiency of regular and VC-backed innovative activity, and thus affects the benefits of VC intermediation (see Appendix E for details). 3.1.2.4 Matching cost |$c_{\iota}$| The parameter |$c_{\iota}$| applying to industries |$\Psi'$| represents the scarce white-collar human capital needed to realize a match between a VC fund and the most productive entrepreneur in an industry. As a result, this parameter is closely related to the minimum value added that a VC fund must generate to cover the costs of intermediation. Covering these costs is required to attract investors’ capital and avoid intermediation breakdowns. I set this parameter to match average management fees of 3% of invested capital, targeting the empirical findings by Hall and Woodward (2007). Using management fees as a moment to inform the parameter |$c_{\iota}$| is motivated by three observations. First, following the logic of Berk and Green (2004), the fixed management fees paid by investors in practice should also imply a lower bound on the value added that a VC fund must generate in order to attract investors’ capital (it is only a lower bound as additional carry fees apply if the fund produces gains for investors ex post). Second, management fees provide the best mapping to the costs associated with the parameter |$c_{\iota}$| in terms of risk properties. Both in the data and in the model, these costs represent nonrisky compensation, a particularly relevant characteristic once I consider imperfect risk sharing in Section 4. Finally, this parameter choice implies infrequent intermediation breakdowns, ensuring plausible time-series properties (see Section 3.2.1). In Section 3.3, I also will evaluate the sensitivity of key equilibrium outcomes with respect to the value of the parameter |$c_{\iota}$|⁠. 3.1.2.5 Decreasing returns to scale parameter |$\eta$| In the model, the combined payoffs of VC managers and entrepreneurs are tightly linked to the decreasing returns to scale parameter |$\eta$|⁠, as this parameter influences the magnitude of rents from VC intermediation. Hall and Woodward (2007) find that fund managers’ total earnings (i.e., management plus carry fees) are 26% of funds invested. The authors further estimate that whereas entrepreneurs historically earned an average of |${\$}$|9.2 million (in 2006 dollars) from each company that attracted VC funding, VC managers received an average of |${\$}$|5.5 million in fee revenue from each company they backed. I set the parameter |$\eta$| such that the ratio of the sum of VC fees and entrepreneurial payoffs to capital commitments in the model is consistent with these empirical estimates. The resultant parameter value of 0.585 also lies within 1 standard deviation of the point estimate Kortum and Lerner (2000) obtain for decreasing returns to scale in the context of VC-backed patenting activity. 3.1.2.6 Network VCs’ bargaining power |$\varrho$| In the model, VC carry fees and entrepreneurial payoffs are bargained fractions |$\varrho$| and |$(1-\varrho$|⁠) of the rents |$\tilde{\pi}_{1}$|⁠, respectively. I choose the parameter |$\varrho$| so that the ratio of carry fees to entrepreneurial payoffs corresponds to the estimates by Hall and Woodward (2007) mentioned above. Overall, the calibration thus ensures that the model matches average aggregate VC carry fees, VC management fees, and entrepreneurial payoffs. 3.1.2.7 Productivity |$\theta$| The distinct productivity processes in the two sets of industries |$\Psi'$| and |$\Psi''$| are differentially exposed to macroeconomic conditions as measured by the state-contingent growth drifts considered in Chen (2010). For industries |$\Psi'$|⁠, I specify a nonlinear relation to these drifts to ensure that the model matches the highly cyclical and skewed distribution of aggregate VC commitments in the data. The caption of Table 2 reports this relation as well as the corresponding values of |$\theta$|⁠, which vary across all nine states. The specification is characterized by three parameters relating to the mean, the volatility, and the skewness of the productivity distribution. Mean productivity is set so that industries |$\Psi'$| exhibit an average rate of creative destruction that is representative of the types of sectors that receive VC funding. In particular, using estimates from Caballero and Jaffe (1993), I match an average rate of creative destruction of 8%. The remaining two parameters are set so that the volatility and the peak levels of aggregate VC commitments in the model match their data counterparts. Empirical peak levels were reached at the end of the internet boom, when commitments scaled by aggregate consumption were more than three standard deviations above their average value. Consistent with existing empirical evidence, the specified relation also ensures that VC activity is strongly correlated with aggregate conditions (see, e.g., Gompers and Lerner 1998; Jeng and Wells 2000; Kaplan and Schoar 2005; Gompers et al. 2008). Table 2 State-contingent outcomes in VC-backed industries |$\Psi'$| A. VC intermediation: |$\bar{\iota}_{1}(v,Z)$| B. Capital commitments: |$\int_{\Psi'}\widetilde{com}(v,Z)dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 1 1 1 |$\mu$| High 0.008 0.010 0.013 Med 1 1 1 Med 0.002 0.002 0.002 Low 0 0 0 Low 0.000 0.000 0.000 C. Creative destruction: |$\bar{h}(v,Z)$| D. VC exit volume: |$\int_{\Psi'} \widetilde{ipo}(v,Z)dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.216 0.308 0.400 |$\mu$| High 0.013 0.018 0.021 Med 0.041 0.042 0.043 Med 0.003 0.003 0.003 Low 0.001 0.000 0.000 Low 0.000 0.000 0.000 E. Productivity: |$\theta(v,Z)$| F. Entry-to-productivity ratio: |$\frac{\bar{h}(v,Z)}{\theta(v,Z)}$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.122 0.151 0.177 |$\mu$| High 1.765 2.039 2.263 Med 0.050 0.051 0.051 Med 0.808 0.828 0.845 Low 0.013 0.008 0.007 Low 0.064 0.038 0.032 A. VC intermediation: |$\bar{\iota}_{1}(v,Z)$| B. Capital commitments: |$\int_{\Psi'}\widetilde{com}(v,Z)dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 1 1 1 |$\mu$| High 0.008 0.010 0.013 Med 1 1 1 Med 0.002 0.002 0.002 Low 0 0 0 Low 0.000 0.000 0.000 C. Creative destruction: |$\bar{h}(v,Z)$| D. VC exit volume: |$\int_{\Psi'} \widetilde{ipo}(v,Z)dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.216 0.308 0.400 |$\mu$| High 0.013 0.018 0.021 Med 0.041 0.042 0.043 Med 0.003 0.003 0.003 Low 0.001 0.000 0.000 Low 0.000 0.000 0.000 E. Productivity: |$\theta(v,Z)$| F. Entry-to-productivity ratio: |$\frac{\bar{h}(v,Z)}{\theta(v,Z)}$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.122 0.151 0.177 |$\mu$| High 1.765 2.039 2.263 Med 0.050 0.051 0.051 Med 0.808 0.828 0.845 Low 0.013 0.008 0.007 Low 0.064 0.038 0.032 The panels report state-contingent outcomes in VC-backed industries |$\Psi'$|⁠. Panel A indicates the states with VC fund intermediation. Panel B reports capital commitments to VC funds scaled by aggregate consumption. Panels C and D list the rates of creative destruction and the aggregate value of VC exits (IPOs and acquisitions) scaled by aggregate consumption. Panels E and F report productivity and the ratio of the arrival rate of successful new entrants to productivity. The state-contingent productivity values follow from the calibrated relation: $ \begin{align} \theta(v,Z) = a + b \cdot (\mu(Z) - \min_{\forall Z}\{\mu(Z)\}))^{c}, \end{align} $ (27) where, |$a=0.0073$|⁠, |$b=30.81$|⁠, |$c=1.95$|⁠, and |$\mu(Z)$| are the aggregate growth drifts from Chen (2010). Open in new tab Table 2 State-contingent outcomes in VC-backed industries |$\Psi'$| A. VC intermediation: |$\bar{\iota}_{1}(v,Z)$| B. Capital commitments: |$\int_{\Psi'}\widetilde{com}(v,Z)dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 1 1 1 |$\mu$| High 0.008 0.010 0.013 Med 1 1 1 Med 0.002 0.002 0.002 Low 0 0 0 Low 0.000 0.000 0.000 C. Creative destruction: |$\bar{h}(v,Z)$| D. VC exit volume: |$\int_{\Psi'} \widetilde{ipo}(v,Z)dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.216 0.308 0.400 |$\mu$| High 0.013 0.018 0.021 Med 0.041 0.042 0.043 Med 0.003 0.003 0.003 Low 0.001 0.000 0.000 Low 0.000 0.000 0.000 E. Productivity: |$\theta(v,Z)$| F. Entry-to-productivity ratio: |$\frac{\bar{h}(v,Z)}{\theta(v,Z)}$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.122 0.151 0.177 |$\mu$| High 1.765 2.039 2.263 Med 0.050 0.051 0.051 Med 0.808 0.828 0.845 Low 0.013 0.008 0.007 Low 0.064 0.038 0.032 A. VC intermediation: |$\bar{\iota}_{1}(v,Z)$| B. Capital commitments: |$\int_{\Psi'}\widetilde{com}(v,Z)dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 1 1 1 |$\mu$| High 0.008 0.010 0.013 Med 1 1 1 Med 0.002 0.002 0.002 Low 0 0 0 Low 0.000 0.000 0.000 C. Creative destruction: |$\bar{h}(v,Z)$| D. VC exit volume: |$\int_{\Psi'} \widetilde{ipo}(v,Z)dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.216 0.308 0.400 |$\mu$| High 0.013 0.018 0.021 Med 0.041 0.042 0.043 Med 0.003 0.003 0.003 Low 0.001 0.000 0.000 Low 0.000 0.000 0.000 E. Productivity: |$\theta(v,Z)$| F. Entry-to-productivity ratio: |$\frac{\bar{h}(v,Z)}{\theta(v,Z)}$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.122 0.151 0.177 |$\mu$| High 1.765 2.039 2.263 Med 0.050 0.051 0.051 Med 0.808 0.828 0.845 Low 0.013 0.008 0.007 Low 0.064 0.038 0.032 The panels report state-contingent outcomes in VC-backed industries |$\Psi'$|⁠. Panel A indicates the states with VC fund intermediation. Panel B reports capital commitments to VC funds scaled by aggregate consumption. Panels C and D list the rates of creative destruction and the aggregate value of VC exits (IPOs and acquisitions) scaled by aggregate consumption. Panels E and F report productivity and the ratio of the arrival rate of successful new entrants to productivity. The state-contingent productivity values follow from the calibrated relation: $ \begin{align} \theta(v,Z) = a + b \cdot (\mu(Z) - \min_{\forall Z}\{\mu(Z)\}))^{c}, \end{align} $ (27) where, |$a=0.0073$|⁠, |$b=30.81$|⁠, |$c=1.95$|⁠, and |$\mu(Z)$| are the aggregate growth drifts from Chen (2010). Open in new tab For industries |$\Psi''$|⁠, I specify a linear relation that is characterized by two parameters determining the mean of productivity and the exposure to macroeconomic conditions (see the caption for Table A5 in the appendix). A linear specification suffices in this case as the data informing these parameter choices do not exhibit skewed behavior of the type witnessed in VC data. In particular, I set the mean of |$\theta$| in industries |$\Psi''$| so that the average rates of creative destruction across all industries |$\Psi$| matches the average rate of creative destruction in the U.S. economy of 4% per year, as estimated by Caballero and Jaffe (1993). The exposure parameter is set so that the standard deviation of white-collar wage-to-output ratios relative to their mean, |$\frac{Std[\tilde{w}]}{E[\tilde{w}]}$|⁠, is consistent with empirical estimates obtained from income shares data for the top 5% of income earners in the United States. Panel B of Table A4 reports the white-collar wage-to-output ratios. I associate income fluctuations for white-collar labor with those for top income earners, as agents from this group are likely to be recruited for innovation- and finance-related jobs in practice.17 Moreover, the specified relation captures the empirical fact that top earners’ income share has substantial positive comovement with the business cycle (see Piketty and Saez 2003; Parker and Vissing-Jorgensen 2009, 2010); white-collar compensation represents a larger fraction of aggregate output in states with high aggregate growth. 3.1.2.8 Relation of the calibration to other financing frictions Given the empirical magnitude of VC commitments relative to economy-wide innovative activity, the calibration implies that an important part of that activity is non-VC intermediated (i.e., regular innovative activity |$n_{2}(v,t)$|⁠). In practice, such non-VC intermediated activity is performed by firms that often rely on other types of financial intermediaries. In particular, levered financial institutions, such as banks, play a prominent role in the funding of small and medium-sized private enterprises and are not in the focus of this paper. Levered financial institutions and firms, in turn, are subject to financial constraints that can affect growth even beyond their impact on innovative activity (see, e.g., Elenev, Landvoigt, and Van Nieuwerburgh 2018). Whereas the model captures regular firms’ activities leading to creative destruction with the endogenous variables |$n_{2}(v,t)$|⁠, additional channels affecting growth are collectively captured in reduced form by the persistent shocks to |$\delta(Z)$| in the model. 3.2 Results of the calibration In this section, I analyze the implications of the calibrated model. First, I discuss aggregate measures of VC intermediation and evaluate the impact of VC investments on the macroeconomy. Second, I analyze the payoffs and returns to various parties involved in the VC process. Third, I examine the properties of expected returns and their effect on exit valuations. Fourth, and finally, I investigate post-IPO failure risk and creative destruction. 3.2.1 VC activity and its macroeconomic impact In the following, I examine the dynamic properties of capital commitments and exit volume, and analyze the impact of VC activity on economic growth and agents’ lifetime utility. 3.2.1.1 Capital commitments and exit volume Panel A of Table 2 indicates the states in which VC funds provide intermediation, and panel B reports the capital commitments the VC industry obtains as a fraction of aggregate consumption. In boom states (high |$\mu$| states), the VC industry attracts larger commitments, with peak levels reaching magnitudes comparable to those observed during the internet boom. In contrast, in the three states with the lowest expected growth (low |$\mu$| states), VC funds are unable to attract capital, as rents from intermediation are insufficient to cover management fees that compensate VC managers’ white-collar human capital. As a result, VC intermediation breaks down in these states. Panel D of Table 2 reports the aggregate market value of VC-backed IPOs and M&A transactions scaled by aggregate consumption. As noted in Section 1.4.3, successful ventures can either undertake an IPO or can be acquired by an existing firm. Consistent with the findings of the empirical literature, the model features large booms and busts in transaction volumes that are strongly correlated with VC fund commitments and aggregate conditions (see, e.g., Gompers and Lerner 1998; Jeng and Wells 2000). The calibration provides a quite remarkable fit of empirical moments relating to IPO and M&A volume, which were not directly targeted in the calibration. On average, transactions amount to 0.5% of aggregate consumption in both the model and the data. The standard deviation of exit volume is 0.6% in the model, whereas it is 0.5% in the data. Exit volume peaked during the internet boom at 2.1% and also reaches that maximum value in the model. This close match of both VC capital commitments (i.e., the value of VC inputs) and VC exit valuations (i.e., the market value of VC output) is reassuring.18 In particular, it lends additional support to the calibrated specification for VC investments’ productivity.19 Moreover, as the calibration matches both total exit valuations and the payoffs to entrepreneurs and VCs, it also yields a good representation of the payoffs to VC investors, which are the remaining third party receiving venture claims. Panels A through C of Figure 2 illustrate the model’s joint dynamics for macroeconomic growth and VC activity based on a time series of model-implied |$Z$|-state realizations. Whereas aggregate VC commitments and exits, once scaled by consumption, are uniquely pinned down by the |$Z$|-state in the model, consumption growth is affected by both the |$Z$|-state and Brownian innovations with volatility |$\sigma(Z)$|⁠. Given the tight link between |$Z$|-states and VC activity in the model, I back out a time series of |$Z$|-state realizations fitted to target only VC data (see the caption for Figure 2 for details). Then, given this time series, I determine the associated model-implied aggregate growth dynamics. In particular, panel C of Figure 2 plots the consumption growth drifts |$\mu(Z)$| and 1-standard-deviation bands associated with the Brownian shocks |$dB$|⁠. The results indicate that the model-implied shocks also provide a good fit for aggregate growth dynamics, even though they were not chosen to match those dynamics. In 23 of 25 years, consumption growth in the data falls within the model-implied one standard deviation bands. While the granularity implied by the nine-state Markov chain structure and the homogeneity of all industries |$\Psi'$| limit the model’s fit for smaller fluctuations, the joint low-frequency movements in aggregate growth and VC activity are captured well. Figure 2 Open in new tabDownload slide Time series of empirical and model-implied outcomes The graphs illustrate the empirical and model-implied time series of aggregate VC commitments scaled by aggregate consumption (panel A), aggregate VC exit valuations scaled by aggregate consumption (panel B), and aggregate consumption growth (panel C). All reported numbers are in percentage points. The model-implied |$Z$|-states for each year are determined by choosing the |$Z$|-state that minimizes the sum of the squared differences between model and data for VC commitments and VC exits in a given year. For the model-implied values of aggregate consumption growth in panel C, I use the resultant time series of |$Z$|-state realizations. I plot the state-contingent values for the drift |$\mu(Z)$| (blue dots) and associated bands of size |$\sigma(Z)$| (gray shaded), highlighting that consumption growth in the model is affected by both the drift |$\mu(Z)$| and Brownian shocks with volatility |$\sigma(Z)$|⁠. Panel D illustrates the associated time series of VC-backed innovations’ growth contributions (left-hand-side axis, light-blue dots) and the value of future VC-backed innovations’ impact on lifetime utility (conditional on the current state |$Z$|⁠) expressed as a fraction of lifetime consumption (right-hand-side axis, dark-blue diamonds). Figure 2 Open in new tabDownload slide Time series of empirical and model-implied outcomes The graphs illustrate the empirical and model-implied time series of aggregate VC commitments scaled by aggregate consumption (panel A), aggregate VC exit valuations scaled by aggregate consumption (panel B), and aggregate consumption growth (panel C). All reported numbers are in percentage points. The model-implied |$Z$|-states for each year are determined by choosing the |$Z$|-state that minimizes the sum of the squared differences between model and data for VC commitments and VC exits in a given year. For the model-implied values of aggregate consumption growth in panel C, I use the resultant time series of |$Z$|-state realizations. I plot the state-contingent values for the drift |$\mu(Z)$| (blue dots) and associated bands of size |$\sigma(Z)$| (gray shaded), highlighting that consumption growth in the model is affected by both the drift |$\mu(Z)$| and Brownian shocks with volatility |$\sigma(Z)$|⁠. Panel D illustrates the associated time series of VC-backed innovations’ growth contributions (left-hand-side axis, light-blue dots) and the value of future VC-backed innovations’ impact on lifetime utility (conditional on the current state |$Z$|⁠) expressed as a fraction of lifetime consumption (right-hand-side axis, dark-blue diamonds). 3.2.1.2 Aggregate impact of VC investments The calibrated model immediately yields estimates for VC investments’ impact on the aggregate economy. In states where VC intermediation obtains, VC-backed innovations contribute an endogenous fraction of agents’ consumption growth. Propositions 1 and 3 imply that these state-contingent growth contributions are given by the product of three quantities: the endogenous arrival rates of innovations |$\bar{h}$| reported in Table 2.C, quality growth per innovation, and the measure of industries featuring VC intermediation in equilibrium. While the resultant average growth contribution is about 3 bps per annum, the fluctuations around this mean are very large, as illustrated by the model-implied time series of growth contributions shown as light-blue dots in panel D of Figure 2. How relevant are these volatile growth contributions for agents’ expected lifetime utility? To answer questions of this type, the macroeconomic literature often determines a so-called “compensation parameter” that allows comparing two stochastic consumption streams in terms of their lifetime utility implications (see, e.g., Lucas 1987, 2003). This measure represents the percentage of lifetime consumption that, if added to one consumption stream, would render agents indifferent to switching to the second consumption stream (see Appendix F for details). This approach reveals that even relatively small growth differences can have significant value due to compounding effects. Consistent with this insight, the model-implied unconditional value of VC-backed innovations is economically significant. It is equivalent to about 1.3% of lifetime consumption. The state-contingent version of this measure, which represents the value of future VC-backed innovations conditional on being in a particular macroeconomic state |$Z$|⁠, is quite stable, as it encodes innovations’ impact on the whole stream of future consumption. The dark-blue diamonds in panel D of Figure 2 illustrate the model-implied time series of these conditional values, which range between 1.2% and 1.5%. These values also can be compared to market-based valuations of VC-backed innovations. Strikingly, they significantly exceed average levels of VC fund commitments and exit valuations, which amount to about 0.3% and 0.5%, respectively, in both the data and the model. The relevance of VC investments is much larger than suggested by data on commitments and exit valuations as the majority of VC-backed innovations’ macroeconomic impact is not reflected in the discounted value of start-ups’ future net payout to investors. In particular, the intangible capital that producers obtain from innovating (patents) generates profits only for a limited time period, until rivals develop superior products. Yet innovations have a permanent positive effect on the economy and agents’ future consumption, as other innovators can build on previously developed knowledge. As Newton (1675) suggested, innovators stand “on the shoulders of giants.” Ceteris paribus, the more rapid creative destruction is, the shorter is the time period during which a producer receives payoffs from a given innovation. However, as highlighted below in Section 3.2.3, the model also reveals a counteracting force: the cyclical properties of creative destruction imply that producer payoffs from intangible capital command low discount rates, which increases their market values. Despite this mitigating effect, exit valuations still only reflect less than half of the total value agents assign to VC-backed innovations. While these estimates reflect the value of innovations that receive VC funding, one may also ask how crucial the VC process itself is for the creation of these innovations. Completely shutting down VC intermediation in all industries and at all times (i.e., forcing |$\iota_{1}(v,t)=0$|⁠, |$\forall v,t$|⁠) reveals that the increased efficiency of the VC process relative to innovative investment by regular firms represents almost half of the estimated baseline value of 1.3%. That is, while the economy would substitute to regular innovative investment under this counterfactual, this activity would be significantly less potent than VC funding. Overall, the magnitudes of these estimates are material when compared to existing evaluations of the impact of other macroeconomic phenomena, such as the costs of business cycles (see Lucas 2003 for an overview). For example, Alvarez and Jermann (2004) estimate that the benefit of eliminating consumption fluctuations corresponding to what is classified as business-cycle frequencies is at most 0.49% of lifetime consumption. These relative magnitudes suggest that VC investments, despite their relatively small scale, have relevant macroeconomic implications. The strong cyclicality of VC activity captured by the calibration substantially affects the above estimates, as risk-averse agents dislike persistent fluctuations in growth. To distill the magnitude of this effect, I also compute the lifetime consumption gain measure under the supposition that VC investments create the same average growth contributions, but without any fluctuations. These constant growth contributions would have a value approximately twice as large as the baseline estimate, indicating the first-order importance of risk adjustments in quantitatively evaluating the impact of venture capital. Furthermore, in Section 3.3, I discuss comparative statics with respect to the preference parameters |$\gamma$| and |$\psi$|⁠, which further emphasize the quantitative relevance of aggregate risk. In addition, to examine the impact of uninsurable idiosyncratic risk, I will analyze in Section 4 how the results are affected when entrepreneurs and VC managers cannot share idiosyncratic venture risk exposures through financial markets. 3.2.2 Venture payoffs and idiosyncratic risk In this section, I evaluate the model’s predictions for payoffs to VC managers and entrepreneurs and for idiosyncratic risk at the venture investment stage. 3.2.2.1 Payoffs to VC managers Panels A and B of Table 3 report management fees and expected carry fees as fractions of commitments. Whereas management fees are paid from fund commitments with certainty, carry fees are stochastic at the fund-level and are only paid when a VC fund generates profits from bringing ventures to the market. As discussed above, fees are set such that fund investors obtain zero “net alpha” in equilibrium (like in Berk and Green 2004). Thus, total fees correspond to what is sometimes referred to as “gross alpha” in the literature. VC funds generate economic profits at the time when new ventures are selected and are still subject to purely idiosyncratic success risk. Total fee revenues are given by the product of commitments reported in Table 2.B, on the one hand, and the sum of the two fee rates |$(f^{man}+f^{car})$|⁠, on the other. While the model is calibrated to match the average level of VC fee revenues relative to contributed capital, it predicts strongly procyclical variation for total fee revenues, consistent with findings of the empirical literature (see, e.g., Ljungqvist and Richardson 2003; Gompers et al. 2008). In addition, in booms, performance-sensitive carry fees represent a larger fraction of total fees. Table 3 VC fees and risk premiums in VC-backed industries |$\Psi'$| A. VC management fees: |$f^{man}(v,Z)$| B. VC carry fees: |$f^{car}(v,Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.010 0.008 0.007 |$\mu$| High 0.239 0.241 0.241 Med 0.038 0.036 0.035 Med 0.223 0.224 0.225 Low n/a n/a n/a Low n/a n/a n/a C. Jump-risk premium: |$rp^{J}(v,Z)$| D. Diffusion-risk premium: |$rp^{D}(v,Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High |$-$|0.055 |$-$|0.097 |$-$|0.141 |$\mu$| High 0.003 0.005 0.008 Med |$-$|0.013 |$-$|0.020 |$-$|0.025 Med 0.003 0.005 0.008 Low |$-$|0.014 |$-$|0.018 |$-$|0.020 Low 0.003 0.005 0.008 A. VC management fees: |$f^{man}(v,Z)$| B. VC carry fees: |$f^{car}(v,Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.010 0.008 0.007 |$\mu$| High 0.239 0.241 0.241 Med 0.038 0.036 0.035 Med 0.223 0.224 0.225 Low n/a n/a n/a Low n/a n/a n/a C. Jump-risk premium: |$rp^{J}(v,Z)$| D. Diffusion-risk premium: |$rp^{D}(v,Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High |$-$|0.055 |$-$|0.097 |$-$|0.141 |$\mu$| High 0.003 0.005 0.008 Med |$-$|0.013 |$-$|0.020 |$-$|0.025 Med 0.003 0.005 0.008 Low |$-$|0.014 |$-$|0.018 |$-$|0.020 Low 0.003 0.005 0.008 Panels A and B list VC management fees and expected carry fee revenues as fractions of VC fund commitments. Panels C and D report jump-risk premiums and diffusion-risk premiums in each of the nine Markov states. Open in new tab Table 3 VC fees and risk premiums in VC-backed industries |$\Psi'$| A. VC management fees: |$f^{man}(v,Z)$| B. VC carry fees: |$f^{car}(v,Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.010 0.008 0.007 |$\mu$| High 0.239 0.241 0.241 Med 0.038 0.036 0.035 Med 0.223 0.224 0.225 Low n/a n/a n/a Low n/a n/a n/a C. Jump-risk premium: |$rp^{J}(v,Z)$| D. Diffusion-risk premium: |$rp^{D}(v,Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High |$-$|0.055 |$-$|0.097 |$-$|0.141 |$\mu$| High 0.003 0.005 0.008 Med |$-$|0.013 |$-$|0.020 |$-$|0.025 Med 0.003 0.005 0.008 Low |$-$|0.014 |$-$|0.018 |$-$|0.020 Low 0.003 0.005 0.008 A. VC management fees: |$f^{man}(v,Z)$| B. VC carry fees: |$f^{car}(v,Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.010 0.008 0.007 |$\mu$| High 0.239 0.241 0.241 Med 0.038 0.036 0.035 Med 0.223 0.224 0.225 Low n/a n/a n/a Low n/a n/a n/a C. Jump-risk premium: |$rp^{J}(v,Z)$| D. Diffusion-risk premium: |$rp^{D}(v,Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High |$-$|0.055 |$-$|0.097 |$-$|0.141 |$\mu$| High 0.003 0.005 0.008 Med |$-$|0.013 |$-$|0.020 |$-$|0.025 Med 0.003 0.005 0.008 Low |$-$|0.014 |$-$|0.018 |$-$|0.020 Low 0.003 0.005 0.008 Panels A and B list VC management fees and expected carry fee revenues as fractions of VC fund commitments. Panels C and D report jump-risk premiums and diffusion-risk premiums in each of the nine Markov states. Open in new tab 3.2.2.2 Payoffs to entrepreneurs Entrepreneurs and network VCs bargain over the surplus from VC intermediation. The calibrated VC bargaining power parameter |$\varrho=0.346$| implies that aggregate entrepreneurial payoffs in each state are about twice as large as aggregate carry fees, which are given by |$\widetilde{com} \cdot f^{car}$|⁠. Aggregate payoffs to entrepreneurs are thus also highly cyclical, just like aggregate carry fee revenues and VC-backed exit volume. Entrepreneurs are compensated with venture ownership stakes, which implies that their payoffs are exposed to idiosyncratic risk. While in the baseline analysis this risk can be shared via financial markets, I analyze in Section 4 the model’s implications when this type of risk sharing is infeasible. 3.2.2.3 Idiosyncratic risk at the venture investment stage At the venture investment stage, idiosyncratic risk is governed by the Poisson processes for innovation arrivals. These Poisson processes imply that over a time interval |$\Delta t$|⁠, the number of venture successes in a given product line has variance |$\bar{h} \Delta t$| and excess skewness |$\frac{1}{\sqrt{\bar{h} \Delta t}}$|⁠, where |$\bar{h}$| represents the state-contingent arrival rates reported in panel C of Table 2 (the relevant states are those where VC intermediation occurs, which are indicated in panel A). As the empirical VC literature focuses on measuring idiosyncratic volatility of returns to venture investments, I analyze model-implied return volatilities in a separate Appendix G. While idiosyncratic return volatilities are infinite over an instant |$dt$|⁠, they also remain large—in excess of 120% annually—over time horizons typically observed in VC data, yielding magnitudes similar to those of comparable estimates by Korteweg and Sorensen (2010). 3.2.3 Asset pricing channels affecting exit valuations Whereas success risk is purely idiosyncratic at the venture investment stage, the cash flows that a venture generates conditional on a success are exposed to aggregate risk. Ventures that are sold in competitive markets, for example in an IPO, thus earn a risk premium. This risk premium is the sum of a jump-risk premium |$rp^{J}$| and a diffusion-risk premium |$rp^{D}$|⁠, which compensate investors for exposures to Markov state shocks |$dZ$| and Brownian shocks |$dB$|⁠, respectively. Panels C and D of Table 3 report these two risk premium components. Consistent with the findings of the existing asset pricing literature, much of the action in total risk premiums is due to compensation for risks associated with low-frequency movements in aggregate growth, which is captured by the jump-risk premiums |$rp^{J}$| in the model (see, e.g., Bansal and Yaron 2004; Parker and Julliard 2005; Hansen, Heaton, and Li 2008). In contrast, diffusion risk premiums are relatively small in magnitude.20 Jump-risk premiums are affected by the industry-specific dynamics of creative destruction, which depend on both productivity and venture funding. The model-implied endogenous exposures to low-frequency risks cause firms to exhibit materially different risk premium dynamics than, for example, a claim to aggregate consumption does. In fact, only 34% of the variation in these risk premiums across states is explained by the risk premiums of a claim to aggregate consumption. Jump-risk premiums are strongly countercyclical and negative, pushing down expected post-IPO returns, which are given by the sum of the two risk premium components and the risk-free rate |$r_{f}$| reported in panel A of Table A4. Conditional expected returns are particularly low during booms (high |$\mu$| states), when they range between minus 1.4% and minus 9.4%. Thus, whereas VC funds’ abnormal returns (which equal fees) spike in booms, reaching about 25% of committed capital, investors’ expected post-IPO returns become locally negative in these times. These low discount rates imply high IPO and acquisition valuations relative to expected future dividends, creating the appearance of a “bubble.” High valuations, in turn, direct large amounts of capital to VC funds. Endogenous declines in discount rates in booms thus play an essential role in rationalizing “lenient” funding standards, in the sense that marginal VC investments yield on average low future dividends. Time-varying funding standards and apparent valuation bubbles are central characteristics of the venture capital “boom-bust-cycle,” which has received significant attention in the literature (see, e.g., Gompers and Lerner 1998; Jeng and Wells 2000; Gompers and Lerner 2003; Korteweg and Sorensen 2010; Kaplan and Lerner 2010). Moreover, poor post-IPO return performance is also consistent with the empirical phenomenon known as the “net issues puzzle” (Loughran and Ritter 1995). While some of the existing literature argues that the strong negative correlation between VC fund inflows and future returns indicates investor irrationality (e.g., Gompers and Lerner 2003) or increased competition between investors in booms (e.g., Gompers and Lerner 2000), the quantitative model presented in this paper illustrates that this pattern also emerges in an environment with rational investors that obtain competitive returns after fees. The fact that jump-risk premiums are lowest in booms indicates that firms in VC-backed industries |$\Psi'$| are particularly good hedges against low-frequency risks (shocks |$dZ$|⁠) in these times. In booms, households are worried about changes in the state |$Z$| that would lead to a persistent slow-down in aggregate growth. However, such shocks are partially good news for ventures that already attracted funding and successfully entered the market during a boom. Although their dividend growth suffers when aggregate growth declines, these incumbents benefit to a first-order degree from a decrease in entry by additional competitors. As venture funding for new firms plummets in downturns, incumbents face a lower threat of displacement, making them relatively safer. This effect on risk premiums is naturally strongest in exactly those industries that have the highest levels of IPOs in booms and exhibit the most extreme busts in entry in ensuing downturns, a pattern typical for young growth industries, such as the IT industry during the internet boom. Procyclical entry is thus self-reinforcing through a risk premium channel: procyclical entry lowers discount rates in booms and raises venture funding in these times. If one classifies VC-backed firms post IPO as “growth firms” in the sense of the asset pricing literature, then empirical analyses of the existing literature lend support to the model’s prediction that these firms have lower exposures to low-frequency risks and therefore low risk premiums (see empirical evidence in Parker and Julliard 2005; Bansal, Dittmar, and Lundblad 2005; Hansen, Heaton, and Li 2008; Bansal, Kiku, and Yaron 2012). Moreover, consistent with the model’s mechanism, IPO volume and VC funding in the data are strongly correlated and procyclical (see, e.g., Gompers and Lerner 1998; Jeng and Wells 2000; Gompers et al. 2008), and lower levels of entry and IPO volume indeed positively affect incumbents’ stock prices and operating performance (see, e.g., Hsu, Reed, and Rocholl 2010). While these empirical observations are consistent with the mechanism and the predictions of the model, I discuss in Section 3.4 several empirical challenges associated with estimating standard risk-based asset pricing models in an environment as the one proposed in this paper. 3.2.4 Post-IPO failure risk and creative destruction The rates of creative destruction |$\bar{h}$| reported in panel C of Table 2 spike during booms when both productivity is high and venture investment increases (see panels E and F). As a result, even those firms that successfully undertook an IPO in a boom are exposed to high failure risk, due to the possibility of being disrupted by newly arriving rivals. The Markov chain dynamics do, however, imply substantial mean reversion, such that annualized exit rates over longer horizons are substantially closer to the 8% average.21 The dynamics of creative destruction in the model are tightly linked to those of IPO volume, consistent with the findings of Jovanovic and Rousseau (2003, 2005), who document spiking rates of entry and declines in the ages of market leaders as central characteristics of technological revolutions, such as the IT boom. While new innovations are important for the process of creative destruction, the timing of when underperforming enterprises are liquidated also affects the reallocation of production factors in practice. Contrary to a classic view going back to Schumpeter (1934) that increased liquidations during crises result in increased restructuring (see, e.g., De Long 1990, for a survey), a more recent literature provides evidence that recessions result in reduced restructuring, and that this stifling of reallocation is costly. For example, Caballero and Hammour (2000, 2005) provide evidence that crises inhibit restructuring due to frictions, such as tight financial market conditions.22 This latter view, in turn, also relates to the intermediation breakdowns that occur in the presented model in recession states. 3.3 Comparative statics In this section, I evaluate how key parameters affecting preferences, matching frictions, and the size of the VC industry influence the results of the analysis. 3.3.1 Risk aversion and elasticity of intertemporal substitution Figures A1 and A2 in the appendix illustrate how equilibrium outcomes change if the parameter values for risk aversion or the elasticity of intertemporal substitution are changed relative to the baseline calibration. Figure A1 reveals the quantitative importance of the risk premium channel for aggregate outcomes. In particular, it shows that higher levels of risk aversion significantly lower discount rates, especially in booms (panel B). The hedging properties of venture equity for low-frequency risks become more important the more averse agents are to risk. As a result, the aggregate value of VC exits is higher and more procyclical (panel A). Moreover, higher market values also attract additional investment, leading to larger growth contributions (panel C). Yet these higher growth contributions are valued less because of their strongly procyclical nature; the lifetime consumption gain measures for |$\gamma=10$| are about 40% lower than they are for |$\gamma=5$| (panel D). Higher levels of risk aversion thus cause ventures’ market prices to be closer to the lifetime consumption gain measures reflecting the total value of VC growth contributions. Comparable adjustments to the elasticity of intertemporal substitution, as considered in Figure A2, also affect discount rates, but significantly less so (panel B). As market values of ventures are hardly affected so are aggregate venture investment and associated growth contributions (panels A and C). Yet the value of those growth contributions is substantially influenced by agents’ elasticity of intertemporal substitution (panel D). Given the fluctuations of VC activity over time, agents assign a higher value to the associated growth contributions if they can more easily substitute consumption intertemporally. 3.3.2 Matching cost Given its fixed-cost nature, the parameter |$c_{\iota}$| affects only the extensive margin of VC intermediation, |$\iota_{1}$|⁠, not the intensive margin, |$n_{1}$| (when holding prices fixed; recall the maximization problem (14)). In the baseline calibration with nine |$Z$|-states, the value for |$c_{\iota}$| implies intermediation breakdowns in the three states with the lowest expected growth |$\mu$|⁠. Here, the costs of creating a match between a VC fund and the most productive entrepreneur exceed the potential value added, that is, |$\tilde{\pi}_{1}<0$|⁠. Yet, in the remaining six states, value added is significantly larger than the costs of creating a match. Moreover, a stabilizing asset pricing effect is at work: the more states exhibit breakdowns of intermediation, the higher are IPO prices in the remaining states with VC activity. As a result, the value of |$c_{\iota}$| needs to be increased very substantially relative to its baseline level before additional states exhibit intermediation breakdowns. Figure A3 in the appendix illustrates such a case by setting |$c_{\iota}=1$|⁠. In this case, intermediation occurs only in high-|$\mu$| states, which reduces the overall value of aggregate VC activity (panel D) and further lowers risk premiums in boom states. In contrast, for a significant range of more moderate upward adjustments to the value of |$c_{\iota}$|⁠, aggregate VC investment dynamics are virtually unchanged relative to those illustrated in Figure 2. Yet, in those cases, the distribution of VC surplus is still affected: as performance-insensitive management fees are increased, lower residual rents go to entrepreneurs and performance-sensitive carry fees. For example, if |$c_{\iota}$| is doubled relative to its baseline value, average management fees roughly double, and the average payoffs to network VCs and entrepreneurs are correspondingly lowered. Finally, lowering the value of |$c_{\iota}$| to zero has relatively minor effects on aggregate outcomes, suggesting that the costs at their baseline level are already quite low. While VC intermediation then does not break down in recession states, productivity in those states is still low, implying low VC activity. As a result, the value of aggregate growth contributions is hardly affected (compare panel D of Figure 2 to Panel D of Figure A3). In other words, the breakdowns of intermediation in recession states are not as costly as one might expect. 3.3.3 Aggregate scale of VC activity I also evaluate the robustness of the results with respect to the aggregate scale of VC activity. Examining this issue is particularly useful if one is concerned that VC activity is not fully recorded in the published VC data. For this analysis, I take the view that the unrecorded activity has the same types of characteristics as the recorded activity, which implies that the only parameter I change relative to the baseline calibration is the relative measure of product lines that may exhibit VC intermediation, |$\Psi'$|⁠. I find that the growth contributions and their value scale with this measure even for significant increases. For example, if aggregate VC commitments were actually twice as large as those matched in the baseline calibration, then the value of growth contributions also would be approximately twice as large. Yet this result naturally hinges on the supposition that unrecorded VC activity is just as potent as the recorded activity. 3.4 Econometric challenges The presented model conceptualizes at least four challenges econometricians face when applying standard risk-based asset pricing models in the context of venture capital, adding to other well-known measurement and selection problems with VC data (see, e.g., Korteweg and Sorensen 2010). First, a typical assumption of standard models, such as the capital asset pricing model (CAPM) or the consumption-based CAPM (CCAPM), are perfectly competitive markets. Yet, in the case of venture capital, investors exhibit strong heterogeneity with regards to their access to investment opportunities. This heterogeneity implies that VC firms may have the ability to make economic profits on their investments. It is generally difficult to cleanly empirically differentiate this profit component from the risk premium component, in particular because the two components have inverse cyclical patterns—whereas abnormal returns peak in booms, conditional risk premiums plummet—and because available VC data typically do not provide reliable market valuations at standard frequencies.23 Second, valuations are affected not only by high-frequency shocks (⁠|$dB$|⁠) but also by low-frequency shocks (⁠|$dZ$|⁠), implying that estimating a single market beta picks up an amalgamation of two risk exposures. In fact, Tables 3.C and 3.D indicate that the two risk premium components associated with high- and low-frequency shocks have opposite signs and opposite cyclical dynamics. As the betas are further correlated with market risk premiums and volatility, estimating unconditional betas also leads to biases (see, e.g., Grant 1977; Jagannathan and Wang 1996). Third, while precisely estimating exposures to low-frequency risks requires long time series of data, VC data spans a relatively short time period, because the industry started to attract significant commitments only in the last three to four decades. Fourth, the model captures the fact that the stock market systematically mismeasures the total wealth portfolio, because it only consists of incumbent firms and misses the value of future entrants. Using a market index, such as the S&P 500, as a proxy for the total wealth portfolio thus can be particularly problematic in times when a large part of future growth is expected to be generated by future entrants. 4. Imperfect Risk Sharing The analysis thus far considered the case in which agents can perfectly share risks. In particular, entrepreneurs and network VCs were able to insure idiosyncratic risks associated with performance-sensitive compensation by selling state-contingent contractual claims in financial markets. In this section, I explore the model’s implications when financial markets are instead incomplete in that risk sharing of this type is infeasible. Exposures to uninsurable idiosyncratic venture income risk can, for example, emerge as the result of optimal contractual arrangements in the presence of agency problems.24 Indeed, the existing empirical literature has documented that while most VC-backed entrepreneurs don’t receive any payoffs associated with IPOs or acquisitions, very few receive over a billion dollars. As a result, Hall and Woodward (2010) conclude that, facing the empirical distribution of these extreme idiosyncratic risks, “an entrepreneur with a coefficient of relative risk aversion of two places a certainty equivalent value only slightly greater than zero” on the interests in her own company. That is, even mildly risk-averse entrepreneurs would apply a very large discount to the almost |${\$}$|6 million in exit payoffs they receive on average. While solving economies with heterogeneous agents and imperfect risk sharing typically requires approximation techniques or computationally intensive methods, the proposed model remains tractable in the presence of these features. As detailed below, this tractability obtains because risk-averse agents in the model are exposed to extremely risky venture payoffs, leading to extreme discounts in certainty equivalent values, similar to the findings of Hall and Woodward (2010) highlighted above. Moreover, the analysis does not require any changes to the specification of the baseline model, except for the introduction of the restriction that entrepreneurs and network VCs cannot enter ex ante contracts that specify payments conditional on the future realizations of their own venture capital opportunities (i.e., network connections or superior ideas). In other words, entrepreneurs and VCs have to stay exposed to the idiosyncratic risk associated with their performance-sensitive compensation until a successful IPO has taken place. To streamline the exposition, I focus on providing intuition and economic insights here in the main text. Appendix H provides the associated detailed formal analysis. Following this analysis, I first conjecture optimal dynamic policies, then discuss agents’ value functions given these policies, and finally show that the conjectured policies are indeed privately optimal. The next three subsections describe these three steps. 4.1 Consumption and investment policies Suppose that at the beginning of each period |$[t+dt)$|⁠, agents trade to invest their tradable wealth in the portfolio of tradable securities at time |$t$|⁠. The set of these tradable securities expands after each period, when successful VCs and entrepreneurs sell their venture stakes that were previously not tradable (pre-IPO). Moreover, conjecture all agents in the economy determine their consumption flow for the period based on the same consumption-to-tradable-wealth ratio. Given these conjectured policies, all agents not experiencing idiosyncratic venture shocks then share the same tradable wealth growth and the same consumption growth. In contrast, the tradable wealth growth of agents idiosyncratically succeeding with ventures is further increased by the IPO equity they can sell at the beginning of the next period. As these agents’ shares of tradable wealth jump up so do their consumption shares. As consumption and tradable wealth shares add up to one, all other agents see their shares decline. In particular, their consumption growth is given by $ \begin{align}\label{eq:ConsumptionDynamicswithDisplacement} \frac{dC_{j}(t)}{C_{j}(t)}&= \mu(Z_{t}) dt + \sigma(Z_{t})dB - \underbrace{ \frac{\int_{\Psi}\bar{\iota}_{1}(v,Z_{t}) \tilde{\pi}_{1}(v,Z_{t}) dv}{\tilde{P}_{T}(Z_{t})}dt}_{\text{Adjustment due to declining wealth share}}, \end{align} $ (28) where |$C_{j}(t)$| denotes the consumption flow rate of an agent identified by the tuple |$j \equiv (S,i_{S}',i_{S}'')$|⁠, the parameters |$\mu$| and |$\sigma$| still represent the drift and local volatility of aggregate output, |$\tilde{P}_{T}$| reflects the value of tradable wealth, and the term |$\int_{\Psi}\bar{\iota}_{1}(v,Z) \tilde{\pi}_{1}(v,Z) dv$| represents the aggregate venture equity claims sold by entrepreneurs and network VCs. Only a measure zero (of order |$dv$|⁠) of agents obtain idiosyncratic shocks in each period, so each agent’s consumption growth is almost surely given by Equation (28).25 4.2 Certainty equivalent value of venture opportunities While the probability of a venture success in a given industry |$v$| is of order |$dt$| over a period |$[t,t+dt)$|⁠, an individual network VC’s and entrepreneur’s chance of being involved in that venture is of order |$dv$|⁠. Agents’ individual opportunities are atomistic relative to the continuum of venture opportunities unfolding over time within an industry. Yet an individual agent’s success probability of order |$dt \cdot dv$| over a period of length |$dt$| is not necessarily negligible, because successes can yield tradable wealth gains of order |$1/dv$| (as agents are atomistic relative to industries so is their initial wealth relative to the market value of an industry incumbent). In particular, a risk-neutral agent’s value function would reflect these expected payoffs of order |$dt$|⁠. Yet Appendix H formally shows that an agent with risk aversion |$\gamma >1$| discounts these extremely risky gains so much that her value function for any given current level of tradable wealth becomes indistinguishable from the one of an agent not expecting any idiosyncratic venture gains in the future (although ex post, the realized dynamics of wealth can differ dramatically across these two types of agents). 4.3 Process for marginal utility and incentives to deviate from conjectured policies These results for both consumption growth and continuation utility are key for agents’ processes for marginal utility, as agents have recursive preferences. The processes for marginal utility, in turn, determine whether agents have incentives to deviate from the conjectured consumption and investment policies. Given the above results, all agents not experiencing idiosyncratic venture shocks have the same growth dynamics for marginal utility. In contrast, agents experiencing an idiosyncratic wealth gain, exhibit a downward jump in marginal utility. Yet this latter scenario has zero probability measure: a probability of order |$(dt)^2$| over a time interval |$d t$|⁠. Moreover, the conditional probability of any tradable asset paying in this agent-specific idiosyncratic state has zero measure, implying no impact on an agent’s willingness to pay for such an asset. At the conjectured allocations, agents assign the same value to all tradable claims, and thus, do not wish to engage in further trades. The equilibrium market prices of all tradable assets are then observationally equivalent to those obtaining in an economy where all agents’ consumption growth follows (28).26 4.4 Summary of reasons for tractability The conjectured consumption and investment decisions are optimal independent of the cross-sectional wealth distribution. Thus, keeping track of the wealth distribution is not necessary to characterize agents’ equilibrium behavior — the only aggregate state variables are still |$Y$| and |$Z$|⁠, just like in the baseline analysis. As outlined above, this result is an implication of agents’ risk aversion in combination with the extreme nature of idiosyncratic risks faced by entrepreneurs and network VCs. 4.5 Implications of uninsurable venture income risk To gauge the impact of imperfect risk sharing on the quantitative results discussed in the previous sections, I re-solve the model given the calibration choices described in Section 3.1. A convenient feature of the model is that all industry-level equilibrium conditions presented in Section 2.1 and the aggregate output dynamics and labor market-clearing conditions in Section 2.2 still hold in the setting with imperfect risk sharing. Solutions to these equilibrium conditions are, however, affected by the fact that consumption dynamics and the corresponding processes for marginal utility are different under incomplete risk sharing. First, I evaluate how much network VCs and entrepreneurs lose due to the increased riskiness of their consumption profile relative to the perfect risk-sharing benchmark. The extreme discount discussed above implies that these agents face potentially dramatic losses to expected lifetime utility, although the magnitudes depend on an agent’s initial tradable wealth. In certainty equivalent terms, effectively all incremental value obtained from future performance-sensitive VC income is lost. Given the extreme risks in the model, this result has an upper-bound character. Yet, actual risk discounts may be not too far from this upper bound, in particular in light of the above-discussed findings by Hall and Woodward (2010). Second, I examine the value agents assign to aggregate VC-backed innovations and find that it is equivalent to 1.2% of the lifetime consumption they receive in the economy with imperfect risk sharing (see the last paragraph of Appendix H for formal details). This percentage value does not differ much from the one obtained in the economy with perfect risk sharing, but it refers to different underlying consumption streams. While at any point in time, the consumption levels for individual agents differ across the two economies, the relative value assigned to future VC-backed innovations is similar, as even under perfect risk sharing, performance-sensitive venture compensation payoffs represent only a small fraction of innovations’ total value. Average aggregate performance-sensitive payoffs are about 15% of the 1.3% baseline estimate; a large fraction of venture stakes is held by diversified investors, and ventures’ market prices reflect less than half of the total value agents obtain from VC-backed innovations. Finally, even under perfect risk sharing, these contractual payoffs are valued at a large discount relative to their average value due to their strongly procyclical nature. Third, I find that SDF dynamics and investment dynamics are not affected significantly. As discussed above, the SDF dynamics relevant for tradable assets are effectively governed by the perturbed consumption growth dynamics shown in Equation (28). Relative to the perfect risk-sharing economy, these consumption dynamics are adjusted only by a term involving the value of venture claims obtained by VCs and entrepreneurs relative to all tradable claims. The average value of this term is 0.25 bps and the maximum value is less than 1 bp. As the total wealth gains of VCs and entrepreneurs are procyclical, imperfect risk sharing then has a very minor dampening effect on the cyclicality of almost all agents’ consumption growth. The dynamics of the SDF pricing tradable assets therefore also remain very similar when compared to the baseline analysis. Facing nearly identical market prices, agents and firms also behave almost the same way. Consequently, the model with imperfect risk sharing is still consistent with the moments targeted in Section 3.1 and generates quantitatively similar predictions as those discussed in Section 3.2. In sum, the analysis reveals that whereas quantities and prices of assets traded in competitive financial markets remain similar, agents’ expected lifetime utility from venture capital income and aggregate VC activity is significantly affected by imperfect risk sharing. 5. Conclusion This paper has developed a model of VC intermediation that explains central empirical facts about the magnitude and cyclicality of VC activity. The framework reveals that accounting for the market structure and the risk properties of venture capital is of first-order importance for both explaining observed empirical regularities and evaluating VC investments’ macroeconomic impact. In particular, boom-bust cycle dynamics of VC activity and uninsurable idiosyncratic risks associated with VC contracts substantially reduce the value risk-averse agents assign to venture capital income and VC-backed innovations’ contributions to consumption growth. Yet, despite these risk discounts, VC investments have a significantly larger macroeconomic impact than that suggested by aggregate VC exit valuations, because producers’ market prices reflect less than half of the total value agents obtain from VC-funded innovations. When compared to estimates of the costs of business cycles (see, e.g., Lucas 2003), the magnitudes suggest that regulatory changes, such as those that appear to have led to the emergence of the VC industry (see footnote 5), can have relevant macroeconomic implications. The proposed dynamic general equilibrium model is highly tractable, even in the presence of imperfect risk sharing, and could be used as a stepping stone to evaluations of policies affecting VC activity and entrepreneurship, including patent regulation and investment mandates for institutional investors. Such endeavors are left for future research. This paper is partially based on my PhD dissertation and was previously circulated under the title “Venture Capital Cycles.” I thank my advisors Lars Peter Hansen (Chair), Stavros Panageas, L̆ubos̆ Pástor, Raghu Rajan, and Pietro Veronesi for their invaluable guidance and support. For helpful comments, I also thank the editor Stijn Van Nieuwerburgh; two anonymous referees; Fernando Alvarez, Doug Breeden, John Cochrane, Doug Diamond, Tarek Hassan, Steven Kaplan, Arthur Korteweg, Gregor Matvos, Marcus Opp, David Robinson, Tom Sargent, Robert Vishny, Jessica Wachter, and Amir Yaron; and seminar participants at Berkeley Haas, European Central Bank, Harvard Business School, MIT Sloan, NBER SI, NYU Economics, Princeton Economics, Stanford GSB, SITE, UBC Sauder, UCLA Anderson, University of Chicago Booth School of Business, and the Wharton School. Appendix A. Definitions Definition A1 (Allocation). An allocation in this economy is given by stochastic processes of agents’ consumption |$[C_{j}(t)]_{j\in \Phi, t=0}^{\infty}$|⁠, where |$j = (S,i_{S}',i_{S}'')$|⁠, quantities of superior and regular innovative ideas implemented in each industry, |$[n_{1}(v,t),n_{2}(v,t)]_{v \in \Psi,t=0}^{\infty}$|⁠, matches established between a VC fund and the best entrepreneur in each industry |$[\iota_{1}(v,t)]_{v \in \Psi,t=0}^{\infty}$|⁠, qualities of leading-edge intermediate goods in each industry |$[q(v,t)]_{v \in \Psi,t=0}^{\infty}$|⁠, and quantities of intermediate goods produced in each industry |$[x(v,t)]_{v \in \Psi,t=0}^{\infty}$|⁠. Definition A2 (Decentralized Equilibrium). An equilibrium in this economy is given by an allocation and stochastic processes for wage rates |$[w_{B}(t),w_{W}(t)]_{t=0}^{\infty}$|⁠, values of venture ownership stakes obtained by network VCs and entrepreneurs in each industry, the stochastic discount factor |$[\xi (t)]_{t=0}^{\infty}$|⁠, and intermediate goods prices denoted by |$[p_{x}(v,t)]_{v \in \Psi,t=0}^{\infty}$|⁠, such that firms’ pricing, production, and funding of regular ideas maximizes their value; VC funds’ intermediation decisions maximize the market value of network VCs’ compensation; agents choose their paths of consumption optimally; labor markets, intermediate good markets, and the final good market clear. Table A1 Consumption growth A. Consumption growth drifts: |$\mu(Z)$| B. Local risk exposures: |$\sigma(Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.041 0.048 0.053 |$\mu$| High 0.021 0.027 0.032 Med 0.018 0.018 0.019 Med 0.021 0.027 0.032 Low |$-$|0.004 |$-$|0.011 |$-$|0.016 Low 0.021 0.027 0.032 A. Consumption growth drifts: |$\mu(Z)$| B. Local risk exposures: |$\sigma(Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.041 0.048 0.053 |$\mu$| High 0.021 0.027 0.032 Med 0.018 0.018 0.019 Med 0.021 0.027 0.032 Low |$-$|0.004 |$-$|0.011 |$-$|0.016 Low 0.021 0.027 0.032 Panels A and B report the state-contingent values of the consumption growth drifts and the local risk exposures, which match the values considered in Chen (2010). Open in new tab Table A1 Consumption growth A. Consumption growth drifts: |$\mu(Z)$| B. Local risk exposures: |$\sigma(Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.041 0.048 0.053 |$\mu$| High 0.021 0.027 0.032 Med 0.018 0.018 0.019 Med 0.021 0.027 0.032 Low |$-$|0.004 |$-$|0.011 |$-$|0.016 Low 0.021 0.027 0.032 A. Consumption growth drifts: |$\mu(Z)$| B. Local risk exposures: |$\sigma(Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.041 0.048 0.053 |$\mu$| High 0.021 0.027 0.032 Med 0.018 0.018 0.019 Med 0.021 0.027 0.032 Low |$-$|0.004 |$-$|0.011 |$-$|0.016 Low 0.021 0.027 0.032 Panels A and B report the state-contingent values of the consumption growth drifts and the local risk exposures, which match the values considered in Chen (2010). Open in new tab Table A2 Stationary distribution of Markov states Probability of Markov state: Pr|$[Z]$| |$\sigma$| Low Med High |$\mu$| High 0.016 0.138 0.015 Med 0.059 0.542 0.062 Low 0.016 0.138 0.015 Probability of Markov state: Pr|$[Z]$| |$\sigma$| Low Med High |$\mu$| High 0.016 0.138 0.015 Med 0.059 0.542 0.062 Low 0.016 0.138 0.015 The table reports the stationary distribution associated with the Markov generator matrix, which matches the one considered in Chen (2010). Open in new tab Table A2 Stationary distribution of Markov states Probability of Markov state: Pr|$[Z]$| |$\sigma$| Low Med High |$\mu$| High 0.016 0.138 0.015 Med 0.059 0.542 0.062 Low 0.016 0.138 0.015 Probability of Markov state: Pr|$[Z]$| |$\sigma$| Low Med High |$\mu$| High 0.016 0.138 0.015 Med 0.059 0.542 0.062 Low 0.016 0.138 0.015 The table reports the stationary distribution associated with the Markov generator matrix, which matches the one considered in Chen (2010). Open in new tab Table A3 State-dependent aggregate growth A. Exogenous drift: |$\delta(Z)$| B. Endogenous drift: |$\log[\kappa] \int_{\Psi} \bar{h}(v,Z) dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 3.53 4.18 4.71 |$\mu$| High 0.54 0.58 0.61 Med 1.35 1.36 1.38 Med 0.47 0.47 0.47 Low 0.88 1.53 2.05 Low 0.45 0.44 0.44 A. Exogenous drift: |$\delta(Z)$| B. Endogenous drift: |$\log[\kappa] \int_{\Psi} \bar{h}(v,Z) dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 3.53 4.18 4.71 |$\mu$| High 0.54 0.58 0.61 Med 1.35 1.36 1.38 Med 0.47 0.47 0.47 Low 0.88 1.53 2.05 Low 0.45 0.44 0.44 Panels A and B report the exogenous and endogenous components of aggregate growth. The endogenous growth components reflect creative destruction across all industries in the economy |$\Psi = \Psi' \cup \Psi''$|⁠. Values are reported in percentage points. Open in new tab Table A3 State-dependent aggregate growth A. Exogenous drift: |$\delta(Z)$| B. Endogenous drift: |$\log[\kappa] \int_{\Psi} \bar{h}(v,Z) dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 3.53 4.18 4.71 |$\mu$| High 0.54 0.58 0.61 Med 1.35 1.36 1.38 Med 0.47 0.47 0.47 Low 0.88 1.53 2.05 Low 0.45 0.44 0.44 A. Exogenous drift: |$\delta(Z)$| B. Endogenous drift: |$\log[\kappa] \int_{\Psi} \bar{h}(v,Z) dv$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 3.53 4.18 4.71 |$\mu$| High 0.54 0.58 0.61 Med 1.35 1.36 1.38 Med 0.47 0.47 0.47 Low 0.88 1.53 2.05 Low 0.45 0.44 0.44 Panels A and B report the exogenous and endogenous components of aggregate growth. The endogenous growth components reflect creative destruction across all industries in the economy |$\Psi = \Psi' \cup \Psi''$|⁠. Values are reported in percentage points. Open in new tab Table A4 Risk-free rate and white-collar wages A. Risk-free rate: |$r_f(Z)$| B. White-collar wages: |$\tilde{w}_W(Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.037 0.039 0.040 |$\mu$| High 0.106 0.108 0.110 Med 0.023 0.021 0.019 Med 0.098 0.097 0.097 Low 0.008 0.001 |$-$|0.005 Low 0.092 0.090 0.088 A. Risk-free rate: |$r_f(Z)$| B. White-collar wages: |$\tilde{w}_W(Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.037 0.039 0.040 |$\mu$| High 0.106 0.108 0.110 Med 0.023 0.021 0.019 Med 0.098 0.097 0.097 Low 0.008 0.001 |$-$|0.005 Low 0.092 0.090 0.088 Panels A and B report the risk-free rate and the white-collar wage-to-output ratio in each of the nine Markov states. Open in new tab Table A4 Risk-free rate and white-collar wages A. Risk-free rate: |$r_f(Z)$| B. White-collar wages: |$\tilde{w}_W(Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.037 0.039 0.040 |$\mu$| High 0.106 0.108 0.110 Med 0.023 0.021 0.019 Med 0.098 0.097 0.097 Low 0.008 0.001 |$-$|0.005 Low 0.092 0.090 0.088 A. Risk-free rate: |$r_f(Z)$| B. White-collar wages: |$\tilde{w}_W(Z)$| |$\sigma$| |$\sigma$| Low Med High Low Med High |$\mu$| High 0.037 0.039 0.040 |$\mu$| High 0.106 0.108 0.110 Med 0.023 0.021 0.019 Med 0.098 0.097 0.097 Low 0.008 0.001 |$-$|0.005 Low 0.092 0.090 0.088 Panels A and B report the risk-free rate and the white-collar wage-to-output ratio in each of the nine Markov states. Open in new tab Table A5 State-contingent productivity in industries |$\Psi''$| Productivity: |$\theta(v,Z)$| |$\sigma$| Low Med High |$\mu$| High 0.068 0.069 0.070 Med 0.066 0.066 0.066 Low 0.063 0.063 0.062 Productivity: |$\theta(v,Z)$| |$\sigma$| Low Med High |$\mu$| High 0.068 0.069 0.070 Med 0.066 0.066 0.066 Low 0.063 0.063 0.062 The table reports the state-contingent values of productivity in industries |$\Psi''$|⁠, which follow from the calibrated relation: $ \begin{align} \theta(v,Z) = a + b \cdot (\mu(Z) - \min_{\forall Z}\{\mu(Z)\}), \end{align} $ (A1) where, |$a=0.062$|⁠, |$b=0.11$|⁠, and where |$\mu(Z)$| are the aggregate growth drifts from Chen (2010). Open in new tab Table A5 State-contingent productivity in industries |$\Psi''$| Productivity: |$\theta(v,Z)$| |$\sigma$| Low Med High |$\mu$| High 0.068 0.069 0.070 Med 0.066 0.066 0.066 Low 0.063 0.063 0.062 Productivity: |$\theta(v,Z)$| |$\sigma$| Low Med High |$\mu$| High 0.068 0.069 0.070 Med 0.066 0.066 0.066 Low 0.063 0.063 0.062 The table reports the state-contingent values of productivity in industries |$\Psi''$|⁠, which follow from the calibrated relation: $ \begin{align} \theta(v,Z) = a + b \cdot (\mu(Z) - \min_{\forall Z}\{\mu(Z)\}), \end{align} $ (A1) where, |$a=0.062$|⁠, |$b=0.11$|⁠, and where |$\mu(Z)$| are the aggregate growth drifts from Chen (2010). Open in new tab Figure A1 Open in new tabDownload slide Comparative statics for risk aversion |$\gamma$| The graphs illustrate the impact of risk aversion |$\gamma$| on equilibrium outcomes. The plots are based on the time series of |$Z$|-state realizations obtained for Figure 2 (see Figure 2’s caption for details). The panels illustrate the model-implied time series of aggregate VC exit valuations scaled by aggregate consumption (panel A), local post-IPO discount rates (risk-free rate plus risk premium) in VC-backed industries |$\Psi'$| (panel B), VC investments’ growth contributions (panel C), and the value of future VC-backed innovations’ impact on lifetime utility (conditional on the current state |$Z$|⁠) expressed as a fraction of lifetime consumption (panel D). Post-IPO discount rates in panel B are plotted only for years where the calibration predicts a positive volume of venture-backed exits (i.e., where |$\bar{\iota}_{1}(v,Z_{t})=1$|⁠). All numbers are reported in percentage points. Figure A1 Open in new tabDownload slide Comparative statics for risk aversion |$\gamma$| The graphs illustrate the impact of risk aversion |$\gamma$| on equilibrium outcomes. The plots are based on the time series of |$Z$|-state realizations obtained for Figure 2 (see Figure 2’s caption for details). The panels illustrate the model-implied time series of aggregate VC exit valuations scaled by aggregate consumption (panel A), local post-IPO discount rates (risk-free rate plus risk premium) in VC-backed industries |$\Psi'$| (panel B), VC investments’ growth contributions (panel C), and the value of future VC-backed innovations’ impact on lifetime utility (conditional on the current state |$Z$|⁠) expressed as a fraction of lifetime consumption (panel D). Post-IPO discount rates in panel B are plotted only for years where the calibration predicts a positive volume of venture-backed exits (i.e., where |$\bar{\iota}_{1}(v,Z_{t})=1$|⁠). All numbers are reported in percentage points. Figure A2 Open in new tabDownload slide Comparative statics for the elasticity of intertemporal substitution |$\psi$|⁠. The graphs illustrate the impact of the elasticity of intertemporal substitution |$\psi$| on equilibrium outcomes. The plots are based on the time series of |$Z$|-state realizations obtained for Figure 2 (see Figure 2’s caption for details). The panels illustrate the model-implied time series of aggregate VC exit valuations scaled by aggregate consumption (panel A), local post-IPO discount rates (risk-free rate plus risk premium) in VC-backed industries |$\Psi'$| (panel B), VC investments’ growth contributions (panel C), and the value of future VC-backed innovations’ impact on lifetime utility (conditional on the current state |$Z$|⁠) expressed as a fraction of lifetime consumption (panel D). Post-IPO discount rates in panel B are plotted only for years where the calibration predicts a positive volume of venture-backed exits (i.e., where |$\bar{\iota}_{1}(v,Z_{t})=1$|⁠). All numbers are reported in percentage points. Figure A2 Open in new tabDownload slide Comparative statics for the elasticity of intertemporal substitution |$\psi$|⁠. The graphs illustrate the impact of the elasticity of intertemporal substitution |$\psi$| on equilibrium outcomes. The plots are based on the time series of |$Z$|-state realizations obtained for Figure 2 (see Figure 2’s caption for details). The panels illustrate the model-implied time series of aggregate VC exit valuations scaled by aggregate consumption (panel A), local post-IPO discount rates (risk-free rate plus risk premium) in VC-backed industries |$\Psi'$| (panel B), VC investments’ growth contributions (panel C), and the value of future VC-backed innovations’ impact on lifetime utility (conditional on the current state |$Z$|⁠) expressed as a fraction of lifetime consumption (panel D). Post-IPO discount rates in panel B are plotted only for years where the calibration predicts a positive volume of venture-backed exits (i.e., where |$\bar{\iota}_{1}(v,Z_{t})=1$|⁠). All numbers are reported in percentage points. Figure A3 Open in new tabDownload slide Comparative statics for matching cost |$c_{\iota}$| The graphs illustrate the impact of the quantity of white-collar human capital needed to realize VC fund matches in industries |$\Psi'$| on equilibrium outcomes. The plots are based on the time series of |$Z$|-state realizations obtained for Figure 2 (see Figure 2’s caption for details). The panels illustrate the model-implied time series of aggregate VC exit valuations scaled by aggregate consumption (panel A), local post-IPO discount rates (risk-free rate plus risk premium) in VC-backed industries |$\Psi'$| (panel B), VC investments’ growth contributions (panel C), and the value of future VC-backed innovations’ impact on lifetime utility (conditional on the current state |$Z$|⁠) expressed as a fraction of lifetime consumption (panel D). Post-IPO discount rates in panel B are plotted only for years where the calibration predicts a positive volume of venture-backed exits (i.e., where |$\bar{\iota}_{1}(v,Z_{t})=1$|⁠). All numbers are reported in percentage points. Figure A3 Open in new tabDownload slide Comparative statics for matching cost |$c_{\iota}$| The graphs illustrate the impact of the quantity of white-collar human capital needed to realize VC fund matches in industries |$\Psi'$| on equilibrium outcomes. The plots are based on the time series of |$Z$|-state realizations obtained for Figure 2 (see Figure 2’s caption for details). The panels illustrate the model-implied time series of aggregate VC exit valuations scaled by aggregate consumption (panel A), local post-IPO discount rates (risk-free rate plus risk premium) in VC-backed industries |$\Psi'$| (panel B), VC investments’ growth contributions (panel C), and the value of future VC-backed innovations’ impact on lifetime utility (conditional on the current state |$Z$|⁠) expressed as a fraction of lifetime consumption (panel D). Post-IPO discount rates in panel B are plotted only for years where the calibration predicts a positive volume of venture-backed exits (i.e., where |$\bar{\iota}_{1}(v,Z_{t})=1$|⁠). All numbers are reported in percentage points. B. Incumbent Optimization and Proof of Proposition 2 First, I show why a firm does not fund additional innovative ideas in a product line where it already owns the best patent. The setup ensures that an incumbent’s maximization problem is separable across industries, because each incumbent is restricted to operate in a finite number of industries |$v$|⁠, which have zero measure. Let |$i \geq 1$| denote the number of innovations that the current incumbent’s best patent is ahead relative to the next-best patent owned by a different firm, that is, the firm’s patent has a quality advantage of a factor |$\kappa \left( v,t\right) =\kappa ^{i}$|⁠. The value the incumbent firm obtains from this patent is given by $ \begin{equation} \label{eq:patentvalue} P_{i}\left( v,t\right) = \mathbb{E}_{t}\left[ \int_{t}^{\tau ^{\ast} }\frac{\xi _{\tau} }{\xi _{t}}Y_{\tau} \left( 1-\frac{1}{\kappa ^{i}}\right) d\tau + \frac{\xi _{s^{\ast} }}{\xi _{t}}P_{i+1}\left( v,s^{\ast} \right) \right] , \end{equation} $ (B1) where the arrival times |$\tau ^{\ast} \equiv \inf \left\{ \tau :M\left( v,\tau \right) >M\left( v,t\right) \right\}$| and |$s^{\ast} \equiv \inf \left\{ s:\kappa \left( v,s\right) >\kappa \left( v,t\right) \right\}$| are affected by the incumbent’s optimal policy for the funding of regular ideas. Conjecture that the associated value function takes the form: $ \begin{equation}\label{eq:patentvaluefunction} P_{i}\left( v,Y_{t},Z_{t}\right) =Y_{t}\cdot \tilde{P}\left( v,Z_{t}\right) \frac{1-\frac{1}{\kappa ^{i}}}{1-\frac{1}{\kappa} }. \end{equation} $ (B2) Incumbents and nonincumbent firms have symmetric access to regular innovative ideas, so the per-idea success rate and the per-idea funding cost are equal for both types of firms. To compare incumbent and nonincumbent firms’ incentives to fund regular ideas it thus suffices to compare each firm type’s gain conditional on a success. If an incumbent firm succeeds in innovating again in the same product line, its scaled value jumps by $ \begin{align}\label{eq:gainfrominnovation} \tilde{P}_{i+1}\left( v,Y,Z\right) - \tilde{P}_{i}\left( v,Y,Z\right) =& \quad Y \cdot \tilde{P}\left( v,Z\right) \frac{ 1-\frac{1}{\kappa ^{i+1}}}{1-\frac{1}{\kappa} } - Y \cdot \tilde{P}\left( v,Z\right) \frac{ 1-\frac{1}{\kappa ^{i}}}{1-\frac{1}{\kappa} } \notag\\ =& \quad Y \cdot \tilde{P}\left( v,Z\right) \frac{1}{\kappa ^{i}} \notag \\ < & \quad Y \cdot \tilde{P}\left( v,Z\right)\!, \end{align} $ (B3) where the last inequality holds because |$\kappa >1$|⁠. Thus, under the conjectured value function, incumbent firms gain less than nonincumbent firms, which gain |$Y \cdot \tilde{P}\left( v,Z\right)$| conditional on a success. As the free-entry condition (8) implies that nonincumbent firms just break even at the equilibrium per-idea success rate and per-idea funding cost, incumbent firms cannot break even when funding innovative ideas in their own product line, as they gain less from innovating (see inequality (B3)). Thus, incumbent firms optimally refrain from funding additional ideas in the product line where they already own the best patent. Finally, I verify the conjectured value function (B2), which also proves Proposition 2. As shown above, under this conjectured value function, incumbents optimally do not fund additional ideas in their own product line, and the arrival rate of new innovations |$\bar{h}$| in the product line is determined as described in Proposition 1. Substituting the conjectured value function (B2) into the HJB equation associated with (B1) yields the following linear system of equations determining the values |$\tilde{P}\left( v,Z\right)$| for all |$Z\in \Omega$|⁠: $ \begin{eqnarray} \label{eq:HJBPatenti} 0 &=&1-\frac{1}{\kappa ^{i}}-\left( r_{f}\left( Z\right) +\gamma \sigma ^{2}\left( Z\right) +\bar{h}\left( v,Z\right) -\mu \left( Z\right) \right) \tilde{P}\left( v,Z\right) \frac{ 1-\frac{1}{\kappa ^{i}}}{1-\frac{1}{\kappa} } \notag \\ &&-\sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }e^{\zeta \left( Z,Z^{\prime} \right)} \left( \tilde{P}\left( v,Z^{\prime} \right) -\tilde{P}\left( v,Z\right) \right) \frac{ 1-\frac{1}{\kappa ^{i}}}{1-\frac{1}{\kappa} }, \end{eqnarray} $ (B4) confirming that the conjecture (B2) is indeed a solution to the firm’s HJB equation. Given that at date |$t=0$| every incumbent’s patent has a quality advantage of a factor |$\kappa$|⁠, the only relevant case is |$\kappa\left( v,t\right) =\kappa$|⁠. Finally, setting |$i=1$| in Equation (B4), rearranging, and defining the diffusion- and jump-risk premiums $ \begin{equation} rp^{D}\left( v,Z\right) \equiv \gamma \sigma ^{2}\left( Z\right)\!, \quad \quad \quad rp^{J}\left( v,Z\right)\equiv \sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }\left( e^{\zeta \left( Z,Z^{\prime} \right)} -1\right) \left(1 -\frac{\tilde{P}\left( v,Z^{\prime }\right)} {\tilde{P}\left( v,Z\right)} \right)\!, \end{equation} $ (B5) yields Equation (20) in Proposition (2). C. Proof of Proposition 3 In an economy where incumbent firms in each industry |$v \in \Psi$| have a quality advantage of a factor |$\kappa$| relative to the next-best producer, incumbents’ profit maximization in combination with the market-clearing condition for blue-collar labor (25) yield a symmetric allocation of blue-collar labor across industries, that is, |$x\left( v,t\right) =1$| for all |$v \in \Psi$|⁠. Plugging this result into the final good production technology yields $ \begin{equation} Y_{t} =A_{t} Q_{t}, \end{equation} $ (C1) where I define the endogenous aggregate quality level: $ \begin{align} Q_{t} & \equiv \exp \left( \int_{\Psi}\log \left[ q\left( v,t\right) \right] dv\right)\!. \end{align} $ (C2) Innovation arrivals in each industry |$v$| follow idiosyncratic Poisson processes with hazard rates |$\bar{h}(v,Z)$| defined in Equation (18). The aggregate quality level |$Q_{t}$| then grows at the state-dependent rate: $ \begin{equation} \frac{\frac{dQ_{t}} {dt}}{Q_{t}} =\log \left[ \kappa \right] \int_{\Psi} \bar{h}\left( v,Z_{t} \right)dv. \end{equation} $ (C3) Thus, final good output follows the stochastic differential equation $ \begin{equation} \frac{dY_{t}}{Y_{t}}=\left(\delta \left( Z_{t}\right) + \log \left[ \kappa \right] \int_{\Psi} \bar{h}\left( v,Z_{t}\right) dv \right) dt+\sigma \left( Z_{t}\right) dB_{t}. \end{equation} $ (C4) D. Proof of Proposition 4 In the baseline analysis of the model, agents can optimally share risks by trading state-contingent claims to future compensation, in particular, white-collar and blue-collar wages, and venture ownership shares provided to network VCs and entrepreneurs. As a result of perfect risk sharing, agents consume constant fractions of aggregate output. If at date 0 agent |$j=(S,i_{S}',i_{S}'')$| has a wealth share |$\varpi_{j}$|⁠, then perfect risk sharing implies that this agent consumes at any time |$t\geq 0$|⁠: $ \begin{align} \label{eq:OptConsRiskSharing} C_{j}(t)=\varpi_{j} Y_{t}. \end{align} $ (D1) The agent’s value function is given by $ \begin{equation} J\left(\varpi_{j}, Y_{t},Z_{t}\right) =\mathbb{E}_{t}\left[ \int_{t}^{\infty} m\left( \varpi_{j} Y_{\tau },J_{\tau} \right) d\tau \right], \end{equation} $ (D2) which yields the associated Hamilton-Jacobi-Bellman (HJB) equation: $ \begin{eqnarray} 0 &=&m\left(\varpi_{j} Y,J\left(\varpi_{j},Y,Z\right) \right) +J_{Y}\left( \varpi_{j},Y,Z\right) Y\mu\left( Z\right) +\frac{1}{2}J_{YY}\left( \varpi_{j},Y,Z\right) Y^{2}\sigma^{2}\left( Z\right) \notag \\ &&+\sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }\left( J\left(\varpi_{j},Y,Z^{\prime }\right) -J\left(\varpi_{j},Y,Z\right) \right)\!. \end{eqnarray} $ (D3) Due to the isoelastic properties of the setup, I conjecture that the solution for |$J$| takes the form $ \begin{equation} \label{eq:HHValueFunctionConj} J\left(\varpi_{j}, Y,Z\right) =F\left( Z\right) \frac{Y^{1-\gamma} }{1-\gamma} \varpi_{j}^{1-\gamma}. \end{equation} $ (D4) Substituting the conjecture (D4) into the HJB equation yields the following system of equations that |$F(Z)$| solves for all |$Z\in \Omega$|⁠: $ \begin{eqnarray} \label{eq:HJB-Equation-F} 0 &=&\left( \frac{\beta (1-\gamma )}{\rho} \left( F\left( Z\right) ^{-\frac{ \rho} {1-\gamma}}-1\right) +(1-\gamma )\mu \left( Z\right) -\frac{1}{2}\gamma (1-\gamma ) \sigma ^{2}\left( Z\right) \right) F\left( Z\right) \notag \\ &&+\sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }\left( F\left( Z^{\prime} \right) -F\left( Z\right) \right)\!. \end{eqnarray} $ (D5) Duffie and Epstein (1992a) show that household maximization implies that a state-pricing process |$\xi _{t}$| may be written as follows: $ \begin{equation} \xi _{t}\equiv \exp \left[ \int_{0}^{t}m_{J}\left( C_{j}(\tau),J_{\tau }\right) d\tau \right] m_{C}\left( C_{j}(t),J_{t}\right)\!. \end{equation} $ (D6) Using the value function (D4) and the consumption policy (D1), I obtain $ \begin{equation} \xi _{t}=(\varpi_{j}Y_{t})^{-\gamma} \beta F\left( Z_{t}\right) ^{1-\frac{\rho} {1-\gamma} }e^{\left\{ \int_{0}^{t}\left( \frac{\beta \left( 1-\gamma -\rho \right)} {\rho} F\left( Z_{\tau} \right) ^{-\frac{\rho} {1-\gamma} }-\frac{\beta (1-\gamma )}{\rho} \right) d\tau \right\}} . \end{equation} $ (D7) Applying Itô’s lemma yields $ \begin{align} \label{eq:SDF-dyn-proof} \frac{d\xi _{t}}{\xi _{t-}} =-r_{f}\left( Z_{t}\right) dt -\gamma\sigma \left( Z_{t}\right) dB_{t} +\sum_{Z^{\prime} \neq Z_{t-}}\left( \left( \frac{F\left( Z^{\prime }\right)} {F\left( Z_{t-}\right)} \right) ^{1-\frac{\rho} {1-\gamma} }-1\right) \left( dN_{t}\left( Z_{t-},Z^{\prime} \right) -\lambda _{Z_{t-}Z^{\prime} }dt\right)\!,\nonumber\\ \end{align} $ (D8) where $ \begin{eqnarray} r_{f}\left( Z\right)=-\frac{\mathbb{E}_{t}\frac{ d\xi _{t}}{dt}}{\xi _{t-}} &=&\frac{\beta (1-\gamma )}{\rho} -\frac{\beta \left(1-\gamma -\rho \right)} {\rho} F\left( Z\right) ^{-\frac{\rho} {1-\gamma} } + \gamma \mu \left( Z\right) -\frac{1}{2} \gamma \left( 1+\gamma \right) \sigma ^{2}\left( Z\right) \notag \\ &&-\sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }\left( \left( \frac{ F\left( Z^{\prime} \right)} {F\left( Z\right)} \right) ^{1-\frac{\rho} { (1-\gamma )}}-1\right)\!. \end{eqnarray} $ (D9) Note that the growth of |$\xi _{t}$| is independent of an agent’s wealth share |$\varpi_{j}$|⁠. Thus, all agents in the economy assign identical values to payoffs in all future states of the world, and have no incentives to deviate from the consumption policy (D1). Defining $ \begin{equation} \zeta \left( Z,Z^{\prime} \right) \equiv \left( 1-\frac{\rho} {1-\gamma} \right) \log \left( \frac{F\left( Z^{\prime} \right)} {F\left( Z\right)} \right)\!. \end{equation} $ (D10) and rearranging (D8) yields the result stated in Proposition 4. E. Details on Data and Empirical Measures Discussed in Section 3 In the following, I provide additional details on data and empirical measures referred to in Section 3.1. Measure of industries |$\Psi'$|⁠. The estimate of the historical average ratio of capital commitments to the VC industry relative to aggregate consumption is based on capital commitments to VC funds from 1990 to 2015 (National Venture Capital Association, 2012, 2017). I scale these numbers by U.S. final consumption expenditure in each year (The World Bank, 2018). To estimate the expected value, I compute the average of this time series. The calibrated model matches this number, that is, |$\mathbb{E}\{\int_{\Psi'}\widetilde{com}(v,Z) dv\}\approx 0.3$|%. Innovation step size |$\kappa$|⁠. The parameter value chosen for |$\kappa$| is also consistent with empirical evidence about IT sector markups estimated in Ward (2013). In the model, the parameter |$\kappa$| pins down the profits-to-sales ratio of an incumbent firm, which is given by |$(1-1/\kappa)$|⁠. Ward (2013) estimates average aggregate IT sector markups from 1972 to 2012 and finds that |$1/(1-\frac{\text{EBITDA}}{\text{Sales}}) -1 =0.142$|⁠. In my model, this markup is obtained by choosing |$\kappa =1.142$|⁠, which is close to the calibrated value of |$\kappa =1.148$|⁠. Efficiency discount |$\phi$|⁠.Kortum and Lerner (2000) estimate a patent production function |$P = (R + b V)^{\alpha}$|⁠, where in their specification |$P$| refers to patenting, |$R$| measures R&D expenditures not funded by VC firms, and |$V$| is R&D funded by VC firms. Based on a variety of empirical tests, the authors obtain an average estimate of the parameter |$b$| of |$3.1$| (see footnote 26 in Kortum and Lerner 2000). This estimate corresponds to a value of |$1/3.1$| for the parameter |$\phi$| in my model. Matching cost |$c_{\iota}$|⁠. In the calibrated model, |$c_{\iota}$| is set such that |$\mathbb{E} [f^{man}|\widetilde{com}>0] \approx 0.03$| in industries |$v\in\Psi'$|⁠. Decreasing returns to scale parameter |$\eta$|⁠. Given the cited numbers from Hall and Woodward (2007), total payoffs to VC managers and entrepreneurs as a fraction of capital commitments are equal to |$0.26 \cdot ( 1+ \frac{9.2}{5.5})\approx 0.69$|⁠. Correspondingly, the calibrated model with |$\eta=0.585$| matches: |$\mathbb{E} [ f^{man} + f^{car} + \frac{(1-\varrho) \tilde{\pi}_{1}}{\widetilde{com}} |\widetilde{com}>0 ] \approx 0.69$|⁠. Using an instrumental variable approach, Kortum and Lerner (2000) estimate a decreasing returns to scale parameter of about |$0.52$|⁠, with a standard error of 0.1 (see panel B of table 4 in Kortum and Lerner (2000)). Network VCs’ bargaining power |$\varrho$|⁠. I choose the parameter |$\varrho$| so that the ratio of carry fees to entrepreneurial payoffs corresponds to the estimates by Hall and Woodward (2007). According to these estimates, carry fees represent on average 23% of provided capital. The ratio of total VC fees to the value of payoffs received by entrepreneurs is |$5.5/9.2$|⁠. Because carry fees account for a fraction |$0.23/0.26$| of total VC fees, the ratio of carry fee payoffs to entrepreneurial payoffs is approximately |$(5.5\cdot0.23/0.26)/9.2$|⁠, which corresponds to the ratio of bargaining powers in the model, |$\frac{\varrho}{1-\varrho}$|⁠, and yields |$\varrho=0.346$|⁠. In the calibrated model, |$\mathbb{E} [f^{car}|\widetilde{com}] >0 ] \approx 0.23$| in industries |$v \in \Psi'$|⁠. Productivity |$\theta$|⁠. Using estimates from Caballero and Jaffe (1993), I match an average rate of creative destruction of 8% in industries |$v \in \Psi'$|⁠. This number is the average estimated rate of creative destruction across the sectors “Drugs,” “Computers and data processing,” “Electronic communications,” and “Medical” reported in Table 5 of Caballero and Jaffe (1993). Scaled commitments are measured as discussed above in the bullet point “Measure of industries |$\Psi'$|⁠.” The standard deviation of these empirical values is 0.3%. Correspondingly, in the calibrated model |$Std[\widetilde{com}] \approx 0.3$|%. The historical peak level of annual commitments was 1.3% of U.S. final consumption expenditure, and was reached in the year 2000. In the model this level is reached in the high-|$\mu$|/high-|$\sigma$| state (see Table 2.B). The exposure parameter for industries |$v \in \Psi''$| is set so that the standard deviation of white-collar wage-to-output ratios relative to their mean, |$\frac{Std[\tilde{w}]}{E[\tilde{w}]}$|⁠, is consistent with empirical estimates obtained from income shares data for the top 5% of income earners in the United States. I use data provided by Alvaredo et al. (2012). After detrending income shares for the top 5% (incl. capital gains), I find that the volatility of income shares relative to their mean was 5.5% during the time period from 1990 to 2010. I focus on this time period, because top income and wage shares display a U-shaped pattern over the last century (see Piketty and Saez 2003). Since the 1980s, the income shares of the top 1% and the top 5% have experienced an upward trend. Details on Section 3.2 titled “Results of the Calibration.” VC Capital commitments and exit volume. Estimates of the average, the standard deviation, and the peak levels of aggregate VC exit volume are based on data published by the National Venture Capital Association (2012, 2017) for the time period from 1990 to 2015. I add the prices of known transactions from venture-backed Mergers & Acquisitions to the post offer values of venture-backed IPOs and subtract new capital raised in the IPOs. I scale these numbers by U.S. final consumption expenditure (The World Bank 2018) in each year. F. The Impact of VC-Backed Innovations on Expected Lifetime Utility We are interested in measuring the expected lifetime utility gains of moving from a counterfactual stochastic consumption stream |$\{\hat{C}_{t}\}_{0}^{\infty}$| that would prevail absent the innovations generated by VC investments to one that includes them, |$\{C_{t}\}_{0}^{\infty}=\{Y_{t}\}_{0}^{\infty}$|⁠, where the latter consumption stream is specified in Proposition 3. Here, the consumption stream |$\{\hat{C}_{t}\}_{0}^{\infty}$| exhibits the dynamics $ \begin{equation} \frac{d\hat{C}_{t}}{\hat{C}_{t}}=\hat{\mu} \left( Z_{t}\right) dt+\sigma \left( Z_{t}\right) dB_{t}, \end{equation} $ (F1) where the local drift |$\hat{\mu}$| takes the following form: $ \begin{equation} \hat{\mu}\left( Z_{t}\right) =\delta \left( Z_{t}\right) +\log \left[ \kappa \right] \int_{\Psi} \left( 1-\bar{\iota}_{1}\left( v,Z_{t}\right) \right) h(v,Z_{t},\phi n_{2}\left(v,Z_{t},0\right) ) dv. \end{equation} $ (F2) That is, the two consumption streams differ in their drift terms (⁠|$\mu(Z)$| vs. |$\hat{\mu}(Z)$|⁠), where |$\hat{\mu}(Z)$| excludes VC-backed innovations’ growth contributions. Following Lucas (1987), the associated welfare gain expressed in units of a percentage of consumption, denoted by the parameter |$\lambda$|⁠, then solves the following equation: $ \begin{align} \label{eq:LucasEquation} \mathbb{E}_{0}\left[ \int_{0}^{\infty} \zeta\left((1+\lambda)\hat{C}_{\tau} ,\hat{J}_{\tau}\right) d\tau \right] = \mathbb{E}_{0}\left[ \int_{0}^{\infty} \zeta\left(C_{\tau} ,J_{\tau}\right) d\tau \right]\!. \end{align} $ (F3) Here, |$\mathbb{E}_{0}$| denotes expectations given a particular initial level of consumption |$C_{0}$| at date |$0$|⁠, but without conditioning on a particular state |$Z$|⁠. Alternatively, one can also solve Equation (F3) using expectations that condition on a particular macroeconomic state |$Z$|⁠, which is, for example, done in Figure 2.D. At date |$0$|⁠, the economy starts from an allocation where agents hold claims to constant fractions of aggregate consumption. Differential consumption growth affects the consumption streams for all |$t>0$|⁠. Given the solution to agents’ value function (D4), Equation (F3) can be rewritten as follows (the consumption shares |$\varpi_{j}$| scale both value functions equally and thus can be canceled out): $ \begin{align} \label{eq:LucasEquationSolution} \mathbb{E}_{0}[\hat{F}(Z)] \frac{((1+\lambda) C_{0} )^{1-\gamma} }{1-\gamma} = \mathbb{E}_{0}[F(Z)] \frac{C_{0}^{1-\gamma} }{1-\gamma}. \end{align} $ (F4) This equation accounts for the fact that both consumption streams start with a consumption level |$C_{0}$|⁠. The function |$\hat{F}(Z)$| encodes the different growth expectations; it solves Equation (D5) with a consumption drift equal to |$\hat{\mu}(Z)$| instead of the drift |$\mu(Z)$|⁠. Under the benchmark calibration presented in Section 3.1, the compensation parameter |$\lambda \approx 1.3\%$| solves Equation (F4). I also evaluate how valuable VC growth contributions would be if they were constant and equal to their unconditional average value under the baseline calibration. To do so, I replace the arrival rates of innovations |$\bar{h}\left( v,Z\right)$| in product lines |$v \in \Psi'$|⁠, which enter the aggregate consumption drift |$\mu \left( Z_{t}\right)$| as stated in Equation (23), with the following terms: $ \begin{equation} \mathbb{E}[ \bar{\iota} _{1}\left( v,Z\right) h(v,n_{1}\left( v,Z\right) )]+\left( 1-\bar{\iota}_{1}\left( v,Z\right) \right) h(v,Z,\phi n_{2}\left(v,Z,0\right) ). \end{equation} $ (F5) These arrival rates deviate from those in the baseline calibration in that the arrival rate of VC-funded innovations is set to be constant and equal to the unconditional value of the baseline calibration. In this case, the compensation parameter |$\lambda$| solving Equation (F4) is approximately twice as large as the baseline value of 1.3%. G. Idiosyncratic Return Volatility In this Appendix, I provide details on VC investments’ return volatility implied by the calibrated model. First, it is useful to note that the quantity |$n_{1}$| in the model is not the equivalent of the number of ventures in the data. Rather, |$n_{1}$| reflects a continuous quantity of real resources allocated to VC-backed investments in a given product line. To gauge how this quantity is related to the typical size of a VC funded start-up, I determine the quantity |$n_{1}$| that matches the success prospects of a typical start-up in the data. Given a quantity |$\hat{n}_{1}$| invested in any of the industries |$v \in \Psi'$|⁠, a successful venture exit (IPO) occurs with hazard rate |$\hat{h} \equiv \hat{n} \frac{h_{1}}{n_{1}}$|⁠, where |$h_{1}$| and |$n_{1}$| are as defined in the main text. Based on data reported by Korteweg and Sorensen (2010, table 1), I consider |$\hat{n}=1.3$| to match a 33% probability of a successful exit (IPO or M&A) over the typical time period for which a venture obtains funding in the data (4.8 years on average). Second, in the model, annualized return volatilities depend on how long a typical venture receives funding. Formally, the annualized return volatility from investing for a time period of length |$\Delta t$| is given by $ \begin{align} \frac{1}{\sqrt{\Delta t}} Std\left[\frac{P \int_{0}^{\Delta t} dM_{t}} { \hat{n} w_{W} \Delta t}\right] = \frac{P\sqrt{\hat{h}} }{\hat{n} w_{W}\Delta t}, \label{eq:IdioRetVol} \end{align} $ (G1) where |$dM_{t}=1$| represents a Poisson jump associated with a success that occurs with intensity |$\hat{h}$|⁠, and where, for notational simplicity, I suppress the dependence of the variables |$P$|⁠, |$w_{W}$|⁠, and |$\hat{h}$| on the state |$Z$|⁠. When considering the funding over an instant |$\Delta t \rightarrow dt$|⁠, the annualized return volatility is infinite, as resources of order |$dt$| (in the denominator) can generate an IPO that is valued at price |$P$|⁠. Volatility declines over longer time periods (for |$\Delta t >0$|⁠), as accumulating independent local Poisson bets over time yields diversification. Extreme returns over short time periods are also features of VC data. For ventures that have an IPO or an acquisition within 1 to 6 months, Cochrane (2005) reports average annualized arithmetic returns of 4.0e |$+$| 10 and a standard deviation of 7.2e + 11 (see table 6 in Cochrane 2005). To compute volatility numbers that are comparable to those measured based on existing VC data, I therefore account for the typical investment horizon of a VC-funded start-up and its cyclical variation. To do so, the time period |$\Delta t$| in Equation (G1) is specified as proportional to the expected time to a successful exit, which is given by |$1/\hat{h}$|⁠. That is, I define |$\Delta t(Z)=\widehat{\Delta t}/\hat{h}(Z)$|⁠, where |$\widehat{\Delta t}=0.39$| is a constant whose value is chosen to ensure that |$\mathbb{E}[\Delta t|\iota =1 ]=4.8$| years. As |$\hat{h}(Z)$| is procyclical, this approach captures the empirical fact that ventures tend to exit faster in booms (see, e.g., Cochrane 2005). Moreover, the constant |$\widehat{\Delta t}=0.38$| implies that the typical venture is terminated before the average time needed to exit successfully has passed (⁠|$\widehat{\Delta t}=1$| would correspond to the average time to a successful exit). This approach yields the following state-contingent return volatilities (expressed in percentage points) computed based on Equation (G1): Idiosyncratic return volatility |$\sigma$| Low Med High |$\mu$| High 142.99 150.95 157.31 Med 121.06 120.47 120.03 Low n/a n/a n/a Idiosyncratic return volatility |$\sigma$| Low Med High |$\mu$| High 142.99 150.95 157.31 Med 121.06 120.47 120.03 Low n/a n/a n/a Open in new tab Idiosyncratic return volatility |$\sigma$| Low Med High |$\mu$| High 142.99 150.95 157.31 Med 121.06 120.47 120.03 Low n/a n/a n/a Idiosyncratic return volatility |$\sigma$| Low Med High |$\mu$| High 142.99 150.95 157.31 Med 121.06 120.47 120.03 Low n/a n/a n/a Open in new tab These numbers are of similar magnitude as the idiosyncratic volatility estimates of Korteweg and Sorensen (2010), who find 41% monthly idiosyncratic volatility, implying 140% annual idiosyncratic volatility. H. Imperfect Risk Sharing For the analysis of the model discussed in Section 4 the contractual space and the logical order of events within a period |$[t,t+dt)$| are specified as follows: Agents complete all financial transactions and choose their consumption for the period. Financial transactions can specify payments conditional on the paths of the variables |$\{Y_{\tau}\}_{\tau \geq t}$|⁠, |$\{Z_{\tau}\}_{\tau \geq t}$|⁠, |$\{\iota_{1}(v,\tau)\}_{v\in \Psi,\tau \geq t}$|⁠, |$\{M(v,\tau)\}_{v\in \Psi, \tau \geq t}$|⁠, and |$\{M_{1}(v,\tau)\}_{v\in \Psi, \tau \geq t}$|⁠, where |$M_{1}(v,t)$| is a counting process keeping track of the number of new patents created from funding the most productive entrepreneurs’ venture ideas in industry |$v$|⁠. VC funds and firms can obtain funding commitments from investors as a function of these variables. In return, investors obtain venture ownership shares. The random variables |$i_{E,t}^{*}$| and |$i_{V,t}^{*}$| are realized, that is, nature chooses the network VCs with connections and the most productive entrepreneurs in the period. Whenever VC intermediation obtains (⁠|$\iota_{1}(v,t)=1$|⁠), the connected network VC and the most productive entrepreneur are compensated with venture equity, as specified in the baseline model. Network VCs that obtained connections choose whether to facilitate the matching process, which determines the realizations of the variables |$\left\{\iota_{1}(v,t)\right\}_{v\in \Psi}$|⁠. State-contingent funding commitments for VC funds are fulfilled (contingent on |$\iota_{1}(v,t)=1$|⁠) and are used to pay for the implementation of venture ideas of the most productive entrepreneurs and for VC management fees. State-contingent funding commitments for regular ideas are fulfilled and can be used to pay for investment costs. Innovation arrivals as reflected by jumps in the variables |$\{M(v,t)\}_{v\in \Psi}$| are realized and patents are allocated to ventures as described in the baseline setup. For the analysis of the economy with incomplete risk sharing, it is useful to first consider a setting where the first element of the double index identifying agents, |$i_{S}'$|⁠, takes values in a discrete set (instead of the continuum |$[0,1]$|⁠). In particular, define |$\Delta i_{S}' \equiv \frac{1}{K+1}$| and let |$i_{S}' \in \{0, \frac{1}{K}, \frac{2}{K}, ...,1\}$|⁠. To establish the solution for the model with double continuums of agents, I take the limit |$K \rightarrow \infty$|⁠, or equivalently, |$\Delta i_{S}' \downarrow 0$|⁠. H.1 Agents’ Probabilities of Receiving Venture Opportunities Consistent with the baseline setup, the first indices |$i_{E}'$| (for entrepreneurs) and |$i_{V}'$| (for network VCs) identifying agents with special venture opportunities are again randomly selected according to a uniform distribution: each possible value of |$i_{E}'$| and |$i_{V}'$| has an equal probability |$\Delta i_{E}'=\Delta i_{V}'$| of being selected in each period. In addition, the specification determining the second element, |$i_{S}''$|⁠, remains exactly as specified in Sections 1.3 and 1.4. Recall that if a particular first index |$i_{E}'$| is selected by nature in a given period, all entrepreneurs with this first index obtain a positive shock in some industry; which entrepreneur obtains this shock in which industry is pinned down by the second index element |$i_{E}''=v$|⁠. An entrepreneur with the second index element |$i_{E}''=v$| then has a probability |$\Delta i_{E}'$| of becoming the most productive entrepreneur in industry |$v$| in a given period. Analogously, a network VC with the second index element |$i_{V}''=v$| has a probability |$\Delta i_{V}'$| of receiving a useful connection in industry |$v$|⁠. H.2 Consumption and Investment Policies I start with the conjecture that, in equilibrium, all agents rebalance their portfolios at the beginning of each period to always invest all their tradable wealth in the market portfolio of all tradable assets, the price of which is denoted by |$P_{T}$|⁠. For example, immediately after experiencing a positive idiosyncratic venture success as an entrepreneur, an agent sells her equity stake to other agents at the market price and rebalances to again hold the market portfolio. Further, conjecture that all agents choose the same consumption-to-tradable-wealth ratios, which implies that the wealth shares also pin down the fraction of output |$Y$| that each agent consumes. Below, I will verify the optimality of these conjectured consumption and investment policies. Let |$\varpi(S,i_{S}',i_{S}'',t)$| denote the share of total tradable wealth owned by agent |$(S,i_{S}',i_{S}'')$| at time |$t$| scaled by |$\frac{4}{\Delta i_{S}' d i_{S}''}$|⁠, such that the actual wealth share is |$\varpi(S,i_{S}',i_{S}'',t)\frac{\Delta i_{S}'di_{S}''}{4}$|⁠. By definition, wealth shares add up to one: $ \begin{align} \sum_{S \in \{B,W,V,E\}} \sum_{i_{S}' \in \{0, \frac{1}{K}, \frac{2}{K}, ...,1\}} \int_{0}^{1} \varpi(S,i_{S}',i_{S}'',t) \frac{\Delta i_{S}'di_{S}''} {4}=1. \end{align} $ (H1) To economize on notation, I will use a subscript |$j$| to refer to an agent identified by a tuple |$(S,i_{S}',i_{S}'')$| going forward. An agent’s tradable wealth at time |$t$| is denoted by |$W_{j}(t)$| and is given by $ \begin{align} W_{j}(t)=P_{T}(t) \varpi_{j}(t) \frac{\Delta i_{S}'di_{S}''}{4}. \end{align} $ (H2) Assets that are part of the tradable market portfolio pay an aggregate dividend that is equal to the aggregate output |$Y$|⁠. Thus, an agent holding a fraction |$\varpi_{j}(t)\frac{ \Delta i_{S}'di_{S}''}{4}$| of the market portfolio obtains a dividend |$Y_{t} \varpi_{j}(t) \frac{\Delta i_{S}' di_{S}''} {4}$|⁠. Identical consumption-to-wealth ratios and market clearing imply that agents consume this dividend, that is, $ \begin{align} \label{eq:ImperfectRiskConsumption} C_{j}(t)= Y_{t} \varpi_{j}(t) \frac{\Delta i_{S}' di_{S}''}{4}. \end{align} $ (H3) H.3 Evolution of Tradable Wealth Shares The set of assets in positive net supply that are tradable changes over time. The venture ownership stakes network VCs and entrepreneurs will obtain through future idiosyncratic shocks are not traded until after ventures succeed and ownership stakes can be sold in the competitive market. It is useful to specify the market values of different vintages of tradable market portfolios. I define |$P_{T,\tau}(t)$| as the time-|$t$| market value of the portfolio of all assets in positive net supply that have been tradable since time |$\tau$|⁠, where |$t \geq \tau$|⁠. Let |$\vartheta_{j}(v,t) \in [0,1]$| denote the equity stake agent |$j$| obtains if she receives a VC-related opportunity in industry |$v$| in period |$[t,t+dt)$|⁠, which is relevant only for entrepreneurs and network VCs (i.e., |$\vartheta_{j}(v,t) =0$| for white-collar and blue-collar agents). Further, let |${\rm 1}\kern-0.24em{\rm I}_{j}(v,t)$| be an indicator variable that takes the value one if agent |$j$| actually obtains a VC-related opportunity in industry |$v$| in period |$[t,t+dt)$|⁠. An agent obtains equity worth |$\vartheta_{j}(v,t) P(v,t) dv$| when succeeding with a VC-backed venture in industry |$v$|⁠. Given the conjectured consumption and portfolio policies an agent’s tradable wealth thus evolves as follows: $ \begin{align} d W_{j}(t) = \varpi_{j}(t) d P_{T,t}(t) \frac{\Delta i_{S}' di_{S}''} {4} + \int_{\Psi} dM_{1}(v,t) {\rm 1}\kern-0.24em{\rm I}_{j}(v,t) \vartheta_{j}(v,t) P(v,t) dv. \end{align} $ (H4) The total fraction of venture equity that entrepreneurs and network VCs will obtain jointly in industry |$v$| at time |$t$| is given by $ \begin{align} \label{eq:defineTotalEquityShare} \vartheta(v,t) &=\begin{cases} \frac{\pi_{1}(v,t)} {h_{1}(v,t) P(v,t)} & \quad \text{ if} h_{1}(v,t) >0,\\ 0 & \quad \text{ otherwise.} \end{cases} \end{align} $ (H5) These equity shares are distributed according to the bargaining power parameter |$\varrho$|⁠. The remainder share of the equity is held by diversified investors and is part of the market portfolio of tradable assets. The dynamics of the value of the market portfolio of tradable assets can be decomposed into the value change of existing tradable assets and the arrival of new venture equity sold by successful entrepreneurs and network VCs: $ \begin{align} d P_{T}(t) &= d P_{T,t}(t) + \int_{\Psi} \vartheta(v,t) dM_{1}(v,t) P(v,t)dv \notag \\ &= d P_{T,t}(t) + \int_{\Psi} \vartheta(v,t) h_{1}(v,t) P(v,t)dv dt \notag \\ &= d P_{T,t}(t) + \int_{\Psi} \pi_{1}(v,t) dv dt, \end{align} $ (H6) where the first step obtains because |$\mathbb{E}[dM_{1}(v,t)]=h_{1}(v,t)dt$|⁠, and the law of large numbers applies (Uhlig 1996),27 and the second step obtains due to the definition of |$\vartheta(v,t)$| provided in (H5) and because |$\pi_{1}(v,t)=0$| when |$h_{1}(v,t)=0$|⁠. Thus, |$\varpi_{j}(t)$| has the following law of motion: $ \begin{align} d\varpi_{j}(t) &= d \left(\frac{W_{j}(t)}{P_{T}(t) \frac{\Delta i_{S}' di_{S}''} {4}}\right) \notag \\ &= \frac{d W_{j}(t)}{P_{T}(t) \frac{\Delta i_{S}' di_{S}''} {4}} -\frac{W_{j}(t)}{P_{T}(t) \frac{\Delta i_{S}' di_{S}''} {4}} \frac{dP_{T}(t)} {P_{T}(t)} \notag \\ &= \frac{ \varpi_{j}(t) d P_{T,t}(t) \frac{\Delta i_{S}' di_{S}''} {4} + \int_{\Psi} dM_{1}(v,t){\rm 1}\kern-0.24em{\rm I}_{j}(v,t) \vartheta_{j}(v,t) P(v,t) dv}{P_{T}(t) \frac{\Delta i_{S}' di_{S}''} {4}}\nonumber\\ &\quad -\varpi_{j}(t) \frac{d P_{T,t}(t) + \int_{\Psi} \pi_{1}(v,t) dv dt} {P_{T}(t)} \notag \\ &= \frac{\int_{\Psi} dM_{1}(v,t) {\rm 1}\kern-0.24em{\rm I}_{j}(v,t) \vartheta_{j}(v,t) P(v,t) dv} {P_{T}(t) \frac{ \Delta i_{S}' di_{S}''} {4}} -\varpi_{j}(t) \frac{ \int_{\Psi} \pi_{1}(v,t) dv} {P_{T}(t)} dt, \end{align} $ (H7) which yields $ \begin{align} \frac{d\varpi_{j}(t)}{\varpi_{j}(t)}= \frac{\int_{\Psi} dM_{1}(v,t){\rm 1}\kern-0.24em{\rm I}_{j}(v,t) \vartheta_{j}(v,t) P(v,t) dv}{\varpi_{j}(t) P_{T}(t) \frac{ \Delta i_{S}' di_{S}''}{4}} -\frac{ \int_{\Psi} \pi_{1}(v,t) dv} {P_{T}(t)} dt. \end{align} $ (H8) Here, the first term reflects idiosyncratic risk. Each agent |$j$| obtains a venture equity share |$\vartheta_{j}(v,t)>0$| in at most one industry |$v$| and thus, may be exposed to venture success risk in that particular industry. The second term reflects the decline in an agent’s wealth share due to the venture successes of other agents (network VCs and entrepreneurs) in the economy. H.4 Consumption Dynamics Given the consumption policy (H3), agent |$j$|’s consumption growth is given by $ \begin{align} \frac{dC_{j}(t)}{C_{j}(t)}= \left(\mu(t) - \frac{ \int_{\Psi} \pi_{1}(v,t) dv}{P_{T}(t)} \right) dt + \sigma(t)dB_{t} + \frac{\int_{\Psi} dM_{1}(v,t){\rm 1}\kern-0.24em{\rm I}_{j}(v,t) \vartheta_{j}(v,t) P(v,t) dv}{\varpi_{j}(t) P_{T}(t) \frac{ \Delta i_{S}' di_{S}''} {4}} . \end{align} $ (H9) H.5 Value Function Under the conjectured consumption policy an agent’s value function is given by $ \begin{equation} J\left(\varpi_{j}(t), Y_{t},Z_{t}\right) =\mathbb{E}_{t}\left[ \int_{t}^{\infty} m\left( C_{j}(\tau),J_{\tau} \right) d\tau \right], \end{equation} $ (H10) which yields the following HJB equation for entrepreneurs with a second index element |$i_{E}''=v$| and network VCs with a second index element |$i_{V}''=v$|⁠: $ \begin{align} 0 =&m\left( C_{j},J\left( \varpi_{j},Y,Z\right) \right) +J_{Y}\left( \varpi_{j},Y,Z\right) Y\mu\left( Z\right) - J_{\varpi_{j}}\left( \varpi_{j},Y,Z\right) \varpi_{j} \frac{ \int_{\Psi}\tilde{\pi}_{1}(v,Z)dv}{\tilde{P}_{T}(Z)} \notag \\ &+\frac{1}{2}J_{Y Y}\left( \varpi_{j},Y,Z\right) Y^{2}\sigma^{2}\left( Z\right)+\sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }\left( J\left( \varpi_{j},Y,Z^{\prime }\right) -J\left( \varpi_{j},Y,Z\right) \right) \notag \\ &+ \left( J\left( \varpi_{j} \cdot \left(1 + \frac{\vartheta_{j}(v,Z) \tilde{P}(v,Z)dv} {\varpi_{j} \tilde{P}_{T}(Z) \frac{\Delta i_{S}' di_{S}''} {4}}\right)\!,Y,Z\right) -J\left( \varpi_{j},Y,Z\right) \right) \bar{\iota}_{1}(v,Z) h_{1}(v,Z) \Delta i_{S}', \end{align} $ (H11) where I use the fact |$\frac{\mathbb{E}[dM_{1}(v,t)|Z]}{dt}=\bar{\iota}_{1}(v,Z) h_{1}(v,Z)$|⁠, and that |$\Pr[{\rm 1}\kern-0.24em{\rm I}_{j}(v,t)=1]=\Delta i_{S}'$|⁠, if agent |$j$| is an entrepreneur or a network VC with second index element |$i_{E}''=v$| and |$i_{V}''=v$|⁠, respectively. Note that for white-collar and blue-collar agents |$\vartheta_{j}(v,Z)=0$|⁠, such that the last term in the previous equation drops out. Conjecture that the solution for |$J$| takes the form: $ \begin{equation} \label{eq:ValueFunctionIdiosyncratic} J\left( \varpi_{j},Y,Z\right) =F\left( Z\right) \frac{(\varpi_{j} Y)^{1-\gamma} }{1-\gamma}. \end{equation} $ (H12) Substituting this conjecture into the HJB equation yields $ \begin{align} \label{eq:HJBZ-imperfectRiskSharing} 0 &=\left( \frac{\beta (1-\gamma )}{\rho} \left( F\left( Z\right) ^{-\frac{ \rho} {1-\gamma}}-1\right) +(1-\gamma ) \left(\mu \left( Z\right) - \frac{ \int_{0}^{1}\tilde{\pi}_{1}(v,Z)dv}{\tilde{P}_{T}(Z)} \right) -\frac{1}{2}\gamma (1-\gamma ) \sigma ^{2}\left( Z\right) \right) F\left( Z\right) \notag \\ &+\sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }\left( F\left( Z^{\prime} \right) -F\left( Z\right) \right)+ \Delta i_{S}' \left(\left(1 + \frac{\vartheta_{j}(v,Z) \tilde{P}(v,Z_{t})dv} {\varpi_{j} \tilde{P}_{T}(Z_{t}) \frac{\Delta i_{S}' di_{S}''} {4}}\right)^{1-\gamma} - 1 \right) F\left( Z\right) \bar{\iota}_{1}(v,Z) h_{1}(v,Z). \end{align} $ (H13) For |$\gamma > 1$|⁠, the value of the term $ \begin{align} \left(\left(1 + \frac{\vartheta_{j}(v,Z)\tilde{P}(v,Z_{t})dv} {\varpi_{j} \tilde{P}_{T}(Z_{t}) \frac{\Delta i_{S}' di_{S}''} {4}}\right)^{1-\gamma}-1 \right) \end{align} $ (H14) is bounded from below by |$-1$| and from above by |$0$|⁠. Thus, in the relevant limiting case $ \begin{align} \lim_{\Delta i_{S}' \downarrow 0} \left\{ \Delta i_{S}' \cdot \left(\left(1 + \frac{\vartheta_{j}(v,Z)\tilde{P}(v,Z_{t})dv} {\varpi_{j} \tilde{P}_{T}(Z_{t}) \frac{\Delta i_{S}' di_{S}''} {4}}\right)^{1-\gamma} - 1 \right) F\left( Z\right) \bar{\iota}_{1}(v,Z) h_{1}(v,Z) \right\} = 0. \end{align} $ (H15) It follows that this term drops from Equation (H13) in the relevant case when |$\lim_{\Delta i_{S}' \downarrow 0}$|⁠, and |$F(Z)$| solves $ \begin{align}\label{eq:HJB-FZ-Idiosyncratic} 0 &=\left( \frac{\beta (1-\gamma )}{\rho} \left( F\left( Z\right) ^{-\frac{ \rho} {1-\gamma}}-1\right) +(1-\gamma ) \left(\mu \left( Z\right) - \frac{ \int_{\Psi}\tilde{\pi}_{1}(v,Z)dv}{\tilde{P}_{T}(Z)}\right)\right. \notag \\ &\left. -\frac{1}{2}\gamma (1-\gamma ) \sigma ^{2}\left( Z\right) \right) F\left( Z\right) +\sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }\left( F\left( Z^{\prime} \right) -F\left( Z\right) \right)\!. \end{align} $ (H16) Note that this equation is independent of an agent’s skill type |$S$| and her wealth share |$\varpi_{j}$|⁠, implying that all agents share the same |$F(Z)$| function, consistent with the conjectured value function. H.6 Verifying the Conjectured Equilibrium for |$\lim_{\Delta i_{S}' \downarrow 0}$| I verify whether any individual agent has an incentive to deviate from the conjectured behavior, given that all other agents behave as conjectured. The marginal utility process of agents following the conjectured investment and consumption policies may be written as follows: $ \begin{equation} \xi _{j,t}\equiv \exp \left[ \int_{0}^{t}m_{J}\left( C_{j}(\tau),J_{\tau }\right) d\tau \right] m_{C}\left( C_{j}(t),J_{t}\right)\!. \end{equation} $ (H17) Using the value function (H12) associated with the conjectured policies, where |$F$| solves Equation (H16), yields $ \begin{equation} \xi _{j,t}=\left(\varpi_{j} Y_{t}\right)^{-\gamma} \beta F\left( Z_{t}\right) ^{1-\frac{\rho} {1-\gamma} }e^{\left\{ \int_{0}^{t}\left( \frac{\beta \left( 1-\gamma -\rho \right)} {\rho} F\left( Z_{\tau} \right) ^{-\frac{\rho} {1-\gamma} }-\frac{\beta (1-\gamma )}{\rho} \right) d\tau \right\}} . \end{equation} $ (H18) Applying Itô’s lemma yields for entrepreneurs with a second index element |$i_{E}''=v$| and network VCs with a second index element |$i_{V}''=v$|⁠: $ \begin{align} -\frac{\mathbb{E}_{t}\frac{ d\xi _{j,t}}{dt}}{\xi _{j,t-}} =&\frac{\beta (1-\gamma )}{\rho} -\frac{\beta \left(1-\gamma -\rho \right)} {\rho} F\left( Z\right) ^{-\frac{\rho} {1-\gamma} } + \gamma \left( \mu \left( Z\right) - \frac{ \int_{\Psi}\tilde{\pi}_{1}(v,Z)dv}{\tilde{P}_{T}(Z)} \right) -\frac{1}{2} \gamma \left( 1+\gamma \right) \sigma ^{2}\left( Z\right) \notag \\ &-\sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }\left( \left( \frac{ F\left( Z^{\prime} \right)} {F\left( Z\right)} \right) ^{1-\frac{\rho} { (1-\gamma )}}-1\right) \notag \\ &- \Delta i_{S}' \left(\left(1 + \frac{\vartheta_{j}(v,Z) \tilde{P}(v,Z_{t})dv} {\varpi_{j} \tilde{P}_{T}(Z_{t}) \frac{\Delta i_{S}' d i_{S}''} {4}}\right)^{-\gamma} - 1 \right) \bar{\iota}_{1}(v,Z) h_{1}(v,Z), \end{align} $ (H19) where, for the relevant limiting case |$\Delta i_{S}' \downarrow 0$|⁠, the last term drops out, because $ \begin{align} \lim_{\Delta i_{S}' \downarrow 0} \left\{ \Delta i_{S}' \cdot \left(\left(1 + \frac{\vartheta_{j}(v,Z)\tilde{P}(v,Z_{t}) dv} {\varpi_{j} \tilde{P}_{T}(Z_{t}) \frac{\Delta i_{S}' di_{S}''} {4}}\right)^{-\gamma} - 1 \right) \bar{\iota}_{1}(v,Z) h_{1}(v,Z) \right\} = 0. \end{align} $ (H20) Note that for white-collar and blue-collar agents |$\vartheta_{j}(v,Z)=0$|⁠, such that the last term also drops out. It follows that all agents agree on the risk-free rate |$r_{f}(Z)=-\frac{\mathbb{E}_{t}\frac{ d\xi _{j,t}}{dt}}{\xi _{j,t-}}$| independent of their skill |$S$| and |$\varpi_{j}$|⁠. For entrepreneurs with a second index element |$i_{E}''=v$| and network VCs with a second index element |$i_{V}''=v$|⁠, the dynamics for |$\xi _{j,t}$| are given by $ \begin{align} \frac{d\xi _{j,t}}{\xi _{j,t-}} =&-r_{f}\left( Z_{t}\right) dt -\gamma\sigma \left( Z_{t}\right) dB_{t} +\sum_{Z^{\prime} \neq Z_{t-}}\left( \left( \frac{F\left( Z^{\prime }\right)} {F\left( Z_{t-}\right)} \right) ^{1-\frac{\rho} {1-\gamma} }-1\right) \left( dN_{t}\left( Z_{t-},Z^{\prime} \right) -\lambda _{Z_{t-}Z^{\prime} }dt\right) \notag \\ & + {\rm 1}\kern-0.24em{\rm I}_{j}(v,t) \left(\left(1 + \frac{\vartheta_{j}(v,Z) \tilde{P}(v,Z_{t})dv} {\varpi_{j}(t) \tilde{P}_{T}(Z_{t}) \frac{\Delta i_{S}' di_{S}''} {4}}\right)^{-\gamma} - 1 \right) dM_{1}(v,t). \end{align} $ (H21) For white-collar and blue-collar agents the last term again drops out. A central feature of the analysis with imperfect risk sharing is that agents cannot trade claims that pay conditional on the joint future realizations of their idiosyncratic venture opportunities (⁠|${\rm 1}\kern-0.24em{\rm I}_{j}(v,t)$| and |$\vartheta_{j}(v,t)$|⁠) and VC-backed venture successes (⁠|$dM_{1}(v,t)$|⁠). Yet consider the set of tradable claims, which can pay a cash flow |$CF$| at some future date |$\tau >t$| contingent on realizations of the aggregate state variables |$Y$| and |$Z$|⁠, the quality ladder counting processes |$\{M(v,\tau)\}_{v\in \Psi}$|⁠, the VC-backed patent counting processes |$\{M_{1}(v,\tau)\}_{v\in \Psi}$|⁠, and the VC intermediation variables |$\{\iota_{1}(v,\tau)\}_{v\in \Psi}$|⁠. Formally, any tradable claim to a cash flow is specified as a general function of these state variables: $ \begin{align} CF_{\tau}=CF(Y_{\tau}, Z_{\tau},\{\iota_{1}(v,\tau)\}_{v\in \Psi},\{M(v,\tau)\}_{v\in [0,1]},\{M_{1}(v,\tau)\}_{v\in [0,1]}) \end{align} $ (H22) Define |$ns$| (for “no success”) as the event where $ \begin{align} \int_{t}^{\tau} \int_{\Psi} {\rm 1}\kern-0.24em{\rm I}_{j}(v,s) \vartheta_{j}(v,Z_{s}) dM_{1}(v,s) = 0, \end{align} $ (H23) that is, agent |$j$| does not obtain idiosyncratic venture equity gains between time |$t$| and time |$\tau$|⁠. Define |$PNS_{j}(t, \tau, Z)$|⁠, as the probability of this event, which will generally depend on the current state |$Z$| and the type of the agent |$j$|⁠. |$PNS_{j}(t, \tau, Z)$| solves for all |$Z$|⁠: $ \begin{align} 0=&\frac{\partial PNS_{j}(t, \tau, Z)}{\partial t} - \frac{1}{dt}\mathbb{E} \left[ \int_{\Psi} {\rm 1}\kern-0.24em{\rm I}_{j}(v,s) \vartheta_{j}(v,Z) d M_{1}(v,t) \right] PNS_{j}(t, \tau, Z) \notag \\ &+ \sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} } ( PNS_{j}(t, \tau, Z') - PNS_{j}(t, \tau, Z)) \end{align} $ (H24) with |$PNS_{j}(\tau, \tau, Z) =1$|⁠. Note that in the relevant case in which |$\Delta i_{S}' \downarrow 0$|⁠, the following result obtains for both entrepreneurs with a second index element |$i_{E}''=v$| and network VCs with a second index element |$i_{V}''=v$|⁠: $ \begin{align} &\lim_{\Delta i_{S}' \downarrow 0}\left\{ \frac{1}{dt}\mathbb{E} \left[ \int_{\Psi} {\rm 1}\kern-0.24em{\rm I}_{j}(v,s) \vartheta_{j}(v,Z)dM_{1}(v,t) \right]\right\} \notag \\ = &\lim_{\Delta i_{S}' \downarrow 0} \left\{\Delta i_{S}' \vartheta_{j}(v,Z) \bar{\iota}_{1}(v,Z) h_{1}(v,Z) \right\}\notag \\ = &0. \end{align} $ (H25) For white-collar and blue-collar agents this term also takes the value zero. It follows that |$PNS_{j}(t, \tau, Z) =1$|⁠. Agents then value a claim to |$CF_{\tau}$| as follows: $ \begin{align} \mathbb{E}_{t}\left[ \frac{\xi _{j,\tau} }{\xi _{j,t}} CF_{\tau} \right] =& \mathbb{E}_{t}\left[ \left. \frac{\xi _{j,\tau} }{\xi _{j,t}} CF_{\tau} \right| ns \right] PNS_{j}(t, \tau, Z_{t})+ \mathbb{E}_{t}\left[\left. \frac{\xi _{j,\tau} }{\xi _{j,t}} CF_{\tau} \right| ns^{c} \right] (1-PNS_{j}(t, \tau, Z_{t})) \notag \\ =& \mathbb{E}_{t}\left[ \left. \frac{\xi _{j,\tau} }{\xi _{j,t}} CF_{\tau} \right| ns \right], \end{align} $ (H26) that is, the tradable claim is valued by all agents as if the evolution of their |$\xi _{j,t}$| was given by $ \begin{align} \frac{d\xi _{j,t}}{\xi _{j,t-}} =&-r_{f}\left( Z_{t}\right) dt -\gamma\sigma \left( Z_{t}\right) dB_{t} \notag \\ &+\sum_{Z^{\prime} \neq Z_{t-}}\left( \left( \frac{F\left( Z^{\prime }\right)} {F\left( Z_{t-}\right)} \right) ^{1-\frac{\rho} {1-\gamma} }-1\right) \left( dN_{t}\left( Z_{t-},Z^{\prime} \right) -\lambda _{Z_{t-}Z^{\prime} }dt\right)\!. \end{align} $ (H27) The prices agents assign to tradable claims are thus independent of their own wealth share |$\varpi_{j}$| and their skill |$S$|⁠. All agents agree on the prices of tradable claims, implying that they have no incentive to deviate from the conjectured investment and consumption policies. As these consumption and investment decisions are optimal independent of the cross-sectional wealth distribution, that distribution has also no incremental relevance for characterizing equilibrium behavior, and the only aggregate state variables are still |$Y$| and |$Z$|⁠, just like in the baseline analysis. H.7 Valuing the Growth Contributions of Aggregate VC Investment The derivations above show that for any agent with a current wealth share |$\varpi_{j}(t)$| the solution for the value function |$J$| is given by (H12), where |$F(Z)$| solves (H16). Under the alternative scenario where starting from date |$t$|⁠, agents do not expect any future innovations from VC investments, but future state-contingent resource allocations remain unchanged, all agents still have the same tradable wealth shares, as all agents hold the market portfolio, and thus, their tradable wealth is equally affected, in percentage terms, by this change in expectations. The current level of output |$Y_{t}$| is also unaffected by this change in expectations. However, the function |$F(Z)$| now solves $ \begin{align} 0 =&\left( \frac{\beta (1-\gamma )}{\rho} \left( F\left( Z\right) ^{-\frac{ \rho} {1-\gamma}}-1\right) +(1-\gamma ) \left(\mu \left( Z\right) - \log \left[ \kappa \right] \int_{\Psi}\bar{\iota}_{1}(v,Z) h_{1}(v,Z)dv\right)\right. \notag \\ &\left. -\frac{1}{2}\gamma (1-\gamma ) \sigma ^{2}\left( Z\right) \right) F\left( Z\right) +\sum_{Z^{\prime} \neq Z}\lambda _{ZZ^{\prime} }\left( F\left( Z^{\prime} \right) -F\left( Z\right) \right)\!, \end{align} $ (H28) which accounts for the lower growth expectations. Overall, each agent’s value function is equally affected, in percentage terms, by the change in expectations, because each agent experiences an identical change in the |$F(Z)$| function. Agents have identical preference parameters, implying that they also assign the same value to the growth contributions from VC investments (expressed as equivalent lifetime consumption gains). Given these results, the equivalent lifetime consumption gains can be computed just like in the baseline setup (see Appendix F for details). Footnotes 1See, for example, Kortum and Lerner (2000) and Samila and Sorenson (2011) for empirical evidence. 2Hochberg, Ljungqvist, and Lu (2007) document the importance of VC managers’ network connections for investment opportunity sets and access to information. Kaplan and Strömberg (2001, 2004) document the importance of VCs’ efforts to evaluate and screen entrepreneurs. Gompers, Kovner, Lerner, and Scharfstein (2008, pp. 2-3) highlight the relevance of “industry-specific human capital,” including a “network of industry contacts to identify good investment opportunities as well as the know-how to manage and add value to these investments.” 3VC activity is strongly correlated with valuations and IPO activity in public equity markets. See, for example, Gompers and Lerner (1998), Kaplan and Schoar (2005), and Gompers et al. (2008) for empirical evidence. In addition, see Greenwood and Jovanovic (1999) and Jovanovic and Rousseau (2003, 2005) for evidence about low-frequency variation in entry rates. 4See Gompers and Lerner (2000, 2001) for evidence about lenient funding standards, high valuations, and high capital commitments during venture capital booms. Korteweg and Sorensen (2010) and Kaplan and Lerner (2010) also highlight the negative relationship between VC fund commitments and future returns. 5The VC industry was able to attract material capital commitments only after regulatory reforms and tax changes were implemented in the late 1970s and early 1980s (see, e.g., Kroszner and Strahan 2014). 6In the equilibrium of the baseline setup, consumption levels will generally differ across agents, but the availability of financial markets allowing for perfect risk sharing will imply that consumption growth is equalized across agents. In contrast, when I study the model under incomplete markets in Section 4, equilibrium consumption growth also will differ across agents. 7This restriction is also customary in the patent-race literature (see, e.g., Tirole 1988; Reinganum 1989). 8After succeeding, ventures and their new patents may be acquired by other existing firms, provided that each of these firms has only a finite number of patents corresponding to a zero measure of industries. 9Ventures either fail or succeed, and the value of a venture, conditional on its success, is known at the beginning of each period, so both equity and debt contracts can achieve the same payoff profile, yielding either zero or a fraction of the successful venture’s value. In practice, VCs and entrepreneurs typically hold a substantial fraction of start-ups’ securities (see, e.g., Barry et al. 1990; Korteweg and Sorensen 2010). 10This assumption is only relevant in cases in which VC funds operate probabilistically in a given industry. Yet this will not be an optimal behavior under the considered calibration. In the case of pure strategies, the equilibrium value of |$n_{1}$| can, in any case, be perfectly anticipated at the beginning of a period. 11When incumbents fund regular ideas to enter new product lines, they solve the same maximization problem as all other firms do, provided that each incumbent is restricted to have the highest-quality good in only a finite number of product lines. Appendix B shows an incumbent firm optimally does not fund additional ideas in product lines where it already has the best product. This result is known in the literature as Arrow’s replacement effect. 12Apart from the factor |$A_{t}$|⁠, the specification for the final good production function (9) is common in the Schumpeterian growth literature. See, for example, Francois and Lloyd-Ellis (2003) and Klette and Kortum (2004). 13This measure is closely related to the generalized market equivalent measure, proposed by Korteweg and Nagel (2014). 14This process approximates the discrete-time model calibration in Bansal and Yaron (2004). I thank Hui Chen for providing the estimated Markov chain generator matrix. 15Panel A of Table A4 in the Appendix reports the risk-free rate in the nine Markov states. Chen (2010) provides a detailed description of the asset pricing implications under this specification for preferences and consumption dynamics, including the properties of risk premiums and price dividend ratios of a levered claim to aggregate consumption. 16Alternatively, the net benefits of VC activity are reduced if the best entrepreneurs have only marginally better ideas than regular entrepreneurs do. The exact channels rendering VC funds’ operation suboptimal in industries |$\Psi''$| is not essential to the analysis, as this paper does not perform counterfactual analyses to ask under which conditions the industries |$\Psi''$| might attract VC activity. 17Although there is no single percentile cutoff in the income space that provides a clear-cut match to white-collar labor in the model, it is worth noting that, empirically, the income share of the top 1% also drives much of the variation in the top 5%. 18Kogan et al. (2017) also use market values as a measure of innovative output, but in the context of established publicly listed firms. 19As discussed below in Section 3.2.3, start-up valuations in the model encode low expected returns, such that high exit valuations are not merely the result of high expected cash flows. 20Incumbent firms have a beta of one with respect to the Brownian innovations |$dB$|⁠. See Chen (2010) for an analysis of levered claims in an environment with the same stochastic discount factor (SDF) dynamics. 21The 5-year survival rate of a firm entering the stock market in the average high-|$\mu$| state falls into the range of estimates that empirical studies typically find for dot-com era entrants, that is, between 40% and 50% (see, e.g., Goldfarb, Kirsch, and Miller 2007; Wagner and Cockburn 2010; Luo and Mann 2011). 22See Eisfeldt and Rampini (2006) for evidence about the cyclical properties of capital reallocation and productivity dispersion. 23VC firms have discretion over setting and reporting valuations over time. This issue also causes sample selection problems. 24See Admati and Pfleiderer (1994) and Holmstrom and Tirole (1997). See also Schmidt (2003) and Casamatta (2003) for papers that consider optimal security design in the context of venture capital. 25These dynamics imply that the aggregate wealth share of the group of entrepreneurs and network VCs converges to one in the long run. As discussed below, the wealth distribution is, however, inconsequential for the equilibrium dynamics of the variables of interest in this study. Thus, counteracting this behavior by introducing additional model features, such as overlapping generations, is not necessary. 26A similar effect arises in Garleanu et al. (2015), who study asset pricing implications of imperfect risk sharing associated with firm entry. 27In the calibration, |$dM_{1}(v,t)$|⁠, with |$v \in \Psi'$| is a collection of identically distributed and pairwise uncorrelated random variables with common finite mean and variance. In industries |$v \in \Psi''$| VC funds are not active, such that |$h_{1}(v,t)=0$| and |$dM_{1}(v,t)=0$| at all times. References Acemoglu, D., Akcigit U., Bloom N., and Kerr W. . 2013 . Innovation, reallocation and growth . Working Paper , The University of Pennsylvania . Google Preview WorldCat COPAC Admati, A. R., and Pfleiderer P. . 1994 . Robust financial contracting and the role of venture capitalists . Journal of Finance 49 : 371 – 402 . Google Scholar Crossref Search ADS WorldCat Aiyagari, S. R. 1994 . Uninsured idiosyncratic risk and aggregate saving . The Quarterly Journal of Economics 109 : 659 – 84 . Google Scholar Crossref Search ADS WorldCat Alvaredo, F., Atkinson A. B., Piketty T., and Saez E. . 2012 . Database . http://gmond.parisschoolofeconomics.eu/topincomes. Last accessed November 7 , 2012 . WorldCat Alvarez, F., and Jermann U. J. . 2004 . Using asset prices to measure the cost of business cycles . Journal of Political Economy 112 : 1223 – 56 . Google Scholar Crossref Search ADS WorldCat Bansal, R., Dittmar R. F., and Lundblad C. T. . 2005 . Consumption, dividends, and the cross section of equity returns . The Journal of Finance 60 : 1639 – 72 . Google Scholar Crossref Search ADS WorldCat Bansal, R., Kiku D., and Yaron A. . 2012 . An empirical evaluation of the long-run risks model for asset prices . Critical Finance Review 1 : 183 – 221 . Google Scholar Crossref Search ADS WorldCat Bansal, R., and Yaron A. . 2004 . Risks for the long run: A potential resolution of asset pricing puzzles . Journal of Finance 59 : 1481 – 509 . Google Scholar Crossref Search ADS WorldCat Barry, C. B., Muscarella C. J., Peavy J. I., and Vetsuypens M. R. . 1990 . The role of venture capital in the creation of public companies: Evidence from the going-public process . Journal of Financial Economics 27 : 447 – 71 . Google Scholar Crossref Search ADS WorldCat Bencivenga, V. R., and Smith B. D. . 1991 . Financial intermediation and endogenous growth . Review of Economic Studies 58 : 195 – 209 . Google Scholar Crossref Search ADS WorldCat Berk, J. B., and Green R. C. . 2004 . Mutual fund flows and performance in rational markets . Journal of Political Economy 112 : 1269 – 95 . Google Scholar Crossref Search ADS WorldCat Bernanke, B., and Gertler M. . 1989 . Agency costs, net worth, and business fluctuations . American Economic Review 79 : 14 – 31 . WorldCat Bewley, T. F. 1986 . Stationary monetary equilibrium with a continuum of independently fluctuating consumers , Amsterdam : North-Holland . Google Preview WorldCat COPAC Brunnermeier, M. K., and Sannikov Y. . 2014 . A macroeconomic model with a financial sector . American Economic Review 104 : 379 – 421 . Google Scholar Crossref Search ADS WorldCat Caballero, R. J., and Hammour M. L. . 2000 . Creative destruction and development: Institutions, crises, and restructuring . Working Paper . WorldCat Caballero, R. J., and Hammour M. L. . 2005 . The cost of recessions revisited: A reverse-liquidationist view . Review of Economic Studies 72 : 313 – 41 . Google Scholar Crossref Search ADS WorldCat Caballero, R. J., and Jaffe A. B. . 1993 . How high are the giants’ shoulders: An empirical assessment of knowledge spillovers and creative destruction in a model of economic growth . In NBER Macroeconomics Annual 1993 , Volume 8 , pp. 15 – 86 . Cambridge, MA : National Bureau of Economic Research . Google Preview WorldCat COPAC Casamatta, C. 2003 . Financing and advising: Optimal financial contracts with venture capitalists . Journal of Finance 58 : 2059 – 86 . Google Scholar Crossref Search ADS WorldCat Chen, H. 2010 . Macroeconomic conditions and the puzzles of credit spreads and capital structure . Journal of Finance 65 : 2171 – 212 . Google Scholar Crossref Search ADS WorldCat Cochrane, J. H. 2005 . The risk and return of venture capital . Journal of Financial Economics 75 : 3 – 52 . Google Scholar Crossref Search ADS WorldCat Comin, D., and Gertler M. . 2006 . Medium-term business cycles . American Economic Review 96 : 523 – 51 . Google Scholar Crossref Search ADS WorldCat De Long, J. B. 1990 . ‘Liquidation’ cycles: Old-fashioned real business cycle theory and the Great Depression . Working Paper . WorldCat DeMarzo, P., Kaniel R., and Kremer I. . 2007 . Technological innovation and real investment booms and busts . Journal of Financial Economics , 85 : 735 – 54 . Google Scholar Crossref Search ADS WorldCat Dow, J., Goldstein I., and Guembel A. . 2011 . Incentives for information production in markets where prices affect real investment . Working Paper . WorldCat Duffie, D., and Epstein L. G. . 1992a . Asset pricing with stochastic differential utility . Review of Financial Studies 5 : 411 – 36 . Google Scholar Crossref Search ADS WorldCat Duffie, D., and Epstein L. G. . 1992b . Stochastic differential utility . Econometrica 60 : 353 – 94 . Google Scholar Crossref Search ADS WorldCat Eisfeldt, A. L., and Rampini A. A. . 2006 . Capital reallocation and liquidity . Journal of Monetary Economics 53 : 369 – 99 . Google Scholar Crossref Search ADS WorldCat Elenev, V., Landvoigt T., and Van Nieuwerburgh S. . 2018 . A macroeconomic model with financially constrained producers and intermediaries . Working Paper . WorldCat Epstein, L. G., and Zin S. E. . 1989 . Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework . Econometrica 57 : 937 – 69 . Google Scholar Crossref Search ADS WorldCat Francois, P., and Lloyd-Ellis H. . 2003 . Animal spirits through creative destruction . American Economic Review 93 : 530 – 50 . Google Scholar Crossref Search ADS WorldCat Garleanu, N., Panageas S., Papanikolaou D., and Yu J. . 2015 . Drifting apart: The pricing of assets when the benefits of growth are not shared equally . Working Paper , Booth School of Business, University of Chicago . Google Preview WorldCat COPAC Garleanu, N. B., Kogan L., and Panageas S. . 2012 . Displacement risk and asset returns . Journal of Financial Economics 105 : 491 – 510 . Google Scholar Crossref Search ADS WorldCat Goldfarb, B., Kirsch D., and Miller D. A. . 2007 . Was there too little entry during the dot com era? Journal of Financial Economics 86 : 100 – 44 . Google Scholar Crossref Search ADS WorldCat Gomes, J., Kogan L., and Zhang L. . 2003 . Equilibrium cross section of returns . Journal of Political Economy 111 : 693 – 732 . Google Scholar Crossref Search ADS WorldCat Gompers, P., Kovner A., Lerner J., and Scharfstein D. . 2008 . Venture capital investment cycles: The impact of public markets . Journal of Financial Economics 87 : 1 – 23 . Google Scholar Crossref Search ADS WorldCat Gompers, P., and Lerner J. . 2000 . Money chasing deals? The impact of fund inflows on private equity valuation . Journal of Financial Economics 55 : 281 – 325 . Google Scholar Crossref Search ADS WorldCat Gompers, P., and Lerner J. . 2001 . The venture capital revolution . Journal of Economic Perspectives 15 : 145 – 68 . Google Scholar Crossref Search ADS WorldCat Gompers, P., and Lerner J. . 2003 . Short-term America revisited? Boom and bust in the venture capital industry and the impact on innovation . In Innovation Policy and the Economy , Volume 3 NBER Chapters , pp. 1 – 28 . Cambridge, MA : National Bureau of Economic Research . Google Preview WorldCat COPAC Gompers, P., and Lerner J. . 2004 . The venture capital cycle . Cambridge : MIT Press . Google Preview WorldCat COPAC Gompers, P., and Lerner J. . 1998 . What drives venture capital fundraising? Brookings Papers on Economic Activity Microeconomics , 1998 : 149 – 204 . Google Scholar Crossref Search ADS WorldCat Grant, D. 1977 . Portfolio performance and the cost of timing decisions . Journal of Finance 32 : 837 – 46 . WorldCat Greenwood, J., and Jovanovic B. . 1990 . Financial development, growth, and the distribution of income . Journal of Political Economy 98 : 1076 – 107 . Google Scholar Crossref Search ADS WorldCat Greenwood, J., and Jovanovic B. . 1999 . The information-technology revolution and the stock market . American Economic Review 89 : 116 – 22 . Google Scholar Crossref Search ADS WorldCat Grossman, G. M., and Helpman E. . 1991 . Quality ladders in the theory of growth . Review of Economic Studies 58 : 43 – 61 . Google Scholar Crossref Search ADS WorldCat Gul, F., and Lundholm R. . 1995 . Endogenous timing and the clustering of agents’ decisions . Journal of Political Economy 103 : 1039 – 66 . Google Scholar Crossref Search ADS WorldCat Gupta, U. 2000 . Done deals: venture capitalists tell their stories . Cambridge, MA : Harvard Business School Press . Google Preview WorldCat COPAC Hall, R. E., and Woodward S. E. . 2007 . The incentives to start new companies: Evidence from venture capital . Working Paper . WorldCat Hall, R. E., and Woodward S. E. . 2010 . The burden of the nondiversifiable risk of entrepreneurship . American Economic Review 100 : 1163 – 94 . Google Scholar Crossref Search ADS WorldCat Hansen, L. P., Heaton J. C., and Li N. . 2008 . Consumption strikes back? Measuring long-run risk . Journal of Political Economy 116 : 260 – 302 . Google Scholar Crossref Search ADS WorldCat He, Z., and Krishnamurthy A. . 2012 . A model of capital and crises . The Review of Economic Studies 79 : 735 – 77 . Google Scholar Crossref Search ADS WorldCat He, Z., and Krishnamurthy A. . 2013 . Intermediary asset pricing . American Economic Review 103 : 732 – 70 . Google Scholar Crossref Search ADS WorldCat Hobijn, B., and Jovanovic B. . 2001 . The information-technology revolution and the stock market: Evidence . American Economic Review 91 : 1203 – 20 . Google Scholar Crossref Search ADS WorldCat Hochberg, Y. V., Ljungqvist A., and Lu Y. . 2007 . Whom you know matters: Venture capital networks and investment performance . Journal of Finance 62 : 251 – 301 . Google Scholar Crossref Search ADS WorldCat Holmstrom, B., and Tirole J. . 1997 . Financial intermediation, loanable funds, and the real sector . The Quarterly Journal of Economics 112 : 663 – 91 . Google Scholar Crossref Search ADS WorldCat Hsu, H.-C., Reed A. V., and Rocholl J. . 2010 . The new game in town: Competitive effects of IPOs . Journal of Finance 65 : 495 – 528 . Google Scholar Crossref Search ADS WorldCat Huggett, M. 1993 . The risk-free rate in heterogeneous-agent incomplete-insurance economies . Journal of Economic Dynamics and Control 17 : 953 – 69 . Google Scholar Crossref Search ADS WorldCat Jagannathan, R., and Wang Z. . 1996 . The conditional CAPM and the cross-section of expected returns . Journal of Finance 51 : 3 – 53 . Google Scholar Crossref Search ADS WorldCat Jeng, L. A., and Wells P. C. . 2000 . The determinants of venture capital funding: evidence across countries . Journal of Corporate Finance 6 : 241 – 89 . Google Scholar Crossref Search ADS WorldCat Johnson, T. C. 2007 . Optimal learning and new technology bubbles . Journal of Monetary Economics 54 : 2486 – 511 . Google Scholar Crossref Search ADS WorldCat Jovanovic, B., and MacDonald G. M. . 1994 . The life cycle of a competitive industry . Journal of Political Economy 102 : 322 – 47 . Google Scholar Crossref Search ADS WorldCat Jovanovic, B., and Rousseau P. L. . 2001 . Why wait? A century of life before IPO . American Economic Review 91 : 336 – 41 . Google Scholar Crossref Search ADS WorldCat Jovanovic, B., and Rousseau P. L. . 2003 . Two technological revolutions . Journal of the European Economic Association 1 : 419 – 28 . Google Scholar Crossref Search ADS WorldCat Jovanovic, B., and Rousseau P. L. . 2005 . General purpose technologies . In Handbook of economic growth , eds. Aghion P. and Durlauf, S. pp. 1181 – 224 . Amsterdam : Elsevier . Google Preview WorldCat COPAC Kaplan, S. N., and Lerner J. . 2010 . It ain’t broke: The past, present, and future of venture capital . Journal of Applied Corporate Finance 22 : 36 – 47 . Google Scholar Crossref Search ADS WorldCat Kaplan, S. N., and Schoar A. . 2005 . Private equity performance: Returns, persistence, and capital flows . Journal of Finance 60 : 1791 – 823 . Google Scholar Crossref Search ADS WorldCat Kaplan, S. N., and Strömberg P. . 2001 . Venture capitalists as principals: Contracting, screening, and monitoring . The American Economic Review 91 : 426 – 30 . Google Scholar Crossref Search ADS WorldCat Kaplan, S. N., and Strömberg P. . 2004 . Characteristics, contracts, and actions: Evidence from venture capitalist analyses . Journal of Finance 59 : 2177 – 210 . Google Scholar Crossref Search ADS WorldCat King, R. G., and Levine R. . 1993a . Finance and growth: Schumpeter might be right . The Quarterly Journal of Economics 108 : 717 – 37 . Google Scholar Crossref Search ADS WorldCat King, R. G., and Levine R. . 1993b . Finance, entrepreneurship and growth: Theory and evidence . Journal of Monetary Economics 32 : 513 – 42 . Google Scholar Crossref Search ADS WorldCat Kiyotaki, N., and Moore J. . 1997 . Credit cycles . Journal of Political Economy 105 : 211 – 48 . Google Scholar Crossref Search ADS WorldCat Klette, T. J., and Kortum S. . 2004 . Innovating firms and aggregate innovation . Journal of Political Economy 112 : 986 – 1018 . Google Scholar Crossref Search ADS WorldCat Kogan, L., Papanikolaou D., Seru A., and Stoffman N. . 2017 . Technological innovation, resource allocation, and growth . The Quarterly Journal of Economics 132 : 665 – 712 . Google Scholar Crossref Search ADS WorldCat Kogan, L., Papanikolaou D., and Stoffman N. . 2015 . Winners and losers: Creative destruction and the stock market . Working Paper . WorldCat Korteweg, A., and Sorensen M. . 2010 . Risk and return characteristics of venture capital-backed entrepreneurial companies . Review of Financial Studies 23 : 3738 – 72 . Google Scholar Crossref Search ADS WorldCat Korteweg, A. G., and Nagel S. . 2014 . Risk-adjusting the returns to venture capital . Research Paper , Stanford University Graduate School of Business . Google Preview WorldCat COPAC Kortum, S., and Lerner J. . 2000 . Assessing the contribution of venture capital to innovation . RAND Journal of Economics 31 : 674 – 92 . Google Scholar Crossref Search ADS WorldCat Kroszner, R., and Strahan P. . 2014 . Economic regulation and its reform: What have we learned? Cambridge, MA : National Bureau of Economic Research . Google Preview WorldCat COPAC Krueger, D., and Lustig H. . 2010 . When is market incompleteness irrelevant for the price of aggregate risk (and when is it not)? Journal of Economic Theory 145 : 1 – 41 . Google Scholar Crossref Search ADS WorldCat Kung, H., and Schmid L. . 2015 . Innovation, growth, and asset prices . Journal of Finance 70 : 1001 – 37 . Google Scholar Crossref Search ADS WorldCat Laitner, J., and Stolyarov D. . 2003 . Technological change and the stock market . American Economic Review 93 : 1240 – 67 . Google Scholar Crossref Search ADS WorldCat Laitner, J., and Stolyarov D. . 2004 . Aggregate returns to scale and embodied technical change: Theory and measurement using stock market data . Journal of Monetary Economics 51 : 191 – 233 . Google Scholar Crossref Search ADS WorldCat Levine, R., and Zervos S. . 1998 . Stock markets, banks, and economic growth . American Economic Review 88 : 537 – 58 . WorldCat Ljungqvist, A., and Richardson M. . 2003 . The cash flow, return and risk characteristics of private equity . Working Paper . WorldCat Loughran, T., and Ritter J. R. . 1995 . The new issues puzzle . Journal of Finance 50 : 23 – 51 . Google Scholar Crossref Search ADS WorldCat Lucas, R. E. 1987 . Models of business cycles . New York : Basil Blackwell . Google Preview WorldCat COPAC Lucas, R. E. 2003 . Macroeconomic priorities . American Economic Review 93 : 1 – 14 . Google Scholar Crossref Search ADS WorldCat Luo, T., and Mann A. . 2011 . Survival and growth of Silicon Valley high-tech businesses born in 2000 . Monthly Labor Review September : 16 – 31 . WorldCat Nash, J. 1950 . The Bargaining Problem . Econometrica 18 : 155 – 62 . Google Scholar Crossref Search ADS WorldCat National Venture Capital Association . 2012 . 2012 NVCA Yearbook . Eagan, MN : Thomson Reuters . WorldCat COPAC National Venture Capital Association . 2017 . NVCA 2017 Yearbook Data Pack . Eagan, MN : Thomson Reuters . WorldCat COPAC Newton, I. 1675 . Letter from Sir Isaac Newton to Robert Hooke . Historical Society of Pennsylvania . Google Preview WorldCat COPAC Parker, J. A., and Julliard C. . 2005 . Consumption risk and the cross section of expected returns . Journal of Political Economy 113 : 185 – 222 . Google Scholar Crossref Search ADS WorldCat Parker, J. A., and Vissing-Jorgensen A. . 2009 . Who bears aggregate fluctuations and how? American Economic Review 99 : 399 – 405 . Google Scholar Crossref Search ADS WorldCat Parker, J. A., and Vissing-Jorgensen A. . 2010 . The increase in income cyclicality of high-income households and its relation to the rise in top income shares . Brookings Papers on Economic Activity 41 : 1 – 70 . Google Scholar Crossref Search ADS WorldCat Pastor, L., and Veronesi P. . 2005 . Rational IPO waves . Journal of Finance 60 : 1713 – 57 . Google Scholar Crossref Search ADS WorldCat Pastor, L., and Veronesi P. . 2009 . Technological revolutions and stock prices . American Economic Review 99 : 1451 – 83 . Google Scholar Crossref Search ADS WorldCat Piketty, T., and Saez E. . 2003 . Income inequality in the United States. 1913–1998 . The Quarterly Journal of Economics 118 : 1 – 39 . Google Scholar Crossref Search ADS WorldCat Rajan, R. G., and Zingales L. . 1998 . Financial dependence and growth . American Economic Review 88 : 559 – 86 . WorldCat Reinganum, J. F. 1989 . The timing of innovation: Research, development, and diffusion . In Handbook of industrial organization , eds. Schmalensee R. and Willig, R. chap. 14 , pp. 849 – 908 . Amsterdam : Elsevier . Google Preview WorldCat COPAC Samila, S., and Sorenson O. . 2011 . Venture capital, entrepreneurship, and economic growth . The Review of Economics and Statistics 93 : 338 – 49 . Google Scholar Crossref Search ADS WorldCat Scharfstein, D. S., and Stein J. C. . 1990 . Herd behavior and investment . American Economic Review 80 : 465 – 79 . WorldCat Schmidt, K. M. 2003 . Convertible securities and venture capital finance . Journal of Finance 58 : 1139 – 66 . Google Scholar Crossref Search ADS WorldCat Schumpeter, J. A. 1934 . The theory of economic development . Cambridge, MA : Harvard University Press . Google Preview WorldCat COPAC Shleifer, A. 1986 . Implementation cycles . Journal of Political Economy 94 : 1163 – 90 . Google Scholar Crossref Search ADS WorldCat Stein, J. C. 1997 . Waves of creative destruction: Firm-specific learning-by-doing and the dynamics of innovation . Review of Economic Studies 64 : 265 – 88 . Google Scholar Crossref Search ADS WorldCat The World Bank . 2018 . World Bank national accounts data . http://data.worldbank.org/indicator/NE.CON.TOTL.CD. WorldCat Tirole, J. 1988 . The theory of industrial organization . Cambridge : MIT Press . Google Preview WorldCat COPAC Uhlig, H. 1996 . A law of large numbers for large economies (*) . Economic Theory 8 : 41 – 50 . Google Scholar Crossref Search ADS WorldCat Wagner, S., and Cockburn I. . 2010 . Patents and the survival of Internet-related IPOs . Research Policy 39 : 214 – 28 . Google Scholar Crossref Search ADS WorldCat Ward, C. 2013 . Is the IT revolution over? An asset pricing view . Working Paper , Carlson School of Management, University of Minnesota . Google Preview WorldCat COPAC Weil, P. 1990 . Nonexpected utility in macroeconomics . The Quarterly Journal of Economics 105 : 29 – 42 . Google Scholar Crossref Search ADS WorldCat © The Author(s) 2019. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
TI - Venture Capital and the Macroeconomy
JF - The Review of Financial Studies
DO - 10.1093/rfs/hhz031
DA - 2019-11-01
UR - https://www.deepdyve.com/lp/oxford-university-press/venture-capital-and-the-macroeconomy-BEVSW7u3hP
SP - 4387
VL - 32
IS - 11
DP - DeepDyve
ER -