Ramsey Meets Cournot by Andrea Colciago and Federico Etro University of Milano, Bicocca, Department of Economics August 14, 2009

Abstract We introduce strategic interactions based on Cournot competition and endogenous entry in an otherwise standard DSGE model with homogenous goods. In the long run, the steady state mark up and the concentration of the market are decreasing in the discount factor and in the labor productivity, and increasing in the rate of business destruction and in the cost of entry. In the short run, a competition e¤ect ampli…es the propagation of the shocks and generates procyclical pro…ts and countercyclical mark ups. We extend the model to di¤erent forms of competition (as imperfect collusion and Stackelberg competition) and accumulation of physical capital for a comparison with RBC models. The analysis of permanent and temporary technology and preference shocks and of the second moments suggests that the model outperforms the RBC framework in many dimensions.

JEL classi…cation: L11, E32. Keywords: Endogenous Market Structure, Firms’ Entry, Business Cycle.

Previously circulated under the title "Real Business Cycles with Cournot Competition and Endogenous Entry " and …rst presented in the winter of 2007. We are grateful to seminar participants at the University of Saint Andrews (Scotland), CERGE-EI (Czech Republic), IMT (Lucca), the University of Rome, the University of Milan, Bicocca and the Catholic University of Leuven for important suggestions. Guido Ascari, Florin Bilbiie, Lilia Cavallari, Alex Cuckierman, Alberto Dalmazzo, Mark Gertler, Fabio Ghironi, Mario Gilli, Stefano Gnocchi, Giammario Impullitti, Pietro Peretto, Tiziano Ropele and Patrizio Tirelli provided insightful discussions on this topic. Further details are available at www.intertic.org. Correspondence : Andrea Colciago, University of Milano-Bicocca, Department of Economics, U6-360, Piazza dell’Ateneo nuovo 1, Milano 20126, Italy.

1

1

Introduction

Most of the business cycle literature is based on models with perfect competition between …rms producing homogenous goods, as in the Real Business Cycles (RBC) tradition started by Kydland and Prescott (1982) or with monopolistic behavior of an exogenous number of …rms as in the New Keynesian literature started by Blanchard and Kiyotaki (1987). The former case leads to marginal cost pricing and to indeterminate market structures where strategic interactions are absent, while the latter case leads to exogenous market structures where monopolistic prices are given by constant mark ups on marginal costs and strategic interactions are, again, neglected. The aim of this paper is to introduce imperfectly competitive markets characterized by strategic interactions in an otherwise standard DSGE model, to endogenize the market structures and to evaluate the business cycle properties of the model. We build on the recent literature on endogenous entry in macroeconomics - in particular on the pathbreaking works by Ghironi and Melitz (2005) and Bilbiie et al. (2007a,b)1 - and we augment it with imperfect competition in quantities à la Cournot in a representative market. The idea is that many realistic markets are characterized by a limited number of …rms producing goods that are highly substitutable, a situation in which strategic interactions are extremely important (while these are less important between …rms of di¤erent markets producing di¤erent goods). In the short run, the endogenous market structure associated with imperfect competition introduces a strong propagation mechanism of exogenous shocks. A temporary expansionary shock generates pro…ts opportunities which attract entry of new competitors. This leads to a positive temporary competition e¤ect on prices, in the form of a reduction in mark ups, which boosts consumption. As a result the e¤ects of shocks are ampli…ed. We analyze both technology shocks and demand shocks (in terms of preference shocks) and we obtain impulse response functions that are empirically plausible and reproduce procyclical pro…ts and entry and countercyclical mark ups. Moreover, the propagation mechanism we emphasize improves the explanation of second moments of the relevant macroeconomic variables with respect to a standard ‡exible prices model. In the long run, the endogenous market structure is fully determined by structural parameters: in particular, we explicitly derive the steady state mark up and the level of concentration of the market. They are both decreasing in the discount factor (which tends to increase savings and investements in business creation) and in the labor productivity (which tends to increase output and demand and to create space for new business creation and stronger competition). 1 The …rst papers to endogenize entry in a DSGE model are due to Chatterjee and Cooper (1993), Devereux et al. (1996) and Wu and Zhang (2001). Chatterjee et al. (1993) have endogenized entry as well, but their focus was on sunspots equilibria in an OLG model. Cooper (1999) surveys this early literature. Cook (2001), Devereux and Lee (2001) and Jaimovich and Floetotto (2008) have extended this class of models to strategic interactions but focusing on free entry in each period (which eliminates any dynamics for the pro…ts). For an introduction to the competition e¤ect emphasized here see Etro and Colciago (2007) and Etro (2007,b, 2009).

2

Finally, the mark up and the concentration level are increasing in the rate of business destruction and in the cost of entry (two factors which reduce the incentives to enter and compete). We use the same framework for multiple extensions. First, we study di¤erent forms of competition, as imperfect collusion, obtained through the conjectural variations approach, and Stackelberg competition. We …nd that competitive frameworks involving a degree of collusion stronger than that implied by the baseline Cournot model could help improving the performance of dynamic general equilibrium models at replicating the features of the business cycle. Moreover, the Stackelberg framework displays an asymmetric response between the individual variables relative to the leader and those relative to the representative follower. Hopefully, these applications will promote further investigations on the interaction between the microeconomic structure of the markets and macroeconomic models. Second, our framework can be augmented with a standard process of capital accumulation to nest both our basic model and the traditional RBC model. In this case, the role of strategic interactions and endogenous entry in improving the performance of the neoclassical model emerges clearly: the performance of the model improves substantially for what concerns the second moments of both output and hours worked. However, since households can use an additional channel, beside investment in new …rms, to smooth consumption, the relative variability of consumption decreases with respect to cases without capital accumulation. In the companion paper Etro and Colciago (2007) we have introduced imperfect substitutability between products. Recent empirical works on the manufacturing sector by Broda and Weinstein (2009) and Bernard et al. (2008) have emphasized the importance of the extensive margin in the process of product creation or innovation. We can easily accommodate this margin by introducing a CES consumption index and deriving the Cournot (or Bertrand) equilibrium between …rms producing di¤erentiated goods. However, it should be emphasized that the entry process in many traditional sectors and especially in the service sectors, which are a large part of developed economies, are mostly associated with new business creation (rather than product innovation). To give an example of our idea, in a boom, pro…table opportunities may lead new businesses to start, new restaurants to open, or new services to be provided, which will increase competition in the respective markets and reduce the mark ups. None of these businesses will a¤ect the extensive margin as de…ned in the statistics in terms of new consumer products, but they will be relevant nevertheless. For this reason, the focus of this paper remains on markets with homogenous goods provided by multiple …rms. In the rest of this Introduction we will discuss the empirical motivations of our approach at the microeconomic and macroeconomic level. A wide industrial organization literature provides theoretical and empirical support for the relevance of the competition e¤ect that we propose as a propagation mechanism. At the theoretical level, there is a crucial di¤erence between models of monopolistic behavior à la Dixit and Stiglitz (1977), recently employed by Bilbiie et 3

0.2

0.15

0.1

0.05

0

-0.05

-0.1

-0.15

-0.2

1950

1955

1960

GDP

1965

1970

Mark up

1975

1980

1985

1990

1995

New Incorporations

Figure 1: Cyclical component of GDP, New Incorporations and Price mark up. Shaded areas represent NBER recessions.

al. (2007a,b) for business cycle analysis, and models with strategic interactions as ours. In the …rst class of models, the mark up and the production of each …rm are constant, while the number of …rms increases linearly with the size of the market. In the second class of models, positive shocks to the size of the market attract entry and strengthen competition in such a way that the mark up decreases, the production of each …rm increases (to cover the …xed costs) and the number of …rms increases less than proportionally. Early works of the New Empirical Industrial Organization literature starting with Bresnahan and Reiss (1987) and more recent works by Manuszak (2002), Campbell and Hopenhayn (2005), Manuszak and Moul (2008) and others have provided convincing evidence in support of the second class of models and of the competition e¤ect on mark ups, …rms’production and number of …rms. There is also macroeconomic evidence in support of the business cycle implications of our approach. Rotemberg and Woodford (1999) and Galì et al. (2007) forcefully document price mark ups countercyclicality. Early references on the procyclicality of …rms’ entry are Chatterjee and Cooper (1993) for the U.S. and Portier (1995) for France. Bilbiee et al. (2007a,b), Lewis (2007) and Etro and Colciago (2007) emphasize, instead, the procyclicality of real pro…ts. To provide further support to these empirical …ndings we constructed a labor share based measure of the price markup for the U.S. along the lines suggested by Rotemberg and Woodford (1999) in their analysis of cyclical markup behav-

4

ior.2 Figure 1 plots the series of detrended GDP, the price markup and the number of new incorporations at a quarterly frequency from 1948:1 to 1995:3.3 In line with the evidence reviewed above net business formation is procyclical and negatively correlated with the price mark up, which is countercyclical. The contemporaneous correlation between GDP and new incorporations is positive and equals 0.11, while that between GDP and the price mark up is -0.42.4 The dynamics delivered by the model in this paper, in response to both demand and supply shocks, are consistent with this evidence. The paper is organized as follows. Section 2 describes the model under Cournot competition and other forms of competition in the simplest setup, with homogenous goods and no capital in the production function of …nal goods. Section 3 studies the impulse response functions for technological shocks and preference shocks. Section 4 studies the second moments of the model. Section 5 introduces capital accumulation in the basic model and compares the performance of the traditional RBC model with perfect competition and a the same model augmented with endogenous market structures. Section 6 concludes. Further details are left in the Appendix.

2

The Model

We develop a simple dynamic stochastic general equilibrium model with imperfect competition and endogenous entry. In our basic speci…cation …rms compete a là Cournot and produce homogenous goods employing labor as the only input, however below we analyze alternative forms of competition and augment the model with capital accumulation. The aim of this model is to introduce the basic mechanism at the basis of our approach, the link between pro…ts, entry, strategic interactions, mark ups and consumption demand.

2.1

Strategic interactions

Let us consider a representative sector characterized by Nt …rms in each period t, all producing the same consumption good. We can think of this as a representative sector of the economy. All our results would go through in the presence of multiple sectors producing di¤erentiated goods and each one with 2 The procedure used to derive our price mark up measure is described in Appendix D together with the data used in the remainder of the paper. 3 Variables have been logged. Since we use these variables in subsequent time series analysis we opt for a polynomial of time to detrend variables instead of the HP …lter, as suggested by Galì et al. (2007). However the signs of contemporaneous correlations between our measures of entry and the price mark up with output are unchanged when variables are detrended using the HP …lter. 4 The contemporaneous correlation between GDP and the index of net business formation is positive too and equals 0.45. We do not report the cyclical component of net business formation in Figure 1 because the number of new incorporations is closer to what we considers as new entrants in the remainder of the paper.

5

the structure of our representative sector.5 Each …rm i produces yt (i) according to a linear production function: yt (i) = At Lct (i)

(1)

where At is exogenous TFP common to all …rms, and Lct (i) is the labor input used for the production of the …nal good. Given the nominal wage Wt , the constant marginal cost of production is Wt =A. Total expenditure in the sector is: PNt yt (j) Et = pt Ct = pt j=1

where pt is the price of the homogenous good and Ct its consumption in period t. Assuming that all …rms take total expenditure as given PNt in each period, their perceived inverse demand function must be pt = Et = j=1 yt (j). Accordingly, the nominal pro…ts of …rm i are: t (i)

= =

Wt A yt (i)Et Nt X yt (j) pt

yt (i) = Wt yt (i) At

(2)

j=1

Assume that in each period, the Nt …rms compete in quantities, choosing their individual production yt (i) to maximize pro…ts taking as given the production of all the other …rms. The Cournot equilibrium generates the symmetric individual output: (Nt 1)Et At (3) yt = Wt Nt2 Substituting in the inverse demand, one obtains the equilibrium price: pt (Nt ) =

Nt Nt 1

Wt At

(4)

which is associated with the mark up t = Nt =(Nt 1). This equilibrium generates individual pro…ts t (Nt ) = Et =Nt2 in nominal terms. Since the equilibrium price of the consumption good is pt , it is convenient to express all the variables in units of consumption, that is in real terms (alternatively one can use the consumption good as the numeraire). Then, real pro…ts t (Nt ) = t (Nt )=pt become: Ct (5) t (Nt ) = Nt2 5 In the companion paper Etro and Colciago (2007), we explicitly extend the model to a continuum of sectors with low elasticity of substitution (imperfect substitutability). Our interest here is therefore on entry of and competition between …rms producing similar goods. Ghironi and Melitz (2005) and Bilbiie et al. (2007a,b) adopt a di¤erent approach and focus on entry of …rms producing new di¤erentiated goods.

6

and the real wage wt = Wt =pt can be derived from the equilibrium pricing relation as: (Nt 1) wt = At (6) Nt When the number of …rms increases, the equilibrium price goes down and the wage goes up, with the former approaching the marginal cost and the latter approaching the marginal productivity of labor only for Nt ! 1. However, the number of …rms in the market is constrained by the presence of …xed costs of entry that endogenously rule the entry of new …rms in the goods market. In every period Nte new …rms enter in the market, and, following Ghironi and Melitz (2005), we assume that a fraction 2 (0; 1) of the (old and new) …rms exits the market for exogenous reasons. Therefore, the number of …rms follows the equation of motion: )(Nt + Nte )

Nt+1 = (1

(7)

The real value of a …rm Vt is the present discounted value of its future expected pro…ts, or in recursive form: Vt = (1

1

)Et (1 + rt+1 )

Vt+1 +

Ct+1 2 Nt+1

where rt+1 is the real interest rate. The number of …rms is endogenous in the sense that in each period entry occurs until the real value of the representative …rm equates the …xed cost of entry. The latter is equal to =At units of labor, where > 0. Given the real wage de…ned above, the endogenous entry condition amounts to: (Nt 1) Vt = (8) Nt

2.2

Consumer choices

The expected lifetime utility of the representative agent reads as: U = Et

1 X

t

log Ct

(9)

t=0

where 2 (0; 1) is the discount factor and Ct is consumption of the homogenous good. Labor supply is exogenously given and normalized to one for simplicity.6 The time t ‡ow budget constraint in real terms is: Bt+1 + Vt (Nt + Nte )xt+1 + Ct = wt + (1 + rt )Bt + [ t (Nt ) + Vt ] Nt xt

(10)

where Bt is net bond holdings and xt is the share of the stock market value of the …rms that are owned by the agent. The agent faces the, usual, problem of 6 In Appendix C we generalize the model to endogenous labor supply. Also we allow for the possibility of preference shocks.

7

maximizing expected lifetime utility subject to (10), and chooses consumption and how much to invest in risk free bonds and risky stocks out of labor and pro…t income. The …rst order conditions with respect to xt+1 and Bt+1 are respectively: Vt (Nt + Nte )Ct

1

= Et [

Ct

1

t+1

= (1 +

1 (Nt+1 ) + Vt+1 ] Nt+1 Ct+1

1 rt+1 )Et Ct+1

(11) (12)

1

where Ct is the time t marginal utility of consumption. Equation (11) is an asset pricing equation for stocks, while equation (12) is a standard Euler equation for bonds. Using these optimality conditions and the equation of motion for the number of …rms (7), it follows that the value of a …rm can be expressed as: ) ( 1 Nt+1 Ct+1 = [ t+1 (Nt+1 ) + Vt+1 ] Vt = Et Ct Nt + Nte ( ) 1 Ct+1 = (1 )Et [Vt+1 + t+1 (Nt+1 )] (13) Ct which corresponds to the present discounted sum of future pro…ts.

2.3

Endogenous market structures in the short run

Using the value of the …rms (7) and the de…nition of pro…ts (5), we can rewrite the endogenous entry condition (8) as: # " 1 Nt 1 Ct+1 (Nt+1 1) Ct+1 (1 )Et = + 2 (14) Ct Nt+1 Nt+1 Nt The ‡ow budget constraint under the equilibrium conditions Bt = 1 xt = 0 for any t, leads to the aggregate resource constraint of the economy: Ct + Vt Nte = wt +

t (Nt )Nt

(15)

which states that the sum of consumption and investment must be equal to total income from labor and pro…ts. Given that Lct + Let = 1, where Let = Nte =At is the quantity of labor used in the production of new …rms, simple manipulations of equation (15) lead to the number of new entrants in each period Nte = (At Ct ) = . Substituting this into the equation of motion for the number of …rms we get: Nt+1 = (1

) Nt +

At

Ct

(16)

Given an exogenous processes for At , the system composed by equations (14) and (16) fully characterizes the dynamic behavior of Ct and Nt , and thus of all the other variables. In particular, it provides the short run characterization of the endogenous market structures in terms of number of …rms, mark ups and production per …rm. 8

2.4

Endogenous market structures in the long run

The deterministic steady state of the economy is characterized by the following two equations:7 N

=

C

=

(1

) (A 1 (1

)

C ) 1 (N

1) N

The …rst steady state relation is a negative relation between consumption and the number of …rms (more consumption implies a lower investment to replace …rms), the second one is a positive and convex relation (more …rms produce more and increase consumption). It follows that the steady state is unique and characterized by a positive number of …rms as depicted in Figure 3. Solving the two steady state conditions for N and C , one can derive that the steady state endogenous market structure is characterized by following the number of …rms: N = where mark up:

p

2 (1

)2 + 4 =

(1

(1

) + ( ; ; A; ) 2 [1 (1 )] ) [1

(1

(17)

)] A, and by the equilibrium

( ; ; A; ) + (1 ) ( ; ; A; ) (1 +2

)

(18)

The mark up is decreasing in the discount factor , in total factor productivity A and increasing in the exit rate and in the entry cost parameter . The level of concentration of the market, which is inversely proportional to the number of …rms,8 follows the same comparative statics. The intuitions for the impact of the four structural parameters determining the endogenous market structure in the long run are the following. Higher productivity leads to more business creation, which increases the steady state number of …rms and enhances competition while reducing the mark ups. The second determinant is the size of the costs of entry: when these are high, pro…tability is low and the long run equilibrium is characterized by high concentration and high mark ups. To the extent that the costs of entry are arti…cial, in the sense that there are barriers to entry due to product market regulations, reforms leading to deregulation reduce concentration and mark ups in the long run. The third factor is the way people discount future utility: when agents are more patient, the interest rate is lower and the discounted sum of future pro…ts is higher, which attracts more entry, 7 An increase in TFP has a positive impact on the number of …rms in steady state. Again, this would disappear if the cost of entry parameter was proportional to the size of the economy, however, notice that the fact that mark ups are decreasing in the number of …rms implies that scale e¤ects become negligible when TFP increases inde…nitely. See Appendix A for details on the steady state. 8 As well known, in a symmetric Cournot equilibrium, the Her…ndahl Index is the inverse of the number of …rms.

9

Ct

C*

N*

Nt

Figure 2: Steady state and saddle path stability

strengthens competition and ultimately reduces the mark ups - therefore more patient agents lead to a higher number of …rms in the steady state. The last element is the rate of business destruction due to exogenous reasons: when the risk of bankruptcy is high, there are only few …rms in the long run (but with a high rate of turnover), and they apply a high mark up to their goods. Summing up, the equilibrium endogenously generates imperfect competition between a positive but limited number of …rms producing the homogenous good. Notice that dynamic ine¢ ciency holds, since a better allocation of resources could be achieved reducing the number of …rms and the waste in …xed costs of production of a homogenous good.9

2.5

Other forms of Competition

Until now we have focused our analysis of the endogenous market structures on Cournot competition. One of the main aim of this paper is to emphasize the need of a deeper investigation of the industrial organization of the markets in macroeconomic models. Therefore, in this section we extend the model to other forms of competition with endogenous entry: one is based on conjectural variations (which is traditionally applied to homogenous goods models), and another is based on Stackelberg competition (for which we have a closed form 9 See Etro and Colciago (2007) for a generalization of this result. Forms of dynamic ine¢ ciency emerged in earlier dynamic general equilibrium models with endogenous market structures as in Etro (2004, 2008,a).

10

solution only in the case of homogenous goods).10 The …rst extension provides a reduced form for a more general model explaining collusion, the second allows us to take into account asymmetries between small and large …rms. 2.5.1

Conjectural Variations Approach

A simple extension of the Cournot model of competition can be obtained assuming general conjectural variations of the …rms. Assuming that each …rm takes as given the di¤erential impact of its output choice on the output choice of the other …rms @xt (j)=@xt (i), the equilibrium price can be obtained as: pt ( ; N t ) =

(Nt

Nt 1)(1

Wt ) At

which nests the case of Cournot competition in quantities for = 0 and tends to the (indeterminate) case of perfect collusion for ! 1. More importantly, intermediate situations with 2 (0; 1) describe cases of imperfect collusion between the …rms which achieve mark ups above the Cournot level but below the perfect collusion level. Then, the real pro…ts are: t(

; Nt ) =

[(Nt

1) + 1] Ct Nt2

Equilibrium dynamics for Nt and Ct are de…ned by (16) and: ( 1 Ct+1 (Nt+1 1) (1 )Nt+1 + Ct+1 Et Ct Nt+1 Nt+1 =

Ct+1 + 2 Nt+1

)

(Nt 1)(1 ) (1 )Nt

whose analysis is similar to the previous one. 2.5.2

Stackelberg Competition

Another relevant case emerges in the presence of heterogeneity between …rms, and is based on the theory of Stackelberg competition. For simplicity, we assume that the leader cannot commit to future strategies and adopts the optimal static strategy in each period.11 This framework can be regarded a …rst attempt to model the micro evidence suggesting that smaller …rms (the followers) make up the majority of entrants and exits. Let us assume that a single leader is always active in the representative sector and Nt followers are active in each period. The Stackelberg equilibrium 1 0 See

Etro (2007,a) on the industrial organization of endogenous market structures. assumption that also the leader cannot commit to a sequence of strategies is crucial here. If the leader could commit, it would engage in aggressive strategies aimed at reducing or deterring entry in the long run. See Etro (2008,b) for a recent analysis of Stackelberg competition with commitment and endogenous entry. 1 1 The

11

is characterized by the following equilibrium price: p(Nt ) =

Nt Wt (Nt 1=2) At

which is lower than under pure Cournot competition in quantities. The pro…ts of the leader and the representative follower are respectively larger and smaller than the pro…ts under Cournot competition: L t (Nt )

=

Ct 4Nt

F t (Nt )

=

Ct 4Nt2

Since we assumed that entry and exit concerns only the followers, whose value is always pinned down by the endogenous entry condition: VtF =

(Nt 1) Nt

the value of the leading …rm must be calculated separately as the expected discounted value of its future pro…ts: VtL = Et (1 + rt+1 )

1

Vt+1 +

Ct+1 4Nt+1

In this case, the equilibrium system is given by (16) and: " # 1 Ct+1 (Nt 1=2) (Nt+1 1=2) Ct+1 Et = + 2 Ct Nt+1 4Nt+1 (1 ) Nt whose analysis is again similar to the one of the Cournot model.

3

Business Cycle Analysis

As customary, we solve the model by log-linearization around the deterministic steady state.12 In the next subsections we consider the behavior of the model in response to both technology and demand shocks. Beside considering the baseline case of pure Cournot competition, we study also the dynamics delivered by the conjectural variations and by the Stackelberg approach. The calibration is conducted on a quarterly basis. The discount factor, , is set to 0.99. As in Bilbiie et al. (2007a,b) the rate of business destruction, , equals 0.025 implying an annual rate equal to 10 percent. We normalize the steady state level of technology A to 1. Similarly we set the entry cost parameter = 1. Notice that none of our results is qualitatively a¤ected by alternative parameterization of . 1 2 Appendix B provides log-linear equilibrium conditions and an analysis of local stability and uniqueness of the rational expectation equilibrium.

12

Consumption

1

Hours: final good

Hours: new firms 1

0

1.08 1.06

0.5 0.9

-0.1

1.04

0 0.8 0

-0.2 20

40

0

Output

20

40

-0.5

Wage 1.1

1.02 0

Number of firms

20

40

0

New entrants

20

40

Mark up 0

1.02

0.4

2

0.3

1

-0.05

1.5

0.2 0.98 0

1

0.1 20

40

0

0

Stock Market Value 0.1

20

Individual output 0.75

40

0.6 0

0

40

0

20

40

Aggregate profits

0.7 0.65

0.2 20

0

0.75

0.4

0.65 20

40

0.6

0.05

0

20

Individual profits

0.7

0

-0.1

0.5 0

40

20

40

0.6 0

20

40

Figure 3: Transitional dynamics after a permanent technology shift under Cornot Competition

3.1

Permanent technology shocks

To shed light on the dynamics implied by the model we initially consider the response to a permanent technology shock. Figure 4 reports the response of the main macroeconomic variables to a one percent permanent increase in technology under Cournot competition.13 The baseline parameterization delivers an endogenous steady state markup equal to 22 percent. This is close to the value which is exogenously calibrated in most studies on the business cycle featuring monopolistic competition. On the vertical axis we report percentage deviations of variables from the initial steady state. Time on the horizontal axis is in quarters. Improved technological conditions increase demand and create large pro…t opportunities for the existing …rms. This pro…t opportunities attract new …rms into the market, which tends to strengthen competition and leads to a reduction of the mark ups. Despite lower markups, pro…ts are above the initial level during the whole transition to the new steady state because consumer demand is permanently larger. As the number of producers and the mark up converge to their new steady state levels, the number of entrants settles down to a perma1 3 The model has been solved using DYNARE a software devoloped by Michele Juillard. In the case of permanent shocks the impulse response functions are actually deterministic simulations. For a given path of the exogenous variable DYNARE provides the response of the whole system assuming that variables are initially at the steady state. This is done by stucking all the equations of the system for all periods and solving the resulting system en bloc using the Newton-Raphson algorithm.

13

nently higher value, which is needed to replace a larger exit of …rms in steady state. Although a higher number of entrants requires a permanently higher level of investment, the expansion in the production possibilities allows to attain permanently higher consumption. Moreover, notice that in the long run the labor force is partly moved from the business creation sector to the production sector, which is more competitive and produces more. C o n s u m p ti o n

H o u rs : fi n a l g o o d

H o u rs : n e w fi rm s

0 .2 1

0

0 .8

-0 .2

0 .6

-0 .4

1 .1

1

1 .0 5 0

1

-1 0

20

40

0

Ou tp u t 1 .1

0 .8

1 .0 5

0 .6

40

0

40

3

20

40

0 0

-0 .0 5

-0 .1 20

0 0

40

In d i vi d u a l o u tp u t

20

40

0

In d i vi d u a l p ro fi ts 1

20

40

A g g re g a te p ro fi ts 0 .9

Leader

Leader

Leader

40

Ma rk u p

1

0 .8 0 .6

20

0

2

0 .2

S to c k Ma rke t V a l u e

0 .4

20

0 .9 5 0

N e w e n tra n ts

0 .4

0 .9 5 0 .9 0

20

N u m b e r o f fi rm s

1

W age

2

0 .8

1 0 .5

Fo l l o w e rs

0 .7

0 .5

Fo l l o w e rs

0 .2

0 .6

Fo l l o w e rs 0 0

20

40

0 0

20

0 0

40

20

40

0 .5 0

20

40

Figure 4: Transitional dynamics after a permanent technology shift under Stackelberg competition and Conjectural Variations At a micro-level the mechanism is as follows. Since the wage jumps on impact by the same amount of the technology shock, the entry cost, which is measured in units of e¤ective labor, is initially unchanged. The increase in the expected pro…ts compared to an unchanged entry cost leads to a sizeable increase in the number of new entrants. As the number of …rms increases, the markup goes down and thus the real wage rises more than proportionally with respect to technology. This determines an increase in the cost of entry, which is mirrored in the dynamic of the stock market value. As the cost of entry reaches its new equilibrium value, investment in new …rms decreases and labor is reallocated to the production of the …nal good. Meanwhile, the number of …rms has increased so as to strengthen competition and push down the markup. Notice that the dynamics of the model are governed by a similar intertemporal substitution mechanism which characterizes the standard RBC model, with investment in new …rms playing an analogous role of investment in capital goods. Nevertheless imperfect competition adds to the traditional framework

14

a new competition e¤ ect which, through its consequences for …rms’ strategic pricing decision, can address basic empirical facts such as countercyclicality of mark ups and procyclicality of aggregate pro…ts. The qualitative dynamics described in the case of Cournot Competition are preserved also under alternative market arrangements. Figure 5 provides transitional dynamics for the conjectural variation model, with = 0:15, together with those delivered by the Stackelberg competition model in the face of a permanent technology shift. The initial mark up in the conjectural variation case amounts to 34 % while it is 15 % under Stackelberg competition. Importantly, both market structures deliver procyclicality of pro…ts together with mark up countercyclicality. As shown in the …gure, the Stackelberg competition framework displays an asymmetric response between the individual variables relative to the leader and those relative to the representative follower. Consider for instance the dynamic path of the stock market value. While the market value of a follower is pinned down on a period-by-period basis by the entry cost, the value of the leader is determined by the discounted value of its futures pro…ts. Since this increases, the leader’s market value jumps on impact.

3.2

Temporary technology shocks

Let us now consider a temporary increase in technology. As usual we assume that the technology shock evolves according to an autoregressive process of the form: log(At =A) = 0:9 log(At 1 =A) + "t where "t is an independently and identically distributed normal random variable with variance 2" . Since it is of interest to compare business cycles generated by our framework with those delivered by the prototype RBC model, we also report the response of variables in the case of endogenous labor supply.14 Figure 6 depicts the impulse response of the economic system to a one percent temporary shock to At . Continuous lines refer to the case with …xed labor supply, while dashed line to that with endogenous labor.15 Similar mechanisms to those emphasized above are at work. The direction of movements of the variable is identical to that described in the case of a permanent shock, the main di¤erence being the absence of a long run e¤ect. As in the standard RBC model the temporary technology shock increases consumption. However, di¤erently from what happens in the afore mentioned framework, the impact of the shock on consumption is enhanced by deeper competition. Entry of new …rms fosters competition and temporarily reduces the mark ups. This provides an extra boost to the intertemporal substitution of future for present consumption with respect to the case in which mark ups 1 4 Appendix

C outlines the model with endogenous labor supply. loss of generality we set utility parameters such that steady state labor supply equals 1. In the latter case the Frish elasticity of labor supply reduces to ', to which we assign a value of 4 as in King and Rebelo (2000). 1 5 Without

15

Hours: variable labor

Consumption

Wag e

Hours: fixed labor 1

10

0.6

5

0.2

2 0 0

20

40

3

New firms

0.4

Ag greg ate hours

2

Final g ood

1

0.6 Final g ood

0.4 0.2

0

0

Output

0.8

New firms

20

40

0

Number of firms

20

40

0.6

1

0.4

0.5

0.2

10

40

-0.05 -0.1

5

-0.15

0 20

40

0

Stock Market Value

20

40

0

Individual output

20

40

0.2

0.2 -0.5

0 20

40

40

0.4

0

0.1

20

Ag greg ate profits 0.6

0.5

0.4 0.05

0

Individual profits

0.6

0.15

0 0

20 Mark up

0

2 1.5

0

0

New entrants

0

20

40

0 0

Variable labor supply

20

40

0

20

40

Fixed labor supply

Figure 5: Impulse response to a temporary technology shock under Cournot competition

are constant or absent. Increased private demand of the …nal good fuels pro…ts which, despite lower mark ups, stay above the baseline for several quarters. The propagation of the shock is much stronger in the case of endogenous labor supply. The higher real wage boosts labor supply which translates into a substantial impact increase in output. This allows higher …rms’entry and thus a stronger countercyclical mark up response.16 To economize on space we do not report the dynamic response to a temporary shock for the conjectural variations and the Stackelberg competition models, however considerations made above extend to these cases.

3.3

Preference shocks

In this section we consider the impact of changes in demand determined by preference shocks. The Appendix introduces preference shocks into the model: in a nutshell, the sub-utility from consumption at time t reads now as log (Ct t ), thus a temporary positive variation in t leads to an urgency to consume by temporarily increasing marginal utility of consumption. Our focus will be on 1 6 The time-pro…le of the correlation of mark ups with the cycle is non-linear: since, in response to shocks, output increases on impact but the number of …rms increases slowly, the mark up falls more in future periods. This means that the correlation of mark ups and output is higher in absolute value when judged againts lagged values of output. This is a well known feature of the data, as shown in the VAR analysis of the introduction.

16

temporary shocks, assuming that the preference shifter autoregressive process: log (

t=

) = 0:9 log (

t=

t

follows the …rst order

) + "t

where is a parameter and "t is an independently and identically distributed random variable. We follow Wen (2006) and assume that at the steady state =C = 0:1.17 Figure 7 depicts the response of key variables to a one percent increase in t under Cournot competition. As in the previous section, continuous lines refer to the case with endogenous labor supply, while dashed line to that with …xed labor. As emphasized by Baxter and King (1992) and Wen (2006), taste shocks in standard general equilibrium models generate countercyclical investment dynamics due to the crowding out e¤ect.18 This is not the case in our framework, as long as we allow for endogenous labor supply. In that case the model replicates the positive comovement between output, consumption and investment which characterizes a typical business cycle; notice also that the comovement between output, aggregate pro…ts and the mark up is in line with the empirical evidence. The intuition for this result goes as follows. When labor supply is …xed, an increase in investment in the aftermath of the shock would require a shift of the …xed quantity of labor from the production of the …nal good to the business creation. In this case, however, the household would not be able to satisfy the desire to consume. Thus, the increase in demand can be satis…ed uniquely through disinvestment. This generates a perverse e¤ect which by reducing the number of new entrants and the overall number of …rms ultimately leads to a higher mark up. On the contrary, when labor supply is endogenous agents have and additional channel through which they can react to the shock. By increasing total hours worked they can set up new …rms without decreasing the production of the …nal good. The increase in hours allows a contemporaneous positive impact variation of investment, consumption and entry (and generates countercyclical mark ups once again).19 Clearly the volatility of variables in response to the shock is low. However our intent is that of proving the capability of the model to generate a standard business cycle in the face of a demand shock and not that of explaining business cycle variability resorting solely on demand shocks. 1 7 Empirical

evidence cannot say much about the steady state ratio =C. For this reason we follow the literature and give it a, conservative, small value. As the value of the ratio increases the impact response of variables, as well as their variability, in the face of a demand shock are ampli…ed. 1 8 Investment increases on impact in the baseline RBC model if one assumes that the taste shock has a near randow walk behavior. Clearly, however, this ad hoc solution does not settle the issue. 1 9 Similar results apply in the cases of Stackelberg competition and Conjectural Variations. Results are available from the authors.

17

Consumption

0.04

Hours: variable labor supply Hours: fixed labor supply 0.08 Final g ood 0 0.06 Ag gregate Hours -0.1 0.04 Final Good

0.02

0.02

0.08 0.06

New firms

0

0 0

20

40

0

New firms

-0.2

40

0

Number of firms

Output

-3

Wage

-3

Mark up

-2 -4 -6

-0.3 20

x 10 0

20

40

0

New entrants

0.08

20

x 10

40

6

0

0

0.06

4

0.04

-0.01

-0.1

0.02

-0.02

-0.2

2 0

-0.3

0 0

20

x 10

40

-3Stock M arket

0

Value

20

40

0

Individual output

20

40

0

Individual profits

20

40

Ag gregate profits

0.08

0.08

0.08

0.06

0.06

0.06

0.04

0.04

0.04

0.02

0.02

0.02

0 -2 -4 -6 0

20

40

0 0

20

0 0

40

Variable labor supply

20

40

0 0

20

40

Fixed labor supply

Figure 6: Impulse response to a temporary preference shock under Cournot competition.

4

Second Moments

To further assess the implications of endogenous market structures for the business cycle, we compute second moments of the key macroeconomic variables. In this exercise we follow the RBC literature by focusing on the case with endogenous labor supply and assuming that the only source of random ‡uctuations are technology shocks. We take the calibration of the parameters characterizing the technology process from King and Rebelo (2000), namely we set the shock persistence to A = 0:979 and its standard deviation to A = 0:0072. We use the same process as in King and Rebelo (2000) for comparison purposes with the RBC literature and with Bilbiie et al. (2007a,b). Jaimovich and Floetotto (2008) have prepared a measure of the TFP based on U.S. data taking into account the mark up variability and then …tting an AR (1) process to the constructed series. While they do not …nd signi…cant di¤erences from King and Rebelo (2000) in the estimated autoregressive coe¢ cient, they estimate a lower standard deviation of TFP innovations. Although our results are marginally a¤ected by a change in A , the main message of the analysis is not altered. We report in the left column of Table 1 the statistics on US data (1947– 1/2007-3) for output Y , consumption C, investment I, labor force L, aggregate pro…ts and the mark up 20 . The right column displays, instead, the theo2 0 Variables

have been logged. We report theoretical moments of HP …ltered variables with

18

V ariables Y C I L

(X) 1:66; 1:36 1:24; 0:87 5:00; 5:86 1:82; 0:57 8:08; 0:70 1:87; 0:09

(X) = (Y ) 1 0:75; 0:64 3:01; 4:31 1:10; 0:42 4:87; 0:51 1:13; 0:07

E (Xt ; Xt 1 ) 0:84; 0:67 0:80; 0:78 0:88; 0:63 0:90; 0:60 0:76; 0:63 0:85; 0:93

Corr(X; Y ) 1 0:76; 0:94 0:79; 0:92 0:88; 0:93 0:67; 0:99 0:27; 0:29

Table 1: Second moments. Left: US data; Right: Cournot Model V ariables Y C I L

(X) 1:29; 1:51 0:88; 0:78 6:72; 5:91 0:47; 0:84 0:71; 0:71 0:07; 0:07;

(X) = (Y ) 1 0:68; 0:51 5:21; 3:91 0:36; 0:56 0:55; 0:46 0:05; 0:05

E (Xt ; Xt 1 ) 0:66; 0:68 0:77; 0:78 0:57; 0:65 0:57; 0:65 0:60; 0:71 0:93; 0:94

Corr (X; Y ) 1 0:95; 0:93 0:90; 0:97 0:90; 0:96 0:98; 0:99 0:32; 0:17

Table 2: Second moments. Left: Stackelberg Competition; Right: Conjectural Variations with lambda=0.15 retical moment produced by our baseline model. The …gures suggest that the Cournot framework with homogeneous goods performs quite well in matching the contemporaneous correlation with output of mark ups and pro…ts, on which the neoclassical model is completely silent. While the volatility of aggregate output and hours is slightly lower with respect to those resulting form the standard RBC model (1.39 and 0.67 respectively, under the same calibration), the latter is outperformed both in terms of absolute and relative variability of consumption (0.6 and 0.43 respectively). This suggests that the competition e¤ect described in the earlier section helps addressing this notorious weakness of the standard RBC model. Notice, instead, that the volatility of real pro…ts and that of the mark up are substantially underestimated. Table 2 accounts for the statistics delivered by the model with Stackelberg competition and for the conjectural variations approach when = 0:15. Again, markup variability allows both models to deliver a stronger transmission of the shock on aggregate consumption. The conjectural variations approach reproduces a volatility of variables closer to that found in the data. We do not further emphasize the good performance of this model because of its somehow ad hoc nature. However, we interpret it as suggesting that competitive frameworks involving a degree of collusion stronger than that implied by the baseline Cournot model could help improving the performance of dynamic general equilibrium models at replicating the features of a smoothing parameter equal to 1600. Pro…ts include both the remunaration of capital and the extra-pro…ts due to market power: while we could not distinguish between the two, future research may try to do it.

19

Consumption

Output

0.8

3

0.6

2

0.4 1

0.2 0

0

10

20

30

0

40

0

10

Hours

20

30

40

Number of firms

3

1

2

0.5

1 0

0 -1

0

10

20

30

-0.5

40

0

10

Mark up

20

30

40

Aggregate profits

0.1

3 2

0

1 -0.1 -0.2

0 0

10

20

30

-1

40 Cournot

0

10

20

30

40

RBC model

Figure 7: Response to a temporary technology shock.

the business cycle. Both Tables 1 and 2 show that variables’autocorrelation is lower with respect to that which shows up in the data. This could be accommodated, as suggested by Bilbiie et al. (2007a,b), by imposing a longer time to build up a new …rm.21

5

Capital Accumulation

In this section we augment the baseline model with endogenous investment in physical capital. Di¤erently from Bilbiie, Ghironi and Melitz (2007,b) we assume that capital is used solely in the production of …nal goods.22 The resulting framework nests, as discussed below, both the standard RBC model and our model with homogenous goods and Cournot competition. Let us assume that …nal goods are produced with a standard Cobb-Douglas production function of the form: 1

yt (i) = At Kt (i) Lct (i) where 0 <

(19)

< 1. New …rms are created with the same technology as before.

2 1 However, this would come together with the cost of imposing more exogenous features on the neat framework presented. 2 2 The afore mentioned authors introduce capital in the business creation sector as well. We could adopt the same approach and con…rm their result for which the equilibrium exibits a cycling path, which is converging for high depreciation and diverging for low depreciation. The simpler approach adopted here avoids these phenomena.

20

The representative household holds the stock of capital and rents it to the producers of the …nal good. The stock of capital evolves according to: k

Kt+1 = 1

Kt + Itk

(20)

where Itk is time-t investment and k is the rate of physical depreciation. The household has a further intertemporal optimality condition with respect to the baseline model, which translates into the following Euler equation: k

k Ct = Et Ct+1 1 + rt+1

(21)

where rtk is the rental rate of physical capital. The symmetric Cournot equilibrium in the market for …nal homogenous goods leads again to the price pt = Nt M Ct = (Nt 1) where M Ct is the nominal marginal cost. The demand of inputs in the good producing sector is obtained by cost minimization, which requires: wt = (1

Kt Lct

)At

and: rtk = At

Kt Lct

M Ct pt 1

M Ct pt

The presence of capital accumulation implies that the new equation governing the dynamic of the number of …rms reads as: " 1 # Ct + Itk 1 At Lt (22) Nt = (1 ) Nt 1 + (1 ) At Kt while equation (14) governing the dynamic of consumption remains unchanged. Therefore (14), (20) and (22) fully determine the dynamics of Ct , Kt , and Nt . This model nests our basic model for = 0 and the standard RBC model when ! 0. Figure 8 reports the impulse response functions to a temporary technology shock delivered by the RBC model and by our augmented model with Cournot competition (we follow King and Rebelo, 2000, calibrating = 1=3 and k = 0:025). The Cournot competition framework can, evidently, generate a larger impact response of output, consumption and labor, together with the usual procyclical responses of entry and pro…ts and the countercyclical response of mark ups. Table 3 reports the second moments produced by the Cournot competition framework with capital accumulation.23 The performance of the model 2 3 The entry cost has been adjusted as to maintain the same mark-up and number of …rms as in the previous section, featuring no physical capital accumulation. Aggregate investment is now de…ned as the sum in investment in physical capital and the investment to create new …rms.

21

improves substantially for what concerns the second moments of both output and hours worked. Since the household can use an additional channel, beside investment in new …rms, to smooth consumption, the relative variability of consumption decreases with respect to cases without capital accumulation. The augmented model delivers a contemporaneous negative correlation between output and markups similar to that found in the data. Low variability of mark-ups and pro…ts remains, instead, an issue to be addressed in future research. V ariables Y C I L

(X) 1:69 0:59 4:58 1:30 1:57 0:11

(X) = (Y ) 1 0:34 2:71 0:77 0:93 0:06

E (Xt ; Xt 0:69 0:77 0:68 0:66 0:61 0:93

1)

Corr (X; Y ) 1 0:90 0:99 0:96 0:95 0:37

Table 3: Second moments. Cournot competition with capital accumulation

6

Conclusions

In this article we have studied a dynamic model with ‡exible prices where the structure of the markets is endogenous and accounts for strategic interactions of di¤erent kinds. Our approach belongs to the emerging literature on endogenous entry in the macroeconomy (Bilbiie et al., 2007,a,b; Etro, 2007, 2009; Etro and Colciago, 2007b; Devereux and Lee, 2001; Jaimovich and Floetotto, 2008) and, as others, it provides some improvements in the explanation of the business cycle compared to the standard real ‡exible prices framework. In particular, when tested against a temporary technology shock our basic model delivers second moments of macroeconomic variables which are in line with those provided by the standard RBC model. Further, it adds to the latter framework an endogenous characterization of the market structure which allows to explain the procyclical variability of pro…ts together with the countercyclical variability of mark ups found in the data. The model could be easily extended in various directions. For example in Etro and Colciago (2007), we extend it to a multisector economy with imperfect substitutability and both competition in quantities and in prices, we evaluate the welfare properties of the equilibrium and study di¤erent kinds of shocks. An analogous model could be employed for the analysis of open economy issues, to study international business cycle as in Ghironi and Melitz (2005) and optimal policy coordination in an open economy context. The simple form of Cournot competition allows one to easily incorporate technological di¤erences between countries and to verify the mechanisms of propagation of the shocks across 22

borders. Finally, one could extend the model to frictions in the labor market to study the dynamics of unemployment in a context where the processes of entry and exit of …rms are endogenous and hiring and …ring of workers can be associated with them (see Etro, 2009, for a wide discussion).

23

References Baxter Marianne and King Robert, 1992, Productive externality and cyclical volatility, Working paper 245, University of Rochester Bernard, Andrew, Stephen Redding and Peter Schott, 2008, Multi-Product Firms and Product Switching, mimeo, Yale University Bilbiie, Florin, Fabio Ghironi and Marc Melitz, 2007a, Monetary Policy and Business Cycles with Endogenous Entry and Product Variety, in NBER Macroeconomic Annual 2007, Edited by D. Acemoglu, Kenneth Rogo¤ and Michael Woodford, 299-353 Bilbiie, Florin, Fabio Ghironi and Marc Melitz, 2007b, Endogenous Entry, Product Variety, and Business Cycles, NBER wp 13646 Blanchard, Olivier and Nobuhiro Kiyotaki, 1987, Monopolistic Competition and the E¤ects of Aggregate Demand, The American Economic Review, 77, 4, 647-66 Bresnahan, Timothy and Peter Reiss, 1987, Do Entry Conditions Vary across Markets?, Brookings Papers on Economic Activity, 3, 833-81 Broda, Christian and David Weinstein, 2009 Product Creation and Destruction: Evidence and Price Implications, mimeo, Columbia University, The American Economic Review, in press Campbell, Je¤rey and Hugo Hopenhayn, 2005, Market Size Matters, Journal of Industrial Economics, 53, 1, 1-25 Chatterjee, Satyajit and Russell W. Cooper, 1993, Entry and Exit, Product Variety and the Business Cycle, NBER Working Paper No. 4562 Cook, David, 2001, Time to Enter and Business Cycles, Journal of Economic Dynamics & Control, 25, 1241-61 Devereux, Michael, Allen Head and Beverly Lapham, 1996, Aggregate Fluctuations with Increasing Returns to Specialization and Scale, Journal of Economic Dynamics and Control, 20, 627-56 Dedola , Luca and Stefano Neri, 2007, What does a Technology Shock do? A VAR Analysis with Model Based sign restrictions, Journal of Monetary Economics, 54, 512–549 Devereux, Michael and Khang Min Lee, 2001, Dynamic Gains from International Trade with Imperfect Competition and Market Power, Review of Development Economics, 5, 2, 239-55 Etro, Federico, 2004, Innovation by Leaders, The Economic Journal, 114, 4, 495, 281-303 Etro, Federico, 2007a, Competition, Innovation, and Antitrust, New York and Berlin, Springer Etro, Federico, 2007b, Endogenous Market Structures and Macroeconomic Theory, Review of Business and Economics, 52, 4, 543-66 Etro, Federico, 2008a, Growth Leaders, Journal of Macroeconomics, 30, 1148-72 Etro, Federico, 2008b, Stackelberg Competition with Endogenous Entry, The Economic Journal, 118, 1670-97 Etro, Federico, 2009, Endogenous Market Structures and the Macroeconomy, New York and Berlin, Springer Etro, Federico and Colciago, Andrea, 2007, Endogenous Market Structures and the Business Cycle, University of Milan, Bicocca, WP 126

24

Galì, Jordi, Mark Gertler and David López-Salido, 2007, Markups, Gaps, and the Welfare Costs of Business Fluctuations, Review of Economics and Statistics, 89, 1, 44-59 Jaimovich, Nir and Max Floetotto, 2008, Firm Dynamics, Markup Variations, and the Business Cycle, Journal of Monetary Economics, in press King, Robert and Sergio Rebelo, 2000, Resuscitating Real Business Cycles, Chapter 14 in Handbook of Macroeconomics, J. B. Taylor & M. Woodford Ed., Elsevier, Volume 1, 927-1007 Kydland, Finn and Edward Prescott, 1982, Time to Build and Aggregate Fluctuations, Econometrica, 50, 6, 1345-70 Lewis, Vivien, 2007, Business Cycle Evidence on Firm Entry, Macroeconomic Dynamics, forthcoming Manuszak, Mark, 2002, Endogenous Market Structure and Competition in the 19th Century American Brewing Industry, International Journal of Industrial Organization, 20, 673-932 Manuszak, Mark and Charles Moul, 2008, Prices and Endogenous Market Structure in O¢ ce Supply Superstores, Journal of Industrial Economics, 56, 94-112 Portier, Frank, 1995, Business Formation and Cyclical Markups in the French Business Cycle, Annales D’Economie et de Statistique, 37/38, 411–440 Ramsey, Frank, 1928, A Mathematical Theory of Saving, The Economic Journal, 38, 152, 543-59 Rotemberg, Julio and Michael Woodford, 2000, The Cyclical Behavior of Prices and Costs, Ch. 16 in Handbook of Macroeconomics, J. B. Taylor & M. Woodford Ed., Elsevier, Volume 1, 1051-135 Solow Robert, 1956, A Contribution to the Theory of Economic Growth, Quarterly Journal of Economics, 70, 65-940 Wen, Yi, 2006, Demand Shocks and Economic Fluctuations, Economics Letters, March 2006, 90, 3, 373-83 Wu, Yangru and Junxi Zhang, 2001, The E¤ects of In‡ation on the Number of Firms and Firm Size, Journal of Money, Credit and Banking, 33 2, 251-71

Appendix A: steady state This appendix shows that the basic model displays, for a given number of …rms, constant steady state output shares of consumption, investment, pro…ts and labor income. We provide details relative to the Cournot case, but identical steps apply to the other competitive frameworks we have considered in the text. The steady state counterpart of the aggregate resource constraint is Y = C + I . Since I = N e V the latter can equivalently be written as 1 = C=Y + N e V =Y . Evaluating at the steady state the Euler equation for bonds delivers (1 + r) = 1, where r is the steady state interest rate. The steady state counterpart of the Euler equation for stock holdings yields the steady state value of a …rm as:

V =

1 r+

25

or, using steady state pro…ts:

V =

1 r+

C N2

The share of pro…ts over consumption is N=C = 1=N , while the share of investment over aggregate consumption reads as:

V Ne =

1 r+

Ne =

N

r+

where we considered that the number of new entrants at the steady state is N=(1 Using previous results, the ratio of investment over total consumption reads as:

N eV = C r+

).

N 1 = C r+ N

To compute the output share of aggregate consumption notice that:

1=

C N eV C C + = Y C Y Y

1+

1 r+ N

thus the share of private consumption over output is:

C 1 = 1+ Y r+ N

1

The output share of investment is instead:

I N eV C 1 C = = = Y C Y r+ N Y (r + ) N + while the share of pro…ts over aggregate output is:

r+ 1 N = Y 1

N eV r + N eV r+ = = Y Y (r + ) N +

Finally notice that the output share of labor income, wL=Y , depends on the steady state number of …rms as follows:

wL =1 Y

r+ (r + ) N +

To fully characterize the steady state it remains to determine the number of producing …rms. This can be obtained by substituting equation (14) into equation (16) and solving for the only root larger than unity, reported as (17) in the text.

Appendix B: local stability and uniqueness This appendix studies local uniqueness and non-explosiveness of the rational expectation equilibrium of the model with endogenous savings and Cournot competition, whose steady state was provided in Appendix A. Given our purpose we can restrict

26

our analysis to a deterministic setting. In this case log-linear approximations around the steady state of equations (16) and (14) read respectively as:

nt+1 = (1

)nt

( + r) (N

1) ct

(23)

and

Et ct+1 =

(1

) 2 ( + r) (N (N 1) (1 )

1)

nt+1

(1 + r) 1) (1

(N

)

nt +

(1 + r) ct (24) (1 )

Rearranging yields a system of two equations into two variables with the following matrix form

Et ct+1 nt+1

ct nt

=B

where B is a 2 2 matrix of coe¢ cients, ct is a forward looking variable and nt+1 is a state variable. In general, the characteristic polynomial of matrix B reads as H ( ) = 2 T race (B) + det (B), and the condition for stability and uniqueness of the equilibrium is H (1) H ( 1) < 0. Matrix B is characterized by the following coe¢ cients: (1+r) B (1; 1) = (1 )

B (1; 2) = B (2; 1) =

( + r) +2 (

+r)( +r) (1 )

1 (1+r) (1 ) N 1 (1 ) + N 1

( + r) (N

1)

(N

1)

2 ( + r)

; B (2; 2) = (1

)

It follows that:

trace (B) =

(1 + r) (1 )

( + r) + 2

( + r) ( + r) (N (1 )

and

det (B) = (1 + r) In this case:

H (1) =

( + r) [r (1 (1 )

and

H ( 1) = 4 + 2r

1) + (1

)

(1 + r) ( + r) (1 ) 2N )

( + r) [r (1 (1 )

2N ] < 0

2N )

2N ] > 0

since N > 1. Therefore the equilibrium is always unique and locally stable. The same result holds under the alternative competitive frameworks we have considered in the text.

Appendix C: Endogenous labor supply and preference shocks

27

The baseline model can be easily extended to endogenous labor supply and taste shocks. In this case the representative agent has expected lifetime utility given by:

U = Et

1 X

(

t

1+1='

log (Ct

t=0

Lt 1 + 1='

t)

)

;'

0

where Lt represents hours worked and t is a random variable which permits us to analyze demand shocks. The parameter ' represents the intertemporal elasticity of substituion in labor supply. As described in Baxter and King (1991) and more recently in Wen (2006), a positive innovation to t by increasing the marginal utility of consumption leads to an urgency to consume. The …rst order conditions with respect to xt+1 and Bt+1 read respectively as:

Vt (Nt + Nte ) (Ct

n = Et [

1

t)

and

(Ct

1

t)

t+1

(Nt+1 ) + Vt+1 ] Nt+1 (Ct+1

= (1 + rt+1 )Et (Ct+1

t+1 )

t+1 )

1

o

1

Notice that with respect to the analysis in the main text we also we have an extra …rst order condition with respect to Lt :

Lt =

(Ct

'

wt

1 t)

Substituting this into the aggregate resource constraint delivers the number of new entrants as: '

Nte =

1

A1+' t

(Nt

1

(Ct

t)

1)

Ct

Nt

The latter in conjunction with the dynamic equation describing the evolution of the number of …rms delivers:

Nt+1 = (1

)Nt +

1

1

A1+' t

(Ct

(Nt t)

1)

'

Ct

Nt

(25)

The equilibrium equation for the dynamic of consumption is now:

Et

"

Ct+1 Ct

1 t+1 t

(Nt+1 1) Ct+1 + 2 Nt+1 Nt+1

#

=

(1

)

Nt 1 Nt

(26)

Equations (25) and (26) together with the exogenous processes for At and t speci…ed in the text determine the equilibrium path of consumption and the number of …rms, from which we can obtain the dynamics of the other relevant variables. Assuming that the steady state ratio C = (= 0.1 in our baseline calibration, as in Wen, 2006), steady state analysis is the same as before with labor supply satisfying:

L=

1 (1

w ) C

28

1 1+ 1 '

As in King and Rebelo (2000) we set the elasticty of labor supply '=4. The steady state number of …rms is the value of N that solves the following equation:

N=

1

A1+' h

(1

(1 (1

)

1+

(N 1)

'

) (1

))N 2

1

(1

i

)

which, given the right hand side is decreasing in N, has a unique solution. Equation (25) takes the following log-linear form:

nt+1 =

n nt

c ct

+

a at

+

t

with the associated coe¢ cients de…ned as:

1) + ' ( + (N 1) = (1 + ') ( + n ) = [' + (1 + ') n ] ( + n) = (1 ) =

n a c

(1

) (N

n)

and where n = ( + r) (N 1). The log-linear version of equation (26) is:

Et ct+1 = zn+1 nt+1

zn nt + zc ct + z Et (

t+1

t)

with coe¢ cients de…ned as:

where

c

= (1

) (1

zn+1

=

zn

=

zc

=

z

=

2 n c (N 1) c n (1 ) (1 + r) 1) c (N n (1 + r) ( + r) c (1 + r) ( + r) c

).

Appendix D: Data and Mark up measure Data on consumption, GDP, labor share, Investment, labor productivity, GDP de‡ator and pro…ts derive from FRED, the Federal Reserve Economic Database of the Federal Reserve Bank of St. Louis. Below, we report in brackets the mnemonics of each series. Compensation of Employees (COE): Billions of Dollars, Quarterly, Seasonally Adjusted Annual Rate (saar), 1947-01-01 2007-07-01.

29

Gross Domestic Product (GDP): Billions of Dollars, Quarterly, saar, 1947-01-01 2007-07-01. Proprietors’ Income with inventory valuation adjustment (IVA) and capital consumption adjustment (PROPINC): Billions of Dollars, Quarterly, saar, 1947-01-01 2007:07-01. Personal Consumption Expenditure (PCEC): Billions of Dollars, saar, 1947-01-01 2007-07-01. Corporate Pro…ts with inventory Valuation Adjustment (IVA) and Capital Consumption Adjustment (CPROFIT): Billions of Dollars, Quarterly, saar, 1947-01-01 2007-04-01. Gross Domestic Product Implicit Price De‡ator (PCEC): Index 2000=100, Quarterly, Seasonally Adjusted (sa), 1947-01-01 2007-07-01. Hours of all Persons, nonfarm business sector (HOANBS): Index 1992=100, Quarterly, sa, 1947-01-01 2007-07-01. Fixed Private Investment (FPI): Billions of Dollars, Quarterly, saar, 1947-01-01 2007-07-01. Output per hours all person (OPHNFB): Index 1992=100, Quarterly, sa, 1947-0101 2007-07-01. The index of net business formation and the series measuring the number of new incorporations are supplied by the Brad&Broadstreet corporation. The net business formation index runs from 1948:1 to 1995:3 (1967=100), for this reason we restrict our empirical analysis to this period. We thank Vivien Lewis for providing us the series on …rms’data. To derive the empirical measure of the markup we proceed as follows Rotemberg and Woodford (1999) assume the existence of overhead labor so that e¤ective labor Lot . Assuming a linear in for the production of consumption goods is Lct = Lt (e¤ective) labor production function, the price markup can be written as: t

=

Lt Yt wt Lt Lt Lot

whose log-linearization is:

bt =

lo 1

lo

^t L

sbt

where lo Lot =Lt represents the average share of overhead labor over total labor input (assumed to be equal to 0.2), st wt Lt =Yt is the labor share of income, and hatted variables indicate percentage deviations from its long run trend.

30