Imperfect Competition andOptimal taxation

Andrea Colciago

Abstract

This paper provides optimal labor and dividend income tax-ation in a general equilibrium model with oligopolistic competi-tion and endogenous firms’entry. In the long run the optimaldividend income tax corrects for ineffi cient entry. The dividendincome tax depends on the form of competition and the nature ofthe sunk entry costs. In particular, it is higher in market struc-tures characterized by competition in quantities with respect tothose characterized by price competition. Oligopolistic competi-tion leads to an endogenous countercyclical price markup. As aresult offsetting the distortions over the business cycle requiresdeviations from full tax smoothing.

1 Introduction

A recent macroeconomic literature emphasizes the importance of the cre-ation of new firms for the propagation of business cycle fluctuations. Bil-biie, Ghironi and Melitz (2012, BGM henceforth), Jaimovich and Floe-totto (2010) and Colciago and Etro (2010 a and b), among others, showthat accounting for firms’dynamics helps improving the performance ofdynamic general equilibrium models at replicating the variability of themain macroeconomic variables in response to exogenous disturbances.1

Most of these studies are characterized by imperfect competition in thegoods market. As a result an ineffi cient number of producers (and prod-ucts) may arise in equilibrium, leading to welfare losses for the society.For this reason, policy measures aimed at removing market distortionscould be desirable.

1Early contributions to this literature are Chattejee and Cooper (1993), Devereuxet al. (1996), Devereux and Lee (2001). More recent development are instead Berginand Corsetti (2008) and Faia (2012).

1

This paper provides optimal Ramsey dividend and labor income tax-ation in a framework characterized by alternative, imperfectly compet-itive, market structures. The economy features distinct sectors, eachone characterized by many firms supplying goods that can be imper-fectly substitutable to a different degree, taking strategic interactionsinto account and competing either in prices (Bertrand competition) orin quantities (Cournot competition). As in BGM (2012) the entry of anew firm in the market amounts to the creation of a new product. Sunkentry costs allow to endogenize entry and the (stock market) value ofeach firm in each sector. Preferences of agents are characterized by lovefor variety, such that spreading a given nominal consumption expendi-ture over a larger number of goods leads to an increase in utility.The degree of market power, as measured by the price markup, de-

pends endogenously on the form of competition, on the degree of sub-stitutability between goods and on the equilibrium number of firms.Importantly, the price markup is countercyclical. During an economicboom profits opportunities attract firms into the market. This strength-ens competition and, via strategic interactions, reduces price markups.An early references on the procyclicality of the number of firms’in theU.S. is Chatterjee and Cooper (1993), while a more recent one is Lewisand Poilly (2012). The countercyclicality of the price markup is consis-tent with the empirical findings by Bils (1987), Rotemberg andWoodford(2000) and Galì et al. (2007). Nevertheless notice that aggregate profitsremain strongly procyclical, as in the data.As emphasize in BGM (2007), the market equilibrium is character-

ized by two distortions: a Labor Distortion and an Entry Distortion.As in other models with an imperfectly competitive goods market, thepresence of a price markup leads to a wedge between the marginal prod-uct of labor and the marginal rate of substitution between consumptionand hours. Oligopolistic competition renders this wedge time varying.The Entry Distortion operates through the intertemporal firms creationmargin and leads to an ineffi cient number of firms in equilibrium. Apositive dividend income tax is optimal in case of excessive entry andviceversa.To minimize the welfare cost associated to these distortions, the

Ramsey optimal fiscal policy, both in the short and in the long run,is provided. The Government levies taxes on dividend income and la-bor income and issues state contingent bonds to finance an exogenousstream of public spending.In the long run the dividend income tax removes the Entry Distor-

tion, as in Chugh and Ghironi (2011). The magnitude, and sign, of thedividend income tax depends on the form of the entry cost and on the

2

form of competition, and it is higher in market structures characterizedby lower competition. In particular, it is higher under Cournot Compe-tition with respect to Bertrand or monopolistic competition. The longrun labor income tax is positive under all market structures considered.As a result the Labor Distortion cannot be removed, and the effi cientlong run allocation cannot be achieved.As emphasized by Chugh and Ghironi (2011), this suggest an analogy

between optimal taxation in the present setting and that in the standardRBC model. As well know, the latter framework is characterized by azero long run capital income tax, whereas the labor tax finances gov-ernment spending and interest payment on debt. Such a policy deliverseffi ciency along the investment margin, but disregards the social costof labor distortion. Similarly, in the current framework the dividendincome tax, which is a form of capital taxation, off-sets the distortionalong the intertemporal (entry) margin.Over the business cycle, Chugh and Ghironi (2011) show that optimal

tax rates are constant under monopolistic competition. This is not thecase in oligopolistic market structures. Due to the countercyclicality ofthe price markup, the distortions affecting the economy are time-varying.Counteracting these distortion requires, thus, non constant tax rates.Finally, notice that besides the form of competition, also the form

of the sunk entry costs matters for optimal taxation purposes. For thisreason, together with alternative forms of competition, two forms of theentry costs are considered. One features a constant entry cost measuredin units of output, the other one features entry costs in terms of labor.As a result the present framework features as special cases two mod-

els in the entry literature which also focus on optimal taxation problems:Coto Martinez et al. (2007) and Chugh and Ghironi (2011). Coto Mar-tinez et al. (2007) consider an environment characterized by monopo-listic competition under constant sunk entry costs. They find that thelong run equilibrium is characterized by an ineffi ciently low number offirms. For this reason it is optimal to subsidize dividend income. Fur-ther they find that tax smoothing is optimal. Chugh and Ghironi (2011)consider a framework with monopolistic competition and sunk entry costin terms of labor.2 In this case the optimal long run dividend incometax is zero and taxes are constant over the business cycle. By neglectingstrategic interactions and considering the appropriate form of the entrycosts the provided framework reduces to either one of these models. Forthis reason it can be regarded as a general framework where to studyoptimal taxation problems under various forms of imperfect competi-

2In their framework setting up a new firm requires workers. As a result the costof entry depends on the real wage.

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tion. Another paper closely related to the present one is that by Lewis(2010). She studies optimal fiscal policy under oligopolistic competitionand endogenous entry, but in a static environment.The reminder of the paper is laid as follows. Section 2 presents the

model; section 3 defines the market equilibrium; section 4 characterizesthe effi cient allocation; section 5 discusses the distortion associated tothe market equilibrium; section 6 provided the Ramsey optimal fiscal;section 7 concludes. Technical details are left in the Appendix.

2 The Model

The economy features a continuum of atomistics sectors, or industries,on the unit interval. Each sector is characterized by different firms pro-ducing a good in different varieties, using labor as the only input. Inturn, the sectoral goods are imperfect substitutes for each other and areaggregated into a final good. Households use the final good for consump-tion and investment purposes. Oligopolistic competition and endogenousfirms’entry is modeled at the sectoral level.At the beginning of each period N e

jt new firms enter into sector j ∈(0, 1), while at the end of the period a fraction δ ∈ (0, 1) of marketparticipants 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 +N ejt) (1)

where N ejt 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 produc-ing at time t+ 1 and that the probability of exit from the market, δ, isindependent of the period of entry and identical across sectors. The as-sumption of an exogenous constant exit rate in adopted for tractability,but it also has empirical support. Using U.S. annual data on manu-facturing, Lee and Mukoyama (2007) find that, while the entry rate isprocyclical, annual exit rates are similar across booms and recessions.Below alternative forms of competition between the firms within each

sector are considered. In particular, the focus is on the traditional mo-nopolistic competition setting and the approach based on oligopolisticcompetition developed by Jaimovich and Floetotto (2008) and Colci-ago and Etro (2010 a and b). As in Ghironi and Melitz (2005) andBGM(2012) who gave new life to an interesting literature on the role ofentry in macroeconomic models, I introduce sunk entry costs to endoge-nize the number of firms in each sector. The nature and the form of the

3As discussed in BGM (2012), if macroeconomic shocks are small enough Nej,t is

positive in every period. New entrants finance entry on the stock market.

4

entry cost will be specified below, where I will also consider alternativespecifications. The household side is standard. They supply labor tofirms and choose how much to save in riskless bonds and in the creationof new firms through the stock market.

2.1 Firms and TechnologyThe final good is produced aggregating a continuum of measure one ofsectoral goods according to the function

Yt =

[∫ 1

0

Yω−1ω

jt dj

] ωω−1

(2)

where Yjt denotes output of sector j and ω is the elasticity of substitu-tion between any two different sectoral goods. The final good producerbehave competitively. In each sector j, there are Njt > 1 firms producingdifferentiated goods that are aggregated into a sectoral good by a CESaggregating function defined as

Yjt =

Njt∑i=1

yjt(i)θ−1θ

θθ−1

(3)

where yjt(i) is the production of good i in sector j, θ > 1 is the elas-ticity of substitution between sectoral goods. As in Colciago and Etro(2010 a), I assume a unit elasticity of substitution between goods be-longing to different sectors. This allows to realistically separate limitedsubstitutability at the aggregated level, and high substitutability at thedisaggregated level. Each firm i in sector j produces a differentiatedgood with the following production function

yjt(i) = Athcjt(i) (4)

where At represents technology which is common across sectors andevolves exogenously over time, while hcjt (i) is the labor input used bythe individual firm for the production of the final good. The unit in-tersectoral elasticity of substitution implies that nominal expenditure,EXPt, is identical across sectors. Thus, the final producer’s demand foreach sectoral good is

PjtYjt = PtYt = EXPt. (5)

where Pjt is the price index of sector j and Pt is the price of the finalgood 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

5

yjt (i) =

(pjtPjt

)−θYjt (6)

where Pjt is defined as

Pjt =

Njt∑i=1

(pjt (i))1−θ

11−θ

(7)

Using (6) and (5) the individual demand of good i can be written as afunction of aggregate expenditure,

yjt (i) =p−θjt

P 1−θjt

EXPt (8)

As technology, the entry cost and the exit probability are identicalacross sectors, in what follows the index j is disregarded to considereda representative sector.

2.2 HouseholdsConsider a representative agent with utility:

U = E0

∞∑t=0

βt

logCt − υ

H1+1/ϕt

1 + 1/ϕ

υ, ϕ ≥ 0 (9)

where β ∈ (0, 1) is the discount factor, Ht are hours worked and Ct isthe consumption of the final good. The representative agent enjoys laborand dividend income. The household maximizes (9) by choosing hoursof work and how much to invest in bonds and risky stocks. The timingof investment in the stock market is as in BGM (2012) and Chugh andGhrioni (2011). At the beginning of period t, the household owns xtshares of a mutual fund of the Nt firms that produce in period t, each ofwhich pays a dividend dt. Denoting with Vt the value of a firm, it followsthat the value of the portfolio held by the household is xtVtNt. Duringperiod t, the household purchases xt+1 shares in a fund of these Nt firmsas well as the N e

t new firms created during period t, to be carried intoperiod t+ 1. Total stock market purchases are thus xt+1Vt(Nt +N e

t ). Atthe very end end of period t, a fraction of these firms disappears fromthe market.4

4Due to the Poisson nature of exit shocks, the household does not know which firmswill disappear from the market, so it finances continued operations of all incumbentfirms as well as those of new entrants.

6

Following production and sales of the Nt varieties in the imperfectlycompetitive goods markets, firms distribute the dividend dt to house-holds. The household’s total dividend income is thus Dt = xtdtNt,which is taxed at the rate τ dt . The variable wt is the market real wage,and τht is the tax rate on labor income. The household’s holdings of thestate-contingent one-period real government bond that pays off in periodt are Bt; and B

jt+1 are end-of-period holdings of government bonds that

pay off in state j in period t+1, which has purchase price 1/Rjt in period

t. The Flow budget constraint of the household is∑j

1

Rjt

Bjt+1+Ct+xt+1Vt(Nt+N

et ) =

(1− τht

)wtHt+Bt+xtVtNt+

(1− τ dt

)xtdtNt

The FOCs (First Order Conditions) for the household problem arerepresented by a standard Euler equation for bonds holdings

1

Ct= βRj

t

1

Cjt+1

an asset pricing equation

Vt = β(1− δ) CtCt+1

Et[(

1− τ dt+1

)πt+1(θ,Nt+1) + Vt+1

](10)

and the condition for optimal labor supply

υCtH1ϕ

t =(1− τht

)wt

2.3 Endogenous EntryUpon entry firms faces a sunk cost, defined as ft. In each period entryis determined endogenously to equate the value of firms to the entrycosts. In what follows I consider two popular forms of the entry costft, defined as Form 1 and Form 2, respectively. Form 1, adopted, interalia, by Jaimovich and Floetotto (2008) and Coto-Martinez et al. (2007),features a constant entry cost measured in units of output, ft = ψ.Form 2, adopted by Bilbiie, Ghironi and Melitz (2007), features an

entry cost equal to η/At units of labor, with η > 0. Notice that, un-der this specification, technology shocks affect the productivity of theworkers that produce goods and also of the workers that create newbusinesses.

2.4 GovernmentThe Government faces an exogenous expenditure stream Gt∞t=0 in realterms. To finance this stream it issues real state contingent bonds, Bj

t ,

7

where the superscript j refer to the state of nature, and collects taxeson labor and dividend income. Its period-by-period budget constraint isgiven by ∑

j

1

Rjt

Bjt+1 + τ t = Gt +Bt (11)

where τ t = τhtwtHt + τ dtNtdt are total tax revenues. The Governmentconsumes the same index of goods faced by the household and opti-mizes the composition of its expenditure across goods.5 Public spendingevolves exogenously over time.

2.5 Strategic InteractionsIn each period, the same expenditure for each sector EXPt is allocatedacross the available goods according to the standard direct demand func-tion derived from the expenditure minimization problem of the house-hold and the Government. It follows that the direct individual demandfaced by a firm, yt (i), can be written as

yt (i) = Yt

(pt (i)

Pt

)−θ=pt (i)−θ

P 1−θt

YtPt =pt (i)−θ EXPt

P 1−θt

i = 1, 2, ..., Nt

(12)Inverting the direct demand functions, the system of inverse demandfunctions can be derived:

pt(i) =yt(i)

− 1θEXPt

Nt∑i=1

yt(i)θ−1θ

i = 1, 2, ..., Nt (13)

which will be useful in the remainder of the analysis. Firms cannotcredibly commit to a sequence of strategies, therefore their behavior isequivalent to maximize current profits in each period taking as giventhe strategies of the other firms. Each good is produced at the constantmarginal cost common to all firms. A main interest of this paper is in theevaluation of the effi ciency of equilibria characterized by popular formsof competition by firms such as competition in prices and quantities.Firms take as given their marginal cost of production and the aggregatenominal expenditure.6 Under different forms of competition we obtain

5Hence It follows that gt (i) =(pt(i)Pt

)−θGt.

6Of course, both of them are endogenous in general equilibrium, but it is reason-able to assume that firms do not perceive marginal cost and aggregate expenditureas affected by their choices.

8

equilibrium prices satisfying

pt (i) = µ(θ,Nt)Wt

At(14)

where Wt

Atis the marginal cost and µ(θ,Nt) > 1 is the markup function.

In the next sections the mark up function under alternative forms ofmarket competition is characterized.

2.5.1 Price Competition

Consider competition in prices. In each period, the gross profits of firmi can be expressed as:

Πt [pt(i)] =

[pt(i)− Wt

At

]pt(i)

−θEXPt[Nt∑j=1

pt (j)1−θ

] (15)

Firms compete by choosing their prices. We consider two alternativeapproaches to this problem. The first one is the traditional monopolis-tic competition approach, which neglects strategic interactions betweenfirms. The second one is the Bertrand approach, where strategic inter-actions are taken into consideration.The outcome of profit maximization under monopolistic competition

is well known. Each firm i chooses the price pt(i) to maximize profitstaking as given the price of the other firms, neglecting the effect of itsprice choice on the sectoral price index. The symmetric equilibriumprice is pt = µMC (θ)Wt/At, which is associated to the constant pricemarkup 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

maximize profits taking as given the price of the other firms. The firstorder condition for any firm i is:

pt(i)−θ − θ

(pt(i)−

Wt

At

)pt(i)

−θ−1 =(1− θ)pt(i)−θ

(pt(i)− Wt

At

)pt(i)

−θ

Nt∑i=1

pt(i)1−θ

Notice that the term on the right hand side is the effect of the price strat-egy of a firm on the price index: higher prices reduce overall demand,therefore firms tend to set higher mark ups compared to monopolisticcompetition. The symmetric equilibrium price pt must satisfy

pt = µB(θ,Nt)Wt

At

9

where the mark up reads as

µB(θ,Nt) =1 + θ(Nt − 1)

(θ − 1)(Nt − 1)(16)

The mark up is decreasing in the degree of substitutability betweenproducts θ and in the number of firms. Importantly, when Nt → ∞ themarkup tends to µMC(θ), the standard one under monopolistic compe-tition.7

2.5.2 Quantity Competition

Consider now competition in quantities in the form of Cournot compe-tition. Using the inverse demand function (13), the profit function of afirm i can be expressed as a function of its output yt(i) and the outputof all the other firms:

Πt [yt(i)] =

[pt(i)−

Wt

At

]y(i) =

=yt(i)

θ−1θ EXPt

Nt∑j=1

yt(j)θ−1θ

− Wtyt(i)

At(17)

Assume now that each firm chooses its production yt(i) taking as giventhe production of the other firms. The first order conditions:(

θ − 1

θ

)yt(i)

− 1θEXPt∑

i yt(i)θ−1θ

−(θ − 1

θ

)yt(i)

θ−2θ EXPt[∑

i yt(i)θ−1θ

]2 =Wt

At

for all firms i = 1, 2, ..., Nt can be simplified imposing symmetry of theCournot equilibrium. This generates the individual output:

yt =(θ − 1)(Nt − 1)AtEXPt

θN2tWt

(18)

7Since total expenditure EXPt is equalized between sectors, we assume that it isalso perceived as given by the firms. Under the alternative hypothesis that the sumbetween public and private consumption, Ct + Gt, is perceived as given, we wouldobtain the higher mark up:

µB(θ,Nt) =θ(Nt − 1)

(θ − 1)(Nt − 1)− 1

which leads to similar qualitative results. This case would correspond to the equilib-rium mark up proposed by Yang and Heijdra (1993).

10

Substituting into the inverse price, one obtains the equilibrium pricept = 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 offirms, the mark up under competition in quantities is always larger thanthe one obtained before under competition in prices, as well known formodels of product differentiation (see for instance Vives, 1999). Noticethat the mark up is decreasing in the degree of substitutability betweenproducts θ and in the number of competitors. In the Cournot equi-librium, the markup remains positive for any degree of substitutability,since even in the case of homogenous goods, we have limθ→∞ µ

Q(θ,Nt) =Nt/(Nt− 1).8 Finally, only when Nt →∞ the markup tends to µMC(θ),the markup under monopolistic competition.

3 Market Equilibrium

This section contains the conditions characterizing the market equilib-rium (ME). Merging the household flow budget constraint with the gov-ernment budget leads to

Y ct +N e

t Vt = wtHt +Ntπt (20)

where Y ct = Ct + Gt denotes the sum between private and public con-

sumption of the final good. Notice that the sum between labor incomeand profits income equals aggregate GDP. The Euler equation for firms’shares reads as

Vt = β(1− δ)Et(Ct+1

Ct

)−1 ((1− τ dt+1

)πt+1 + Vt+1

)(21)

while the set of Euler equations for bond holdings provide the defin-

ition of the stochastic discount factor as βEt(Ct+1Ct

)−1

.The real wage can be derived from the equilibrium pricing relation

as

wt =Atpt

µ(θ,Nt)Pt=ρtµtAt

where in the symmetric equilibrium ρt = ptPt

= N1/(θ−1)t . The first order

condition for labor supply is

υCtH1ϕ

t =(1− τ lt

) ρtµtAt (22)

8This allow to consider the effect of strategic interactions in an otherwise standardsetup with perfect substitute goods within sectors. See Colciago and Etro (2010 b)

11

also we must consider the equation determining the dynamics of thenumber of firms

Nt+1 = (1− δ) (Nt +N et ) (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 entrycost.Form 1. When the latter is measured in, constant, units of output,

the labor input is entirely employed for the production of the final good,thus the clearing of the labor market requires Ht = Nth

ct . The demand

faced by firm i reads as yt = YtNtρt

. In this case firm i’s profits are

πt =

(1− 1

µt

)ρtyt =

(1− 1

µt

)YtNt

Aggregating profits over firms and summing to labor income deliversGDP as ρtAtHt. Since the entry condition is simply Vt = ψ the resourceconstraint (24) becomes

Y ct +N e

t ψ = ρtAtHt (24)

Notice that GDP here coincides with the production of the final good.The Euler equation for firms share translates into

ψ = β(1− δ)Et(Ct+1

Ct

)−1((1− τ dt+1

)(1− 1

µt+1

)YtNt+1

+ ψ

)(25)

Form 2. When the entry cost is measured in units of labor, theeconomy amounts to one which features two sectors, one where N e

tηAt

units of labor are used to produce new firms, the other one where Nthct =

Ht−N etηAtunits of labor are used to produce the final good. This implies

that the set up of a new firm reduces the labor input available for theproduction of the final good. In this setting the individual demand facedby firms i is yt =

Y ctNtρt

and firm level profits can be written as9

πt =

(1− 1

µt

)ρtyt =

(1− 1

µt

)Y ct

Nt

9To see this notice that

Pt (Ct +Gt) =

∫ Nt

0

pityitdi

in a symmetric equilibrium it follows that

Pt (Ct +Gt) = Ntptyt

thus

yt =Ptpt

(Ct +Gt)

Nt=(Ct +Gt)

ρtNt.

12

Aggregating profits over firms and summing to labor income deliversGDP as GDPt =

(1− 1

µt

)Y ct + ρt

µtAtHt. Also recall that the entry

condition implies Vt = ft = ηAtwt = η ρt

µt. In this case, the resource

constraint reads asY ct +N e

t ηρt = ρtAtHt (26)

The Euler equation (21) for the value of the firm reduces, instead, to

ηρtµt

= β(1− δ)Et(Ct+1

Ct

)−1((1− τ dt+1

)(1− 1

µt+1

)Y ct+1

Nt+1

+ ηρt+1

µt+1

)(27)

Definition 1 (Market Equilibrium) Given the exogenous processesAt, Gt∞t=0 and processes

τ dt , τ

lt

∞t=0, the Market Equilibrium (ME) con-

sists of an allocation Ct, Ht, Nt, Net , Bt+1∞t=0 which satisfies the first

order condition for labor supply, equation (22), the Government budget,equation (11), the dynamics of the number of firms, equation (23), thedefinition of the price markup and the definition of the love for variety towhich we must add the resource constraint (24) and the Euler equation(25) in the case of entry costs in Form 1, and the resource constraint(26) together with the Euler equation (27) in the case of entry costs inForm 2.

4 Effi cient Equilibrium

This section outlines a scenario where a benevolent Social Planner (SP)maximizes households’lifetime utility by choosing quantity directly. Indoing this, the SP is subject to the same technological constraints de-scribed in the previous sections. The SP maximizes (9) with respect toCt, Nt+1, N

et , Ht∞t=0. The choice is subject to two constraints. The first

one is given by the dynamics of the number of firms, equation (23), thesecond one is the resource constraint, which is represented by equation(26) in the case of Form 2 of the entry costs and by (24) when the en-try cost is measured in terms of output. The social planner takes intoaccount 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 isindependent of the form of the entry cost and reads as

χCtH1ϕ

t = ρtAt (28)

this condition simply states that the Planner equates the marginalrate of substitution between hours and consumption, the left hand side,to the marginal product of labor, the right hand side, which depends onthe number of varieties in the economy.

13

The Planner’s Euler equation depends instead on the form of theentry cost. In the case of Form 1, it reads as

1

Ctψ = βEt (1− δ) 1

Ct+1

[ε (Nt+1)

Yt+1

Nt+1

+ ψ

](29)

where ε (Nt+1) =ρN,t+1Nt+1

ρt+1is the benefit of variety in elasticity form

and ρN,t+1 is the derivative of the benefit of variety with respect to thenumber of firms. Under CES preferences ε (Nt+1) is a constant equal to

1ε−1, hence, in the remainder, we simply denote it with ε.Condition (29) states that the social planner will create firms up to

the point where the utility cost of creating a new firm, 1Ctψ, equals the

benefit from the creation of an additional variety. The latter is givenby the discounted utility value of the sum between the additional utilitythat obtains from increasing time t+1 number of varieties measured inunits of the final good, namely ε Yt+1

Nt+1, and the continuation value of the

marginal firm, that is ψ.10

The planner’s Euler equation in the case of entry costs in form 2reads as

ηρt = βEt (1− δ) CtCt+1

[εY ct+1

Nt+1

+ ηρt+1

](30)

The interpretation of equation (30) is similar to that provided forequation (29), with two differences. The first one is that the value of anew firm in terms of the final good is not constant over time (and statesof nature), but depends on the benefit of variety ρt. The reason is thatthe love for variety affects the marginal productivity of labor and thusthe opportunity cost of creating a new firm in terms of the final good.This also implies that whenever the number of varieties differs from theeffi cient one the value of a firm is distorted with respect to the effi cientone. The second difference is that at time t+1 the creation of N e

t+1

new firms requires ηAtN et+1 units of labor, which are diverted from the

production of the final good. As a result the additional output that themarginal firm creates depends on the sum between private and publicconsumption, Y c, and not on GDP.

Definition 2 (Effi cient Equilibrium) The Effi cient Equilibrium con-sists of an allocation Ct, ht, Nt, N

et ∞t=0 satisfying the dynamics of the

number of firms, equation (23), the first order condition for labor supply,equation (28), together with equations (26) and (30) in the case of entrycost in terms of labor, and equations (24) and (29) in the case of entrycosts in terms of output, for given N0 and At, Gt∞t=0.

10Notice that ε (Nt+1) is an elasticity, thus is a pure number.

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5 Market Distortions

The market allocation features two distortions. To identify them it isconvenient to set, for the moment being, fiscal instruments to zero, τ dt =τht = 0. In the next sections we reintroduce fiscal instruments and designthem in order to minimize the welfare losses associated to the distortionsthat we are about to discuss.The first distortion is reffered to as to the Labor Distortion. In the

competitive equilibrium labor is supplied up to the point that the fol-lowing condition is satisfied

χCtH1/ϕt =

ρtµtAt (31)

A comparison between equation (28) and equation (31) reveals that inthe decentralized equilibrium the marginal rate of substitution betweenhours and consumption, χCth

1/ϕt , is lower than the marginal rate of

transformation between hours and output, ρtAt. As in other modelswith an imperfectly competitive goods market, as for example in BGM(2012), this wedge is due to the presence of a price markup. Oligopolisticcompetition renders this wedge time varying.The second distortion involved in the decentralized allocation is an

Entry Distortion. This wedge operates through the intertemporal firmscreation margin and could lead to an ineffi cient number of firms in equi-librium. The number of firms could be ineffi cient for the following reason,as argued by Coto Martinez et al. (2007). The entry decision of the in-dividual firm ignores the welfare gains associated to increased variety.This leads to ineffi ciently low entry. Also the potential entrant ignoresthe negative effects of increased competition on the profits of other firmsalready in the market. The latter, known as business steasling effect,leads to excessive entry. As a result the equilibrium number of firmscould be either inneffi ciently high or low, depending on which external-ity predominates.To illustrate the entry distortion I will compare the Euler equations

in the decentralized economy with those obtained in the effi cient equi-librium. Again, it is convenient to distinguish according to the form ofthe entry cost and to consider a non-stochastic version of the economy.Form 1. Consider the right hand side of the Euler equation in the

decentralized equilibrium, equation (25). The latter equals the righthand side of the corresponding equation in the centralized equilibrium,equation (29), if (

1− 1

µt+1

)= ε (32)

15

Thus, effi ciency holds if the so called Lerner index equals the benefit ofvariety in elasticity form.11 In this case the private incentive to enterinto the market,

(1− 1

µt+1

), is identical to the social one, and the two

externality 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−1the condition above requires the price markup to be

constant over time and states of nature, µt = µ. As a result it cannotbe satisfied under oligopolistic competition. Notice also that, althoughmonopolistic competition leads to a constant markup µ = θ

θ−1, the con-

dition for effi ciency is not satisfied since 1− 1µ< ε.

Form 2. Comparing the Euler equation in the decentralized equilib-rium, equation (27) with the corresponding Euler equation in the SocialPlanner equilibrium, equation (30), shows that two conditions need tobe imposed to reinstate effi ciency, namely

µt

(1− 1

µt+1

)= ε, (33)

andµtµt+1

= 1.

Combining conditions (33) and (??) we recover those emphasized byBGM (2007), that is

ε = µt+1 − 1 andµtµt+1

= 1

which imply that the price mark up should be constant over timeand that the benefit of variety in elasticity form should equal the marketpower as measured by the net markup. Under oligopolistic competitionboth conditions fail. Contrary to the previous case, these conditions aresatisfied under monopolistic competition, as emphasized by BGM (2007)and Chugh and Ghironi (2011). Thus, under monopolistic competitionwith entry costs in form 2 the entry margin is not distorted.Notice that the economic environments in Chugh and Ghironi (2011)

and Coto Martinez at al. (2007), can be considered as special cases of

11The Lerner Index is defined as follows

LI =p (i)−MC (i)

p (i)=

p(i)P −

MC(i)P

p(i)P

=

=ρ− ρ

µ

ρ= 1− 1

µ

16

the one outlined here. In particular, the present setup collapses to thatconsidered by Coto Martinez et al. (2007) when entry costs are in form 1and the strategic interactions between firms are neglected. It coincides,instead, with that considered by Chugh and Ghironi (2011) under entrycosts in form 2 and strategic interaction are neglected.The next section outlines the fiscal policy aimed at minimizing wel-

fare losses due to the market distortions when the Government can raiserevenues solely by imposing distortionary taxes on labor and dividendsand issuing state contingent bonds.

6 Calibration

Since part of the following analysis is numerical, the calibration of struc-tural parameters follows. The time unit is meant to be a quarter. Thediscount 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 U.S.empirical level of 10 per cent business destruction a year. Steady stateproductivity is equal to A = 1. The baseline value for the entry cost isset to η = ψ = 1. This leads to a share of investment in GDP around12 percent in our models.The baseline value for the intrasectoral elasticity of substitution is

θ = 6, which is in line with the typical calibration for monopolisticcompetition and delivers markups levels belonging to the empiricallyrelevant range.12

Turning to fiscal parameters, the ratio of Government spending overGDP equals 0.22, as estimated by Schmitt-Grohè and Uribe (2005). Forthe sake of calibration, we set the steady state labor income tax to 20percent and the steady state dividend income tax to 30 percent andassume that the government has lump sum taxes available to balanceits budget. These values represent the mean labor income tax rate andthe mean dividend income tax rate in the US over the period 1947:Q1-2009:Q4, as reported by Chugh and Ghironi (2011). When computingthe Ramsey steady state, and Ramsey optimal policy, we assume thatlump sum taxes are not available and fix the ratio of Government debtto output equals 0.5 on an annual basis, in line with the U.S. postwaraverage.In what follows equilibrium allocations under Cournot, Bertrand and

Monopolistic Competition for each entry cost configuration are com-

12Oliveira Martins and Scarpetta (1999) provide estimates of price mark ups forUS manufacturing industries over the period 1970-1992. In broad terms most of thesectoral markups defined over value added are in the range 30-60 per cent, whilewhen defined over gross output they are in the range 5-25 per cent. In the lattercase, high mark ups, over 40 per cent, are observed in few sectors.

17

pared. This is done holding parameters fixed, in order to understandthe role of the different market structures. To this end, the followingcalibration strategy for the utility parameter υ is adopted. The value ofυ is such that steady state labor supply is equal to one under monopolis-tic competition. In this case the Frish elasticity of labor supply reducesto ϕ, to which we assign a value of four as in King and Rebelo (2000).Next the values of υ and ϕ are held constant under both Bertrand andCournot competition.There are two exogenous processes in the economy, that for Govern-

ment spending and that for technology. They are both assumed to beAR (1) processes in log deviations from the steady state:

logGt

G= ρg log

Gt−1

G+ εgt

logAtA

= ρa logAt−1

A+ εat

The autoregressive coeffi cient for the technology process is ρa = 0.979and the standard deviation of the disturbance, σa, is 0.0072, as in theRBC model by King and Rebelo (2000). The parameterization of theGovernment spending process follows Chari and Kehoe (1999) in settingρg = 0.97 and σg = 0.027.

7 Ramsey Optimal Fiscal policy

In this section we study the second best tax policy in an economy withoutlump sum taxes. Consider the Euler equation for firms’shares, which inits general form reads as

ft = β(1− δ)Et(Ct+1

Ct

)−1 ((1− τ dt+1

)πt+1 + ft+1

)Expected future dividend income taxes cannot be removed from this

equation using other equilibrium conditions. As a result a pure primalapproach cannot be applied to solve for the optimal policy. As empha-sized by Chugh and Ghironi (2011) the set of allocations that the Plannercan select cannot uniquely be characterized by means of the so calledimplementability constraint. Further, computing a first order conditionwith respect to Etτ dt+1 would leave τ

dt+1 indeterminate from the point of

view of time-t.To resolve these indeterminacy issue I adopt the solution proposed by

Chugh and Ghironi (2011). In particular, it is assumed that the plannerchooses a state contingent schedule for the time t+1 dividend income taxrate τ dj,t+1,where j indexes the state of the economy. This schedule is in

18

the time t information set. Also we assume that the Planner commits tothe schedule, meaning that the state contingent tax rate is implementedwith certainty at time t+1.13

Definition 3 (Ramsey Equilibrium) Given the government expen-diture Gt∞t=0 and the initial conditions b0, N0 the allocations asso-ciated to the optimal fiscal policy

τht , τ

dj,t+1

∞t=0

are derived by solving

maxE0

∞∑t=0

βt

logCt − υ

h1+1/ϕt

1 + 1/ϕ

The choice variables are Ct, Nt, Ht, N et , and τ

dt/t+1. The allocation is

restricted by four constraints, two of them depend on the form of theentry cost. The constraints which are independent of the form of theentry cost are equation (23), determining the dynamic of the number offirms, and the implementability constraint, which reads as

E0

∞∑t=0

βt[1− υh

1ϕ

t

]=

b0

C0

+1

C0

[(1− τ d0

)d0 + V0

]N0s0

The allocation is further restricted by equations (24) and (25) in the caseof entry cost of form 1 and by equations (26) and (27) in the case of entrycost in form 2.

The first order conditions for the Ramsey problem are reported inthe Appendix, for both forms of the entry cost. To compute the Ramseyallocation I follow Linneman and Shabert (2011). Namely, throughoutthe analysis, it is assumed that the policy maker can credibly commithimself, but the initial period (t = 0) is ignored. In deriving the Ramseypolicy, the problem that the policy maker’s decision rules will be differ-ent for the first period in which the policy is implemented is neglected.This is justified by the fact that the interest is that of making state-ments about the deterministic steady state as well as about businesscycle fluctuations around it, while the transition path from the initialvalues towards the steady state is not analyzed. As it is common inthe Ramsey literature, see Khan, King and Wolman (2003), it is furtherassumed that the initial values of the predetermined variables are equalto their values in the deterministic Ramsey steady state. This amountto adopt to so called Timeless Perspective, popularized by Woodford(2003).

13In other words τdt+1 is chosen at time t. The selected value is then implementedwith certainty at time t+1.

19

Dynamics in response to shocks under the Ramsey policy are ob-tained by solving a first order approximation to the Ramsey first orderconditions. As shown in various contributions by Schmitt-Grohè andUribe, in models characterized by more frictions than the present one, afirst order approximation to the first order conditions delivers dynamicsthat are very close to the exact ones.

7.1 Ramsey Steady StateBy using the dividend income tax, the Ramsey Planner removes theineffi ciency along the entry margin. This result has first been identifiedby Chugh and Ghironi (2011). In what follows it is shown that the levelof the optimal dividend income tax rate differs according to the form ofcompetition and to the form of the entry cost. Proposition 1 providesthe main result of this section.

Proposition 4 (Optimal long-run dividend income taxes) Optimallong-run dividend income taxes depend on the form of the entry costs

(i) Form 1: τ d = 1− ε

1− 1µ

(ii) Form 2: τ d = 1− ε

µ− 1

Proof. In the Appendix.

Consider condition (i). As mentioned above when the private in-centive to create a new firm, as measured by the Lerner Index, that is(

1− 1µ

), 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.Notice that the number of firms is not the optimal one even if the in-effi ciency along the entry margin is removed. The reason is that theRamsey Planner cannot remove the labor distortion. As a result steadystate hours will be lower in the Ramsey steady state with respect to thethose observed in the long run allocation reached by the Social Plan-ner. In other words the Ramsey Planner disregards the social cost ofthe labor distortion. However, as shown below, the Planner implementsthe effi cient level of units of effective labor per firm, ρH

N. Notice that

in the standard neoclassical growth model the Ramsey Planner wouldtarget the effi cient ratio between capital and hours. This further empha-sizes the analogy between the stock of capital in the neoclassical modeland the stock of firms in the economy we have outlined. The optimaldividend income tax rate under Bertrand competition is

20

0.5 1 1.50

20

40Number of firms

0.5 1 1.51

1.5

2Hours

0.5 1 1.50

0.2

0.4Units of effective labor per firm

0.5 1 1.50

0.2

0.4Tax rate on labor income

0.5 1 1.5­0.5

0

0.5Tax rate on dividend income

Cournot Bertrand Social Planner Monopolistic

Figure 1: Ramsey Steady State under Entry Costs in Form 1. Entrycost (ψ) on the horizontal axis.

τ dBertrand =1

N− 1

θ − 1

while under Cournot competition we obtain

τ dCournot =1

N− 1

(θ − 1)2

Notice that the optimal tax rate in the case of monopolistic competitionis

τ dMonopolistic = − 1

θ − 1,

Under oligopolistic competition the optimal dividend income tax couldtake the form of a subsidy, depending on the parameterization of themodel. On the contrary, removing the entry distortion under monopo-listic competition requires a subsidy no matter the parameterization ofthe model. This means that monopolistic competition always leads to asuboptimal steady state number of firms. Figure 3 displays the Ramseysteady state number of firms, hours, output per firm and optimal taxrates as a function of the entry cost under Bertrand, Cournot and mo-nopolistic competition. Allocations are compared to the effi cient ones.First, for any given value of the entry cost, the Ramsey Planner im-plements the same allocation for hours and the number of firms acrossmarket structures.These differ from the effi cient ones. As mentioned above, since the

Planner cannot remove the labor distortion, hours are too low with re-spect to the effi cient level, this implies lower output and thus less firms

21

0.5 1 1.50

5

10

15Number of firms

0.5 1 1.50.5

1

1.5Hours

0.5 1 1.50

0.2

0.4Output per firm

0.5 1 1.5­0.5

0

0.5Tax rate on labor income

0.5 1 1.50

0.5

1Tax rate on div idend income

Cournot Bertrand Social Planner Monopolis tic

Figure 2: Ramsey Steady State under Entry Costs in Form 2. Entrycost (η) on the horizontal axis.

with respect to the effi cient level. The Ramsey Planner targets the effi -cient level of units of effective labor per firm.14 This, however, requiresa different combination of optimal taxes across market structures. Inthe case of Cournot competition the market allocation would lead toan ineffi ciently large number of firms, as a result restoring effi ciencyalong the entry margin requires a positive dividend income tax. Onthe contrary, as in Coto-Martinez et al. (2007), the number of firmsunder monopolistic competition would be too low and dividend incomeshould be subsidized to promote entry. Bertrand competition falls be-tween these two case. As stated by Vives (1984), Bertrand competitioncan be regarded as a more competitive market structure with respect toCournot and for this reason it is judged as more effi cient.15 This analysissuggests that, for given entry costs, the dividend income tax should belower in markets characterized by more competitive market structures.The opposite instead holds for the labor income tax. Due to the needof financing the exogenous level of public expenditure the optimal laborincome tax is instead higher under monopolistic competition, where div-idend income was more heavily subsidized, with respect to other marketstructures.14This is in anology with what would happen in an RBC model where the Planner

targets the ratio K/L.15Vives (1984) provide the following intuitive explanation to support this view. In

Cournot competition each firm expects the others to cut prices in response to pricecuts, while in Bertrand competition the firm expects the others to maintain theirprices; therfore Cournot penalizes price cutting more. One should expect Cournotprices to be higher than Bertrand prices.

22

Condition (ii) is isomorphic to that obtained by BGM (2007) andChugh and Ghironi (2011) under the case of monopolistic competition.In this case profits should be taxed whenever the net markup exceedsthe benefit of variety in elasticity form. However, under monopolisticcompetition condition (ii) is automatically satisfied, since the benefitof variety in elasticity form equals the net markup and markups areconstant, and the market equilibrium displays effi ciency along the entrymargin. This is not the case under oligopolistic competition. UnderBertrand competition the optimal tax rate is

τ dBertrand =1

N

while under Cournot competition is

τ dCournot =θ

N + θ − 1

Hence entry costs in form 2 always imply a positive dividend incometax rate under oligopolistic competition. Figure 4 displays the RamseySteady State number of firms and hours together with optimal tax rateas a function of the entry cost under entry cost in form 2. In thiscase allocations slightly differ across market structures. Hours and thenumber of firms are lower with respect to their effi cient counterpart, butoutput per firm is equal to its effi cient level. It is interesting to notethat under Cournot competition the tax rate on labor income is alwayslower than that on dividend income.16

7.2 Ramsey Dynamics and Optimal Tax VolatilityThis section shows the business cycle implications of the Ramsey optimalpolicy. As mentioned above, the dynamics under optimal policy are ob-tained by solving a first order approximation to the Ramsey equilibriumconditions.16As 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− τst ) ft. It can beshown that optimal subsidies depend on the form of the entry cost as follows

(i) Form 1: τs = 1−1− 1

µ

ε

(ii) Form 2: τs = 1− µ− 1ε

The Ramsey Planner will resort to an entry tax in the case of excessive entry or toa subsidy in the case of ineffi ciently low entry.

23

0 20 400

0.5

1

1.5GDP

0 20 40­0.5

0

0.5Hours

0 20 400

2

4

6New entrants

0 20 400

0.5

1Number of firms

1 20 40­0.2

­0.1

0Markup

0 20 400

0.5

1

1.5Aggregate profits

Effic ient Dynamics Bertrand Cournot Monopolis tic Competition

0 20 400

0.5

1Consumption

0 20 40­0.05

0

0.05

0.1Labor income tax

0 20 40­0.5

0

0.5Dividend income tax

Figure 3: Entry Costs in form 1. Response of the main macroeconomicvariables to a one standard deviation shock to technology. Solid linesrefer to the social planner allocation, dashed and dash-dotted lines re-fer to Ramsey dynamics under Bertrand and the Cournot respectivelyand dotted lines to the case of Ramsey dynamics under monopolisticcompetition.

Figures 3 depicts percentage deviations from the steady state of keyvariables in response to a one standard deviation technology shock un-der entry cost in form 1. For tax rates we report deviations from thesteady state level in percentage points. Time on the horizontal axis isin quarters. Solid lines refer to the Effi cient (Social Planner) allocation,dashed and dash-dotted lines refer to Ramsey dynamics under Bertrandand the Cournot competition respectively, and dotted lines to the caseof Ramsey dynamics under monopolistic competition. The technologyshock creates expectations of future profits which lead to the entry ofnew firms in the market. This is so under both the Ramsey and theeffi cient equilibrium. Under all market structures the dynamics of out-put, consumption, hours and the number of firms are very close to theeffi cient ones. In the non stochastic economy we showed that a neces-sary condition for effi ciency is the costancy of the price markup. Thefollowing arguments suggests that a similar result applied to the sto-chastic case. Under monopolistic competition the markup is constantalong the business cycle. As can be seen from Figure 3, this results inconstant tax rates. This is not the case under Oligopolistic competition.Under Bertrand and Cournot competition the entry of new firms leadsto higher competition which, in turn, leads to a countercyclical pricemarkup. The price markup variability entails a deviation from optimal

24

0 20 400

0.5

1

1.5GDP

0 20 40­0.5

0

0.5Hours

0 20 400

0.5

1Consumption

0 20 400

2

4

6New entrants

0 20 400

0.5

1Number of firms

0 20 40­0.2

­0.1

0Markup

0 20 400

0.5

1Aggregate profits

0 20 40

0

0.1

0.2Labor income tax

0 20 40­0.5

0

0.5Dividend income tax

Effic ient Dynamics Bertrand Cournot Monopolis tic Competition

Figure 4: Entry Costs in form 2. Response of the main macroeconomicvariables to a one standard deviation shock to technology. Solid linesrefer to the social planner allocation, dashed and dash-dotted lines re-fer to Ramsey dynamics under Bertrand and the Cournot respectivelyand dotted lines to the case of Ramsey dynamics under monopolisticcompetition.

dynamics which is offset by the Ramsey Planner adjusting the tax rates.In particular, we observe an increase in the labor income tax coupledwith a decrease in the dividend income tax.Changes in the tax rates are mild, but stronger under Cournot com-

petition where the markup is characterize by a higher elasticity to thenumber of firms with respect to Bertrand. The Ramsey policy underentry costs in Form 1 is, thus, characterized by a countercyclical laborincome tax rate and by a procyclical dividend income tax.Figure 4 displays the percentage changes in response to a technol-

ogy shock under entry costs in form 2. Lines have the same meaningas in the Figure 3. Previous considerations extend to this case. A rele-vant difference is the impact increase in the dividend income tax, whichis reverted after few periods. Notice that similar dynamics, althoughquantitatively less sizeable, can be observed in the case of a governmentspending shock.17

Next the variability of the main macroeconomic variables in responseto the technology and government spending shocks is analyzed. Table1 displays the mean (x), the coeffi cient of variation (σ (x) /x) and thecorrelation with output (Cor(Y t, xt)) of a number of variables of interest

17Notice, however, that aggregate consumption drops in response to a Governemtspending shock under all the market structures considered.

25

Y C H N Ne τh τ d

Monopolistic competitionx 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 0Cor(Y t, xt) 1 0.60 0.87 0.4 0.95 0 0

Bertrand Competitionx 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 0.07Cor(Y t, xt) 1 0.60 0.87 0.4 0.95 0.11 -0.4

Cournot Competitionx 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 0.17Cor(Y t, xt) 1 0.60 0.87 0.4 0.95 0.12 -0.4

Table 1: Mean, Standard deviations and correlations with output ofmain-macro variables under alternative market structures. Shocks areto productivity and Government spending. Entry Costs in Form 1

under the Ramsey dividend and labor income taxation policy, underentry costs in form 1. For tax rates it is reported the standard deviationin percentage points (σ (x)). Table 2 reports the same statistics for thecase of entry costs in form 2.Under entry cost in form 1 the variability of the main macroeconomic

variables under the optimal policy is identical across market structures.However, this is reached by means of a different fiscal policy. As expectedfrom Figure 3, while under monopolistic competition taxes are constantthis is not the case under oligopolistic competition. Tax rates are morevolatile under Cournot competition with respect to Bertrand, with thedividend income tax more variable than the labor income tax rate. Recallthat the elasticity of the price markup to the number of firms is higherunder Cournot. As a result minimizing the welfare cost of the distortionsover the business cycle requires more variable taxes when firms competein quantities.Under entry costs in form 2 allocations and volatilities are no longer

identical across market structures. Interestingly, while the overall vari-ability characterizing the economy in response to shocks, as measured bythe 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 par-ticular the standard deviation of the dividend income tax under Cournotcompetition is sizeable.

26

x Y C H N Ne τh τ d

Monopolistic competitionx 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 0Cor(Y t, xt) 1 0.60 0.85 0.50 0.93 0 0

Bertrand Competitionx 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 0.26Cor(Y t, xt) 1 0.60 0.85 0.50 0.93 0.21 0.60

Cournot Competitionx 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 0.59Cor(Y t, xt) 1 0.60 0.85 0.53 0.91 0.24 0.72

Table 2: Mean, Standard deviations and correlations with output ofmain-macro variables under alternative market structures. Shocks areto productivity and Government spending. Entry costs in form 2

8 Conclusions

This paper proposes an economy where the degree of market power, asmeasured by the price markup, depends endogenously on the form ofcompetition, on the degree of substitutability between goods and on thenumber of firms. Imperfect competition leads to distortions in both thegoods and the labor market and in both the short and the long run. Theoptimal long run dividend income corrects for ineffi cient entry, and it ishigher in market structures characterized by lower competition. In par-ticular it is higher under Cournot Competition with respect to Bertrandor monopolistic competition. Whereas optimal taxes over the businesscycle are constant under monopolistic competition, this is not the casein an oligopolistic market structure. Also the effect of alternative formsof sunk entry costs for the design of optimal taxation has been consid-ered. The resulting framework features as special cases two models inthe entry literature which also focus on optimal taxation problems inthe case of an endogenous dynamics of the number of firms. Coto Mar-tinez et al (2007) consider an environment characterized by monopolisticcompetition under constant sunk entry costs. Chugh and Ghironi (2011)consider a framework with monopolistic competition and sunk entry costin terms of labor. By neglecting strategic interactions and consideringthe appropriate form of the entry costs our model reduces to either oneof these models. For this reason it can be regarded as a general frame-work where to study optimal taxation problems under various form of

27

imperfect competition.

28

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and Optimum Product Diversity: Comment, The American EconomicReview, 83, 1, 295-301.Woodford, M. (2003). Interest and Prices: Foundations of a Theory

of Monetary Policy.Princeton: Princeton University Press.

Appendix

Appendix A. Steady State of the Market Equilib-riumEntry Costs in Form 1

The steady state number of entrants is

N e =δ

(1− δ)N

30

The Euler equation for shares implies

V =β(1− δ)

(1− τ d

)[1− β(1− δ)] π

Steady state profits are given by18

π = ρy − wh = ρ

(1− 1

µ

)y = ρ

(1− 1

µ

)Y

Nρ=

(1− 1

µ

)Y

N,

hence the share of profits over output reads as

πN

Y= 1− 1

µ

and the value of firms over output is

NV

Y=β(1− δ)

(1− τ d

)[1− β(1− δ)]

(1− 1

µ

)Investment over output is

V N e

Y=V

Y

δ

(1− δ)N =V N

Y

δ

(1− δ) = δβ(1− τ d

) (1− 1

µ

)[1− β(1− δ)]

The share of consumption over output is

C

Y= 1− gy −

N eψ

Y

Finally to get the ratio of labor income over output recall that

wH

Y= 1− πN

Y

In order to fix υ we assume that H=1.19 In this case

υ =(1− τh

) wYCY

where both ratios are known. To compute the number of firms noticethat

V =β(1− δ)

(1− τ d

)[1− β(1− δ)] π

18Notice that this is the main difference wrt to cost 2 since in that case profitsdepend on Y c.19As mentioned in the section on calibration I fix H=1 under monopolistic com-

petition and obtain the corresponding value of υ. Under oligopolistic competition Iconsider the value of υ so obtained and compute the corresponding value of H.

31

Imposing the entry condition and substituting for individual profits

N =β(1− δ)

(1− τ d

) (1− 1

µ

)ρ

(1− τ s) [1− β(1− δ)]ψ AH

the solution to this equation delivers the number of firms at the steadystate. This allows to compute all the other variables. For a given Hthe number of firms at the steady state is larger the higher the markup,hence is larger under oligopolistic competition.

Entry Costs in Form 2

In this case the steady state level of individual profits is

π = ρy−wh =(ρ− w

A

)y = ρ

(1− 1

µ

)y =

(1− 1

µ

)(C +G)

N=

(1− 1

µ

)Y c

N

As a resultπN

Y c=

(1− 1

µ

)To obtain the share of investment over consumption output notice that

V =β(1− δ)

(1− τ d

)[1− β(1− δ)]

(1− 1

µ

)Y c

N

and

V N e

Y c=β(1− δ)

(1− τ d

)[1− β(1− δ)]

(1− 1

µ

)N e

N=(1− τ d

) δβ

[1− β(1− δ)]

(1− 1

µ

)=(1− τ d

) δ (µ− 1)

µ (r + δ)

To compute shares over aggregate output recall that

1 =Y c

Y+N eV

Y c

Y c

Y=Y c

Y

[1 + δ

β(1− τ d

)[1− β(1− δ)]

(1− 1

µ

)]

thusY c

Y=

[1 +

(1− τ d

)δ

β

[1− β(1− δ)]

(1− 1

µ

)]−1

which implies that the share of private consumption over output is

C

Y=Y c

Y− gy

32

Since WHY

+ ΠY

= 1 we can compute the ratio between labor income andGDP as

WH

Y= 1− πN

Y c

Y c

Y

Given H, the latter leads to

υ =

(1− τh

)wH

CH1+ 1ϕ

=

(1− τh

)wHY

CYH1+ 1

ϕ

Labor market equilibrium requires

H =HCt +HE

t = Nh+η

AN e = N

y

A+η

AN e

=Y c

ρA+η

AN e

thus

N e =AH

η− Y c

ρη

and

N = (1− δ)(N +

AH

η− Y c

ηρ

)or

N =(1− δ)δ

AH

η− (1− δ)

δ

Y c

ηρ

Next consider

V =β(1− δ)

(1− τ d

)[1− β(1− δ)]

(1− 1

µ

)Y c

N

substituting for the entry condition delivers

Y c

ηρ=

(1− τ s)µ

[1− β(1− δ)]β(1− δ) (1− τ d)

(1− 1

µ

)NSubstituting the latter into the equation of motion for the number of

firms delivers an equation that can be solved for N

N =

(1−δ)δ

AHη(

1 + (1−τs)δ

[1−β(1−δ)]β(1−τd)(µ−1)

)Aa above a higher markups leads to a higher number of firms in equilib-rium for any given H.

33

Appendix B. Effi cient EquilibriumEntry Costs in Form 1

The social Planner problem reads as

maxCt,Nt+1,Ne

t ,Ht∞t=0

E0

∞∑t=0

βt

logCt − υ

H1+1/ϕt

1 + 1/ϕ

s.t.

Ct +Gt +N et ψ = ρtAtHt

andNt+1 = (1− δ) (Nt +N e

t )

We attach the Lagrange Multiplier λt to the first constraint and themultiplier σt to the second one. First order conditions are as follows

Ct :1

Ct= λt

Nt+1 : σt = βEtλt+1ρN,t+1AtHt + βEtσt+1 (1− δ)N et : λtψ = (1− δ)σt

Ht : υH1/ϕt = λtρtAt

Combining the first and the third condition delivers

1

Ct

ψ

(1− δ) = σt

Substituting the latter into the third condition we obtain

ψ = (1− δ) βEtCtCt+1

(ρN,t+1AtHt + ψ

)which can be written as

ψ = (1− δ) βEtCtCt+1

(εYtNt

+ ψ

)finally the FOC with respect to hours can be written as

υH1/ϕt Ct = ρtAt

To obtain the steady state we have the following equations

C +G+N eψ = ρAH

N = (1− δ) (N +N e)

34

ψ = (1− δ) β(εY

N+ ψ

)υH1/ϕC = ρA

Consider the resource constraint

C

Y= 1− gy −

N eψ

Y= 1− gy −

ψ

Y

δ

1− δN

orC

Y= 1− gy − ψ

δ

1− δN

Y

From the third equation

Y

N=

(1− (1− δ) β)ψ

(1− δ) βε

Combining we obtain

C

Y= 1− gy −

δ

1− δ(1− δ) βε

(1− (1− δ) β)

Next consider equation

H1+1/ϕ =1

υCρAH =

Y

υC

hence we have H as

H =

(Y

υC

) 11+1/ϕ

Next we want to compute N. Notice that

N

ρ=

(1− δ) βε(1− (1− δ) β)ψ

AH

given ρ = N1θ−1 it follows

Nθ−2θ−1 =

(1− δ) βε(1− (1− δ) β)ψ

AH

or

N =

[(1− δ) βε

(1− (1− δ) β)ψAH

] θ−1θ−2

35

Entry Costs in form 2

The Social Planner problem can be written as follows

maxCt,Nt+1,Ne

t ,Ht∞t=0

E0

∞∑t=0

βt

logCt − υ

H1+1/ϕt

1 + 1/ϕ

s.t.

Ct +Gt +N et ηρt = ρtAtHt

andNt+1 = (1− δ) (Nt +N e

t )

We attach the Lagrange Multiplier λt to the first constraint and themultiplier σt to the second one. First order conditions are as follows

Ct :1

Ct= λt

Nt+1 : σt = βEtλt+1ρN,t+1 (AtHt −N et η) + βEtσt+1 (1− δ)

N et : λtηρt = (1− δ)σtHt : υH

1/ϕt = 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 equationswe are left with

ηρt = (1− δ) βEtCtCt+1

[ρN,t+1 (AtHt −N e

t η) + ηρt+1

](34)

andυH

1/ϕt Ct = ρtAt

Since AtHt −N et η = Ct+Gt

ρt, equation (34) can be rewritten as

ηρt = (1− δ) βEtCtCt+1

[ρN,t+1

Ct +Gt

ρt+ ηρt+1

]or

ηρt = (1− δ) βEtCtCt+1

[εCt+1 +Gt+1

Nt+1

+ ηρt+1

]To find the steady state we can consider the following equations

ηρ = (1− δ) β[εY c

N+ ηρ

]36

υH1/ϕC = ρA

Y c +N eηρ = ρAH

N = (1− δ) (N +N e)

From the first one

Y c

ηρ=

(1− (1− δ) β)

(1− δ) βε N

The aggregate resource constraint implies

Y c

ηρ=AH

η− δ

1− δN

Combining

N =

AHη

(1−(1−δ)β)(1−δ)βε + δ

1−δ

we get N as a function of H. Notice that we have repeatedly used thesteady state version of the equation of motion for the number of firms.Consider again the aggregate resource constraint

C +G+N eηρ = ρAH

orC

ρAH= 1− gy −

N eηρ

ρAH= 1− gy − ηρ

δ

1− δN

ρAH

then

υH1/ϕ C

ρAH=

ρA

ρAH

delivers H implicitly as a function of N

υH1/ϕ

(1− gy − ηρ

δ

1− δN

ρAH

)=

1

H

The latter is equivalent to

υH1+1/ϕ

(1− gy − ηρ

δ

1− δN

ρAH

)= 1

Next substitute for N as a function of H in the round bracket and

H =

υ

[1− gy −

δ

1− δ

((1− (1− δ) β)

(1− δ) βε +δ

1− δ

)−1]− ϕ

(1+ϕ)

37

Appendix C. The Implementability ConstraintThis Appendix follows closely Arsenau and Chugh (2012) and Chughand Ghironi (2012). Consider the household flow budget constraint (inthe symmetric equilibrium)∑j

1

Rjt

Bjt+1+Vt(Nt+N

et )xt+1+Ct =

(1− τ lt

)wtHt+Bt+

[(1− τ dt

)dt + Vt

]Ntxt

Multiply both sides by βtuc (ct)∑j

βtuc (ct)1

Rjt

Bjt+1 + βtuc (ct)Vt(Nt +N e

t )xt+1 + βtuc (ct)Ct

= βtuc (ct)(1− τ lt

)wtHt + βtuc (ct) bt + βtuc (ct)

[(1− τ dt

)dt + Vt

]Ntxt

and sum over dates starting from t=0, where all term are understood asin expectation as of time 0

∞∑t=0

∑j

βtuc (ct)1

Rjt

Bjt+1 +

∞∑t=0

βtuc (ct)Vt(Nt +N et )xt+1 +

∞∑t=0

βtuc (ct)Ct

=∞∑t=0

βtuc (ct)(1− τht

)wtHt +

∞∑t=0

βtuc (ct) bt +∞∑t=0

βtuc (ct)[(

1− τ dt)dt + Vt

]Ntxt

The euler equation for bonds implies uc (Ct) = βRjtuc(Cjt+1

), using this

in the first term on the LHS

∞∑t=0

∑j

βt+1uc(Cjt+1

)Bjt+1 +

∞∑t=0

βtuc (Ct)Vt(Nt +N et )xt+1 +

∞∑t=0

βtuc (Ct)Ct

=

∞∑t=0

βtuc (ct)(1− τht

)wtHt +

∞∑t=0

βtuc (Ct) bt +

∞∑t=0

βtuc (Ct)[(

1− τ dt)dt + Vt

]Ntxt

Notice that the term∑

j uc(Cjt+1

)Bjt+1 can be understood as the

payoffof a risk free bond. As such we can cancel out the first summationon the LHS with the respective terms in the second summation in theRHS, leaving just time 0 terms

∞∑t=0

βtuc (ct)Vt(Nt +N et ) +

∞∑t=0

βtuc (ct)Ct

=

∞∑t=0

βtuc (ct)(1− τht

)wtHt + uc (c0) b0 +

∞∑t=0

βtuc (ct)[(

1− τ dt)dt + Vt

]Nt

38

Notice that the clearing of the asset market implies xt = 1 at all t.Considering that uh(ht)

uc(ct)= −

(1− τht

)wt leads to

∞∑t=0

βtuc (ct)Vt(Nt +N et ) +

∞∑t=0

βtuc (ct)Ct +

∞∑t=0

βtuh (ht)Ht

=uc (c0) b0 +∞∑t=0

βtuc (ct)[(

1− τ dt)dt + Vt

]Nt

Next consider

uc (Ct)Vt = Etβ (1− δ)uc (Ct+1)[(

1− τ dt+1

)dt+1 + Vt+1

]and plug it into the first summation in the LHS

∞∑t=0

βt+1 (1− δ)uc (Ct+1)[(

1− τ dt+1

)dt+1 + Vt+1

](Nt +N e

t )xt+1 +

+∞∑t=0

βtuc (ct)Ct +∞∑t=0

βtuh (ht)ht

=uc (c0) b0 +∞∑t=0

βtuc (ct)[(

1− τ dt)dt + Vt

]Ntxt

considering that

(Nt +N et ) =

Nt+1

1− δit follows

∞∑t=0

βt+1uc (Ct+1)[(

1− τ dt+1

)dt+1 + Vt+1

]Nt+1 +

+

∞∑t=0

βtuc (ct)Ct +

∞∑t=0

βtuh (ht)ht

=uc (c0) b0 +∞∑t=0

βtuc (ct)[(

1− τ dt)dt + Vt

]Nt

Simplifying the first summation on the LHS with the second in the RHSdelivers the implementability constraint

E0

∞∑t=0

βt [uc (Ct)Ct + uh (Ht)Ht] = uc (C0)B0+uc (C0)[(

1− τ d0)d0 + V0

]N0

where we reintroduced the expectation operator.

39

Appendix D. The Ramsey ProblemEntry Costs in form 1. Includes proof of result (i) in Proposi-tion 1.

The Ramsey problem reads as

max E0

∞∑t=0

βtu (Ct, Ht)

subject toNt+1 = (1− δ) (Nt +N e

t ) : λ1t

Ct +Gt +N et ψ = ρtAtHt : λ2t

ψuct = β(1− δ)Etuct+1

((1− τ dt+1/t

)(1− 1

µ

)Ct+1 +Gt+1

Nt+1

+ ψ

): λ3t

E0

∞∑t=0

βtE0 [uctCt + uhtHt] = uc0B0 + uc0[(

1− τ d0)d0 + V0

]N0 : ξ

Where λit define the Lagrange multipliers respectively attached toeach constraint and ξ is the (constant) lagrange multiplier attached tothe implementability constraint.The choice variables areCt, Nt,Ht,N e

t , and τdt/t+1. Following Ljungqvist

and Sargent (2004) I define

V (Ct, Ht, ξ) = u (Ct, Ht) + ξ (uctCt + uhtHt)

andΩ = uc0B0 + uc0

[(1− τ d0

)d0 + V0

]N0

As a result the Lagrangian function can be written as

L = E0

∞∑t=0

βt

V (Ct, Ht, ξ) + λ1t [(1− δ) (Nt +N e

t )−Nt+1]+λ2t (ρtAtHt − Ct −Gt −N e

t ψ) +

λ3t

[ψuct − β(1− δ)Etuct+1

((1− τ dt+1/t

)(1− 1

µt+1

)ρt+1At+1Ht+1

Nt+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− τ dt/t−1

)(1− 1

µt

)ρtAtHt

Nt

+ ψ

)= 0

Nt+1 : λ1t + β(1− δ)λ3t

Etuct+1

(1− τ dt+1/t

)At+1Ht+1

Nt+1

[(1− 1

µt+1

)ρNt+1Nt+1−ρt+1

Nt+1

+µNt+1ρt+1

µ2t+1

]= β (1− δ)Etλ1t+1 + βEtλ2t+1ρNt+1At+1Ht+1

40

τ dt+1/t : (1− δ)βt+1λ3tE0uct+1

(1− 1

µt+1

)ρt+1At+1Ht+1

Nt+1

= 0

N et : λ1t (1− δ) = λ2tψ

Ht : Vh (Ct, Ht, ξ)+λ2tρtAt−λ3t−1(1−δ)uct(1− τ dt/t−1

)(1− 1

µt

)ρtAtNt

= 0

SinceV (Ct, Ht, ξ) = u (Ct, Ht) + ξ (uctCt + uhtHt)

it follows

Vc (Ct, Ht, ξ) = uc (C,H) + ξucctCt + ξuct =1

Ct− ξ 1

C2t

Ct + ξ1

Ct=

1

Ct

and

Vh (Ct, Ht, ξ) = −υH1/ϕt

[ξ

(1 + ϕ

ϕ

)+ 1

]Consider now the steady state. The FOC with respect to τ dt+1/t reads

as

τ dt+1/t : (1− δ)βt+1λ3

(1− 1

µ

)C +G

Nuc = 0

The latter implies that at the steady state λ3 is equal to zero. In theRamsey steady state the firms entry condition does not restrict the al-location. As a result we can write the steady state version of the FOCsas

Ct : λ2 = uc

Ht : Vh (C,H, ξ) + uctρA = 0

N et : 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]

SinceρN =

1

θ − 1

ρ

N= ε

ρ

N

it follows

ψ = β (1− δ)[ερAH

N+ ψ

]

41

The euler equation for asset implies that

ψ = β(1− δ)((

1− τ d)(

1− 1

µ

)ρAH

N+ ψ

)For the latter two equations to be consistent with each other it has tobe the case that

ε =(1− τ d

)(1− 1

µ

)or

τ d = 1− ε(1− 1

µ

)which proves point (i) in proposition 1. Importantly, the dividend in-come tax differs from zero also under monopolistic competition. Sub-stituting the optimal dividend income tax into the Euler equation forshares we get

ψ = β(1− δ)(εY

N+ ψ

)Which implies

Y

N=

[1− β (1− δ)]β (1− δ) ε ψ

Notice thatC

Y= 1− gy −

δ

1− δψN

Y

Next consider the implementability constraint, which can be writtenas

Ω =Y

C

BY

+ε(

1− 1µ

) Π

Y+V N

Y

where B

Yis exogenously given and C

Yhas been computed above. The

Euler equation with respect to assets implies

V N

Y=

β (1− δ)[1− β (1− δ)]ε

and also we knowΠ

Y=

(1− 1

µ

)Ω =

Y

C

(B

Y+ ε+

V N

Y

)H =

[1− (1− β) Ω

υ

] ϕ1+ϕ

42

Finally given H and recalling that

Y

N=

[1− β (1− δ)] (θ − 1)

β (1− δ) ψ

it follows

N =

[β (1− δ) ε

[1− β (1− δ)]ψAH] θ−1θ−2

Entry Costs in Form 2. Includes proof of result (ii) in Propo-sition 2.

In this case the Lagrangian is

L = E0

∞∑t=0

βt

V (Ct, Ht, ξ) + λ1t [(1− δ) (Nt +N e

t )−Nt+1]+λ2t (ρtAtHt − Ct −Gt −N e

t ηρt) +

λ3t

[η ρtµtuct+

−β(1− δ)uct+1

((1− τ dt+1/t

)(1− 1

µt+1

)Ct+1+Gt+1

Nt+1+ η

ρt+1µt+1

)]−ξΩ

As above, the choice variables are Ct, Nt, Ht, N et , and τ

dt/t+1. The

first order conditions are

Ct : Vc (Ct, Ht, ξ) + λ3tηρtµucct

−λ3t−1

(1− δ)ucct((

1− τ dt/t−1

)(1− 1

µ

)Ct+GtNt

+ η ρtµ

)+

+(1− δ)uct(

1− τ dt/t−1

)(1− 1

µ

)1Nt

=λ2t

Nt+1 : λ1t +

β(1− δ)λ3tEtuct+1

+

(1− τ dt+1/t

)µt+1

(µN ,t+1

µt+1

− µt+1 − 1

Nt+1

)Y ct+1

Nt+1

+βη(1− δ)λ3tEtuct+1

(ρNt+1µt+1 − µNt+1ρt+1

µ2t+1

)= β (1− δ)Etλ1t+1 + βEtλ2t+1ρNt+1

(At+1Ht+1 −N e

t+1η)

+

+βηEtλ3t+1uct+1

(ρNt+1µt+1 − µNt+1ρt+1

µ2t+1

)

Ht : Vh (Ct, Ht, ξ) + λ2tρtAt = 0

43

N et : λ1t (1− δ) = λ2tηρt

τ dt+1/t : (1− δ)βt+1Etuct+1λ3t

(1− 1

µt+1

)Ct+1 +Gt+1

Nt+1

= 0

Notice thatρtN =

1

θ − 1

ρtNt

µCt =θNt

(θ − 1)(Nt − 1)

µCNt =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

µBt =1 + θ(Nt − 1)

(θ − 1)(Nt − 1)

µBNt =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 −N eη)]

Ht : Vh (C,H, ξ) + λ2ρA = 0

N et : 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 defini-

tions of Vc and Vh we get

1

β= (1− δ)

[1 + ε

1

ηρ

Y c

N

]Evaluating the Euler equation for assets at the steady state implies

1

β= (1− δ)

((1− τ d

)(µ− 1)

ηρ

Y c

N+ 1

)For the two to be consistent is has to be the case that(

1− τ d)

(µ− 1) = ε

which proves point (ii) in Proposition 1. Notice that in the monopolisticcompetition case this implies τ d = 0. The Euler equation for assetsevaluated at the steady state reads as

1

β= (1− δ)

[1 +

ε

ρη

Y c

N

]44

thenY c

ηρ=

1− β (1− δ)(1− δ) βε N

The aggregate resource constraint implies

Y c

ηρ=AH

η−N e

using the equation for the dynamics of the number of firms

Y c

ηρ=AH

η− δ

1− δN

Combing the latter two equations

N =

AHη

(1−(1−δ)β)(1−δ)βε + δ

1−δ

=

(1−δ)δ

AHη

1 + (1−(1−δ)β)δβε

we get N as a function of H. This also implies that we can computethe 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 =ρ

µAH; Π =

(1− 1

µ

)Y c

it follows

Y =ρ

µAH +

(1− 1

µ

)Y c

using the aggregate resource constraint Y c = ρAH − ηρN e we obtain

Y =ρ

µAH +

(1− 1

µ

)(ρAH − ηρN e)

= ρAH − ρA(µ− 1

µ

)η

AN e

Also notice

1 =

(1− 1

µ

)Y c

Y+

1

µ

ρAH

Y

To compute Y c

Yand C

Yconsider the euler equation for assets

V =β (1− δ) ε

µ

1− β (1− δ)Y c

N

45

which implies

V N e

Y c=

β (1− δ) εµ

1− β (1− δ)N e

N=

δβ εµ

1− β (1− δ)

this allows to compute Y c

Yas follows

Y c +N eV = Y

thenY c

Y= 1− N eV

Y c

Y c

Y

and finallyY c

Y=

(1 +

V N e

Y c

)−1

From the latter we get CYas

C

Y=Y c

Y− G

Y

Knowing Y c

Ywe can determine ρAH

Y

ρAH

Y= µ

[1−

(1− 1

µ

)Y c

Y

]The FOC for hours

−Vh (C,H, ξ)

Vc (C,H, ξ)= ρA

substituting the definitions of variables

υH1/ϕt

[ξ(

1+ϕϕ

)+ 1]

1C

= ρA

or

H1/ϕ =1υρAC[

ξ(

1+ϕϕ

)+ 1]

Multiplying both sides by H, the latter is equivalent to

H =

ρAHY

YC

υ[ξ(

1+ϕϕ

)+ 1]

ϕ1+ϕ

46

Hence H is both a function of H and ξ. Next consider the imple-mentability constraint

Ω =Y

C

B

Y+(1− τ d

) πNC

+V N

C

As a result

Ω =Y

C

(B

Y+(1− τ d

) Π

Y+V N

Y

)=Y

C

(B

Y+

ε

(µ− 1)

Π

Y+V N

Y

)where B

Yis given and C

Yis a function of H. Also from

V =β (1− δ) ε

µ

1− β (1− δ)Y c

N

we getV N

Y=

β (1− δ) εµ

1− β (1− δ)Y c

Y

andΠ

Y=

(1− 1

µ

)Y c

Y

Hence we can compute Ω as a function of H. Next using the steadystate version of the implementability constraint we get

1

1− β[1− υH1+1/ϕ

]= Ω

or1− υH1+1/ϕ = (1− β) Ω

which implies

H =

[1− (1− β) Ω

υ

] ϕ1+ϕ

which is a function solely of H and can be solved numerically. Given thevalue H we can determine the lagrange multiplier ξ

ξ =ϕ

1 + ϕ

[1υρAHY

YC

H1+ϕϕ

− 1

]

Recall that N can be computed as

N =

Aη

1−δδ

1 + (1−(1−δ)β)δβε

H

47

which allows to compute the price markup at thus the Ramsey steadystate. Also it implies a a value for ρ. Since

ηρ

µA =

β (1− δ) εµ

1− β (1− δ)Y c

N

we get

Y c =ηρAN

µ

1− β (1− δ)β (1− δ) ε

µ

In particular notice that

τ d = 1− ε

(µ− 1)

and

τh = 1− υCH1ϕ

w

48