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The Review of Economic Studies
, Volume Advance Article – Apr 23, 2018

43 pages

/lp/ou_press/financial-intermediary-capital-vKRZlgxj3d

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2018. Published by Oxford University Press on behalf of The Review of Economic Studies Limited.
- ISSN
- 0034-6527
- eISSN
- 1467-937X
- D.O.I.
- 10.1093/restud/rdy020
- Publisher site
- See Article on Publisher Site

Abstract We propose a dynamic theory of financial intermediaries that are better able to collateralize claims than households, that is, have a collateralization advantage. Intermediaries require capital as they have to finance the additional amount that they can lend out of their own net worth. The net worth of financial intermediaries and the corporate sector are both state variables affecting the spread between intermediated and direct finance and the dynamics of real economic activity, such as investment, and financing. The accumulation of net worth of intermediaries is slow relative to that of the corporate sector. The model is consistent with key stylized facts about macroeconomic downturns associated with a credit crunch, namely, their severity, their protractedness, and the fact that the severity of the credit crunch itself affects the severity and persistence of downturns. The model captures the tentative and halting nature of recoveries from crises. 1. Introduction The capitalization of financial intermediaries is arguably critical for economic fluctuations and growth. We provide a dynamic theory of financial intermediaries that have a collateralization advantage, that is, are better able to collateralize claims than households. Financial intermediaries require net worth as their ability to refinance their loans to firms by borrowing from households is limited, as intermediaries need to collateralize their promises as well. Importantly, since both intermediaries and firms are subject to collateral constraints, the net worth of both plays a role in our model, in contrast to most previous work, and these two state variables jointly determine the dynamics of economic activity, investment, financing, and loan spreads. A key feature of our model is that the accumulation of the net worth of intermediaries is slow relative to that of the corporate sector. The slow-moving nature of intermediary capital results in economic dynamics that are consistent with key stylized facts about macroeconomic downturns associated with a credit crunch, namely, their severity, their protractedness, and the fact that the severity of the credit crunch itself affects the severity and persistence of downturns. Most uniquely, the model captures the tentative and halting nature of recoveries from crises. In our model, firms need to finance investment and can raise financing from both intermediaries and households. Firm financing needs to be collateralized with tangible assets and is subject to two types of collateral constraints, one for loans from households and one for loans from intermediaries. Firms require net worth as collateral constraints limit financing. Intermediaries are better able to enforce collateralized claims and thus can lend more to firms than households can. However, intermediaries have to collateralize their promises as well, and can borrow against their corporate loans only to the extent that households themselves can collateralize the assets backing these loans. Thus, intermediaries need to finance the additional amount that they can lend out of their own net worth, giving a role to financial intermediary capital. We show that these collateral constraints can be derived from an economy with limited enforcement that constrains firms’ and intermediaries’ ability to make credible promises.1 We focus our analysis on the deterministic version of the economy and start by analysing the steady state. In the steady state, intermediaries are essential in our economy in the sense that allocations can be achieved with financial intermediaries, which cannot be achieved otherwise. Since intermediary net worth is limited, intermediated finance commands a positive spread over the interest rate charged by households. Moreover, in the steady state, the equilibrium capitalization of both the representative firm and intermediary are positive. Steady state firm net worth is determined by the fraction of tangible assets that firms cannot pledge to intermediaries or households and thus have to finance internally, while steady state intermediary net worth is determined by the fraction of investment that intermediaries have to finance due to their collateralization advantage, that is, by the difference in the ability to enforce collateralized claims between intermediaries and households. Away from the steady state, the equilibrium spread on intermediated finance is determined by firm and intermediary net worth jointly. Intermediary net worth increases intermediated loan supply and hence reduces the spread all else equal. In contrast, firm net worth has two opposing effects on intermediated loan demand: on the one hand, firm net worth increases investment, lowering the levered marginal product of capital, reducing firms’ willingness to pay, and lowering the spread; on the other hand, firm net worth, by increasing investment, increases firms’ collateralizable assets, which in turn raises loan demand, raising the spread. Hence, the spread can be high or low when firm net worth is low as it depends on the relative capitalization of firms and intermediaries. When intermediary net worth is relatively scarce, the collateral constraint on intermediated finance is slack and firm net worth reduces the spread. When firm net worth is relatively scarce instead, the collateral constraint on intermediated finance binds and firm net worth increases the spread as it increases firms’ ability to pledge and hence loan demand. This interaction of loan supply and demand results in rich and subtle dynamics for intermediated finance and its spread. In the dynamics of our model, two state variables, the net worth of firms and intermediaries, jointly determine the dynamic supply and demand for intermediated loans and the equilibrium intermediary interest rate. A key feature of the equilibrium dynamics is that intermediary net worth accumulation is slow relative to corporate net worth accumulation, at least early in a recovery; the reason being that intermediary net worth grows at the intermediary interest rate, which is at most the marginal levered product of capital, and may be lower than that when the collateral constraint on intermediated finance binds; in contrast, firms accumulate net worth at the average levered product of capital, which exceeds the marginal levered product of capital. Several aspects of the dynamic response of the economy to drops in the net worth of firms, intermediaries, or both are noteworthy. First, consider the response of the economy to a drop in corporate net worth only. In such a downturn, corporate investment drops and the collateral constraints on intermediated finance implies a reduction in firms’ loan demand for intermediated loans. Facing reduced corporate loan demand, intermediaries respond by paying dividends and may lend to households at an interest rate less than their rate of time preference, that is, hold “cash”; thus, initially, the intermediary interest rate drops. The reason that intermediaries conserve net worth despite temporarily low interest rates is that, since firms reaccumulate net worth faster, corporate loan demand is expected to recover relatively quickly and intermediary capital becomes scarce again, raising the intermediary interest rate. Secondly, the recovery from a drop in intermediary net worth (a “credit crunch”) is relatively slow making such episodes protracted. In a credit crunch, the spread on intermediary finance rises and investment drops even if the corporate sector remains well capitalized, as firms are forced to delever due to the limited supply of intermediated loans and thus have to finance a larger part of their investment internally. Indeed, deleveraging may mean that firms temporarily accumulate more net worth then they retain in the steady state. Moreover, and importantly, a credit crunch can have persistent real effects as corporate investment may not recover for a prolonged period of time, due to the slow recovery of intermediary capital. Thirdly, a drop in both corporate and intermediary net worth (a downturn associated with a credit crunch) makes the downturn more severe and the recovery more protracted, featuring a higher spread on intermediated finance. Moreover, the recovery can stall, after an initial relatively swift recovery, when firm net worth has partially recovered while intermediaries are yet to recover. When the economy stalls, firms may seem to be well capitalized because they are paying dividends, but the economy nevertheless has not fully recovered. The severity of the credit crunch itself significantly affects the depth and protractedness of the macroeconomic downturn and the spike in the spread on intermediary finance; indeed, the recovery from a downturn associated with a more severe credit crunch is especially slow and halting, with output depressed and the spread elevated for a prolonged period of time. Finally, our theory predicts that, in a bank-oriented economy, downturns associated with a credit crunch are more severe and more protracted, with longer stalls of the recovery at lower levels of investment. Thus, the recovery from crises in bank-oriented economies may be more sluggish than in economies with more market-oriented financial systems. We revisit the evidence on the effect of financial crises from the vantage point of our theory. There are three main stylized facts about downturns associated with financial crises that emerge from prior empirical work: (1) downturns associated with financial crises are more severe; (2) recoveries from financial crises are protracted and often tentative; and (3) the severity of the financial crises itself affects the severity and protractedness of the downturn. Consistent with this evidence, our model predicts that the effects of a credit crunch on economic activity is protracted due to the slow accumulation of intermediary net worth. But perhaps most uniquely, our model captures the tentative and halting nature of recoveries from such episodes emphasized by Reinhart and Rogoff (2014) and allows the analysis of the severity of the credit crunch itself on the recovery. Thus, our model implies empirically plausible dynamics. Few extant theories of financial intermediaries provide a role for intermediary capital. Notable is in particular Holmström and Tirole (1997) who model intermediaries as monitors that cannot commit to monitoring and hence need to have their own capital at stake to have incentives to monitor. In their static analysis, firm and intermediary capital are exogenous and the comparative statics with respect to these are analysed. Holmström and Tirole conclude that “[a] proper investigation ... must take into account the feedback from interest rates to capital values. This will require an explicitly dynamic model, for instance, along the lines of Kiyotaki and Moore [1997].” We provide a dynamic model in which the joint evolution of firm and intermediary net worth and the interest rate on intermediated finance are endogenously determined. Diamond and Rajan (2000) and Diamond (2007) model intermediaries as lenders which are better able to enforce their claims due to their specific liquidation or monitoring ability in a similar spirit to our model, but do not consider equilibrium dynamics. In contrast, the capitalization of intermediaries plays essentially no role in liquidity provision theories of financial intermediation (Diamond and Dybvig 1983), in theories of financial intermediaries as delegated, diversified monitors (Diamond 1984; Ramakrishnan and Thakor 1984; Williamson 1986) or in coalition based theories (Townsend 1978; Boyd and Prescott 1986). Dynamic models in which net worth plays a role, such as Bernanke and Gertler (1989) and Kiyotaki and Moore (1997), typically consider the role of firm net worth only, although dynamic models in which intermediary net worth matters have recently been considered (see, e.g., Gertler and Kiyotaki 2010, and Brunnermeier and Sannikov 2014).2 However, to the best of our knowledge, we are the first to consider a dynamic contracting model in which both firm and intermediary net worth are critical and jointly affect the dynamics of financing, spreads, and economic activity. In Section 2, we describe the model with two types of collateral constraints, for intermediated and direct finance, respectively, and discuss how these collateral constraints can be derived in an economy with limited enforcement and limited participation. Section 3 shows that intermediation is essential in our economy and determines the capitalization of intermediaries and spreads on intermediated finance in the steady state. The dynamics of intermediary capital are analysed in Section 4, focusing on the dynamic interaction between corporate and intermediary net worth, the two state variables in the model; specifically, we consider the effects of a downturn, a credit crunch, and a downturn associated with a credit crunch. In Section 5, we use the model to revisit three main stylized facts about downturns associated with financial crises. Section 6 considers risk management of financial intermediary capital. Section 7 concludes. All proofs are in Appendix A. 2. Collateralized Finance with Intermediation We propose a dynamic model of financial intermediaries that have a collateralization advantage, that is, are better able to collateralize claims than households. In this model firms can borrow from both intermediaries and households, and all financing needs to be collateralized. Firm financing is subject to two types of collateral constraints, one for loans from households and one for loans from intermediaries. Since intermediaries are better able to enforce collateralized claims, they can lend more than households, but the additional amount that they can lend has to be financed out of their own net worth, giving a role to financial intermediary capital. Thus, the net worth of both intermediaries and firms are state variables and jointly determine economic activity. We show that these collateral constraints can be derived from an economy with limited enforcement that constrains firms’ and intermediaries’ ability to make credible promises. Intermediaries, but not households, participate in markets at all times which affords intermediaries with an advantage in enforcing claims. This economy with limited enforcement and limited participation is equivalent to our economy with collateral constraints. 2.1. Environment Time is discrete and the horizon infinite. We focus on a deterministic environment here and study a stochastic environment in Section 6. There are three types of agents: entrepreneurs, financial intermediaries, and households; we discuss these in turn. There is a continuum of entrepreneurs or firms with measure one which are risk neutral and subject to limited liability and discount the future at rate $$\beta\in(0,1)$$. We consider an environment with a representative firm. The representative firm (which we at times refer to simply as the firm or the corporate sector) has limited net worth $$w_0>0$$ at time 0 and has access to a standard neoclassical production technology; an investment of capital $$k_t$$ at time $$t$$ yields output $$A'f(k_t)$$ at time $$t+1$$ where $$A'>0$$ is the total factor productivity and $$f(\cdot)$$ is the production function, which is strictly increasing and strictly concave and satisfies the Inada condition $$\lim_{k\rightarrow 0} f_k(k)=+\infty$$. Capital depreciates at rate $$\delta\in(0,1)$$. The firm can raise financing from both intermediaries and households as we discuss below. There is a continuum of financial intermediaries with measure one which are risk neutral, subject to limited liability, and discount future payoffs at $$\beta_i\in (0,1)$$. We consider the problem of a representative financial intermediary with limited net worth $$w_{i0}>0$$ at time 0.3 Intermediaries can lend to and borrow from firms and households as described in more detail below. There is a continuum of households with measure one which are risk neutral and discount future payoffs at a rate $$R^{-1}\in(0,1)$$. Households have a large endowment of funds and collateral in all dates, and hence are not subject to enforcement problems but rather are able to commit to deliver on their promises. They are willing to provide any claim at a rate of return $$R$$ as long as the claims satisfy the firms’ and intermediaries’ collateral constraints. We assume that $$\beta<\beta_i<R^{-1}$$, that is, households are more patient than intermediaries which in turn are more patient than the firms. Since firms and intermediaries are financially constrained, they would have an incentive to accumulate net worth and save themselves out of their constraints. Assuming that firms and intermediaries are impatient relative to households is a simple way to ensure that their net worth matters even in the long run. Moreover, assuming that intermediaries are somewhat more patient than firms implies that the net worth of both the corporate sector and the intermediary sector are uniquely determined in the long run, too. We think these features are desirable properties of a dynamic model of intermediation and are empirically plausible. Financial intermediaries in this economy have a collateralization advantage. Specifically, intermediaries are better able to collateralize claims than households; intermediaries can seize up to fraction $$\theta_i\in(0,1)$$ of the (resale value of) collateral backing promises issued to them whereas households can seize only fraction $$\theta<\theta_i$$, where $$\theta>0$$. One interpretation of the environment is that there are three types of capital, working capital, equipment (fraction $$\theta_i-\theta$$), and structures (fraction $$\theta$$) (see Figure 1). Firms have to finance working capital entirely out of their own net worth. Only intermediaries can lend against equipment, but both households and intermediaries can lend against structures. Equipment loans have to be extended by intermediaries and have to be financed out of financial intermediary capital. We refer to these loans as intermediated finance. In contrast, structure loans can be provided by either intermediaries or households. We assume that these loans are provided by households and refer to such loans as direct finance. This is without loss of generality and we could equivalently assume that all corporate loans are extended by the intermediary who in turn borrows from households, which we refer to as the indirect implementation. However, we focus on the (equivalent) direct implementation in which households extend all structure loans directly throughout as it simplifies the notation and analysis.4 Figure 1 View largeDownload slide This figure shows, at the top, the extent to which one unit of capital can be collateralized by households (fraction $$\theta$$, interpreted as structures) and intermediaries (fraction $$\theta_i$$, interpreted to include equipment), in the middle, the collateral value next period after depreciation, and at the bottom, the maximal amount that households and intermediaries can finance, as well as the minimum amount of internal funds required. Figure 1 View largeDownload slide This figure shows, at the top, the extent to which one unit of capital can be collateralized by households (fraction $$\theta$$, interpreted as structures) and intermediaries (fraction $$\theta_i$$, interpreted to include equipment), in the middle, the collateral value next period after depreciation, and at the bottom, the maximal amount that households and intermediaries can finance, as well as the minimum amount of internal funds required. We assume that loans are one-period and the economy has markets in two types of one-period ahead claims, claims provided by intermediaries and claims provided by households, each subject to a collateral constraint. These collateral constraints are similar to the ones in Kiyotaki and Moore (1997), except that there are different collateral constraints for promises to pay intermediaries and households. Here we simply assume that there are only one-period ahead claims and that intermediaries provide the equipment loans, and only the equipment loans, and must finance these out of their own net worth. In Section 2.3, we provide an environment with limited enforcement and limited participation which is equivalent to the economy with collateral constraints described here. In that environment each period has two subperiods, morning and afternoon, and equipment can serve as collateral only in the morning. The key assumption affording intermediaries an enforcement advantage is that intermediaries, but not households, participate in markets at all times; thus, equipment loans must be provided by intermediaries. Moreover, limited enforcement of intermediaries’ liabilities implies that intermediaries must finance such loans out of their own funds. Thus, the properties we assume here are in fact endogenous properties of optimal dynamic contracts. 2.2. Economy with collateral constraints We write the firm’s and intermediary’s problems recursively by defining an appropriate state variable, net worth, for the firm ($$w$$) and intermediary ($$w_i$$).5 The state of the economy $$z\equiv \{w,w_i\}$$ comprises these two endogenous state variables, the net worth of the corporate sector $$w$$ and of the intermediary sector $$w_i$$. The interest rate on intermediated finance $$R_i'$$ depends on the state $$z$$ of the economy, as shown below, but we suppress the argument for notational simplicity. The firm’s problem stated recursively is, for given net worth $$w$$ and aggregate state $$z$$, to maximize the discounted expected value of future dividends by choosing a dividend payout policy $$d$$, capital $$k$$, promises $$b'$$ and $$b'_i$$ to households and intermediaries, and net worth $$w'$$ for the next period, taking the interest rate on intermediated finance $$R'_i$$ and its law of motion as given, to solve \begin{equation} \label{eqn:obj} v(w,z) = \max_{\{d,k,b',b'_i,w'\}} d+ \beta v(w',z') \end{equation} (1) subject to the budget constraints for the current and next period \begin{eqnarray} w & \ge & d + k - b' - b'_i, \label{eqn:bc} \\ \end{eqnarray} (2) \begin{eqnarray} A' f\left(k\right) + k(1-\delta) & \ge & w'+Rb'+ R_i'b_i', \label{eqn:bcp} \end{eqnarray} (3) the collateral constraints for loans from intermediaries and households \begin{eqnarray} (\theta_i-\theta) k (1-\delta) & \ge & R_i' b_i', \label{eqn:cci} \\ \end{eqnarray} (4) \begin{eqnarray} \theta k (1-\delta) & \ge & R b', \label{eqn:cc} \end{eqnarray} (5) and the non-negativity constraints \begin{equation} d, k, b_i' \ge 0. \label{eqn:nn} \end{equation} (6) Next period the firm repays $$Rb'$$ to households and $$R'_ib'_i$$ to financial intermediaries as the budget constraint for the next period, equation (3), shows.6 While equation (3) is stated as an inequality, which allows for free disposal, it binds at an optimal solution, and hence we can define the net worth of the firm (next period) as $$w' \equiv A' f\left(k\right) + k(1-\delta) - Rb' - R_i'b_i'$$, that is, cash flows plus assets (net of depreciation) minus liabilities. The budget constraint for this period, equation (2), states that current net worth can be spent on dividends and purchases of capital net of the proceeds of the loans from households and intermediaries. The interest rate on loans from households $$R$$ is constant as discussed above. In the direct implementation which we focus on, equipment loans are provided by intermediaries and all structure loans are provided by households. Our economy has two types of collateral constraints (4) and (5), illustrated in Figure 1; these state that repayments to intermediaries and households cannot exceed the residual value of equipment and structures, respectively.7 These collateral constraints are inequality constraints and may or may not bind. The intermediary’s problem stated recursively is, for given net worth $$w_i$$, to maximize the discounted value of future dividends by choosing a dividend payout policy $$d_i$$, loans to households $$l'$$, intermediated loans to firms $$l'_i$$, and net worth $$w'_i$$ next period to solve \begin{equation} \label{eqn:obji} v_i(w_i,z)=\max_{\{d_i,l',l'_i,w'_i\}} d_i+\beta_i v_i(w'_i,z') \end{equation} (7) subject to the budget constraints for the current and next period \begin{eqnarray} w_i & \ge & d_i+l'+l'_i, \label{eqn:bci} \\ \end{eqnarray} (8) \begin{eqnarray} Rl'+R'_il'_i & \ge & w_i', \label{eqn:bcpi} \end{eqnarray} (9) and the non-negativity constraints \begin{equation} d_i, l', l_i' \ge 0. \label{eqn:nni} \end{equation} (10) We can define the net worth of the intermediary (next period) as $$w_i'\equiv Rl'+R'_il'_i$$, that is, the sum of the proceeds from loans to households and firms (as equation (9) binds at an optimal solution). Recall that we focus on the direct implementation in which the intermediary only lends the additional amount that it can take as collateral from firms to simplify the analysis (but this is without loss of generality). In this direct implementation, the intermediary can lend to households but not borrow from them ($$l'\ge 0$$); thus, the intermediary’s collateral constraint reduces to a short-sale constraint.8 We now define an equilibrium for our economy which determines both aggregate economic activity and the cost of intermediated finance. Definition 1 (Equilibrium) An equilibrium is an allocation $$x\equiv[d,k,b^{\prime},b_i^{\prime},w^{\prime}]$$ for the representative firm and $$x_i\equiv[d_i,l^{\prime},l_i^{\prime},w_i^{\prime}]$$ for the representative intermediary for all dates and an interest rate process $$R'_i$$ for intermediated finance such that (1) $$x$$ solves the firm’s problem in equations (1)–(6) and $$x_i$$ solves the intermediary’s problem in equations (7)–(10) and (2) the market for intermediated finance clears in all dates, that is, $$l'_i = b'_i.$$ Note that equilibrium promises are default free, as the promises satisfy the collateral constraints (4) and (5), which ensure that neither firms nor financial intermediaries are able to issue promises on which it is not credible to deliver. The first-order conditions of the firm’s problem in equations (1)–(6), which are necessary and sufficient, can be written as \begin{eqnarray} \mu & = & 1+\nu_d, \label{eqn:dfoc} \\ \end{eqnarray} (11) \begin{eqnarray} \mu & = & \beta\left(\mu'\left[A' f_k\left(k\right) + (1-\delta)\right]+\left[\lambda'\theta+\lambda'_i(\theta_i-\theta)\right](1-\delta)\right), \label{eqn:kpfoc} \\ \end{eqnarray} (12) \begin{eqnarray} \mu & = & R \beta\mu'+R\beta\lambda', \label{eqn:bpfoc} \\ \end{eqnarray} (13) \begin{eqnarray} \mu & = & R'_i\beta\mu'+R'_i\beta\lambda'_i-R'_i\beta\nu'_i, \label{eqn:bpifoc} \\ \end{eqnarray} (14) \begin{eqnarray} \mu' & = & v_w(w',z'), \label{eqn:wpfoc} \end{eqnarray} (15) where the multipliers on the constraints (2) through (5) are $$\mu$$, $$\beta\mu'$$, $$\beta\lambda'$$, and $$\beta\lambda'_i$$, and $$\nu_d$$ and $$R'_i\beta\nu'_i$$ are the multipliers on the non-negativity constraints on dividends and intermediated borrowing.9 The envelope condition is $$v_w(w,z)=\mu,$$ the marginal value of firm net worth, which by equation (11) exceeds 1 and equals 1 when the firm pays dividends, as does the marginal value of net worth next period, denoted $$\mu'$$ (see equation (15)). Define the down payment$$\wp$$ when the firm borrows the maximum amount it can from households only as $$\wp=1-R^{-1}\theta(1-\delta)$$. Similarly, define the down payment when the firm borrows the maximum amount it can from both households (at interest rate $$R$$) and intermediaries (at interest rate $$R'_i$$) as $$\wp_i(R'_i)=1-[R^{-1}\theta+(R'_i)^{-1}(\theta_i-\theta)](1-\delta)$$ (illustrated at the bottom of Figure 1). Note that the down payment, at times referred to as the margin requirement, is endogenous in our model. Using this definition and equations (12) through (14) the firm’s investment Euler equation can be written as \begin{equation} 1 \ge \beta\frac{\mu'}{\mu}\frac{A' f_k\left(k\right) +(1-\theta_i)(1-\delta)}{\wp_i(R'_i)}, \label{eqn:iee} \end{equation} (16) which obtains whether or not the collateral constraints bind. The first-order conditions of the intermediary’s problem in equations (7)–(10), which are necessary and sufficient, can be written as \begin{eqnarray} \mu_i & = & 1+\eta_d, \label{eqn:difoc} \\ \end{eqnarray} (17) \begin{eqnarray} \mu_i & = & R \beta_i\mu'_i+R\beta_i\eta', \label{eqn:lpfoc} \\ \end{eqnarray} (18) \begin{eqnarray} \mu_i & = & R'_i\beta_i\mu'_i+R'_i\beta_i\eta'_i, \label{eqn:lpifoc} \\ \end{eqnarray} (19) \begin{eqnarray} \mu'_i & = & v_{i,w}(w'_i,z'), \label{eqn:wpifoc} \end{eqnarray} (20) where the multipliers on the constraints (8) and (9) are $$\mu_i$$ and $$\beta_i\mu'_i$$, and $$\eta_d,$$$$R\beta_i\eta',$$ and $$R'_i\beta_i\eta'_i$$ are the multipliers on the non-negativity constraints on dividends and direct and intermediated lending. The envelope condition is $$v_{i,w}(w_i,z)=\mu_i,$$ the marginal value of intermediary net worth, which exceeds 1 by equation (17) and equals 1 when dividends are paid, as does the marginal value of net worth next period, denoted $$\mu'_i$$ (see equation (20)). Equations (18) and (19) imply that intermediaries lend to households only when $$R_i'=R$$. 2.3. Deriving collateral constraints from limited enforcement This section describes an economy with limited enforcement which is equivalent to the economy with collateral constraints described above. We first describe the environment with limited enforcement, allowing for long-term contracts, and then sketch our equivalence result which we formally state and prove in Appendix B.10 This equivalence is significant for three reasons; it shows that (1) intermediaries must provide equipment loans, that is, loans against the additional amount of collateral they can seize; (2) intermediaries must finance these loans out of their own net worth; and (3) the restriction to one-period ahead contracts is without loss of generality. Thus, the economy with limited enforcement endogenizes three key properties of the model with collateral constraints that we have simply assumed so far. That said, a reader, who is primarily interested in the dynamic implications of our model, may choose to skip this derivation and proceed directly to Section 3. Suppose that the environment is as before, but that each period has two subperiods which we refer to as morning and afternoon. The economy has limited participation by households. All types of agents participate in markets in the afternoon. In the morning, however, only entrepreneurs and intermediaries participate in markets but not households. This is the key assumption affording intermediaries an enforcement advantage. The economy has limited enforcement in the spirit of Kehoe and Levine (1993) except that firms or intermediaries that default cannot be excluded from participating in financial and real asset markets going forward. Rampini and Viswanathan (2010, 2013) study this class of economies but consider an economy with only one type of lender with deep pockets and hence take the interest rate as given. We build on their work by considering an economy with two types of lenders, intermediaries and households, of which one has limited net worth, and extend their analysis by determining the interest rates on intermediated finance in dynamic general equilibrium with aggregate fluctuations. Specifically, enforcement is limited as follows: Firms can abscond both in the morning and in the afternoon. In the morning, after cash flows are realized, firms can abscond with all cash flows and a fraction $$1-\theta_i$$ of depreciated capital, where $$\theta_i\in(0,1)$$. In the afternoon, firms can abscond with cash flows net of payments made in the morning and a fraction $$1-\theta$$ of depreciated capital, where $$\theta\in(0,1)$$. Critically, we assume that $$\theta_i>\theta$$, which means that firms can abscond with less capital in the morning than in the afternoon. Intermediaries, too, can abscond in both subperiods, although there is no temptation for intermediaries to do so in the morning, as they will at best receive payments, and so we can ignore this constraint and focus just on the afternoon. In the afternoon, intermediaries can abscond with any payments received in the morning. To reiterate, neither firms nor intermediaries are excluded from markets after default. The timing is summarized as follows (see Figure 2): Each afternoon, firms and intermediaries first decide whether to make their promised payments or default. Then, firms, intermediaries, and households consume, invest, and borrow and lend. The next morning, cash flows are realized. Firms decide whether to make their promised morning payments or default. Firms carry over the cash flows net of payments made and intermediaries carry over any funds received until the afternoon. Figure 2 View largeDownload slide This figure shows the time line of the firm’s problem in the afternoon of the current period including the repayment decisions in the morning (on loans from intermediaries $$b_i'$$ due then) and in the afternoon of the next period (on loans from intermediaries $$b_a'$$ and from households $$b'$$ due then). Figure 2 View largeDownload slide This figure shows the time line of the firm’s problem in the afternoon of the current period including the repayment decisions in the morning (on loans from intermediaries $$b_i'$$ due then) and in the afternoon of the next period (on loans from intermediaries $$b_a'$$ and from households $$b'$$ due then). Loans backed by the additional amount of collateral that can be seized in the morning, that is, $$\theta_i-\theta$$, must be repaid in the morning, as by the afternoon firms can abscond with that additional amount of capital and these payments are no longer enforceable. This implies property (1); such loans must be extended by intermediaries, as only they participate in markets in the morning when the claims need to be enforced. Moreover, it means that intermediaries must finance such loans out of their own net worth, that is, property (2), as they cannot in turn finance them by borrowing from households because they could simply default on promises to repay the households in the afternoon and abscond with the payments received in the morning.11 In contrast, financial intermediaries could refinance corporate loans that they make to firms, which are repaid in the afternoon, up to a fraction $$\theta$$ of collateral by borrowing from households. Loans beyond that, for fraction $$\theta_i-\theta$$, have to be financed out of financial intermediary capital. The limits on enforcement for the three types of capital (see Figure 1) are as follows: firms can always abscond with working capital; firms cannot abscond with equipment in the morning, but can abscond with equipment in the afternoon; and firms can never abscond with structures. Structure loans can be provided by either intermediaries or households. In contrast, equipment loans have to be extended by intermediaries, have to be repaid in the morning, and have to be financed out of intermediary net worth. Finally, let us sketch our main equivalence result (see Appendix B for the formal statement and proof). The economic intuition for the equivalence of the economy with limited enforcement and the economy with collateral constraints, both described in detail in Appendix B, is based on two main insights. First, limited enforcement implies that the present value of any sequence of promises, that is, long-term contract, can never exceed the current value of collateral, as otherwise delivering on these promises would not be optimal and the borrower would default. Indeed, limited enforcement constraints are equivalent to a type of collateral constraint on the present value of sequences of promises (see Theorem B.1). Secondly, any sequence of promises satisfying these collateral constraints on present values can be implemented with one-period ahead morning and afternoon claims subject to collateral constraints for the firm and the intermediary (see Theorem B.2). Hence, the economy with collateral constraints is tractable, in part because we can restrict attention, without loss of generality, to complete markets in one-period ahead morning and afternoon Arrow securities, that is, property (3). This economy with limited enforcement and limited participation therefore endogenizes three key properties that we previously simply assumed in the economy with collateral constraints. Henceforth, we work with the equivalent, recursive formulation of the economy with collateral constraints. 3. Intermediary Capital and Steady State Intermediary capital is scarce in the model. We first show that, as a consequence, intermediated finance carries a premium or spread and that this spread affects investment and real economic activity. We then show that intermediaries are essential in our economy, that is, allow the economy to achieve allocations that would not be achievable in their absence. Finally, we show that, in a steady state, intermediary finance carries a positive spread over direct finance and determine the steady state capitalization of intermediaries. 3.1. Cost of intermediated finance Internal funds and intermediated finance are both scarce in our model and command a premium as collateral constraints drive a wedge between the cost of different types of finance. Since the firm would never be willing to pay more for intermediated finance than the shadow cost of internal funds, the premium on internal finance is higher than the premium on intermediated finance. Define the premium on internal funds $$\rho$$ as $$1/(R+\rho)\equiv \beta \mu'/\mu$$, where the right-hand side is the firm’s discount factor. Define the premium on intermediated finance $$\rho_i$$ as $$1/(R+\rho_i)\equiv (R_i')^{-1}$$, so $$\rho_i=R'_i-R$$, the spread over the household interest rate. In equilibrium, intermediaries lend at the intermediary interest rate $$R'_i$$ and thus, using equations (18) and (19), it must be that $$R_i' \ge R$$, and since firms borrow from intermediaries, equation (14) implies that $$R'_i\le (\beta\mu'/\mu)^{-1}$$; therefore: Proposition 1 (Premia on internal and intermediated finance) The premium on internal finance $$\rho$$ (weakly) exceeds the premium on intermediated finance $$\rho_i$$, that is, $$\rho \ge \rho_i \ge 0$$, and the two premia are equal, $$\rho=\rho_i$$, iff the collateral constraint for intermediated finance does not bind, that is, $$\lambda_i'=0$$. Moreover, the premium on internal finance is strictly positive, $$\rho>0$$, iff the collateral constraint for direct finance binds, that is, $$\lambda'>0$$. When all collateral constraints are slack, there is no premium on either type of finance, but typically the inequalities are strict and both premia are strictly positive, with the premium on internal finance strictly exceeding the premium on intermediated finance. The scarcity of internal and intermediated finance affects investment and in turn real economic activity. To see this, we can adapt Jorgenson’s (1963) definition of the user cost of capital to our model with intermediated finance, and rewrite the investment Euler equation (16) as $$u=R\beta\frac{\mu'}{\mu}A'f_k(k)$$, where we define the user cost of capital $$u$$ as \begin{equation} \label{eqn:ucc} u \equiv r+\delta+\frac{\rho}{R+\rho}(1-\theta_i)(1-\delta)+\frac{\rho_i}{R+\rho_i}(\theta_i-\theta)(1-\delta), \end{equation} (21) where $$r+\delta$$ is the frictionless user cost derived by Jorgenson and $$r\equiv R-1$$. The user cost of capital exceeds the user cost in the frictionless model, because part of investment needs to be financed with internal funds at premium $$\rho$$ (the second term on the right-hand side) and part is financed with intermediated finance at premium $$\rho_i$$ (the last term on the right-hand side). The premium on intermediated finance thus affects investment; scarcer intermediary capital reduces corporate investment. 3.2. Intermediation is essential Intermediary capital is positive in equilibrium, that is, intermediaries always keep strictly positive net worth and never choose to pay out their entire net worth as dividends. Proposition 2 (Positive intermediary net worth) Financial intermediaries always have strictly positive net worth in equilibrium. The intuition is that if intermediary net worth went to zero, the marginal value of intermediary net worth in equilibrium would go to infinity, because intermediaries would earn a positive spread forever; thus, intermediaries never pay out all their net worth. Since intermediaries always have positive net worth, in equilibrium the intermediary interest rate $$R_i'$$ must be such that the representative firm would never want to lend at that rate (that is, $$\nu'_i=0$$ in equation (14)), as otherwise there would be no demand for intermediated finance, as the following lemma shows: Lemma 1 In any equilibrium, (1) the cost of intermediated funds (weakly) exceeds the cost of direct finance, that is, $$R'_i\ge R$$; (2) the multiplier on the collateral constraint for direct finance (weakly) exceeds the multiplier on the collateral constraint for intermediated finance, that is, $$\lambda'\ge\lambda'_i$$; (3) the constraint that the representative firm cannot lend at $$R_i'$$ never binds, that is, $$\nu'_i=0$$ w.l.o.g.; (4) the constraint that the representative intermediary cannot borrow at $$R_i'$$ never binds, that is, $$\eta'_i=0$$; and (5) the collateral constraint for direct financing always binds, that is, $$\lambda'>0$$. We define the essentiality of intermediaries as follows: Definition 2 (Essentiality of intermediaries) Intermediaries are essential if an allocation can be supported with financial intermediaries but not without.12 The above results together imply that financial intermediaries must always be essential. First, note that firms are always borrowing the maximal amount from households, since direct finance is relatively cheap. If firms moreover always borrow a positive amount from intermediaries, then they must achieve an allocation that would not otherwise be feasible. If $$R_i'=R$$, then the firm must be collateral constrained in terms of intermediated finance, too, that is, borrow a positive amount. If $$R_i'>R$$, then intermediaries lend all their funds to the corporate sector and in equilibrium firms must be borrowing from intermediaries. We have proved that intermediation always plays a role in our economy: Proposition 3 (Essentiality of intermediaries) Financial intermediaries are always essential in equilibrium. 3.3. Intermediary capitalization and spreads in steady state Consider a steady state defined as follows: Definition 3 (Steady state) A steady state equilibrium is an equilibrium with constant allocations, that is, $$x^*\equiv[d^*,k^{*},b^{\prime *},b_i^{\prime *},w^{\prime *}]$$ and $$x^*_i\equiv[d^*_i,l^{\prime *},l_i^{\prime *},w_i^{\prime *}]$$, and a constant interest rate on intermediated finance $$R_i'^*$$. In a steady state, intermediary capital and the spread on intermediated finance are positive: Proposition 4 (Steady state) There exists a unique steady state with the following properties: Intermediaries are essential, have positive net worth, and pay positive dividends. The spread on intermediated finance is strictly positive: $$\rho_i^*\equiv R_i^{\prime *}-R=\beta^{-1}_i-R > 0.$$ Firms’ collateral constraint for intermediated finance binds. The relative (ex dividend) intermediary capitalization is \[ \frac{w_i^{*}}{w^{*}}=\frac{\beta_i(\theta_i-\theta)(1-\delta)}{\wp_i(\beta^{-1}_i)}. \] The relative (ex dividend) intermediary capitalization, that is, the ratio of the representative intermediary’s net worth (ex dividend) relative to the representative firm’s net worth (ex dividend), is the ratio of the intermediary’s financing (per unit of capital) to the firm’s down payment requirement (per unit of capital). In a steady state, the shadow cost of internal funds of the firm is $$\beta^{-1}$$ while the interest rate on intermediated finance $$R_i^{\prime *}=\beta^{-1}_i$$, the shadow cost of internal funds of the intermediary. Since $$\beta_i>\beta$$, intermediated finance is cheaper than internal funds for firms in the steady state, and firms borrow as much as they can from intermediaries. The spread on intermediated finance is strictly positive in the steady state because intermediaries are less patient than households. In the analysis of the equilibrium dynamics in the next section, we find that the spread on intermediated finance depends on the net worth of both firms and intermediaries, and can be higher or lower than the steady state spread. In a steady state equilibrium, financial intermediaries have positive capital and pay out the steady state interest income as dividends $$d^*_i=(R_i^{\prime *}-1)l_i^{\prime *}.$$ Both firms and intermediaries have positive net worth in the steady state despite the fact that their rates of time preference differ and both are less patient than households. The reason is that firms have access to investment opportunities, but face collateral constraints and hence need to finance part of their investment internally, and intermediaries can finance part of firms’ investment more cheaply, but face collateral constraints themselves. The determinants of the capital structure of firms and intermediaries are distinct. In a steady state, firm leverage, that is, the total value of debt relative to total tangible assets, is $$1-\wp_i(R_i^{\prime *})=(R^{-1}\theta+(R_i^{\prime *})^{-1}(\theta_i-\theta))(1-\delta)$$ and is determined by the extent to which the firm can collateralize tangible assets, as emphasized in Rampini and Viswanathan (2013). In contrast, intermediary leverage can be defined in our indirect implementation as the value of total direct finance divided by the total value of debt, that is, $$R^{-1}\theta(1-\delta)$$ divided by $$(R^{-1}\theta+(R_i^{\prime *})^{-1}(\theta_i-\theta))(1-\delta)$$, which is approximately equal to $$\theta/\theta_i$$. Intermediary leverage is therefore determined by the relative enforcement ability of households and intermediaries. The substantial difference in leverage between firms and intermediaries in practice may simply be a consequence of their different determining factors. Thus, the model provides consistent guidance on the financial structure of firms and intermediaries. Financial intermediaries are essential in our economy. Intermediated finance is costly and the spread on intermediated finance affects investment and aggregate economic output. Equilibrium determines the capitalization of both firms and intermediaries as well as the spread on intermediated finance; in a steady state equilibrium financial intermediary capital is positive as is the spread on intermediated finance. Next we consider the dynamics of our economy with intermediated finance, including the dynamics of firm and intermediary net worth and the spread on intermediated finance. 4. Dynamics of Intermediary Capital Our model allows the analysis of the joint dynamics of the capitalization of the corporate and intermediary sector. The net worth of firms and intermediaries are the key state variables determining dynamic intermediated loan demand and supply and the interest rate on intermediated finance. The interaction between firms and intermediaries which are both subject to financial constraints leads to subtle dynamics with several compelling features. For example, spreads on intermediated finance are high when both firms’ and intermediaries’ net worth is low and intermediaries are poorly capitalized even relative to firms. A key feature is that intermediary capital accumulation is slow relative to corporate net worth accumulation, at least early in a recovery. One reflection of this is that the recovery from a credit crunch, that is, a drop in intermediary net worth, is relatively slow. Another reflection is that a simultaneous drop in the net worth of both firms and intermediaries, that is, a downturn associated with a credit crunch, results in an especially slow recovery, and that such recoveries can stall, with firm investment and output remaining depressed for an extended period of time. We relate the dynamic properties of our economy to stylized facts in the next section. 4.1. Dynamics of intermediary capital and spreads To characterize the deterministic dynamics in an equilibrium converging to the steady state in general, consider the recovery of the economy from an initial, low level of net worth of firms and/or intermediaries, say after a downturn or credit crunch. We show that the equilibrium dynamics evolve in two main phases, an initial one in which the corporate sector pays no dividends and a second one in which the corporate sector pays dividends. Intermediaries do not pay dividends until the steady state is reached, except for an initial dividend, if they are initially well capitalized relative to the corporate sector. Before stating these results formally (see Proposition 5 below, illustrated in Figure 3), we provide an intuitive discussion. Suppose both firms and intermediaries are constrained, that is, the marginal value of net worth strictly exceeds 1; then neither firms nor intermediaries pay dividends (Region ND in the proposition below). If the firms’ collateral constraint on intermediated finance is slack, the intermediary interest rate equals firms’ marginal levered return on capital (and exceeds the corporate discount rate $$\beta^{-1}$$), that is, \[ R'_i=\frac{A'f_k\big(\frac{w+w_i}{\wp}\big)+(1-\theta)(1-\delta)}{\wp}, \] Figure 3 View largeDownload slide Contours of the regions describing the dynamics of firm and financial intermediary net worth (see Proposition 5). Region ND, in which firms pay no dividends, is to the left of the solid line and Region D, in which firms pay positive dividends, is to the right of the solid line. The point where the solid line reaches the dotted line is the steady state $$(w^*,w_i^*)$$. The kink in the solid line is the point $$(\bar{w},\bar{w}_i)$$ where $$R_i'=\beta^{-1}$$ and the collateral constraint just binds. The solid line segment between these two points is $$\bar{w}(w_i)=\wp k(w_i)-w_i$$ (with $$R_i'\in (\beta_i^{-1},\beta^{-1})$$). The solid line segment sloping down is $$\bar{w}(w_i)=\wp \bar{k}-w_i$$ (with $$R_i'=\beta^{-1}$$). Region ND is divided by two dash dotted lines: below the dash dotted line through $$(\bar{w},\bar{w}_i)$$$$R_i'>\beta^{-1}$$; between the two dash dotted lines $$R_i'\in (\beta_i^{-1},\beta^{-1})$$; and above the dash dotted line through $$(w^*,w_i^*)$$$$R_i'<\beta_i^{-1}$$. The parameter values are: $$\beta=0.90$$, $$R=1.05$$, $$\beta_i=0.94$$, $$\delta=0.10$$, $$\theta=0.60$$, $$\theta_i=0.80$$, $$A'=0.20$$, and $$f(k)=k^{\alpha}$$ with $$\alpha=0.80$$. Figure 3 View largeDownload slide Contours of the regions describing the dynamics of firm and financial intermediary net worth (see Proposition 5). Region ND, in which firms pay no dividends, is to the left of the solid line and Region D, in which firms pay positive dividends, is to the right of the solid line. The point where the solid line reaches the dotted line is the steady state $$(w^*,w_i^*)$$. The kink in the solid line is the point $$(\bar{w},\bar{w}_i)$$ where $$R_i'=\beta^{-1}$$ and the collateral constraint just binds. The solid line segment between these two points is $$\bar{w}(w_i)=\wp k(w_i)-w_i$$ (with $$R_i'\in (\beta_i^{-1},\beta^{-1})$$). The solid line segment sloping down is $$\bar{w}(w_i)=\wp \bar{k}-w_i$$ (with $$R_i'=\beta^{-1}$$). Region ND is divided by two dash dotted lines: below the dash dotted line through $$(\bar{w},\bar{w}_i)$$$$R_i'>\beta^{-1}$$; between the two dash dotted lines $$R_i'\in (\beta_i^{-1},\beta^{-1})$$; and above the dash dotted line through $$(w^*,w_i^*)$$$$R_i'<\beta_i^{-1}$$. The parameter values are: $$\beta=0.90$$, $$R=1.05$$, $$\beta_i=0.94$$, $$\delta=0.10$$, $$\theta=0.60$$, $$\theta_i=0.80$$, $$A'=0.20$$, and $$f(k)=k^{\alpha}$$ with $$\alpha=0.80$$. where we use the investment Euler equation (16), that is, $$1=\beta\frac{\mu'}{\mu}\frac{A'f_k(k)+(1-\theta_i)(1-\delta)}{\wp_i(R_i')}$$, and substitute out the discount factor using equation (14) and the fact that the collateral constraint for intermediated finance is slack, which implies that $$\beta\frac{\mu'}{\mu}=(R_i')^{-1}$$, and then rearrange. This case obtains when corporate net worth is sufficiently high so that firms’ loan demand exceeds intermediaries’ loan supply, which is constrained by intermediary net worth. Intermediaries then lend their entire net worth $$w_i$$ to firms, which in turn use their own net worth $$w$$ plus loans from intermediaries to finance the fraction of investment not financed by households, that is, $$k=\frac{w+w_i}{\wp}$$. We observe that in this case the intermediary interest rate decreases in both firm and intermediary net worth since increased investment reduces the marginal return on capital. If the firms’ collateral constraint on intermediated finance binds instead, that constraint determines the interest rate which is then strictly lower than firms’ marginal levered return on capital; specifically, \[ R'_i=(\theta_i-\theta)\frac{\frac{w}{w_i}+1}{\wp}(1-\delta), \] where we use the collateral constraint (4) at equality, that is, $$R_i'b_i'=(\theta_i-\theta)k(1-\delta)$$ together with the fact that $$k=\frac{w+w_i}{\wp}$$ and in equilibrium $$b_i'=l_i'=w_i$$. Notice that in this case, the ratio of the net worth of firms relative to intermediaries matters, and remarkably the intermediary interest rate increases in firms’ net worth keeping intermediary net worth the same; the economic intuition is that higher firm net worth raises investment and thus the collateral firms are able to pledge, increasing the equilibrium interest rate. If the ratio $$\frac{w}{w_i}$$ is sufficiently low, loan demand can be so low that the interest rate on intermediated finance is below not just the firms’ discount rate but also intermediaries’ discount rate ($$\beta_i^{-1}$$). Indeed, the interest rate on intermediated finance can be as low as $$R$$, the discount rate of households; this can happen when intermediaries save net worth by lending to households, because current corporate loan demand is very low but expected to increase as firms recover. Throughout Region ND, firm net worth must accumulate faster than intermediary net worth because firms’ net worth grows at their average levered return on capital (which exceeds their marginal levered return on capital) whereas intermediaries accumulate net worth at the intermediary interest rate, which, as just argued, is weakly below firms’ marginal levered return on capital. Suppose now that firms pay dividends but not intermediaries (Region D in the proposition below). If the firms’ collateral constraint on intermediated finance is slack, the intermediary interest rate must again equal firms’ marginal levered return on capital which in this case equals firms’ discount rate, that is, \[ R_i'=\Big(\beta\frac{\mu'}{\mu}\Big)^{-1}=\beta^{-1}=\frac{A'f_k(\bar{k})+(1-\theta)(1-\delta)}{\wp}, \] as $$\mu=\mu'=1$$ since firms pay dividends. This case obtains when firms’ net worth is relatively high while intermediaries’ net worth does not suffice to meet the corporate loan demand at the intermediary interest rate $$R_i'=\beta^{-1}$$. In this phase, investment is constant at $$\bar{k}$$ and financed with firms’ ex dividend net worth $$w_{ex}$$ and intermediary loans, that is, $$\wp\bar{k}=w_{ex}+w_i$$; as intermediaries accumulate net worth with the law of motion $$w_i'=\beta^{-1}w_i$$ and progressively meet the corporate loan demand, firms gradually relever and draw down their (ex dividend) net worth by paying dividends. Therefore, $$\frac{w_i'}{w_i}=\beta^{-1}>1>\frac{w_{ex}'}{w_{ex}}$$, as intermediaries accumulate net worth while firms draw it down; this is the time when financial intermediaries are “catching up”. If firms’ collateral constraint binds, which happens once intermediaries’ net worth is sufficient to meet loan demand at $$\beta^{-1}$$, the collateral constraint (4) and firms’ investment Euler equation (16) jointly determine the intermediary interest rate, and, as intermediary net worth increases, the intermediary interest rate falls and investment increases. From the collateral constraint (4), $$R_i'=(\theta_i-\theta)\frac{\frac{w_{ex}}{w_i}+1}{\wp}(1-\delta)$$, we see that as the intermediary interest rate falls, the (ex dividend) net worth of firms relative to intermediaries must fall, too. Thus, in this phase, while firms’ and intermediaries both accumulate net worth, intermediaries accumulate net worth faster than firms, as firms continue to relever; intermediaries continue to “catch up” until the steady state is reached. Intermediaries do not pay dividends until the steady state is reached with one exception. If the initial corporate net worth is so low, that intermediaries are well capitalized relative to the corporate sector and the interest rate is below intermediaries’ discount rate due to the limited corporate loan demand, then intermediaries may pay an initial dividend if they expect corporate loan demand to be depressed for an extended period of time. But after such an initial dividend, intermediaries do not resume payout until such time as the steady state is reached. We emphasize, however, that, in contrast, firms do initiate payout before the economy reaches the steady state. The firms’ and intermediary’s first-order conditions for intermediated borrowing and lending, respectively, equations (14) and (19), imply that as long as $$R'_i>\beta^{-1}$$, $$\mu\ge R'_i\beta\mu'>\mu'\ge 1$$, so $$\mu>1$$, and similarly when $$R'_i>\beta_i^{-1}$$, $$\mu_i>1$$; thus, firms do not pay dividends until the intermediary interest rate reaches $$\beta^{-1}$$, while intermediaries wait to pay dividends until the intermediary interest rate reaches $$\beta_i^{-1}<\beta^{-1}$$. The following proposition and lemma state these results formally and Figure 3 illustrates the pertinent regions of firm net worth $$w$$ and intermediary net worth $$w_i$$: Proposition 5 (Dynamics) Given $$w$$ and $$w_i$$, there exists a unique deterministic dynamic equilibrium which converges to the steady state characterized by a no dividend (ND) region and a dividend (D) region (which is absorbing) as follows: Region ND $$w_i\le w^*_i$$ (w.l.o.g.) and $$w<\bar{w}(w_i),$$ and (1) $$d=0$$ ($$\mu>1$$), (2) the cost of intermediated finance is \[ R'_i=\max\left\{R,\min\left\{(\theta_i-\theta)\frac{\frac{w}{w_i}+1}{\wp}(1-\delta),\frac{A'f_k\left(\frac{w+w_i}{\wp}\right)+(1-\theta)(1-\delta)}{\wp}\right\}\right\}, \] (3) investment $$k=(w+w_i)/\wp$$ if $$R'_i>R$$ and $$k=w/\wp_i(R)$$ if $$R'_i=R,$$ and (4) $$w'/w_i'>w/w_i$$, that is, firm net worth increases faster than intermediary net worth. Region D $$w\ge\bar{w}(w_i)$$ and (1) $$d>0$$ ($$\mu=1$$). For $$w_i\in(0,\bar{w}_i)$$, (2) $$R'_i=\beta^{-1}$$, (3) $$k=\bar{k}$$ which solves $$1=\beta[A'f_k(\bar{k})+(1-\theta)(1-\delta)]/\wp,$$ (4) $$w'_{ex}/w'_i<w_{ex}/w_i,$$ that is, firm net worth (ex dividend) increases more slowly than intermediary net worth, and (5) $$\bar{w}(w_i)=\wp \bar{k}-w_i.$$ For $$w_i\in[\bar{w}_i,w^*_i)$$, (2) $$R'_i=(\theta_i-\theta)(1-\delta)k/w_i$$, (3) $$k$$ solves $$1=\beta[A'f_k(k)+(1-\theta)(1-\delta)]/(\wp-w_i/k),$$ (4)