TY - JOUR
AU - Colciago,, Andrea
AB - Abstract This article provides optimal labour and dividend income taxation in a general equilibrium model with oligopolistic competition and endogenous firms’ entry. In the long run, the optimal dividend income tax corrects for inefficient entry. The dividend income tax depends on the form of competition and the nature of the sunk entry costs. In particular, it is higher in market structures characterised by competition in quantities rather than those characterised by price competition. Oligopolistic competition leads to an endogenous countercyclical price markup. As a result, offsetting the distortions over the business cycle requires deviations from full tax smoothing. Recent macroeconomic literature emphasises the importance of the creation of new firms for the propagation of business cycle fluctuations. Colciago and Etro (2010), Etro and Colciago (2010), Jaimovich and Floetotto (2010) and Bilbiie et al. (2012, BGM henceforth), among others, show that accounting for firms’ dynamics helps to improve the performance of dynamic general equilibrium models in replicating the variability of the main macroeconomic variables in response to exogenous disturbances.1 Most of these studies are characterised by imperfect competition in the goods market. As a result, an inefficient number of producers (and products) may arise in equilibrium, leading to welfare losses for the society. For this reason, policy measures aimed at removing market distortions could be desirable. This article provides optimal Ramsey dividend and labour income taxation in a framework characterised by alternative, imperfectly competitive, endogenous market structures (EMSs). Market structures are said to be endogenous since the number of producers and price markups are determined in each period. The issue of capital and corporate taxation has been at the forefront of policy discussions. Some countries, including Spain, France, Sweden and the US have, in recent years, reduced capital gains taxes or corporate taxes. While this study provides an analysis of the dynamic inefficiencies characterising an economy featuring imperfect competition with endogenous entry and endogenous labour supply, the resulting framework is a suitable environment in which to evaluate the effects of changes in corporate taxation on firm creation. The economy features distinct sectors, each one characterised by many firms supplying goods that can be imperfectly substitutable to a different degree, taking strategic interactions into account and competing either in prices (Bertrand competition) or in quantities (Cournot competition). As in BGM (2012), the entry of a new firm into the market amounts to the creation of a new product. Sunk entry costs allow the endogenisation of the entry and the (stock market) value of each firm in each sector. The preferences of agents are characterised by love for variety, such that spreading a given nominal consumption expenditure over a larger number of goods leads to an increase in utility. The degree of market power, as measured by the price markup, depends endogenously on the form of competition, on the degree of substitutability between goods and on the equilibrium number of firms. Importantly, the price markup is countercyclical. During an economic boom profit opportunities attract firms into the market. This strengthens competition and, via strategic interactions, reduces price markups. An early reference on the procyclicality of the number of firms in the US is Chatterjee and Cooper (1993), while a more recent one is Lewis and Poilly (2012). The countercyclicality of the price markup is consistent with the empirical findings by Bils (1987), Rotemberg and Woodford (1999) and Galì et al. (2007). Nevertheless, aggregate profits remain strongly procyclical as in the data. As emphasised by BGM (2007), the market equilibrium is characterised by two distortions: a Labour Distortion and an Entry Distortion. As in other models with an imperfectly competitive goods market, the presence of a price markup leads to a wedge between the marginal product of labour and the marginal rate of substitution between consumption and hours. Oligopolistic competition renders this wedge time varying. The Entry Distortion operates through the intertemporal firms creation margin and leads to an inefficient number of firms in equilibrium. A positive dividend income tax is optimal in the case of excessive entry and vice versa. To minimise the welfare cost associated with these distortions, the Ramsey optimal fiscal policy, both in the short and long run, is provided. The government levies taxes on dividend income and labour income and issues state contingent bonds to finance an exogenous stream of public spending. In the long run, the dividend income tax removes the entry distortion, as in Chugh and Ghironi (2011). The dividend income tax can be positive, negative or zero depending on the form of the entry cost and on the form of competition. A clear message emerging from the analysis is that the optimal dividend income tax rate is higher in market structures characterised by lower competition. In particular, it is higher under Cournot competition than under Bertrand or monopolistic competition. The long‐run labour income tax is positive under all market structures considered. As a result, the labour distortion cannot be removed and the long‐run Ramsey equilibrium is inefficient. This suggests an analogy between dividend income taxation in our framework and capital income taxation in the standard growth model with imperfect competition. Adopting the neoclassical growth model and assuming full commitment with time‐varying taxes, Judd (1985) and Chamley (1986) show that the optimal policy should eliminate all distortions on investment decisions by suppressing capital taxes in the long run. This conclusion is robust to many different extensions.2 Judd (1997, 2002) augments the standard growth model to allow for imperfectly competitive product markets with a fixed number of firms. He shows that the optimal long‐run capital income tax can be negative. In particular, he argues that tax policy can be used as a substitute for antitrust policy to remove the inefficiencies delivered by a monopoly in terms of capital accumulation and the output level. Labour income taxation or consumption taxation should be used to provide the long‐run subsidy to capital, income and to finance government spending. Such a policy delivers efficiency along the investment margin but disregards the social cost of labour distortion. Similarly, in the framework analysed in this study, the dividend income tax, which is a form of capital taxation, offsets the distortion along the intertemporal (entry) margin. Guo and Lansing (1999) extend the analysis in Judd (1997) to argue that the magnitude and sign of the capital income tax crucially depend on the degree of monopoly power. In a similar vein, my analysis suggests that higher dividend income taxes should be implemented in less competitive market structures. Coto‐Martinez et al. 2007, CGS henceforth) consider an environment characterised by endogenous entry and monopolistic competition. They argue that a key assumption in Judd’s result about negative long‐run capital income tax was that the number of firms was fixed. They show that with endogenous entry a capital income subsidy, which is equivalent to a negative dividend income tax in this article, could lead to too many firms in equilibrium. When both endogenous entry and endogenous labour supply are considered, it is critical whether factors and dividend income can be taxed at different rates. CGS (2007) show that when this is the case, one tax instrument is used to control for the markup inefficiency and the other tax instrument controls firms’ entry. Over the business cycle, Chari et al. (1991) show, in an environment similar to that in Chamley (1986), that the capital income tax rate is close to zero on average. Chugh and Ghironi (2011) show that the optimal dividend and labour income tax rates are constant in a framework with endogenous entry and monopolistic competition. This is not the case in oligopolistic market structures. Due to the countercyclicality of the price markup, the distortions affecting the economy are time varying. Thus, counteracting these distortions requires non‐constant tax rates. In particular, the dividend income tax rate is procyclical, meaning that it decreases during expansions, whereas the labour income tax rate is countercyclical. These findings are consistent with the discussions about the cyclicality of optimal subsidies in BGM (2007) and Etro (2009). Etro (2009), in particular, characterises the cyclicality of optimal ad valorem sales subsidies and optimal labour income subsidies in a model similar to the one adopted here. However, both BGM (2007) and Etro (2009) assume that the government has access to lump sum taxes to finance the optimal subsidies. This possibility is ruled out in my analysis. Since revenues are raised with distortionary taxes, the long‐run inefficiency of the Ramsey equilibrium is unavoidable and the level of activity is below the efficient one. Besides the form of competition, the form of the sunk entry costs also matters for optimal taxation purposes. For this reason, together with alternative forms of competition, two forms of the entry costs are considered. One features a constant entry cost measured in units of output, and the other one features entry costs in terms of labour. In the latter framework, building a new firm requires hiring workers. As a result, the entry cost depends on the real wage. Since the latter is set as a markdown over the marginal product of labour, with the markup depending on the degree of competition in the goods market, changes in the extent of competition also affect the magnitude of the entry costs. The present framework features two models in the entry literature as special cases: CGS (2007) and Chugh and Ghironi (2011). CGS (2007) consider an environment characterised by monopolistic competition under constant sunk entry costs. As mentioned above, they show that endogenous entry fundamentally affects the optimal dividend tax policy. Further, they find that tax smoothing is optimal. Chugh and Ghironi (2011) consider a framework with monopolistic competition and sunk entry cost in terms of labour. In this case, the optimal long‐run dividend income tax is zero and taxes are constant over the business cycle. By neglecting strategic interactions and considering the appropriate form of the entry costs, the framework provided reduces to either one of these models. For this reason, it can be regarded as a general framework for studying optimal taxation problems under various forms of imperfect competition in a dynamic setting. It should be noted that in all the versions of the model I consider, a reduction in dividend income taxes spurs an increase in entry rates in the short run which, in turn, leads to an increase in the steady‐state number of producers. This is consistent with the empirical evidence in Da Rin et al. (2011), who consider a firm‐level database with entry data of several million European companies between 1997 and 2004. They find an economically sizeable and statistically significant positive effect of a reduction in corporate taxation on entry rates. Colciago and Rossi (2011) introduce EMSs in the Mortensen and Pissarides (1994) model of search in the labour market. This allows a simultaneous evaluation of the effects of corporate taxation on the creation of jobs and firms and the quantification of the macroeconomic effects of structural reforms in product and labour markets, an issue of central interest during the Great Recession. To conclude, it should be noted that this work extends to a dynamic general equilibrium setting a well‐established literature on imperfect competition and optimal taxation in static contexts. Myles (1989) studies the design of optimal commodity taxes under imperfect competition, while Myles (1996) studies the combination of ad valorem and specific taxes that induce Ramsey pricing. De Nicolò and Matteuzzi (2000) consider homogenous Cournot oligopolies and show that ad valorem taxes are, in the case of constant marginal costs, welfare dominant with respect to specific taxes. More recently, Lewis and Winkler (2015) studies the design of optimal labour and dividend income taxes under oligopolistic competition and endogenous entry in a static environment. She shows how optimal tax rates depend upon the degree of substitutability between goods. The remainder of the article is as follows. Section 1 presents the model, Section 2 defines the market equilibrium, Section 3 characterises the efficient allocation, Section 4 provides the Ramsey optimal fiscal policy and Section 5 concludes. The technical details are in the Appendix. 1. The Model The economy features a continuum of atomistic sectors, or industries, on the unit interval. Each sector is characterised by different firms producing a good in different varieties and using labour as the only input. In turn, the sectoral goods are imperfect substitutes for each other and are aggregated into a final good. Households use the final good for consumption and investment purposes. Oligopolistic competition and endogenous firms’ entry are modelled at the sectoral level. At the beginning of each period, Njte new firms enter sector j ∈ (0,1), while at the end of the period a fraction δ ∈ (0,1) of market participants exits from the market for exogenous reasons.3 As a result, the number of firms in a sector Njt ⁠, follows the equation of motion: Njt+1=(1−δ)(Njt+Njte),(1) where Njte is the number of new entrants in sector j at time t. Following BGM (2012), I assume that new entrants at time t will only start producing at time t + 1 and that the probability of exit from the market, δ, is independent of the period of entry and identical across sectors. The assumption of an exogenous constant exit rate is adopted for tractability but it also has empirical support. Using US annual data on manufacturing, Lee and Mukoyama (2007) find that, although the entry rate is procyclical, annual exit rates are similar across booms and recessions. Alternative forms of competition between the firms within each sector are considered below. In particular, the focus is on the traditional monopolistic competition setting and the approach based on oligopolistic competition developed by Jaimovich and Floetotto (2010) and Etro and Colciago (2010). As in Ghironi and Melitz (2005) and BGM (2012), who gave new life to the interesting literature on the role of entry in macroeconomic models, sunk entry costs are introduced to endogenise the number of firms in each sector. The nature and form of the entry costs will be specified below, where alternative specifications will also be considered. The household side is standard. They supply labour to firms and choose how much to save in riskless bonds and in the creation of new firms through the stock market. 1.1. Firms and Technology The final good is produced according to the function Yt=∫01Yjtω−1ωdjωω−1,(2) where Yjt denotes the output of sector j and ω is the elasticity of substitution between any two different sectoral goods. The final good producer behaves competitively. In each sector j, there are Njt>1 firms producing differentiated goods that are aggregated into a sectoral good by a CES (constant elasticity of substitution) aggregating function defined as: Yjt=∑i=1Njtyjt(i)θ−1θθθ−1,(3) where yjt(i) is the production of good i in sector j and θ > 1 is the elasticity of substitution between sectoral goods. As in Etro and Colciago (2010), a unit elasticity of substitution between goods belonging to different sectors was assumed. This allows realistic separation of limited substitutability at the aggregate level and high substitutability at the disaggregate level. Each firm i in sector j produces a differentiated good with the following production function: yjt(i)=Athjtc(i),(4) where At represents technology that is common across sectors and evolves exogenously over time, while hjtc(i) is the labour input used by the individual firm for the production of the final good. The unit intersectoral elasticity of substitution implies that nominal expenditure, EXPt, is identical across sectors. Thus, the final producer’s demand for each sectoral good is PjtYjt=PtYt=EXPt,(5) where Pjt is the price index of sector j and Pt is the price of the final good at period t. Denoting with pjt(i) the price of good i in sector j, the demand faced by the producer of each variant is yjt(i)=pjtPjt−θYjt,(6) where Pjt is defined as: Pjt=∑i=1Njt(pjt(i))1−θ11−θ.(7) Using (5) and (6), the individual demand of good i can be written as a function of aggregate expenditure: yjt(i)=pjt−θPjt1−θEXPt.(8) As technology, the entry cost and the exit probability are identical across sectors, in what follows the index j is disregarded to consider a representative sector. 1.2. Households Consider a representative agent with utility: U=E0∑t=0∞βtlogCt−υHt1+1/φ1+1/φυ,φ≥0,(9) where β ∈ (0,1) is the discount factor, Ht are the hours worked and Ct is the consumption of the final good. The representative agent enjoys labour and dividend income. The household maximises (9) by choosing hours of work and how much to invest in bonds and risky stocks. The timing of investment in the stock market is as in Chugh and Ghironi (2011) and BGM (2012). At the beginning of period t, the household owns xt shares of a mutual fund of the Nt firms that produce in period t, each of which pays a dividend dt. Denoting the value of a firm with Vt, it follows that the value of the portfolio held by the household is xtVtNt ⁠. During period t, the household purchases xt+1 shares in a fund of these Nt firms as well as the Nte new firms created during period t, to be carried into period t + 1. Total stock market purchases are thus xt+1Vt(Nt+Nte) ⁠. At the very end of period t, a fraction of these firms disappears from the market.4 Following the production and sales of the Nt varieties in the imperfectly competitive goods markets, firms distribute the dividend dt to households. The household’s total dividend income is thus Dt=xtdtNt ⁠, which is taxed at the rate τtd ⁠. The variable wt is the market real wage, and τtl is the tax rate on labour income. The household’s holdings of the state‐contingent one‐period real government bond that pays off in period t are Bt and Bt+1j are the end‐of‐period holdings of government bonds that pay off in state j in period t + 1, which has a purchase price of 1/Rtj in period t. The Flow budget constraint of the household is: ∑j1RtjBt+1j+Ct+xt+1Vt(Nt+Nte)=(1−τtl)wtHt+Bt+xtVtNt+(1−τtd)xtdtNt. The first‐order conditions (FOCs) for the household problem are represented by a standard Euler equation for bond holdings: 1Ct=βRtj1Ct+1j, an asset pricing equation Vt=β(1−δ)EtCtCt+1j[(1−τt+1d)πt+1(θ,Nt+1)+Vt+1],(10) and the condition for optimal labour supply υCtHt1φ=(1−τtl)wt. 1.3. Endogenous Entry Upon entry firms face a sunk cost, defined as ft. In each period, entry is determined endogenously to equate the value of firms to the entry costs. In what follows two popular forms of the entry cost, ft, are considered. Form 1, adopted, inter alia, by CGS (2007) and Jaimovich and Floetotto (2010), features a constant entry cost measured in units of output, ft = ψ. Form 2, adopted by BGM (2007), features an entry cost equal to η/At units of labour, with η > 0. Note that under this specification, technology shocks affect the productivity of the workers that produce goods and also of the workers that create new businesses. 1.4. Government The government faces an exogenous expenditure stream {Gt}t=0∞ in real terms. To finance this stream it issues real state contingent bonds, Btj ⁠, where the superscript j refers to the state of nature and collects taxes on labour and dividend income. Its period‐by‐period budget constraint is given by: ∑j1RtjBt+1j+τt=Gt+Bt,(11) where τt=τtlwtHt+τtdNtdt are total tax revenues. The government consumes the same index of goods faced by the household and optimises the composition of its expenditure across goods.5 Public spending evolves exogenously over time. 1.5. Strategic Interactions In each period, the same expenditure for each sector EXPt is allocated across the available goods according to the standard direct demand function derived from the expenditure minimisation problem of the household and the government. It follows that the direct individual demand faced by a firm, yt(i), can be written as: yt(i)=Ytpt(i)Pt−θ=pt(i)−θPt1−θYtPt=pt(i)−θEXPtPt1−θi=1,2,…,Nt.(12) Inverting the direct demand functions, the system of inverse demand functions can be derived: pt(i)=yt(i)−1θEXPt∑i=1Ntyt(i)θ−1θi=1,2,…,Nt,(13) which will be useful in the remainder of the analysis. Firms cannot credibly commit to a sequence of strategies, therefore, their behaviour is equivalent to maximise current profits in each period taking as given the strategies of the other firms. A main interest of this study is in the evaluation of the efficiency of equilibria characterised by popular forms of competition by firms, such as competition in prices and quantities. Firms take as given their marginal cost of production and the aggregate nominal expenditure.6 Under different forms of competition, I obtain equilibrium prices satisfying: pt(i)=μ(θ,Nt)WtAt,(14) where Wt/At is the marginal cost and μ(θ,Nt)>1 is the markup function. In the following Sections, the markup functions under alternative forms of market competition are characterised. 1.5.1. Price competition Consider competition in prices. In each period, the gross profits of firm i can be expressed as: Πt[pt(i)]=pt(i)−WtAtpt(i)−θEXPt∑j=1Ntptj1−θ.(15) Firms compete by choosing their prices. We consider two alternative approaches to this problem. The first one is the traditional monopolistic competition approach, which neglects strategic interactions between firms. The second one is the Bertrand approach, where strategic interactions are taken into consideration. The outcome of profit maximisation under monopolistic competition is well known. Each firm i chooses the price pt(i) to maximise profit taking as given the price of the other firms, neglecting the effect of their price choice on the sectoral price index. The symmetric equilibrium price is pt = μMC(θ)Wt/At, which is associated with the constant price markup μMC(θ) = θ/(θ−1). The latter does not depend on the extent of competition but just on the elasticity of substitution between goods. Under Bertrand competition, each firm i chooses the price pt(i) to maximise profit taking as given the price of other firms. The first‐order condition for any firm i is: pt(i)−θ−θpt(i)−WtAtpt(i)−θ−1=(1−θ)pt(i)−θpt(i)−WtAtpt(i)−θ∑i=1Ntpt(i)1−θ. Note that the term on the right‐hand side is the effect of the price strategy of a firm on the price index: higher prices reduce overall demand, therefore firms tend to set higher markups compared to monopolistic competition. The symmetric equilibrium price pt must satisfy: pt=μB(θ,Nt)WtAt, where the markup reads as μB(θ,Nt)=1+θ(Nt−1)(θ−1)(Nt−1).(16) As discussed in more detail in Etro and Colciago (2010), the markup is decreasing in the degree of substitutability between products θ and in the number of firms. Importantly, when Nt→∞ the markup tends to μMC(θ), the standard one under monopolistic competition.7 1.5.2. Quantity competition Consider now competition in quantities in the form of Cournot competition. Using the inverse demand function (13), the profit function of a firm i can be expressed as a function of its output yt(i) and the output of all the other firms: Πt[yt(i)]=pt(i)−WtAty(i)=yt(i)θ−1θEXPt∑j=1Ntyt(j)θ−1θ−Wtyt(i)At.(17) Assume now that each firm chooses its production yt(i) taking as given the production of the other firms. The first‐order conditions: θ−1θyt(i)−1θEXPt∑iyt(i)θ−1θ−θ−1θyt(i)θ−2θEXPt∑iyt(i)θ−1θ2=WtAt for all firms i=1,2,…,Nt can be simplified imposing the symmetry of the Cournot equilibrium. This generates the individual output: yt=(θ−1)(Nt−1)AtEXPtθNt2Wt.(18) Substituting into the inverse price, one obtains the equilibrium price pt=μC(θ,Nt)Wt/At ⁠, where μC(θ,Nt)=θNt(θ−1)(Nt−1)(19) is the markup under competition in quantities. For a given number of firms, the markup under competition in quantities is always larger than the one obtained before under competition in prices, as is well known for models of product differentiation (see, for instance, Vives, 1999). Note that the markup is decreasing in the degree of substitutability between products θ and in the number of competitors. Finally, only when Nt→∞ the markup tends to μMC(θ), the markup under monopolistic competition. 2. Market Equilibrium This Section contains the conditions characterising the market equilibrium (ME). Merging the household flow budget constraint with the government budget leads to: Ytc+NteVt=wtHt+Ntπt,(20) where Ytc=Ct+Gt denotes the sum between private and public consumption of the final good. Note that the sum between labour income and profits income equals aggregate GDP. The Euler equation for firms’ shares reads as: Vt=β(1−δ)EtCt+1Ct−1[(1−τt+1d)πt+1+Vt+1],(21) while the set of Euler equations for bond holdings provides the definition of the stochastic discount factor as βEt(Ct+1/Ct)−1. The real wage can be derived from the equilibrium pricing relation as: wt=Atptμ(θ,Nt)Pt=ρtμtAt, where in the symmetric equilibrium ρt=pt/Pt=Nt1/(θ−1) ⁠. The first‐order condition for labour supply is: υCtHt1φ=(1−τtl)μtρtAt.(22) Also I must consider the equation determining the dynamics of the number of firms: Nt+1=(1−δ)(Nt+Nte).(23) It remains to impose the entry condition and the clearing of the market. To do so, the analysis differentiates according to the form of the entry cost. Form 1. When the latter is measured in constant units of output, the labour input is entirely employed for the production of the final good; thus, the clearing of the labour market requires Ht=Nthtc ⁠. The demand faced by firm i reads as yt=Yt/Ntρt ⁠. In this case, firm i’s profits are: πt=1−1μtρtyt=1−1μtYtNt. Aggregating profits over firms and summing to labour income delivers GDP as ρtAtHt. Since the entry condition is simply Vt = ψ, the resource constraint (24) becomes: Ytc+Nteψ=ρtAtHt.(24) Note that GDP here coincides with the production of the final good. The Euler equation for firms’ share translates into: ψ=β(1−δ)EtCt+1Ct−11−τt+1d1−1μt+1YtNt+1+ψ.(25)Form 2. When the entry cost is measured in units of labour, the economy amounts to one that features two sectors, one where Nte(η/At) units of labour are used to produce new firms and the other one where Nthtc=Ht−Nte(η/At) units of labour are used to produce the final good. This implies that the set‐up of a new firm reduces the labour input available for the production of the final good. In this setting, the individual demand faced by firms i is yt=Ytc/Ntρt and firm level profits can be written as:8 πt=1−1μtρtyt=1−1μtYtcNt. Aggregating profits over firms and summing to labour income delivers GDP as GDPt=[1−(1/μt)]Ytc+(ρt/μt)AtHt ⁠. Also, recall that the entry condition implies Vt = ft = (η/At)wt = η(ρt/μt). In this case, the resource constraint reads as: Ytc+Nteηρt=ρtAtHt.(26) The Euler equation (22) for the value of the firm reduces, instead, to ηρtμt=β(1−δ)EtCt+1Ct−1(1−τt+1d)1−1μt+1Yt+1cNt+1+ηρt+1μt+1.(27) Appendix A reports the full set of equilibrium conditions defining the ME for both forms of the entry costs. 3. Efficient Equilibrium This Section outlines a scenario where a benevolent social planner (SP) maximises households’ lifetime utility by choosing quantities directly. In doing this, the SP is subject to the same technological constraints described in the previous Sections. The SP maximises (9) with respect to {Ct,Nt+1,Nte,Ht}t=0∞ ⁠. The choice is subject to two constraints. The first one is given by the dynamics of the number of firms, (23), and the second one is the resource constraint, which is represented by (26) in the case of Form 2 of the entry costs and by (24) when entry costs are measured in terms of output. The SP takes into account the effect of the number of varieties, Nt ⁠, on the relative price, ρt, which is a primitive of the problem. The FOC with respect to Ht is independent of the form of the entry cost and reads as: χCtHt1φ=ρtAt.(28) This condition simply states that the SP equates the marginal rate of substitution between hours and consumption, the left‐hand side, to the marginal product of labour, the right‐hand side, which depends on the number of varieties in the economy. The SP’s Euler equation depends instead on the form of the entry cost. In the case of Form 1, it reads as: ψ=βEt(1−δ)CtCt+1ϵ(Nt+1)Yt+1Nt+1+ψ,(29) where ϵ(Nt+1) = ρN,t+1Nt+1/ρt+1 is the benefit of variety in elasticity form and ρN,t+1 is the derivative of the benefit of variety with respect to the number of firms. Under CES preferences ϵ(Nt+1) is a constant equal to 1/(θ − 1); hence, in the remainder, we simply denote it with ϵ ⁠. Condition (29) states that the SP will create firms up to the point where the utility cost of creating a new firm, (1/Ct)ψ, equals the benefit from the creation of an additional variety. The latter is given by the discounted utility value of the sum between the additional utility that obtains from increasing time t + 1 number of varieties measured in units of the final good, namely ϵ(Yt+1/Nt+1), and the continuation value of the marginal firm, that is ψ.9 The SP’s Euler equation in the case of entry costs in Form 2 reads as: ηρt=βEt(1−δ)CtCt+1ϵYt+1cNt+1+ηρt+1.(30) The interpretation of (30) is similar to that provided for (29), with two differences. The first one is that the value of a new firm in terms of the final good is not constant over time (and states of nature) but depends on the benefit of variety ρt. The reason is that love for variety affects the marginal productivity of labour and thus the opportunity cost of creating a new firm in terms of the final good. This also implies that whenever the number of varieties differs from the efficient one the value of a firm is distorted with respect to the efficient one. The second difference is that at time t + 1 the creation of Nt+1e new firms requires (η/At)Nt+1e units of labour, which are diverted from the production of the final good. As a result, the additional output that the marginal firm creates depends on the sum between private and public consumption, Yc, and not on GDP. Appendix B reports details concerning the computation of the efficient equilibrium for both forms of the entry costs. 3.1. Market Distortions The market allocation features two distortions. To identify them it is convenient, for the time being, to set fiscal instruments to zero, τtd=τtl=0 ⁠. In the next Sections I reintroduce fiscal instruments and design them in order to minimise the welfare losses associated with the distortions that we are about to discuss. The first distortion is referred to as to the Labour Distortion. In the competitive equilibrium, labour is supplied up to the point where the following condition is satisfied: χCtHt1/φ=ρtμtAt.(31) A comparison between (28) and (31) reveals that in the decentralised equilibrium the marginal rate of substitution between hours and consumption, χCtht1/φ ⁠, is lower than the marginal rate of transformation between hours and output, ρtAt. As in other models with an imperfectly competitive goods market, as for example in BGM (2012), this wedge is due to the presence of a price markup. Importantly, oligopolistic competition renders this wedge time varying. The second distortion involved in the decentralised allocation is an Entry Distortion. This wedge operates through the intertemporal firms creation margin and could lead to an inefficient number of firms in equilibrium. The number of firms could be inefficient for the following reason, as argued by CGS (2007). The entry decision of the individual firm ignores the welfare gains associated with increased variety. This leads to inefficiently low entry. Also, the potential entrant ignores the negative effects of increased competition on the profits of other firms already in the market. The latter, known as the business stealing effect, leads to excessive entry. As a result, the equilibrium number of firms could be either inefficiently high or low, depending on which externality predominates. To illustrate the entry distortion, the Euler equations in the decentralised economy will be compared with those obtained in the efficient equilibrium. Again, it is convenient to distinguish according to the form of the entry costs and to consider a non‐stochastic version of the economy. Form 1. Consider the right‐hand side of the Euler equation in the decentralised equilibrium, (25). The latter equals the right‐hand side of the corresponding equation in the centralised equilibrium, (29), if: 1−1μt+1=ϵ.(32) Thus, efficiency holds if the so‐called Lerner index equals the benefit of variety in elasticity form.10 In this case the private incentive to enter the market, [1 − (1/μt+1)], is identical to the social one and the two externalities described above cancel out. If the Lerner index is larger (lower) than the social incentive there will be excessive (too low) entry. Since ϵ = 1/(θ − 1) the condition above requires the price markup to be constant over time and states of nature, μt = μ. As a result, it cannot be satisfied under oligopolistic competition. Note also that although monopolistic competition leads to a constant markup μ = θ/(θ − 1), the condition for efficiency is not satisfied since 1 − (1/μ) < ϵ ⁠. Form 2. Comparing the Euler equation in the decentralised equilibrium, (27), with the corresponding Euler equation in the SP equilibrium, (30), shows that two conditions need to be imposed to reinstate efficiency, namely μt1−1μt+1=ϵ,(33) and μtμt+1=1.(34) Combining conditions (33) and (34), I recover those emphasised by BGM (2007), that is: ϵ=μt+1−1andμtμt+1=1, which imply that the price markup should be constant over time and that the benefit of variety in elasticity form should equal the market power as measured by the net markup. Under oligopolistic competition both conditions fail. Contrary to the previous case, these conditions are satisfied under monopolistic competition, as emphasised by BGM (2007) and Chugh and Ghironi (2011). Thus, under monopolistic competition, with entry costs in Form 2, the entry margin is not distorted. Note that the economic environments in CGS (2007) and Chugh and Ghironi (2011) can be considered as: special cases of the one outlined here. In particular, the present set‐up collapses to that considered by CGS (2007) when entry costs are in Form 1 and strategic interactions between firms are neglected. It coincides, instead, with that considered by Chugh and Ghironi (2011) when entry costs are in Form 2 and strategic interaction are neglected. The next Section outlines the fiscal policy aimed at minimising welfare losses due to the market distortions when the government can raise revenues solely by imposing distortionary taxes on labour and dividends and issuing state contingent bonds. 4. Ramsey Optimal Fiscal Policy In this Section, we study the second‐best tax policy in an economy without lump sum taxes. Consider the Euler equation for firms’ shares, which in its general form reads as: ft=β(1−δ)EtCt+1Ct−1[(1−τt+1d)πt+1+ft+1]. Expected future dividend income taxes cannot be removed from this equation using other equilibrium conditions. As a result, a pure primal approach cannot be applied to solve for the optimal policy. As emphasised by Chugh and Ghironi (2011), the set of allocations that the SP can select cannot uniquely be characterised by means of the so‐called implementability constraint. Further, computing a first‐order condition with respect to Etτt+1d would leave τt+1d indeterminate from the point of view of time t. To resolve this indeterminacy issue, the solution proposed by Chugh and Ghironi (2011) is adopted. In particular, it is assumed that the SP chooses a state contingent schedule for the time t + 1 dividend income tax rate τj,t+1d ⁠, where j indexes the state of the economy. This schedule is in the time t information set. Also, I assume that the Ramsey Planner commits to the schedule, meaning that the state contingent tax rate is implemented with certainty at time t + 1.11 The Ramsey allocation is derived by solving: maxE0∑t=0∞βtlogCt−υht1+1/φ1+1/φ. The choice variables are Ct, Nt ⁠, Ht, Nte and τt/t+1d ⁠. The allocation is restricted by four constraints; two of them depend on the form of the entry cost. The constraints, which are independent of the form of the entry cost, are (23), determining the dynamic of the number of firms and the implementability constraint, which reads as: E0∑t=0∞βt1−υht1φ=b0C0+1C0[(1−τ0d)d0+V0]N0s0. The allocation is further restricted by (24) and (25) in the case of the entry cost of Form 1 and by (26) and (27) in the case of the entry cost in Form 2. Government expenditure {Gt}t=0∞ is exogenously given. The first‐order conditions for the Ramsey problem are reported in the Appendix for both forms of the entry costs. To compute Ramsey allocations I follow Linnemann and Shabert (2012). That is, throughout the analysis it is assumed that the policy maker can credibly commit himself but the initial period (t = 0) is ignored. In deriving the Ramsey policy, the problem that the policy maker’s decision rules will be different for the first period in which the policy is implemented is neglected. This is justified by the fact that the interest is in making statements about the deterministic steady state as well as about business cycle fluctuations around it, while the transition path from the initial values towards the steady state is not analysed. As is common in the Ramsey literature (Khan et al., 2003), it is further assumed that the initial values of the predetermined variables are equal to their values in the deterministic Ramsey steady state. This amounts to adopting the so‐called Timeless Perspective, popularised by Woodford (2003). The dynamics in response to shocks under the Ramsey policy are obtained by solving a first‐order approximation to the Ramsey first‐order conditions. As shown in various contributions by Schmitt‐Grohè and Uribe (2005), in models characterised by more frictions than are in the present one, a first‐order approximation to the first‐order conditions delivers dynamics that are very close to the exact ones. 4.1. Calibration Since part of the following analysis is numerical, the calibration of structural parameters follows. The time unit is a quarter. The discount factor, β, is set to the standard value for quarterly data, 0.99, while the rate of business destruction, δ, equals 0.025 to match the US empirical level of 10% business destruction per year. Steady‐state productivity is equal to A = 1. The baseline value for the entry cost is set to η = ψ = 1. This leads to a share of investment in GDP of around 12% in our models. The baseline value for the intrasectoral elasticity of substitution is θ = 6, which is in line with the typical calibration for monopolistic competition and delivers markup levels within the empirically relevant range.12 Turning to fiscal parameters, the ratio of government spending over GDP equals 0.22, as estimated by Schmitt‐Grohè and Uribe (2005). For the sake of calibration, we set the steady state labour income tax rate to 20% and the steady state dividend income tax rate to 30% and assume that the government has lump sum taxes available to balance its budget. These values represent the mean labour income tax rate and the mean dividend income tax rate in the US over the period 1947:Q1–2009:Q4, as reported by Chugh and Ghironi (2011). When computing the Ramsey steady state and Ramsey optimal policy, we assume that lump sum taxes are not available and fix the ratio of government debt to output equal to 0.5 on an annual basis, in line with the US post‐war average. In what follows, equilibrium allocations under Cournot, Bertrand and monopolistic competition for each entry cost configuration are compared. This is done while holding parameters fixed in order to understand the role of the different market structures. To this end, the following calibration strategy for the utility parameter υ is adopted. The value of υ is such that steady state labour supply is equal to one under monopolistic competition. In this case, the Frish elasticity of labour supply is reduced to φ, to which I assign a value of four, as in King and Rebelo (1999). Next, the values of υ and φ are held constant under both Bertrand and Cournot competition. In all the versions of the models I consider, a reduction in dividend income taxes which spurs an increase in entry rates in the short run, which, in turn, leads to an increase in the steady state number of producers. Under the baseline calibration, a 1% reduction in the dividend income tax rate leads to an impact increase in the entry rate of about 0.4% in all the versions of the model considered.13 The elasticity of the long‐run number of producers to the long‐run dividend income taxes ranges from 1% in the case of Cournot competition under entry costs in Form 2 to 1.7% under Bertrand competition under entry costs in Form 1. These features of the model are consistent with the empirical evidence in Da Rin et al. (2011), who consider a firm‐level database with entry data of several million European companies between 1997 and 2004. They find an economically sizeable and statistically significant positive effect of a reduction of corporate taxation on entry rates.14 There are two exogenous processes in the economy, namely those for government spending and for technology. They are both assumed to be AR (1) processes in log deviations from the steady state: logGtG=ρglogGt−1G+εtg, logAtA=ρalogAt−1A+εta. The autoregressive coefficient for the technology process is ρa = 0.979 and the standard deviation of the disturbance, σa, is 0.0072, as in the real business cycle (RBC) model by King and Rebelo (1999). The parameterisation of the government spending process follows Chari and Kehoe (1999) in setting ρg = 0.97 and σg = 0.027. 4.2. Ramsey Steady State As discussed above, the market equilibrium is cursed by two distortions. The Labour Distortion is due to a wedge between the within period marginal rate of substitution between hours and consumption and the marginal rate of transformation between hours and output. Considering the tax rate on labour income, the steady state wedge is given by (1 − τl)/μ.15 Efficiency requires a labour income subsidy such that 1 − τl = μ. However, without lump sum taxes and with the need to finance Government spending, the subsidy cannot be provided.16 As a result the Ramsey Planner cannot remove the labour distortion, and hours will be lower in the Ramsey steady state with respect to those worked in the long‐run allocation reached by the SP. On the contrary, by using the dividend income tax, the Ramsey Planner removes the Entry Distortion. This result was first identified by Chugh and Ghironi (2011). In what follows it is shown that the level of the optimal dividend income tax rate differs according to the form of competition and to the form of the entry cost. Proposition 1 provides the main result of this Section. Proposition 1. Optimal long‐run dividend income taxes depend on the form of the entry costs (i)Form1:τd=1−ϵ1−1μand (ii)Form2:τd=1−ϵμ−1 Proof. In the Appendix. Consider condition (i). As mentioned above when the private incentive to create a new firm, as measured by the Lerner Index, that is [1 − (1/μ)], is larger than the social incentive to introduce a new variety, ϵ ⁠, there would be excessive entry. In this case it is optimal to tax profits. A profit income subsidy is, instead, optimal in the opposite situation. In the Appendix it is shown that under the optimal dividend income tax policy, the steady state version of the Planner Euler equation for the number of firms is: ψ=β(1−δ)ϵρAHN+ψ, which is identical to the steady state counterpart of (29), that is the Euler equation obtained in the efficient equilibrium. This implies that the Planner implements the efficient level of units of effective labour per firm, ρAH/N. Recall, however that the Ramsey Planner cannot remove the labour distortion. For this reason steady state hours will be lower in the Ramsey steady state with respect to the those observed in the long‐run allocation reached by the SP. As a result, the number of firms is lower than the optimal one even if the inefficiency along the entry margin is removed. Notice that in the standard neoclassical growth model the Ramsey Planner would target the efficient ratio between capital and hours. This further emphasises the analogy between the stock of capital in the neoclassical model and the stock of firms in the economy we have outlined. The optimal dividend income tax rate under Bertrand competition is: τBertrandd=1N−1θ−1, while under Cournot competition I obtain τCournotd=1N−1(θ−1)2. Notice that the optimal tax rate in the case of monopolistic competition is τMonopolisticd=−1θ−1. Under oligopolistic competition the optimal dividend income tax could take the form of a subsidy, depending on the parameterisation of the model. On the contrary, removing the entry distortion under monopolistic competition requires a subsidy no matter what the parameterisation of the model. This means that monopolistic competition always leads to a suboptimal steady state number of firms. Figure 1 displays, in the case of entry costs in Form 1, the Ramsey steady state number of firms, hours, output per firm and optimal tax rates as a function of the entry cost under Bertrand, Cournot and monopolistic competition. Allocations are compared to the efficient ones. First, for any given value of the entry cost, the Ramsey Planner implements the same allocation for hours and the number of firms across market structures. Fig. 1. Open in new tabDownload slide Ramsey Steady State under Entry Costs in Form 1. Entry cost (ψ) on the horizontal axis Fig. 1. Open in new tabDownload slide Ramsey Steady State under Entry Costs in Form 1. Entry cost (ψ) on the horizontal axis These differ from the efficient ones. As mentioned above, since the Planner cannot remove the labour distortion, hours and the number of firms are lower than their efficient counterpart. The Ramsey Planner targets the efficient level of units of effective labour per firm. This, however, requires a different combination of optimal taxes across market structures. In the case of Cournot competition, the market allocation would lead to an inefficiently large number of firms, as a result restoring efficiency along the entry margin requires a positive dividend income tax. On the contrary, as in CGS (2007), the number of firms under monopolistic competition would be too low and dividend income should be subsidised to promote entry. Bertrand competition falls between these two cases. As stated by Vives (1999), Bertrand competition can be regarded as a more competitive market structure with respect to Cournot and for this reason it is judged as more efficient.17 This analysis suggests that, for given entry costs, the dividend income tax should be lower in markets characterised by more competitive market structures. The opposite instead holds for the labour income tax. Due to the need of financing the exogenous level of public expenditure, the optimal labour income tax is instead higher under monopolistic competition, where dividend income was more heavily subsidised, with respect to other market structures. Condition (ii) is isomorphic to that obtained by BGM (2007) and Chugh and Ghironi (2011) under the case of monopolistic competition. In this case, profits should be taxed whenever the net markup exceeds the benefit of variety in elasticity form. However, under monopolistic competition condition (ii) is automatically satisfied, since the benefit of variety in elasticity form equals the net markup and markups are constant, and the market equilibrium displays efficiency along the entry margin. This is not the case under oligopolistic competition. Under Bertrand competition the optimal tax rate is: τBertrandd=1N, while under Cournot competition it is τCournotd=θN+θ−1. Hence, entry costs in Form 2 always imply a positive dividend income tax rate under oligopolistic competition. Figure 2 displays, for the case of entry costs in Form 2, the Ramsey steady state number of firms and hours together with optimal tax rate as a function of the entry cost. In this case, allocations differ across market structures.18 The reason is that, under entry costs in Form 2, the price markup affects the market value of a firm. As a result the initial stock of wealth, which affects the implementability constraint, depends on the form of the market structure.19 As above, since the Labour distortion cannot be removed, hours and the number of firms are lower with respect to their efficient counterpart. In the Appendix, it is shown that the Ramsey Planner targets the efficient level of output per firm, Yc/N.20 Fig. 2. Open in new tabDownload slide Ramsey Steady State under Entry Costs in Form 2. Entry cost (η) on the horizontal axis Fig. 2. Open in new tabDownload slide Ramsey Steady State under Entry Costs in Form 2. Entry cost (η) on the horizontal axis 4.3. Ramsey Dynamics and Optimal Tax Volatility This subsection compares the efficient equilibrium allocation to the Ramsey allocation and shows the business cycle implications of the Ramsey optimal policy in response to technology shocks. As mentioned above, dynamics are obtained by solving a first‐order approximation to the equilibrium conditions. Consider variable x. Define x~t=xteff−xtRxeff as the deviation between the value assumed by xt in the efficient equilibrium, xteff ⁠, and that assumed by the same variable in the Ramsey equilibrium, xtR ⁠, where xeff is the value of x at the efficient steady state. Deviations are expresses as a percentage of efficient steady state values.21 Figure 3 depicts the dynamics of H~t ⁠, N~t ⁠, C~t and N~te in response to a one standard deviation technology shock under entry costs in Form 1. Dashed and dash‐dotted lines refer to Bertrand and Cournot competition respectively and dotted lines to the case of monopolistic competition. Horizontal continuous lines in each panel represent deviations between the first‐best steady state and the Ramsey steady state as a percentage of the first best steady state, that is x~ss=(xeff−xR/xeff) ⁠, where xR is the value of x at the Ramsey steady state. Fig. 3. Open in new tabDownload slide Entry Costs in Form 1. Percentage deviations between efficient and second best dynamics in response to a technology shock. Horizontal lines refer to steady state percentage deviations Fig. 3. Open in new tabDownload slide Entry Costs in Form 1. Percentage deviations between efficient and second best dynamics in response to a technology shock. Horizontal lines refer to steady state percentage deviations Notice that deviations between the efficient and the Ramsey equilibria, as well as steady state deviations, are identical across market structures. The Figure shows that the bulk of the difference between the first and second best allocation is due to the steady state difference, which is not affected by the shock. Since the Planner must raise revenues using distortionary taxes, the long‐run inefficiency of the Ramsey equilibrium is unavoidable, and the level of activity is below the efficient one. While the deviations of the main macroeconomic variables from their first‐best values do not depend on the form of the market structure, the dynamics of tax rates in response to the technology shock do. To show this, and to further understand the transmission of technology shocks implied by the model economy, Figure 4 depicts percentage deviations from the respective steady states of key variables in response to a one standard deviation technology shock under entry costs in Form 1. For tax rates, I report deviations from the steady state level in percentage points. Continuous lines refer to the efficient equilibrium, dashed and dash‐dotted lines refer to Ramsey dynamics under Bertrand and Cournot competition respectively finally dotted lines to the case of Ramsey dynamics under monopolistic competition. Fig. 4. Open in new tabDownload slide Entry Costs in Form 1. Response of the main macroeconomic variables to a one standard deviation shock to technology. Notes. Solid lines refer to the SP allocation, dashed and dash‐dotted lines refer to Ramsey dynamics under Bertrand and Cournot respectively and dotted lines to the case of Ramsey dynamics under monopolistic competition. Fig. 4. Open in new tabDownload slide Entry Costs in Form 1. Response of the main macroeconomic variables to a one standard deviation shock to technology. Notes. Solid lines refer to the SP allocation, dashed and dash‐dotted lines refer to Ramsey dynamics under Bertrand and Cournot respectively and dotted lines to the case of Ramsey dynamics under monopolistic competition. The technology shock creates expectations of future profits which lead to the entry of new firms in the market. This is so under both the Ramsey and the efficient equilibrium. Under monopolistic competition, the markup is constant along the business cycle. As can be seen from Figure 4, this results in constant tax rates. This is not the case under Oligopolistic competition. Under Bertrand and Cournot competition, the entry of new firms leads to higher competition which, in turn, leads to a countercyclical price markup. A non‐constant price markup implies that the inefficiency wedges in the economy are also non‐constant. The Ramsey Planner partially offsets the effects of changes in the price markup on the inefficiency wedges by adjusting the tax rates. To see this, consider the optimal response of the labour income tax rate, τtl ⁠. As mentioned above, in the ME there is an inefficiency wedge between the marginal rate of substitution between hours and consumption and the corresponding marginal rate of transformation. Considering the tax rate on labour income, this is given by (1−τtl/μt) ⁠.22 Efficiency requires 1−τtl=μt ⁠, that is a labour income subsidy which, in response to shocks, moves in opposite direction with respect to the price markup. However, without lump sum taxes and with the need to finance Government spending, the subsidy cannot be provided. Nevertheless, the analysis shows that τtl offsets the effects of changes in the price markup on the inefficiency wedge. In particular, when the markup decreases, due to higher competition, the labour income tax rate increases. As discussed in Bilbiee et al. (2007) and Etro (2009) a lower tax rate should be adopted in periods/states with a lower number of producers.23 The Figure shows that the optimal response of the labour income tax rate is symmetric, with respect to the horizontal axis, to the response of the price markup. Similar reasoning explains the dynamics of the optimal dividend income tax rate. As competition increases the price markup reduces, leading to a lower private incentive to create a new firm. To align the private to the social incentive, which is constant, the Ramsey Planner reduces the dividend income tax rate. Changes in the tax rates are mild but stronger under Cournot competition where the markup is characterise by a higher elasticity to the number of firms with respect to Bertrand. The Ramsey policy under entry costs in Form 1 is, thus, characterised by a countercyclical labour income tax rate and by a procyclical dividend income tax. Figure 5 displays the dynamics of H~t ⁠, N~t ⁠, C~t and N~te in response to a one standard deviation technology shock under entry costs in Form 2. Lines have the same meaning as in Figure 3. With respect to the case of entry costs in Form 1, allocations differ across market structures.24 For both hours and consumption, the largest deviation with respect to the first best can be observed under monopolistic competition. This is due to the fact that the deviation between the Ramsey steady state and the efficient steady state is larger under monopolistic competition with respect to other market structures. Fig. 5. Open in new tabDownload slide Entry Costs in Form 2. Percentage deviations between efficient and second best dynamics in response to a technology shock Note. Horizontal lines refer to steady state percentage deviations. Fig. 5. Open in new tabDownload slide Entry Costs in Form 2. Percentage deviations between efficient and second best dynamics in response to a technology shock Note. Horizontal lines refer to steady state percentage deviations. Figure 6 shows percentage deviations from the respective steady state of key variables in response to a one standard deviation technology shock under entry costs in Form 2. Lines have the same meaning as in Figure 4. Previous considerations extend to this case. A relevant difference is that offsetting the distortion along the entry margin requires an initial increase in the dividend income tax rate which is reverted after few periods. Fig. 6. Open in new tabDownload slide Entry Costs in Form 2. Response of the main macroeconomic variables to a one standard deviation shock to technology Notes. Solid lines refer to the social planner allocation, dashed and dash‐dotted lines refer to Ramsey dynamics under Bertrand and the Cournot respectively and dotted lines to the case of Ramsey dynamics under monopolistic competition. Fig. 6. Open in new tabDownload slide Entry Costs in Form 2. Response of the main macroeconomic variables to a one standard deviation shock to technology Notes. Solid lines refer to the social planner allocation, dashed and dash‐dotted lines refer to Ramsey dynamics under Bertrand and the Cournot respectively and dotted lines to the case of Ramsey dynamics under monopolistic competition. Notice that similar dynamics, although quantitatively less sizeable, can be observed in the case of a Government spending shock.25 The variability in the main macroeconomic variables when the system is contemporaneously perturbed by a technology and a government spending shock is analysed next. Table 1 displays the mean (⁠ x¯ ⁠), the coefficient of variation (⁠ σ(x)/x¯ ⁠) and the correlation with output (Cor(Yt,xt)) of a number of variables of interest under the Ramsey dividend and labour income taxation policy in the case of entry costs in Form 1. For tax rates, the standard deviation in percentage points (σ(x)) is reported instead of the coefficient of variation. Table 2 reports the same statistics for the case of entry costs in Form 2. Under entry costs in Form 1, the variability in the main macroeconomic variables under the optimal policy is identical across the market structures. However, this is reached by means of a different fiscal policy. As expected from Figure 4, although under monopolistic competition taxes are constant, this is not the case under oligopolistic competition. Tax rates are more volatile under Cournot competition than under Bertrand, with the dividend income tax being more variable than the labour income tax rate. Recall that the elasticity of the price markup to the number of firms is higher under Cournot. As a result, minimising the welfare cost of the distortions over the business cycle requires more variable taxes when firms compete in quantities. Under entry costs in Form 2, allocations and volatilities are no longer identical across market structures. Interestingly, while the overall variability characterising the economy in response to shocks, as measured by the standard deviation of output, is higher under entry cost in Form 1, the variability of tax rates is higher under entry costs in Form 2. In particular, the standard deviation of the dividend income tax under Cournot competition is sizeable. Table 1 Mean, Standard Deviations and Correlations with Output of Main‐macro Variables under Alternative Market Structures. Shocks are to productivity and government spending. Entry costs in Form 1 x . Y . C . H . N . Ne . τtl . τd . Monopolistic competition x¯ 1.58 1.01 1.02 8.83 0.22 0.32 −0.2 σ(x)/x¯ 1.79 0.88 1.15 0.61 7.49 0 0 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0 0 Bertrand competition x¯ 1.58 1.01 1.02 8.83 0.22 0.31 −0.08 σ(x)/x¯ 1.79 0.88 1.15 0.61 7.49 0.02 0.07 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0.11 −0.4 Cournot competition x¯ 1.58 1.01 1.02 8.83 0.22 0.24 0.23 σ(x) 1.79 0.88 1.15 0.61 7.49 0.06 0.17 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0.12 −0.4 x . Y . C . H . N . Ne . τtl . τd . Monopolistic competition x¯ 1.58 1.01 1.02 8.83 0.22 0.32 −0.2 σ(x)/x¯ 1.79 0.88 1.15 0.61 7.49 0 0 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0 0 Bertrand competition x¯ 1.58 1.01 1.02 8.83 0.22 0.31 −0.08 σ(x)/x¯ 1.79 0.88 1.15 0.61 7.49 0.02 0.07 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0.11 −0.4 Cournot competition x¯ 1.58 1.01 1.02 8.83 0.22 0.24 0.23 σ(x) 1.79 0.88 1.15 0.61 7.49 0.06 0.17 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0.12 −0.4 Open in new tab Table 1 Mean, Standard Deviations and Correlations with Output of Main‐macro Variables under Alternative Market Structures. Shocks are to productivity and government spending. Entry costs in Form 1 x . Y . C . H . N . Ne . τtl . τd . Monopolistic competition x¯ 1.58 1.01 1.02 8.83 0.22 0.32 −0.2 σ(x)/x¯ 1.79 0.88 1.15 0.61 7.49 0 0 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0 0 Bertrand competition x¯ 1.58 1.01 1.02 8.83 0.22 0.31 −0.08 σ(x)/x¯ 1.79 0.88 1.15 0.61 7.49 0.02 0.07 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0.11 −0.4 Cournot competition x¯ 1.58 1.01 1.02 8.83 0.22 0.24 0.23 σ(x) 1.79 0.88 1.15 0.61 7.49 0.06 0.17 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0.12 −0.4 x . Y . C . H . N . Ne . τtl . τd . Monopolistic competition x¯ 1.58 1.01 1.02 8.83 0.22 0.32 −0.2 σ(x)/x¯ 1.79 0.88 1.15 0.61 7.49 0 0 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0 0 Bertrand competition x¯ 1.58 1.01 1.02 8.83 0.22 0.31 −0.08 σ(x)/x¯ 1.79 0.88 1.15 0.61 7.49 0.02 0.07 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0.11 −0.4 Cournot competition x¯ 1.58 1.01 1.02 8.83 0.22 0.24 0.23 σ(x) 1.79 0.88 1.15 0.61 7.49 0.06 0.17 Cor(Yt,xt) 1 0.60 0.87 0.4 0.95 0.12 −0.4 Open in new tab Table 2 Mean, Standard Deviations and Correlations with Output of Main‐macro Variables under Alternative Market Structures. Shocks are to productivity and government spending. Entry costs in Form 2 x . Y . C . H . N . Ne . τtl . τd . Monopolistic competition x¯ 1.27 0.85 0.95 4.64 0.11 0.28 0 σ(x)/x¯ 1.60 0.91 1.10 0.61 7.90 0 0 Cor(Yt,xt) 1 0.60 0.85 0.50 0.93 0 0 Bertrand competition x¯ 1.27 0.86 0.96 4.66 0.12 0.24 0.21 σ(x)/x¯ 1.57 0.91 1.09 0.61 7.90 0.02 0.26 Cor(Yt,xt) 1 0.60 0.85 0.50 0.93 0.21 0.60 Cournot competition x¯ 1.25 0.87 0.96 4.68 0.12 0.07 0.62 σ(x) 1.46 0.91 1.08 0.61 7.91 0.15 0.59 Cor(Yt,xt) 1 0.60 0.85 0.53 0.91 0.24 0.72 x . Y . C . H . N . Ne . τtl . τd . Monopolistic competition x¯ 1.27 0.85 0.95 4.64 0.11 0.28 0 σ(x)/x¯ 1.60 0.91 1.10 0.61 7.90 0 0 Cor(Yt,xt) 1 0.60 0.85 0.50 0.93 0 0 Bertrand competition x¯ 1.27 0.86 0.96 4.66 0.12 0.24 0.21 σ(x)/x¯ 1.57 0.91 1.09 0.61 7.90 0.02 0.26 Cor(Yt,xt) 1 0.60 0.85 0.50 0.93 0.21 0.60 Cournot competition x¯ 1.25 0.87 0.96 4.68 0.12 0.07 0.62 σ(x) 1.46 0.91 1.08 0.61 7.91 0.15 0.59 Cor(Yt,xt) 1 0.60 0.85 0.53 0.91 0.24 0.72 Open in new tab Table 2 Mean, Standard Deviations and Correlations with Output of Main‐macro Variables under Alternative Market Structures. Shocks are to productivity and government spending. Entry costs in Form 2 x . Y . C . H . N . Ne . τtl . τd . Monopolistic competition x¯ 1.27 0.85 0.95 4.64 0.11 0.28 0 σ(x)/x¯ 1.60 0.91 1.10 0.61 7.90 0 0 Cor(Yt,xt) 1 0.60 0.85 0.50 0.93 0 0 Bertrand competition x¯ 1.27 0.86 0.96 4.66 0.12 0.24 0.21 σ(x)/x¯ 1.57 0.91 1.09 0.61 7.90 0.02 0.26 Cor(Yt,xt) 1 0.60 0.85 0.50 0.93 0.21 0.60 Cournot competition x¯ 1.25 0.87 0.96 4.68 0.12 0.07 0.62 σ(x) 1.46 0.91 1.08 0.61 7.91 0.15 0.59 Cor(Yt,xt) 1 0.60 0.85 0.53 0.91 0.24 0.72 x . Y . C . H . N . Ne . τtl . τd . Monopolistic competition x¯ 1.27 0.85 0.95 4.64 0.11 0.28 0 σ(x)/x¯ 1.60 0.91 1.10 0.61 7.90 0 0 Cor(Yt,xt) 1 0.60 0.85 0.50 0.93 0 0 Bertrand competition x¯ 1.27 0.86 0.96 4.66 0.12 0.24 0.21 σ(x)/x¯ 1.57 0.91 1.09 0.61 7.90 0.02 0.26 Cor(Yt,xt) 1 0.60 0.85 0.50 0.93 0.21 0.60 Cournot competition x¯ 1.25 0.87 0.96 4.68 0.12 0.07 0.62 σ(x) 1.46 0.91 1.08 0.61 7.91 0.15 0.59 Cor(Yt,xt) 1 0.60 0.85 0.53 0.91 0.24 0.72 Open in new tab 5. Conclusions This study proposes an economy where the degree of market power, as measured by the price markup, depends endogenously on the form of competition, on the degree of substitutability between goods and on the number of firms. Imperfect competition leads to distortions in both the goods and the labour market and in both the short and the long run. The optimal long‐run dividend income corrects for inefficient entry and it is higher in market structures characterised by lower competition. In particular, it is higher under Cournot competition than under Bertrand or monopolistic competition. The labour distortion cannot be removed by the Ramsey Planner. As a result, hours worked and the number of firms/products in the Ramsey steady state are lower than their efficient counterparts. As mentioned in the text, the resulting steady‐state distortion is large. Whereas optimal taxes over the business cycle are constant under monopolistic competition, this is not the case in an oligopolistic market structure. Also, the effect of alternative forms of sunk entry costs for the design of optimal taxation has been considered. The resulting framework features as special cases two models in the entry literature that also focus on optimal taxation problems in the case of endogenous dynamics of the number of firms. CGS (2007) consider an environment characterised by monopolistic competition under constant sunk entry costs. Chugh and Ghironi (2011) consider a framework with monopolistic competition and sunk entry cost in terms of labour. By neglecting strategic interactions and considering the appropriate form of the entry costs our model reduces to either one of these models. For this reason, it can be regarded as a general framework to study optimal taxation problems under various forms of imperfect competition. The analysis could be extended in various dimensions. One aspect I neglect is the asymmetry between market competitors in terms of both size and the probability of exit from the market. Haltiwanger et al. (2010) show that younger firms are more likely to exit from the market than more mature firms. Implementing these features in the model and studying their effects on optimal taxation is left for future research. Appendix A. Market Equilibrium A.1. Entry Costs in Form 1 Given the exogenous processes {At,Gt}t=0∞ and processes {τtd,τtl}t=0∞ , the Market Equilibrium (ME) consists of an allocation {Ct,Ht,Nt,Nte,Bt+1}t=0∞ which satisfies the first order condition for labour supply, (22), the government budget, (11), the dynamics of the number of firms, (23), the definition of the price markup and the definition of the love for variety, the resource constraint (24) and the Euler equation (25). To compute the steady state of the ME, notice that the steady state number of entrants is Ne = [δ/(1−δ)]N and that the Euler equation for shares implies V = β(1−δ)(1−τd)π/[1−β(1−δ)]. Steady state profits are given by:26 π=ρy−wh=ρ1−1μy=ρ1−1μYNρ=1−1μYN, hence the share of profits over output reads as πN/Y = 1 − (1/μ), and the value of firms over output is: NVY=β(1−δ)(1−τd)[1−β(1−δ)]1−1μ. Investment over output is: VNeY=VYδ(1−δ)N=VNYδ(1−δ)=δβ(1−τd)1−1μ[1−β(1−δ)]. The share of consumption over output and that of labour income over output are: CY=1−gy−NeψYandwHY=1−πNY. In order to fix υ we assume that H = 1.27 In this case υ=(1−τl)[(w/Y)(C/Y)] ⁠, where both ratios are known. To compute the number of firms notice that V=β(1−δ)(1−τd)[1−β(1−δ)]π. Imposing the entry condition and substituting for individual profits N=β(1−δ)(1−τd)1−1μρ[1−β(1−δ)]ψAH. The solution to this equation delivers the number of firms at the steady state. This allows me to compute all the other variables. For a given H the number of firms at the steady state is larger the higher the markup, hence is larger under oligopolistic competition. A.2. Entry Costs in Form 2 In the case of entry costs in form 2 the definition of the ME differs from that provided in the case of entry costs in form 1 for two equations. The first one is the aggregate resource constraint, which in this case is given by (26), the second one is the Euler equation, which is given by (27). The steady state level of individual profits is: π=ρy−wh=ρ−wAy=ρ1−1μy=1−1μC+GN=1−1μYcN. As a result πNYc=1−1μ. To obtain the share of investment over consumption output notice that V=β(1−δ)1−τd1−β(1−δ)1−1μYcN and VNeYc=(1−τd)δβ[1−β(1−δ)]1−1μ=(1−τd)δ(μ−1)μ(r+δ). To compute shares over aggregate output recall that: 1=YcY+NeVYcYcY=YcY1+δβ(1−τd)[1−β(1−δ)]1−1μ, thus YcY=1+(1−τd)δβ[1−β(1−δ)]1−1μ−1, which implies that the share of private consumption over output is CY=YcY−gy. Since (WH/Y) + (Π/Y) = 1 we can compute the ratio between labour income and GDP as: WHY=1−πNYcYcY. Given H, the latter leads to: υ=(1−τl)wHCH1+1φ=(1−τl)wHYCYH1+1φ. Labour market equilibrium requires: H=HtC+HtE=Nh+ηANe=NyA+ηANe=YcρA+ηANe, thus Ne = (AH/η) − (Yc/ρη) and N=(1−δ)δAHη−(1−δ)δYcηρ. Next consider V=β(1−δ)(1−τd)[1−β(1−δ)]1−1μYcN, substituting for the entry condition delivers Ycηρ=1μ[1−β(1−δ)]β(1−δ)(1−τd)1−1μN. Substituting the latter into the equation of motion for the number of firms delivers an equation that can be solved for N: N=(1−δ)δAHη1+1δ[1−β(1−δ)]β(1−τd)(μ−1). As above a higher markups leads to a higher number of firms in equilibrium for any given H. B. Appendix Efficient Equilibrium B.1. Entry Costs in Form 1 The social Planner problem reads as: max{Ct,Nt+1,Nte,Ht}t=0∞E0∑t=0∞βtlogCt−υHt1+1/φ1+1/φ, s.t. Ct+Gt+Nteψ=ρtAtHt and Nt+1=(1−δ)(Nt+Nte). I attach the Lagrange Multiplier λt to the first constraint and the multiplier σt to the second one. First order conditions are as follows: Ct:1Ct=λt, Nt+1:σt=βEtλt+1ρN,t+1AtHt+βEtσt+1(1−δ), Nte:λtψ=(1−δ)σt, Ht:υHt1/φ=λtρtAt. Combining the first and the third condition delivers: 1Ctψ(1−δ)=σt. Substituting the latter into the third condition I obtain: ψ=(1−δ)βEtCtCt+1(ρN,t+1AtHt+ψ), which can be written as ψ=(1−δ)βEtCtCt+1ϵYtNt+ψ, finally the FOC with respect to hours can be written as υHt1/φCt=ρtAt. To obtain the steady state I have the following equations: C+G+Neψ=ρAH, N=(1−δ)(N+Ne), ψ=(1−δ)βϵYN+ψ, υH1/φC=ρA. Consider the resource constraint: CY=1−gy−NeψY=1−gy−ψYδ1−δN or CY=1−gy−ψδ1−δNY. From the third equation: YN=[1−(1−δ)β]ψ(1−δ)βϵ. Combining we obtain: CY=1−gy−δ1−δ(1−δ)βϵ[1−(1−δ)β]. Next consider equation: H1+1/φ=1υCρAH=YυC, hence I have H as H=YυC11+1/φ. Next I want to compute N. Notice that: Nρ=(1−δ)βϵ[1−(1−δ)β]ψAH, given ρ = N1/(θ − 1) it follows that Nθ−2θ−1=(1−δ)βϵ[1−(1−δ)β]ψAH or N=(1−δ)βϵ[1−(1−δ)β]ψAHθ−1θ−2. B.2. Entry Costs in Form 2 The Social Planner problem can be written as follows max{Ct,Nt+1,Nte,Ht}t=0∞E0∑t=0∞βtlogCt−υHt1+1/φ1+1/φ s.t. Ct+Gt+Nteηρt=ρtAtHt and Nt+1=(1−δ)(Nt+Nte). I attach the Lagrange Multiplier λt to the first constraint and the multiplier σt to the second one. First order conditions are as follows: Ct:1Ct=λt, Nt+1:σt=βEtλt+1ρN,t+1(At+1Ht+1−Nt+1eη)+βEtσt+1(1−δ), Nte:λtηρt=(1−δ)σt, Ht:υHt1/φ=λtρtAt. Substituting the first condition into the third delivers 1(1−δ)Ctηρt=σt. Substituting the latter and the definition of λt into the other equations I am left with: ηρt=(1−δ)βEtCtCt+1ρN,t+1At+1Ht+1−Nt+1eη+ηρt+1,(B1) and υHt1/φCt=ρtAt. Since AtHt−Nteη=(Ct+Gt)/ρt ⁠, (B1) can be rewritten as: ηρt=(1−δ)βEtCtCt+1ρN,t+1Ct+1+Gt+1ρt+1+ηρt+1 or ηρt=(1−δ)βEtCtCt+1ϵCt+1+Gt+1Nt+1+ηρt+1. To find the steady state I can consider the following equations: ηρ=(1−δ)βϵYcN+ηρ, υH1/φC=ρA, Yc+Neηρ=ρAH, N=(1−δ)(N+Ne). From the first one Ycηρ=[1−(1−δ)β](1−δ)βϵN. The aggregate resource constraint implies: Ycηρ=AHη−δ1−δN. Combining N=AHη[1−(1−δ)β](1−δ)βϵ+δ1−δ, I get N as a function of H. Notice that I have repeatedly used the steady state version of the equation of motion for the number of firms. Consider again the aggregate resource constraint C+G+Neηρ=ρAH, or CρAH=1−gy−NeηρρAH=1−gy−ηρδ1−δNρAH, then υH1/φCρAH=ρAρAH delivers H implicitly as a function of N: υH1/φ1−gy−ηρδ1−δNρAH=1H. The latter is equivalent to: υH1+1/φ1−gy−ηρδ1−δNρAH=1. Next substitute for N as a function of H in the round bracket and H=υ1−gy−δ1−δ[1−(1−δ)β](1−δ)βϵ+δ1−δ−1−φ(1+φ). C. Appendix The Implementability Constraint This Appendix follows closely Arseneau and Chugh (2012) and Chugh and Ghironi (2011). Consider the household flow budget constraint (in the symmetric equilibrium): ∑j1RtjBt+1j+Vt(Nt+Nte)xt+1+Ct=(1−τtl)wtHt+Bt+[(1−τtd)dt+Vt]Ntxt. Multiply both sides by βtuc(ct): ∑jβtuc(ct)1RtjBt+1j+βtuc(ct)Vt(Nt+Nte)xt+1+βtuc(ct)Ct=βtuc(ct)(1−τtl)wtHt+βtuc(ct)bt+βtuc(ct)[(1−τtd)dt+Vt]Ntxt and sum over dates starting from t = 0, where all term are understood as in expectation as of time 0: ∑t=0∞∑jβtuc(ct)1RtjBt+1j+∑t=0∞βtuc(ct)Vt(Nt+Nte)xt+1+∑t=0∞βtuc(ct)Ct=∑t=0∞βtuc(ct)(1−τtl)wtHt+∑t=0∞βtuc(ct)bt+∑t=0∞βtuc(ct)[(1−τtd)dt+Vt]Ntxt. The Euler equation for bonds implies uc(Ct)=βRtjuc(Ct+1j) ⁠, using this in the first term on the LHS: ∑t=0∞∑jβt+1uc(Ct+1j)Bt+1j+∑t=0∞βtuc(Ct)Vt(Nt+Nte)xt+1+∑t=0∞βtuc(Ct)Ct=∑t=0∞βtuc(ct)(1−τtl)wtHt+∑t=0∞βtuc(Ct)bt+∑t=0∞βtuc(Ct)[(1−τtd)dt+Vt]Ntxt Notice that the term ∑juc(Ct+1j)Bt+1j can be understood as the payoff of a risk free bond. As such I can cancel out the first summation on the LHS with the respective terms in the second summation in the RHS, leaving just time 0 terms: ∑t=0∞βtuc(ct)Vt(Nt+Nte)+∑t=0∞βtuc(ct)Ct=∑t=0∞βtuc(ct)(1−τtl)wtHt+uc(c0)b0+∑t=0∞βtuc(ct)[(1−τtd)dt+Vt]Nt. Notice that the clearing of the asset market implies xt = 1 at all t. Considering that uh(ht)/uc(ct)=−(1−τtl)wt leads to: ∑t=0∞βtuc(ct)Vt(Nt+Nte)+∑t=0∞βtuc(ct)Ct+∑t=0∞βtuh(ht)Ht=uc(c0)b0+∑t=0∞βtuc(ct)[(1−τtd)dt+Vt]Nt. Next consider: uc(Ct)Vt=Etβ(1−δ)uc(Ct+1)[(1−τt+1d)dt+1+Vt+1] and substitute it into the first summation in the LHS ∑t=0∞βt+1(1−δ)uc(Ct+1)[(1−τt+1d)dt+1+Vt+1](Nt+Nte)xt+1+∑t=0∞βtuc(ct)Ct+∑t=0∞βtuh(ht)ht=uc(c0)b0+∑t=0∞βtuc(ct)[(1−τtd)dt+Vt]Ntxt, considering that (Nt+Nte)=Nt+11−δ it follows that ∑t=0∞βt+1uc(Ct+1)[(1−τt+1d)dt+1+Vt+1]Nt+1+∑t=0∞βtuc(ct)Ct+∑t=0∞βtuh(ht)ht=uc(c0)b0+∑t=0∞βtuc(ct)[(1−τtd)dt+Vt]Nt. Simplifying the first summation on the LHS with the second in the RHS delivers the implementability constraint: E0∑t=0∞βt[uc(Ct)Ct+uh(Ht)Ht]=uc(C0)B0+uc(C0)[(1−τ0d)d0+V0]N0, where I reintroduced the expectation operator. D. Appendix The Ramsey Problem D.1. Entry Costs in Form 1. Includes Proof of Result (i) in Proposition 1 The Ramsey problem reads as: maxE0∑t=0∞βtu(Ct,Ht), subject to Nt+1=(1−δ)(Nt+Nte):λ1t, Ct+Gt+Nteψ=ρtAtHt:λ2t, ψuct=β(1−δ)Etuct+11−τt+1/td1−1μCt+1+Gt+1Nt+1+ψ:λ3t, E0∑t=0∞βtE0[uctCt+uhtHt]=uc0B0+uc0[(1−τ0d)d0+V0]N0:ξ, where λit define the Lagrange multipliers respectively attached to each constraint and ξ is the (constant) Lagrange multiplier attached to the implementability constraint. The choice variables are Ct, Nt ⁠, Ht, Nte ⁠, and τt/t+1d ⁠. Following Ljungqvist and Sargent (2004) I define: V(Ct,Ht,ξ)=u(Ct,Ht)+ξ(uctCt+uhtHt) and Ω=uc0B0+uc0[(1−τ0d)d0+V0]N0. As a result the Lagrangian function can be written as: L=E0∑t=0∞βtV(Ct,Ht,ξ)+λ1t[(1−δ)(Nt+Nte)−Nt+1]+λ2t(ρtAtHt−Ct−Gt−Nteψ)+λ3tψuct−β(1−δ)Etuct+11−τt+1/td1−1μt+1ρt+1At+1Ht+1Nt+1+ψ−ξΩ. The first order conditions for periods t ≥ 1 are: Ct:Vc(Ct,Ht,ξ)−λ2t+λ3tucctψ−λ3t−1(1−δ)ucct(1−τt/t−1d)1−1μtρtAtHtNt+ψ=0, Nt+1:λ1t+β(1−δ)λ3tEtuct+1(1−τt+1/td)At+1Ht+1Nt+11−1μt+1ρNt+1Nt+1−ρt+1Nt+1+μNt+1ρt+1μt+12=β(1−δ)Etλ1t+1+βEtλ2t+1ρNt+1At+1Ht+1, τt+1/td:(1−δ)βt+1λ3tE0uct+11−1μt+1ρt+1At+1Ht+1Nt+1=0, Nte:λ1t(1−δ)=λ2tψ, Ht:Vh(Ct,Ht,ξ)+λ2tρtAt−λ3t−1(1−δ)uct(1−τt/t−1d)1−1μtρtAtNt=0. Since V(Ct,Ht,ξ)=u(Ct,Ht)+ξ(uctCt+uhtHt), it follows that Vc(Ct,Ht,ξ)=uc(C,H)+ξucctCt+ξuct=1Ct−ξ1Ct2Ct+ξ1Ct=1Ct and Vh(Ct,Ht,ξ)=−υHt1/φξ1+φφ+1. Consider now the steady state. The FOC with respect to τt+1/td reads as τt+1/td:(1−δ)βt+1λ31−1μC+GNuc=0 The latter implies that at the steady state λ3 is equal to zero. In the Ramsey steady state the firms entry condition does not restrict the allocation. As a result I can write the steady state version of the FOCs as: Ct:λ2=uc, Ht:Vh(C,H,ξ)+uctρA=0, Nte:λ1=ψ(1−δ)uc, Nt+1:βρNAH=[1−β(1−δ)]ψ(1−δ). The FOC with respect to N can be written as: ψ=β(1−δ)[ρNAH+ψ], since ρN=1θ−1ρN=ϵρN, it follows that ψ=β(1−δ)ϵρAHN+ψ. The Euler equation for asset implies that: ψ=β(1−δ)(1−τd)1−1μρAHN+ψ. For the latter two equations to be consistent with each other it has to be the case that: ϵ=(1−τd)1−1μ or τd=1−ϵ1−1μ, which proves point (i) in Proposition 1. Importantly, the dividend income tax differs from zero also under monopolistic competition. Substituting the optimal dividend income tax into the Euler equation for shares I get: ψ=β(1−δ)ϵYN+ψ which implies YN=[1−β(1−δ)]β(1−δ)ϵψ. Notice that CY=1−gy−δ1−δψNY. Next consider the implementability constraint, which can be written as: Ω=YCBY+ϵ1−1μΠY+VNY, where B/Y is exogenously given and C/Y has been computed above. The Euler equation with respect to assets implies: VNY=β(1−δ)[1−β(1−δ)]ϵ and also I know ΠY=1−1μ, Ω=YCBY+ϵ+VNY, H=1−(1−β)Ωυφ1+φ. Finally given H and recalling that: YN=[1−β(1−δ)](θ−1)β(1−δ)ψ, it follows that N=β(1−δ)ϵ[1−β(1−δ)]ψAHθ−1θ−2. D.2. Entry Costs in Form 2. Includes Proof of Result (ii) in Proposition 2 In this case the Lagrangian is: L=E0∑t=0∞βtV(Ct,Ht,ξ)+λ1t[(1−δ)(Nt+Nte)−Nt+1]+λ2t(ρtAtHt−Ct−Gt−Nteηρt)+λ3tηρtμtuct+−β(1−δ)uct+1(1−τt+1/td)1−1μt+1Ct+1+Gt+1Nt+1+ηρt+1μt+1−ξΩ. As above, the choice variables are Ct, Nt ⁠, Ht, Nte and τt/t+1d ⁠. The first order conditions are: Ct:Vc(Ct,Ht,ξ)+λ3tηρtμucct−λ3t−1(1−δ)ucct1−τt/t−1d1−1μCt+GtNt+ηρtμ+(1−δ)uct1−τt/t−1d1−1μ1Nt=λ2t, Nt+1:λ1t+β(1−δ)λ3tEtuct+1+(1−τt+1/td)μt+1μN,t+1μt+1−μt+1−1Nt+1Yt+1cNt+1+βη(1−δ)λ3tEtuct+1ρNt+1μt+1−μNt+1ρt+1μt+12=β(1−δ)Etλ1t+1+βEtλ2t+1ρNt+1(At+1Ht+1−Nt+1eη)+βηEtλ3t+1uct+1ρNt+1μt+1−μNt+1ρt+1μt+12. Ht:Vh(Ct,Ht,ξ)+λ2tρtAt=0, Nte:λ1t(1−δ)=λ2tηρt, τt+1/td:(1−δ)βt+1Etuct+1λ3t1−1μt+1Ct+1+Gt+1Nt+1=0. Notice that: ρtN=1θ−1ρtNt, μtC=θNt(θ−1)(Nt−1), μNtC=θ(θ−1)(Nt−1)−(θ−1)θNt(θ−1)2(Nt−1)2=θ(Nt−1)−θNt(θ−1)(Nt−1)2=−θ(θ−1)(Nt−1)2, μtB=1+θ(Nt−1)(θ−1)(Nt−1), μNtB=θ(θ−1)(Nt−1)−(θ−1)[1+θ(Nt−1)](θ−1)2(N−1)2=−1(θ−1)(N−1)2. Given λ3 = 0 we can write the steady state system as above: Ct:Vc(C,H,ξ)=λ2, Nt+1:λ1=β[(1−δ)λ1+λ2ρN(AH−Neη)], Ht:Vh(C,H,ξ)+λ2ρA=0 Nte:λ1(1−δ)=λ2ηρ Since λ1=ηρ(1−δ)Vc(C,H,ξ) and −Vh(C,H,ξ)Vc(C,H,ξ)=ρA and given the definitions of Vc and Vh I get 1β=(1−δ)1+ϵ1ηρYcN. Evaluating the Euler equation for assets at the steady state implies: 1β=(1−δ)1+(1−τd)(μ−1)ηρYcN. For the two to be consistent it has to be the case that: (1−τd)(μ−1)=ϵ, which proves point (ii) in Proposition 1. Notice that in the monopolistic competition case this implies τd = 0. The Euler equation for assets evaluated at the steady state reads as: 1β=(1−δ)1+ϵρηYcN, then Ycηρ=1−β(1−δ)(1−δ)βϵN. The aggregate resource constraint implies: Ycηρ=AHη−Ne using the equation for the dynamics of the number of firms Ycηρ=AHη−δ1−δN. Combing the latter two equations N=AHη[1−(1−δ)β](1−δ)βϵ+δ1−δ=(1−δ)δAHη1+[1−(1−δ)β]δβϵ we get N as a function of H. This also implies that we can compute the markup, under both Cournot and Bertrand, as a function of H. Recall that it has to be the case that: Y=wH+Π, since wH=ρμAH;Π=1−1μYc it follows that Y=ρμAH+1−1μYc. Using the aggregate resource constraint Yc = ρAH − ηρNe we obtain: Y=ρμAH+1−1μ(ρAH−ηρNe)=ρAH−ρAμ−1μηANe. Also notice that 1=1−1μYcY+1μρAHY. To compute Yc/Y and C/Y consider the euler equation for assets V=β(1−δ)ϵμ1−β(1−δ)YcN, which implies VNeYc=β(1−δ)ϵμ1−β(1−δ)NeN=δβϵμ1−β(1−δ), this allows to compute Yc/Y as follows: Yc+NeV=Y, then YcY=1−NeVYcYcY and finally YcY=1+VNeYc−1. From the latter we get C/Y as: CY=YcY−GY. Knowing Yc/Y we can determine ρAH/Y: ρAHY=μ1−1−1μYcY. The FOC for hours: −Vh(C,H,ξ)Vc(C,H,ξ)=ρA, substituting the definitions of variables υHt1/φξ1+φφ+11C=ρA or H1/φ=1υρACξ1+φφ+1. Multiplying both sides by H, the latter is equivalent to: H=ρAHYYCυξ1+φφ+1φ1+φ, hence H is both a function of H and ξ. Next consider the implementability constraint Ω=YCBY+(1−τd)πNC+VNC. As a result: Ω=YCBY+(1−τd)ΠY+VNY=YCBY+ϵ(μ−1)ΠY+VNY, where B/Y is given and C/Y is a function of H. Also from V=β(1−δ)ϵμ1−β(1−δ)YcN, I get VNY=β(1−δ)ϵμ1−β(1−δ)YcY and ΠY=1−1μYcY. Hence I can compute Ω as a function of H. Next using the steady state version of the implementability constraint I get: 1−υH1+1/φ=1−βΩ, which implies H=1−(1−β)Ωυφ1+φ, which is a function solely of H and can be solved numerically. Given the value H we can determine the Lagrange multiplier ξ: ξ=φ1+φ1υρAHYYCH1+φφ−1. Recall that N can be computed as: N=Aη1−δδ1+[1−(1−δ)β]δβϵH, which allows me to compute the price markup and thus the Ramsey steady state. Also it implies a value for ρ. Since ηρμA=β(1−δ)ϵμ1−β(1−δ)YcN, we get Yc=ηρANμ1−β(1−δ)β(1−δ)ϵμ. In particular notice that τd=1−ϵ(μ−1)andτl=1−υCH1φw. Footnotes 1 " Early contributors to this literature are Chatterjee and Cooper (1993), Devereux et al. (1996) and Devereux and Khang (2001). More recent developments have come from Bergin and Corsetti (2008) and Faia (2012). 2 " For a review of the extensions, see Chari and Kehoe (1999) and Ljungqvist and Sargent (2004). 3 " As discussed in BGM (2012), if macroeconomic shocks are small enough, Nj,te is positive in every period. New entrants finance their entry on the stock market. 4 " Due to the Poisson nature of exit shocks, the household does not know which firms will disappear from the market, so it finances the continued operations of all incumbent firms as well as those of the new entrants. 5 " Hence, it follows that gt(i) = [pt(i)/Pt]−θGt. 6 " Of course, both of them are endogenous in general equilibrium but it is reasonable to assume that firms do not perceive marginal costs and aggregate expenditure as being affected by their choices. 7 " Since total expenditure EXPt is equalised between sectors, I assume that it is also perceived as given by the firms. Under the alternative hypothesis that the sum between public and private consumption, Ct + Gt , is perceived as given, we would obtain the higher markup: μ~B(θ,Nt)=θ(Nt−1)(θ−1)(Nt−1)−1, which leads to similar qualitative results. This case would correspond to the equilibrium markup proposed by Yang and Heijdra (1993). 8 " To see this, note that Pt(Ct+Gt)=∫0Ntpityitdi in a symmetric equilibrium it follows that Pt(Ct+Gt)=Ntptyt, thus yt=Ptpt(Ct+Gt)Nt=(Ct+Gt)ρtNt. 9 " Note that ϵ(Nt+1) is an elasticity; thus, it is a pure number. 10 " The Lerner index is defined as follows: LI=p(i)−MC(i)p(i)=p(i)P−MC(i)Pp(i)P=ρ−ρμρ=1−1μ. 11 " In other words, τt+1d is chosen at time t. The selected value is then implemented with certainty at time t + 1. 12 " Oliveira Martins and Scarpetta (1999) provide estimates of price markups for US manufacturing industries over the period 1970–92. In broad terms, most of the sectoral markups defined over value added are in the range of 30–60%; when they are defined over gross output they are in the range of 5–25%. In the latter case, high markups of over 40% are observed in a few sectors. 13 " The entry rate is measured by the number of entering firms as a percentage of the total number of firms in each period. The impact change of the entry rate in the face of a reduction in the dividend income tax rate is obtained numerically, simulating the effect of a 1% unexpected reduction in the tax rate. 14 " Specifically, they find that a 10% reduction in taxation implies a 2.64% increase in the entry rate. 15 " To see this, consider the first‐order condition for labour supply in the ME, that is, (22). 16 " A labour income subsidy τl < 0 can only occur if the Government holds assets to the extent that taxes are not necessary to finance public spending. 17 " Vives (1999) provides the following intuitive explanation to support this view. In Cournot competition, each firm expects the others to cut prices in response to price cuts, while in Bertrand competition the firm expects the others to maintain their prices; therefore Cournot penalises price cutting more. One should expect Cournot prices to be higher than Bertrand prices. 18 " Differences across market structures are small in absolute terms. In the next Sections, for selected variables, I plot the differences between the SP steady state values and the Ramsey steady state values as a percentage of the SP steady state values. In that case differences across market structures will be, also visually, sizeable. 19 " Recall that under the Timeless Perspective the initial values of the predetermined variables are equal to their values in the deterministic Ramsey steady state. 20 " As shown by Chugh and Ghironi (2011) the Ramsey Planner can also remove the entry distortion by using an entry subsidy instead of the dividend income tax. Define τs the entry subsidy such that the net entry cost is (1−τts)ft ⁠. It can be shown that optimal subsidies depend on the form of the entry cost as follows: (i)Form 1:τs=1−1−1μϵand (ii)Form 2:τs=1−μ−1ϵ. The Ramsey Planner will resort to an entry tax in the case of excessive entry or to a subsidy in the case of inefficiently low entry. 21 " Deviations are exact up to a first order approximation. 22 " This can be understood by considering the first order condition for labour supply in the ME, that is (22). 23 " Etro (2009) also considers a tax rate on sales. He argues that the tax rate should be countercyclical. 24 " In the previous subsection it was suggested that, under entry costs in Form 2, the initial stock of wealth differs across market structures. The initial stock of wealth affects the implementability constraint, leading to a difference in the allocations across market structures. 25 " Dynamics in response to a Government spending shock are not reported. Notice, however, that aggregate consumption drops in response to a Government spending shock under all the market structures considered. 26 " Notice that this is the main difference wrt to cost 2 since in that case profits depend on Yc. 27 " As mentioned in the Section on calibration I fix H = 1 under monopolistic competition and obtain the corresponding value of υ. Under oligopolistic competition I consider the value of υ so obtained and compute the corresponding value of H. References Arseneau , D.M. and Chugh , S.K. ( 2012 ). ‘ Tax smoothing in frictional labor markets ’, Journal of Political Economy , vol. 120 ( 5 ), pp. 926 – 985 . Google Scholar Crossref Search ADS WorldCat Bergin , P. and Corsetti G. ( 2008 ). ‘ The extensive margin and monetary policy ’, Journal of Monetary Economics , vol. 55 ( 7 ), pp. 1222 – 37 . Google Scholar Crossref Search ADS WorldCat Bilbiee , F. , Ghironi , F. and Melitz , M. ( 2007 ). ‘ Monopoly power and endogenous product variety: distortions and remedies ’, Working Paper No. 14383, NBER. Bilbiie , F. , Ghironi , F. and Melitz , M. ( 2012 ). ‘ Endogenous entry, product variety, and business cycles ’, Journal of Political Economy , vol. 120 ( 2 ), pp. 304 – 45 . Google Scholar Crossref Search ADS WorldCat Bils , M. ( 1987 ). ‘ The cyclical behavior of marginal cost and price ’, The American Economic Review , vol. 77 ( 5 ), pp. 838 – 55 . OpenURL Placeholder Text WorldCat Chamley , C. ( 1986 ). ‘ Optimal taxation of capital income in general equilibrium with infinite lives ’, Econometrica, Econometric Society , vol. 54 ( 3 ), pp. 607 – 22 . Google Scholar Crossref Search ADS WorldCat Chari V.V. , and Kehoe P.J. ( 1999 ). ‘Optimal fiscal and monetary policy’, in ( J.B. Taylor and M. Woodford, eds.), Handbook of Macroeconomics , vol. 1C, pp. 1671 – 745 , North‐Holland : Elsevier Science . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Chari , V.V. , Christiano , L.J. and Kehoe , P.J. ( 1991 ). ‘ Optimal fiscal and monetary policy: some recent results ’, Journal of Money, Credit and Banking , vol. 23 ( 3 ), pp. 519 – 39 . Google Scholar Crossref Search ADS WorldCat Chatterjee , S. and R. Cooper 1993 . ‘ Entry and exit, product variety and the business cycle ’, Working Paper No. 4562, NBER. Chugh , S. and Ghironi , F. ( 2011 ), ‘ Optimal fiscal policy with endogenous product variety ’, Working Paper. Colciago , A. and Etro , F. ( 2010 ). ‘ Real business cycle with cournot competition and endogenous entry ’, Journal of Macroeconomics , vol. 32 ( 4 ), pp. 1101 – 17 . Google Scholar Crossref Search ADS WorldCat Colciago , A. and Rossi L. ( 2011 ). ‘ Firms entry, oligopolistic competition and labor market dynamics ’, Working Paper No. 465, De Nederlandsche Bank. Coto‐Martinez , J. , Garriga , C. and Sanchez‐Losada , F. ( 2007 ). ‘ Optimal taxation with imperfect competition and aggregate returns to specialization ’, Journal of the European Economic Association , vol. 5 ( 6 ), pp. 1269 – 99 . Google Scholar Crossref Search ADS WorldCat Da Rin , M. , Di Giacomo , M. and Sembenelli , A. ( 2011 ). ‘ Entry dynamics and the taxation of corporate profits: evidence from firm‐level data ’, Journal of Public Economics , vol 95 ( 9–10 ), pp. 1048 – 66 . Google Scholar Crossref Search ADS WorldCat De Nicolò , V. and Matteuzzi M. ( 2000 ). ‘ Specific and ad valorem taxation in asymmetric Cournot oligopolies ’, International Tax and Public Finance , vol. 7 ( 3 ), pp. 335 – 42 . Google Scholar Crossref Search ADS WorldCat Devereux , M. and Khang M.L. ( 2001 ). ‘ Dynamic gains from international trade with imperfect competition and market power ’, Review of Development Economics , vol. 5 ( 2 ), pp. 239 – 55 . Google Scholar Crossref Search ADS WorldCat Devereux , M. , Allen H. and Lapham B. ( 1996 ). ‘ Aggregate fluctuations with increasing returns to specialization and scale ’, Journal of Economic Dynamics and Control , vol. 20 ( 4 ), pp. 627 – 56 . Google Scholar Crossref Search ADS WorldCat Etro , F. ( 2009 ). Endogenous Market Structures and the Macroeconomy , Berlin : Springer‐Verlag . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Etro , F. and Colciago , A. ( 2010 ). ‘ Endogenous market structures and the business cycle ’, Economic Journal , vol. 129 ( 549 ), pp. 1201 – 33 . Google Scholar Crossref Search ADS WorldCat Faia , E. ( 2012 ). ‘ Oligopolistic competition and optimal monetary policy ’, Journal of Economic Dynamics and Control , vol. 36 ( 11 ), pp. 1760 – 74 . Google Scholar Crossref Search ADS WorldCat Galì , J. , Gertler , M. and López‐Salido , J.D. ( 2007 ). ‘ Markups, gaps, and the welfare costs of economic fluctuations ’, Review of Economics and Statistics , vol. 89 ( 1 ), pp. 44 – 59 . Google Scholar Crossref Search ADS WorldCat Ghironi , F. and Melitz , M. ( 2005 ). ‘ International trade and macroeconomic dynamics with heterogenous firms ’, Quarterly Journal of Economics , vol. 120 ( 3 ), pp. 865 – 915 . OpenURL Placeholder Text WorldCat Guo J.T. and Lansing K.J. ( 1999 ). ‘ Optimal taxation of capital income with imperfectly competitive product markets ’, Journal of Economic Dynamics and Control , vol. 23 ( 7 ), pp. 967 – 95 . Google Scholar Crossref Search ADS WorldCat Haltiwanger , J.C. , Jarmin , R.S. and Miranda , J. ( 2010 ). ‘ Who creates jobs? Small vs. large vs. young ’, Working Paper No. 16300, NBER. Jaimovich , N. and Floetotto , M. ( 2010 ). ‘ Firm dynamics, mark up variations, and the business cycle ’, Journal of Monetary Economics , vol. 55 ( 7 ), pp. 1238 – 52 . Google Scholar Crossref Search ADS WorldCat Judd , K.L. ( 1985 ). ‘ Redistributive taxation in a simple perfect foresight model ’, Journal of Public Economics , vol. 28 ( 1 ), pp. 59 – 83 . Google Scholar Crossref Search ADS WorldCat Judd , K.L. ( 1997 ). ‘ The optimal tax rate for capital income is negative’, Working Paper No. 6004, NBER. Judd , K.L. ( 2002 ). ‘ Capital‐income taxation with imperfect competition ’, American Economic Review, Papers and Proceedings , vol. 92 ( 2 ), pp. 417 – 21 . Google Scholar Crossref Search ADS WorldCat Khan , A. , King R.G. and Wolman A.L. ( 2003 ). ‘ Optimal monetary policy ’, Review of Economic Studies , vol. 70 ( 4 ), pp. 825 – 860 . Google Scholar Crossref Search ADS WorldCat King , R.G. and Rebelo , S. ( 1999 ). ‘Resuscitating real business cycles’, in ( J.B. Taylor and M. Woodford, eds.), Handbook of Macroeconomics , vol. 1B, pp. 927 – 1007 , Amsterdam : North‐Holland . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Lee Y. , and Mukoyama , T. ( 2007 ). ‘ Entry, exit and plant‐level dynamics over the business cycle ’, Working Paper No. 07‐18, Cleveland Fed. Lewis , V. and Winkler , R. ( 2015 ). ‘ Product diversity, demand structures and optimal taxation ’, Economic Inquiry , vol. 53 ( 2 ), pp. 979 – 1003 . Google Scholar Crossref Search ADS WorldCat Lewis V. , and Poilly , C. ( 2012 ). ‘ Firm entry and the monetary transmission mechanism ’, Journal of Monetary Economics , vol. 59 ( 7 ), pp. 670 – 85 . Google Scholar Crossref Search ADS WorldCat Linnemann , L. and Shabert , A. ( 2012 ). ‘ Optimal government expenditure and unemployment ’, Journal of Economic Dynamics and Control , vol. 36 ( 5 ), pp. 795 – 811 . Google Scholar Crossref Search ADS WorldCat Ljungqvist , L. and Sargent , T.J. ( 2004 ). Recursive Macroeconomic Theory , 2nd edn. Cambridge MA : MIT Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Mortensen , D. and Pissarides , C. ( 1994 ). ‘ Job creation and job destruction in the theory of unemployment ’, Review of Economic Studies , vol. 61 ( 3 ), pp. 397 – 415 . Google Scholar Crossref Search ADS WorldCat Myles , G.D. ( 1989 ). ‘ Ramsey tax rules for economies with imperfect competition ’, Journal of Public Economics , vol. 38 ( 1 ), pp. 95 – 115 . Google Scholar Crossref Search ADS WorldCat Myles G.D. ( 1996 ). ‘ Imperfect competition and the optimal combination of ad valorem and specific taxation ’, International Tax and Public Finance , vol. 3 ( 1 ), pp 29 – 44 . Google Scholar Crossref Search ADS WorldCat Oliveira Martins , J. and Scarpetta , S. ( 1999 ). ‘ The level and cyclical behavior of mark‐ups across countries and market structures ’, Working Paper No. 213, OECD. Rotemberg , J. and Woodfrod , M. ( 1999 ). ‘The cyclical behavior of prices and costs’, in ( J.B. Taylor and M. Woodford, eds.), Handbook of Macroeconomics, vol. 1B, pp. 1051 – 135 , Amsterdam : North‐Holland . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Schmitt‐Grohè , S. and Uribe , M. ( 2005 ). ‘ Optimal fiscal and monetary policy in a medium scale macroecomic model ’, NBER Macroeconomics Annual. Vives , X. ( 1999 ). Oligopoly Pricing. Old Ideas and New Tools , Cambridge MA : MIT Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Woodford , M. ( 2003 ). Interest and Prices: Foundations of a Theory of Monetary Policy , Princeton, NJ : Princeton University Press . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Yang , X. and Heijdra B.J. ( 1993 ). ‘ Monopolistic competition and optimum product diversity: comment ’, American Economic Review , vol. 83 ( 1 ), pp. 295 – 301 . OpenURL Placeholder Text WorldCat Author notes " The article previously circulated with the title ‘Imperfect Competition and Optimal Taxation’ (DNB Working Paper No. 383). I thank Guido Ascari, Martin Ellison, Jordi Galì, Pierre Laforcade, Andreas Shabert, Ctirad Slavik, Patrizio Tirelli, two anonymous referees and seminar participants at the Dutch National Bank, Tilburgh University, the University of Lauven and the University of Lancaster for useful comments. The article has also benefited from the comments of participants at the TM2 conference in Lausanne, the meeting of the Society for Computational Economics in Vancouver and the 10th Dynare conference in Paris. I thank the UniCredit & Universities Knight of Labour Ugo Foscolo Foundation for their financial support and the Bank of Finland for their hospitality during the summer of 2011. © 2015 Royal Economic Society
TI - Endogenous Market Structures and Optimal Taxation
JO - The Economic Journal
DO - 10.1111/ecoj.12293
DA - 2016-08-01
UR - https://www.deepdyve.com/lp/oxford-university-press/endogenous-market-structures-and-optimal-taxation-sDebvmHIG4
SP - 1441
VL - 126
IS - 594
DP - DeepDyve
ER -