New-Keynesian Phillips curve with Bertrand competition and endogenous entry

Journal of Economic Dynamics & Control 51 (2015) 318–340

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New-Keynesian Phillips curve with Bertrand competition and endogenous entry Federico Etro a,1, Lorenza Rossi b,n a b

Department of Economics, University of Venice Ca’ Foscari, Sestiere Cannaregio, 30121, Fond.ta S.Giobbe 873, Venice, Italy Department of Economics and Management, University of Pavia, Via San Felice, 5, I-27100 Pavia, Italy

a r t i c l e in f o

abstract

Article history: Received 30 May 2014 Received in revised form 15 October 2014 Accepted 21 October 2014 Available online 12 November 2014

We derive a New Keynesian Phillips curve under Calvo staggered pricing and endogenous market structures with Bertrand competition. Both strategic interactions and endogenous business creation strengthen the nominal rigidities. Price adjusters change their prices less when there are more direct competitors that do not adjust, which reduces the slope of the Phillips curve. Current and future firms entering in the markets decrease current inflation because they reduce markups and the welfare-based price index. Endogenous entry amplifies the impact of both monetary and supply shocks. We also characterize the optimal social planner allocation, that can be replicated with a labor subsidy and a dividend tax (both decreasing in the number of firms) and zero producer price inflation. The optimal Ramsey allocation implies zero inflation tax in steady state. & 2014 Elsevier B.V. All rights reserved.

JEL classification: E3 E4 E5 Keywords: Bertrand competition Endogenous entry New Keynesian Phillips curve Staggered prices Inflation Optimal monetary policy

1. Introduction We examine the role of strategic interactions between a small and endogenous number of firms in the New Keynesian model with sticky prices à la Calvo (1983).2 Firms produce differentiated goods and compete à la Bertrand in each market rather than ignoring strategic interactions as under monopolistic competition.3 Moreover, following the recent literature on dynamic entry (Ghironi and Melitz, 2005; Bilbiie et al., 2008, 2012, 2014; Etro and Colciago, 2010; Faia, 2012) we endogenize the number of firms active in each market. We find that both strategic interactions and endogenous business creation

n

Corresponding author. E-mail addresses: [email protected] (F. Etro), [email protected] (L. Rossi). 1 Tel.: þ 39 0412349172. 2 The standard version of the model with a continuum of firms and monopolistic competition was developed in Yun (1996), King and Wolman (1996) and Woodford (2003). Recent extensions concern nominal wage rigidities (see Galì, 2008), firm-specific capital (Altig et al., 2011), open economies (Galì, 2008; Lagoa, forthcoming) and more. 3 A large body of empirical evidence in industrial organization (Campbell and Hopenhayn, 2005), trade (Feenstra and Weinstein, 2010) and macroeconomics (see the discussion in Rotemberg and Woodford, 1999 and Faia, 2012) suggests that common markets are characterized by relevant competition effects due to imperfect competition. http://dx.doi.org/10.1016/j.jedc.2014.10.009 0165-1889/& 2014 Elsevier B.V. All rights reserved.

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modify the standard New Keynesian Phillips curve (NKPC) and the reaction of the economy to shocks in a way that is more in line with the empirical evidence. Thanks to the strategic complementarities between price-setting firms, the modified NKPC is characterized by a smaller elasticity of inflation with respect to changes in the marginal costs4 compared to the standard New Keynesian model: under Bertrand competition price adjusters change their prices less when there are more firms that do not adjust. Accordingly, the slope of the NKPC decreases in the level of concentration of the markets and in the inter-sectoral elasticity of substitution between goods. Thanks to this lower slope, our model contributes to reconcile the micro-evidence of frequent price adjustments (Bils and Klenow, 2004; Nakamura and Steinsson, 2008) with the macroeconomic data indicating that inflation is rather inertial (see Altig et al., 2011). By endogenizing the evolution of the number of firms active we introduce an additional source of inflation inertia. We show that the impact of aggregate shocks is much stronger when taking into account endogenous business creation and that high substitutability between goods and high inter-sectoral concentration are essential for strategic interactions to amplify monetary shocks. On one side, the process of endogenous entry magnifies the effect of technology shocks by attracting more entry and strengthening competition, which in turn promotes consumption and labor supply and induces a hump-shaped pattern of inflation. On the other side, in front of an expansionary monetary shock both the real rigidities due to strategic interactions and the gradual creation of new differentiated varieties contribute to stabilize inflation so as to magnify the real effects beyond what happens in the baseline New Keynesian model. In this framework we also characterize the optimal monetary policy under different conditions. First, we characterize the unconstrained social planner problem and determine the required inflation and subsidies/taxes (financed with lump sum taxation) necessary to replicate such an efficient allocation: the first best allocation can be obtained through (1) a countercyclical labor subsidy that neutralizes the distortions associated with strategic interactions, (2) a countercyclical dividend tax that optimizes the process of business creation and (3) a countercyclical consumer price inflation which implies zero producer price inflation eliminating the distortion associated with price dispersion (the last results matches the one obtained under monopolistic competition and Rotemberg pricing by Bilbiie et al., 2008). Given the complexity of this optimal policy, we move to consider the Ramsey problem where the monetary authority can control only one instrument, the inflation rate, and none of the fiscal instruments above is available. Following the methodology proposed by SchmittGrohé and Uribe (2011) we focus on the steady state of such a constrained optimal allocation and confirm that zero producer price inflation remains optimal in the long run (as in Faia, 2012, under Rotemberg pricing).5 Other works, at least since Ball and Romer (1990), have already stressed the role of strategic complementarities between firms' prices as a source of real rigidities (see Nakamura and Steinsson, 2013, for a survey), but we are not aware of any formalization of Bertrand competition with price staggering as the natural source of strategic complementarities. A first approach to microfound real rigidities, due to Basu (1995), relies on the fact that each firm employs all the other goods as intermediate inputs. A second one, adopted by Woodford (2003) and Altig et al. (2011), relies on firm-specific inputs. In both cases marginal costs depend on firms’ own relative prices, so as to generate optimal prices increasing in the price index. A third approach, advanced by Kimball (1995), relies on a demand elasticity that is increasing in the relative price, generating again strategic complementarity between prices. Recent applications of this approach by Dotsey and King (2005), Levin et al. (2007) and Sbordone (2007) have been based on a generalization of the Dixit–Stiglitz aggregator to obtain elasticities increasing in prices and in the number of goods, but ignoring strategic interactions between price-setters.6 The recent research on dynamic entry has been mostly focused on standard monopolistic competition (Bilbiie et al., 2008, 2012, 2014). Strategic interactions and Bertrand competition have been explicitly introduced in a flexible price model by Etro and Colciago (2010), and the first applications in the NK framework have been developed by Faia (2012) to analyze the choice of the optimal state contingent inflation tax rates and Lewis and Poilly (2012) for estimation purposes.7 However, all these models neglect price staggering and adopt a price adjustment cost à la Rotemberg (1982), which implies that all firms (rather than a fraction of them) adjust prices simultaneously and identically in each period without price dispersion.8 Introducing time-dependent staggered pricing à la Calvo we obtain a different form of real rigidity associated with the substitutability between goods: if few firms in a market do not change prices after a cost shock, the price adjustment of the others are smaller the more substitutable are the goods. Contrary to this, Rotemberg pricing delivers higher adjustments when substitutability increases, in the same way under monopolistic and Bertrand competition.9 The paper is organized as follows. Section 2 presents the DSGE model with Bertrand Competition and endogenous business creation deriving the modified NKPC and the main results on the dynamics of the model. Section 3 discusses optimal monetary policy issues. Section 4 concludes. 4

Notice that this elasticity is proportional to the elasticity with respect to the output gap in this class of models. See also Bilbiie et al. (2014) on monopolistic competition with different homothetic preferences. Bergin and Feenstra (2000) have replaced CES preferences with translog preferences, that are homothetic and deliver an elasticity of demand increasing in a (finite) number of goods. Nevertheless, they focus on different issues and, again, neglect the role of strategic interactions. 7 See also Cecioni (2010) for an empirical assessment of the Bertrand model and Benigno and Faia (2010) on open economy issues. Colciago and Rossi (2011) have introduced labor market rigidities, while Colciago and Etro (2010) is about Cournot competition. 8 See also Auray et al. (2012). Cavallari (2013) has adopted Calvo pricing but focusing on monopolistic competition and ignoring strategic interactions. 9 Notice that the empirical analysis of Lewis and Poilly (2012) has found a small competition effect in their model with Rotemberg pricing, but this is not surprising since, besides various differences between setups, they have focused on a relatively high number of firms and low inter-sectoral substitutability. 5 6

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2. The model We analyze a decentralized economy with a continuum of sectors, each one characterized by Bertrand competition between a finite and endogenous number of firms. Consider a representative household with utility: (Z ) 1 1 υL1t þ ϕ t U 0 ¼ E0 ∑ β log C ιt dι  υ; ϕ Z0 ð1Þ 1þϕ 0 t¼0 where β A ð0; 1Þ is the discount factor, Lt is labor supply and Et ½ is the expectation operator at time t. Here C ιt represents the Dixit–Stiglitz consumption index for each sector ι A ½0; 1: " #θ=ðθ  1Þ nιt

∑ C ιt ðjÞðθ  1Þ=θ

C ιt ¼

ð2Þ

j¼1

where nιt A ½2; 1Þ is the number of producers of differentiated varieties in sector ι and the price index is 1  θ 1=ð1  θÞ ιt P ιt ¼ ½∑nj ¼  for any sector ι. Substitutability between goods is low across sectors, namely unitary given the 1 pιt ðjÞ log utility, but high within sectors, according to the constant elasticity of substitution θ A ð1; 1Þ. In each period t the optimality condition for consumption across sectors requires the same expenditure I t ¼ P ιt C ιt for any sector ι. Accordingly, symmetry between sectors allows us to consider a representative sector with consumption Ct, a price index P t and a number of firms nt. The household maximizes intertemporal utility choosing how much to work and to save in shares of a mutual fund of firms and in a complete contingent claims market. All profits are distributed as dividends and a fraction st of the mutual fund entitles to receive a corresponding fraction of the dividends of all the firms, whose expected value is Vt. Bt denotes bond holding and it is their nominal interest rate. Therefore, the budget constraint expressed in real terms is     W t Lt ð1 þ it  1 ÞP t  1 Bt  1  C t þ V t nt þnet st þ 1 þBt ¼ ð1 þ τt Þ þ þ 1  τD t dt þV t nt st T t Pt Pt

ð3Þ

e

where nt and nt are the active firms in period t and the new entrants at the end of the period in the representative sector, Wt D is the nominal wage, τt is a labor income subsidy, τt a capital income tax and Tt are lump sum taxes needed to finance the subsidy under budget balance. Households are subject to a standard borrowing limit to avoid Ponzi games. The FOCs derived from the household problem imply: ϕ

Lt : υLt ¼

ð1 þ τt Þwt Ct

ð4Þ

( ) ½ð1  τD V t ðnt þ net Þ t þ 1 Þdt þ 1 þ V t þ 1 nt þ 1 ¼ β Et st þ 1 : Ct Ct þ 1 Bt þ 1 :

  1 1 þ it ¼ βEt Ct ð1 þ π t þ 1 ÞC t þ 1

ð5Þ

ð6Þ

where wt ¼ W t =P t is the real wage and π t þ 1 ¼ P t þ 1 =P t  1 the consumer price inflation rate. Labor is the only input, and the output of firm i is yt ðiÞ ¼ At lt ðiÞ

ð7Þ

where At is total factor productivity at time t, and lt(i) is total labor employed by firm i. The real marginal cost is mct ¼ wt =At . Following the macroeconomic literature on endogenous entry (Ghironi and Melitz, 2005), the number of firms active in the representative sector follows the law of motion: nt ¼ ð1  δÞðnt  1 þnet  1 Þ

ð8Þ e

where δ A ð0; 1Þ is an exogenous exit probability and nt is the endogenous number of entrants in period t. This allows us to rewrite (5) as a standard asset price equation: ( )     C t þ 1  1  D V t ¼ β 1  δ Et 1  τt þ 1 dt þ 1 þV t þ 1 ð9Þ Ct We assume that the creation of a new firm requires a fixed investment of η in units of output, so that an increase in productivity affects the technologies for producing both goods and firms. The endogenous entry condition in each period sets Vt ¼ η

ð10Þ

as long as net 4 0, as we will assume to hold around the steady state. Using all this, we can rewrite the equilibrium conditions as

υLϕt C t ¼ ð1 þ τt Þmct At

ð11Þ

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" !#   ð1  τD t þ 1 Þdt þ 1 C t 1 ¼ β 1  δ Et C tþ11 1 þ

ð12Þ

η

Market clearing in the asset markets requires Bt ¼ 0 and st ¼ 1, therefore (3) delivers the following aggregate resource constraint: Y t ¼ C t þnet V t ¼ nt dt þ wt Lt :

ð13Þ

where Yt is total output. Since the demand of good i at time t is yt ðiÞ ¼ ½pt ðiÞ=P t    nt nt nt pt ðjÞ  θ ∑ yt ðjÞ ¼ At ∑ lt ðjÞ ¼ At Lt ¼ ∑ Y t  Δt Y t Pt j¼1 j¼1 j¼1

θ

Y t , from (7) it must hold that

and thus the aggregate output is   nt At Lt pt ðjÞ  θ Yt ¼ with Δt ¼ ∑ Pt Δt j¼1

ð14Þ

where the dispersion aggregator Δt depends on both price dispersion and the number of goods. Notice that with constant 1=ð1  θÞ producer prices we would have Δt ¼ nt , which is decreasing in the number of goods: while price dispersion increases the amount of labor needed to produce the composite final good, a higher number of goods available reduces the labor needed for it. 2.1. Calvo pricing and the modified NKPC Following Calvo (1983), we assume that in each period there is a fraction λ of firms across all sectors that cannot adjust the nominal price and maintain their pre-determined price, and a fraction 1  λ that can reoptimize the nominal price at the new level pt, so as to maximize the discounted value of future profits.10 In this environment, we derive a new version of NKPC that takes into account the effects of strategic interactions and business creation. For simplicity, we assume (as in Bilbiie et al., 2008) that newly created firms are randomly assigned the same price of the existing firms, and they have the same chance to adjust or not their prices (later we will consider the case in which all new entrants can adopt the optimal price initially). Remembering that there is a continuum of identical sectors, the average price index can be expressed as P t ¼ ½λP 1t 1θ þ ð1  λÞnt p1t  θ 1=ð1  θÞ

ð15Þ P 1t  θ

θ1

Elevating both sides to the power of 1  θ and dividing by we have 1 ¼ λð1 þ π t Þ the optimal relative price pnt  pt =P t . Solving for the latter we obtain !1=ð1  θÞ 1  λð1 þ π t Þθ  1 pnt ¼ ð1  λÞnt

n1  θ

þð1  λÞnt pt

which is increasing in the inflation rate and in the number of firms. To characterize the evolution of price dispersion in this model, we explicit the expression for    θ  n nt  1 pt  1 ðjÞ  θ  p Δt ¼ λ t ∑ þ 1  λ nt t Pt nt  1 j ¼ 1 Pt   nt ¼λ ð1 þ π t Þθ Δt  1 þ 1  λ nt pnt  θ nt  1

, where we defined

ð16Þ

Δt as follows:

ð17Þ

Replacing (16), we reach

Δt ¼ λ

nt ð1 þ π t Þθ Δt  1 þ ½ð1  λÞnt 1=ð1  θÞ ½1  λð1 þ π t Þθ  1 θ=ðθ  1Þ nt  1

ð18Þ

which is the equation of motion of the price dispersion aggregator depending on the inflation rate, on the number of firms and on the rate of change of the number of firms.11 The log-linearization of (15) around a zero inflation steady state, which satisfies P 1  θ ¼ np1  θ , provides   1λ b t ¼ λP bt  1 þ 1  λ p bt  P n^ θ1 t

ð19Þ

which depends also on the rate of change of the number of firms. This and the price-setting rule allow us to determine the path of consumer price inflation. Since its derivation is quite complex, in what follows we provide the main insights and leave the full derivation to the Appendix. 10 As well known, the alternative microfoundation of nominal rigidities advanced by Taylor (1979) relies on prices predetermined for a fixed number of periods. In principle Bertrand competition between a fraction of price-setting firms would lead to the same qualitative implications as those obtained under Calvo pricing. 11 This equation of motion generalizes what found by Yun (2005).

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In every period t, each optimizing firm i chooses the new price pt to maximize the expected discounted profits until the next adjustment, taking into account the probability that in each subsequent period the firm will be active ð1  δÞ and not adjusting its price (λ):   W pt  t þ k pt θ I t 1 At þ k ð20Þ maxEt ∑ ½λβð1  δÞk 1θ t pt ∑nj ¼ k¼0 1 pt þ k ðjÞ As customary under Bertrand competition, the prices of the intersectoral competitors in the initial period and in all the future periods are taken as given. The FOC of this problem, after simplifying and imposing symmetry of all the adjusted prices, reduces to     Wt þk Wt þk k k γ p  γ p  t t 1 1 1   γk At þ k At þ k pt E t ∑ 1  θ  θ E t ∑ ¼ 1  θ p1t  θ Et ∑ ð21Þ 1θ  θÞ Pt þ k P t2ð1 k ¼ 0 Pt þ k k¼0 k¼0 þk where γ  λβð1  δÞ. As shown in the Appendix, this can be used to derive a more compact expression for the optimal relative price: pnt F t ¼ K t þ ðpnt Þ2  θ Gt ðpnt Þ1  θ H t

ð22Þ

where Ft, Kt, Gt and Ht can be written in recursive form as F t ¼ 1 þ λβEt fð1 þ π t þ 1 Þθ  1 F t þ 1 g; Gt ¼ 1 þ λβEt fð1 þ π t þ 1 Þ2ðθ  1Þ Gt þ 1 g

Kt ¼

θ mc þ λβEt fð1 þ π t þ 1 Þθ K t þ 1 g θ1 t

and

H t ¼ mct þ λβEt fð1 þ π t þ 1 Þ2θ  1 H t þ 1 g

Due to the novel terms Gt and Ht, which derive from the strategic interactions, (22) cannot be solved explicitly for pnt . Nevertheless, we can log-linearize (21) around a zero inflation steady state and obtain 1

1

1

k¼0

k¼0

k¼0

b ct þ k  A b t þ k þ a2 Et ∑ γ k ðW a0 p^ t ∑ γ k þ a1 Et ∑ γ k P t þ kÞ ¼ 0

ð23Þ

where the coefficients a0 ¼  ½θ  2 þ n  a1 ðn  1Þa2 =ðn  1Þ, a1 ¼ ðθ  1Þ2 =ðθn þ 1  θÞ and a2 ¼ ðθ  1Þðn 1Þ=n are derived in the Appendix. The pricing equation (23) combined with the average price index (19) introduces two new mechanisms that modify the standard NKPC: (i) the strategic interaction mechanism and (ii) the endogenous business creation mechanism. It is then useful to disentangle the role played by each of the two mechanisms in modifying the standard NKPC. With Bertrand competition when the number of firms is constant, the only new mechanism at work is one of the strategic interaction. Thus, in what follows we first derive the NKPC with Bertrand competition and a constant number of firms. Then, to analyze the additional effect due to the entry process, we reintroduce an endogenous number of firms. 2.1.1. The role of strategic interactions We now derive the modified NKPC by assuming that the number of firms active in each sector is exogenous and constant, namely n^ t ¼ 0 and nt ¼ n for any period (and sector) and δ ¼0. b b t  1 Þ=ð1  λÞ. Replacing (23) with mc b t  λP ct þ k  A b c t þk ¼ W In this case, from (19) we derive p^ t ¼ ðP t þ k  P t þ k , and solving for p^ t in a recursive form we can obtain   b t þ ð1  λβÞðn  1Þ½θðn  1Þ þ 1mc c t þ λβEt p^ t þ 1 ; p^ t ¼ 1  λβ P ½θ 1 þ θnðn 1Þ

ð24Þ

bt  P b t  1 ¼ ð1  λÞðp^  P^ t  1 Þ we obtain Using the expression for consumer price inflation π t ¼ P t

π t ¼ β Et π t þ 1 þ

κ ðn  1Þ½θðn  1Þ þ 1 c mc λ½θ 1 þ θnðn  1Þ t

ð25Þ

where κ  ð1  λÞð1  λβÞ=λ is the traditional slope of the Phillips curve with respect to changes in the marginal cost under monopolistic competition. First of all, notice that the modified coefficient coincides with the traditional one in two cases: for n arbitrarily large, since we are back to the case in which each firm is negligible in its own market, and when θ-1, since firms tend to produce independent goods and strategic interactions disappear even between a small number of firms. In all the other cases, the slope of the NKPC is smaller than the standard one and, at most, it becomes a third of it in case of duopolies. This reduces drastically the impact of the changes in the real marginal costs on inflation, a result which is desirable from an empirical point of view since, as known in the literature, the basic NKPC implies an excessive reaction of inflation to changes in real marginal costs. Usual estimates from macrodata for the coefficient of the NKPC on the marginal cost change range between 0.03 and 0.05 (as for example in Levin et al., 2007). The standard model with monopolistic competition with λ ¼0.67 (price adjustments every 3 quarters) generates a coefficient equal to 0.166, but this does not fit with the mentioned macroevidence

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323

on the small reaction of inflation to changes in marginal costs. Instead, Bertrand competition generates a smaller coefficient, with a lower bound of 0.055, which is much closer to the macroevidence, mentioned above.12 Recently, Altig et al. (2011) have estimated a coefficient as low as 0.014. To match such an estimate, the standard Calvo model with monopolistic competition and β ¼0.99 requires a value of λ ¼0.9. This implies that prices adjust on average every 10 quarters, much more than what appears to be reasonable.13 The introduction of strategic interactions reduces the implied degree of nominal frictions up to λ ¼0.82, which implies price adjustments on average every 5.8 quarters: thus, it almost halves the average period of price adjustment, being more in accordance with the microeconomic evidence: indeed, Bils and Klenow (2004) find that half of prices last 5.5 months excluding temporary changes due to “sales”,14 but Nakamura and Steinsson (2008) increase this estimate to 12–13 months (more than 4 quarters) excluding both sales and product substitutions.15 The determinants of the modified slope of the Phillips curve are also important. Under Bertrand competition, and contrary to the standard case of monopolistic competition, the degree of concentration of markets (inversely related to n) and the substitutability between goods within sectors (increasing in θ) do affect the relation between changes in real marginal costs and inflation. In particular, it is immediate to derive the following comparative statics: Proposition 1. Under Bertrand competition and Calvo pricing the NKPC becomes flatter when the elasticity of substitution among goods increases or the number of firms decreases. The slope of the NKPC with respect to marginal cost changes depends on the substitutability between goods. This is quite important since most firms compete mainly with few rivals whose products are close substitutes. And in this case high values of the demand elasticity are associated with smaller price adjustments. When θ increases, firms become less prone to change prices, because their demand is more sensitive to price differentials. As a consequence, monetary shocks have smaller effects on inflation and therefore larger effects on the real economy. The limit behavior for (almost) homogenous goods is limθ-1 κ ðθ; nÞ ¼ κ ðn  1Þ2 = ½1 þnðn  1Þ, which represents the lower bound of the slope of the Phillips curve. The role of the number of firms is also crucial. Any change in real marginal costs implies a smaller reaction of current inflation in more concentrated markets because firms tend to adjust less their prices. This is going to amplify the real impact of monetary shocks and downplay the impact of technology shocks. Another implication of this result is that in the long run entry of firms increases the slope of the Phillips curve: this is in line with the idea that globalization (larger markets) and deregulation (lower entry barriers) increase the pass through of shocks on prices (see Benigno and Faia, 2010). However, a complete analysis of the role of entry requires us to endogenize the number of firms, which is our next step. 2.1.2. The role of endogenous entry In this subsection we derive the new version of NKPC by taking into account also the effects of endogenous business creation. In this case (19) provides p^ t ¼

b t  λP bt  1 n^ t P þ 1λ θ1

ð26Þ

Combining this with (23) we obtain (see the Appendix) the generalized NKPC as stated in the proposition that follows: Proposition 2. Under Bertrand competition with endogenous entry and Calvo pricing the consumer price inflation rate satisfies: 



ctþ π t ¼ β 1  δ Et π t þ 1 þ κ mc

βð1  δÞð1  λÞ ^ ð1  λÞ Et n t þ 1  n^ θ1 λðθ 1Þ t

ð27Þ

with

κ¼

ð1  λÞ½1  λβð1  δÞðn  1Þ½θðn  1Þ þ 1 λ½θ 1 þ θnðn  1Þ

ð28Þ

where n is the steady state number of firms. First of all, notice that the semi-elasticity of inflation with respect to real marginal costs and that with respect to expected inflation are substantially the same as in the model with a fixed number of firms reported in the previous section, 12 Notice that Smets and Wouters (2007) have estimated their model for the U.S. economy by replacing the Dixit–Stiglitz aggregator with the Kimball aggregator. The latter implies that the price elasticity of demand becomes increasing in the firm's price. Therefore, prices in the model become more rigid and respond by smaller amounts to shocks, for a given frequency of price changes λ. Using their macro-model, they obtain a much smaller estimate of the Calvo parameter, about λ ¼0.67, which implies that price contracts last 3 quarters on average, more in accordance with US microeconomic evidence (see Maćkowiak and Smets, 2008). Our setup suggests an alternative way to reconcile the micro with the macro-evidence. Needless to say, the different sources of real rigidity can be complementary. 13 Under Calvo pricing the average price duration is given by 1=ð1  λÞ. Since the period is a quarter, a value of λ equal to 0.9 means that prices adjust on average every 10 quarters (i.e. every 30 months). 14 Our model implies an even better fit to Euro-area data, where as emphasized by many authors the median consumer price lasts about 3.7 quarters (see Maćkowiak and Smets, 2008). 15 A change of the slope of the NKPC has important implications for the analysis of the optimal monetary policy aimed at minimizing a welfare loss in front of shocks (see Etro and Rossi, 2014).

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but now n refers to the steady state number of firms. The only difference is that for both the elasticities, the parameter β is now multiplied by the firms survival probability ð1  δÞ. This implies that, the higher the probability of future exit δ, the lower will be the overall firms' discount factor, that is β ð1  δÞ. As a consequence, a higher exit probability reduces the relevance of future inflation and increases the relevance of the current real marginal cost on price adjustments.16 Despite these differences, Proposition 1 holds as before: under Bertrand competition with endogenous entry and Calvo pricing the NKPC becomes flatter when the elasticity of substitution among goods increases or the steady state number of firms decreases. Endogenizing the number of firms introduces two new determinants of consumer price inflation: the current entry rate and the future entry rate.17 To verify the total impact of endogenous business creation on inflation, let us iterate forward the NKPC imposing zero expected inflation and zero net entry in the long run. We can easily obtain 1

c t þk  π t ¼ κ Et ∑ ½βð1  δÞk mc k¼0

1 ð1  λÞ2 Et ∑ ½βð1  δÞk n^ t þ k λðθ  1Þ k ¼ 0

ð29Þ

which shows that consumer price inflation depends positively on the present discounted value of future changes in the marginal cost and negatively on a discounted measure of the expected number of firms entering in the economy. Current and future firms entering in the markets contribute to reduce the current rate of inflation because they reduce markups and the welfare-based price index, with an impact that is inversely proportional to the elasticity of substitution between goods produced by the competitors; only when firms produce almost homogenous goods changes in the number of firms do not have any impact on consumer price inflation. Therefore, the endogeneity of entry does not affect the slope of the NKPC in a substantial way, but generates an additional source of rigidity of inflation, which is going to amplify monetary shocks even more than with a fixed number of firms. A quantitative analysis of these effects is our next task. 2.2. Impulse response functions The equilibrium equations of the model described above and their log-linearization are summarized in the Appendix. Technology shocks are introduced assuming that the log-deviation of productivity A^ t evolves according to a standard AR(1) process A^ t ¼ ρa A^ t  1 þ εa;t where ρa A ½0; 1Þ and εa;t is a white noise. To close the model we adopt a standard Taylor rule for monetary policy determining deviations of the nominal interest rate from its steady state as follows: ı^t ¼ γ π π t þ γ y yt þ μt ;

ð30Þ

where γ π 40 and γ y 4 0 are the coefficients on logdeviations of inflation and output from their steady state values, and μt is a stationary monetary policy shock, following an AR(1) process μt ¼ ρμ μt  1 þ εμ;t with ρμ A ½0; 1Þ and εμ;t white noise. To assess both the quantitative and the qualitative implications of our model, we now study its dynamic reaction to technology and monetary shocks. We adopt the following standard parameterization: β ¼0.99, ϕ ¼1/4 , λ ¼ 0.67, and δ ¼0.025.18 Moreover, since our framework focuses on a continuum of sectors producing goods with limited substitutability and on high inter-sectoral substitutability, we adopt a value of the elasticity of substitution that is in the high range of usual calibrations, namely θ ¼30.19 It is important to remark that this is consistent with non-negligible markups under Bertrand competition because strategic interactions increase markups compared to monopolistic competition and the number of rivals within each sector is assumed to be low in steady state: the markup is still 7% in duopolies, 5% with three competitors, 4.3% with five firms, 3.8% with ten firms. The cost parameter η is set to obtain three firms per sector in the steady state of the baseline model (with Bertrand competition and endogenous entry). This calibration implies that, on average, in each quarter a firm out of three does adjust its own price.20 The persistence of technology and monetary policy shocks are respectively ρa ¼ 0:9 and ρμ ¼ 0:5. The two shocks are calibrated to have a 1% standard deviation. We consider the case in which the monetary authority implements the standard Taylor rule, and therefore we set γ π ¼ 1:5 and γ y ¼ 0:125, while fiscal policy is absent, with τnt ¼ τtnD ¼ T t ¼ 0 at any time t. In Figs. 1 and 2 we show the impulse response functions to 1% technology and monetary shocks of three models with Calvo pricing: (i) our Bertrand model with endogenous entry (solid line); (ii) a model with monopolistic competition and endogenous entry of firms (croix line); (iii) a standard NK model with monopolistic competition and a fixed number of firms (dotted line). This allows to disentangle the additional effects of endogenous entry and strategic interactions compared to the traditional model. Let us consider first an expansionary productivity shock (Fig. 1). The reduction in firms marginal costs leads to price reductions only for the firms that can adjust their prices. Of course, this is common to all models embedding Calvo pricing. 16 Moreover, the long run Phillips curve becomes flatter compared to the standard model, since κ =½1  βð1  δÞ is decreased by the reduction in κ and the presence of δ 40. 17 Remarkably, notice that the steady state number of firms affects the impact of changes in the marginal cost on inflation, but is neutral on the impact of business creation on inflation. 18 None of these parameters affect the qualitative results of the model. 19 Qualitative results are not affected by the adoption of lower value of θ. 20 One can reinterpret the model assuming a higher number of goods produced by multiproduct firms (Minniti and Turino, 2013).

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Bertrand with Endogenous Entry

8

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Monopolistic Competition with Endogenous Entry

6

8

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Standard Monopolistic Competition

Fig. 1. IRFs to 1% technology shock.

In our model however, we have an additional source of real rigidity: strategic interactions reduce even further the endogenous size of the adjustments of the price-setters. As shown in Fig. 1, the inflation rate decreases by a much smaller amount compared to the standard model with monopolistic competition and a fixed number of firms, but also compared to the model with monopolistic competition and endogenous entry (the gap with the latter is entirely due to the strategic interactions). In spite of the competition effect induced by the shock, which would reduce markups in a flexible price setup (Etro and Colciago, 2010), the markups remain pro-cyclical in front of supply shocks due to the nominal rigidity: the stronger this is, the higher is the increase on impact of the average markup, that is the gap between average prices and lower marginal costs. Higher profitability attracts more entry generating a hump shaped pattern for the number of active firms and contributing to boost labor supply and output. The impact on output is so large as to induce a contractionary reaction of the monetary policy that partly crowds out the expansionary effect due to the negative response of inflation. As shown in the figure, this is in contrast with the expansionary reaction of monetary policy in the baseline NK model. In spite of this, the quantitative impact is largely different compared to such baseline model without both entry and strategic interactions, which generates a much smaller impact on output. Let us now consider the case of an expansionary monetary shock (Fig. 2). As usual, a reduction of the nominal interest rate leads to a reduction of the real interest rate which affects consumption and investment opportunities, boosting output and generating inflation. As usual, nominal rigidities imply a limited reaction of inflation which generates a positive response of output. However, strategic interactions due to Bertrand competition generate additional real rigidities, which induce an even smaller reaction of inflation and a much higher response of output. The average markups are countercyclical in all the three models in reaction to the monetary shock, but a key difference between the model with monopolistic competition and endogenous entry and the one with Bertrand competition and endogenous entry, is that the nominal interest rate cut induces business destruction in the former and business creation in the latter, thus overcoming an important drawback characterizing the model of Bilbiie et al. (2008). This implies that our model generates a much larger impact of the monetary intervention on the output level compared to the standard NK model, and even more compared to models with endogenous entry and monopolistic competition. Therefore, the presence of strategic interactions under staggered price setting strongly magnifies the impact of the monetary shock on output.

2.3. Calvo vs. Rotemberg pricing The first NK models with endogenous entry by Bilbiie et al. (2008, 2014) and Faia (2012) neglected price staggering but assumed sticky prices with adjustment costs à la Rotemberg (1982), and it is therefore useful to compare those models with ours. A recent work by Lewis and Poilly (2012) has augmented such models with other extensions for estimation purposes. In these models, the marginal profitability of a price adjustment increases with competition, therefore an increase in substitutability between goods has the opposite effect compared to our model: it increases the extent to which (all) firms

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8

10

12

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4

6

8

Standard Monopolistic Competition

Fig. 2. IRFs to 1% monetary policy shock.

adjust their prices. However, like in our setup, a reduction in the number of firms increases the markups and reduces the extent to which firms adjust their prices. Adopting a nominal price adjustment cost as ðχ =2Þ½pt ðiÞ=pt  1 ðiÞ  12 pt ðiÞC t ðiÞ for each firm i facing demand Ct(i) at time t, with χ 40, the price setting decisions lead to the following NKPC in terms of the consumer price inflation: 



π t ¼ β 1  δ Et π t þ 1 þ

ðθ  1Þðn  1Þ βð1  δÞEt nb t þ 1 þ nb t  1 ctþ mc  ϱðnÞn^ t χn θ 1

ð31Þ

where ϱðnÞ is a parameter derived in the Appendix. The introduction of strategic interactions reduces the slope of the NKPC under both Calvo and Rotemberg pricing (though for different reasons), but an increase in substitutability between goods θ makes the NKPC steeper under Rotemberg pricing and less steep under Calvo pricing. Moreover, the impact of past, current and future number of firms on inflation is qualitatively different in the two models.21 The empirical analysis of Lewis and Poilly (2012) has found a small competition effect in their model with strategic interactions. This is not surprising since (besides various differences between setups) they focus on a relatively high number of firms and low inter-sectoral substitutability.22 It would be interesting to estimate the present model focusing on concentrated markets with high inter-sectoral substitutability. 2.4. Product substitution The model with staggered pricing can be extended in many directions to enrich the propagation mechanism and augment the persistence of inflation. Besides traditional directions, such as introducing capital accumulation, consumption habits or wage stickiness, we will point out one that is peculiar to the current setup. As Nakamura and Steinsson (2008) have shown, a relevant portion of price changes is actually associated with the introduction of new products. We can account for this by assuming that the new entrants adopt right away the optimal price for their new products: this implies that an expansionary monetary shock gradually attracts new price-setting firms and induces a growing inflationary pressure. In each period, a fraction λ of the ð1  δÞnt  1 previously active firms cannot adjust the nominal price and has to maintain the pre-determined price, and the remaining fraction 1  λ plus all the new active entrants nt  ð1  δÞnt  1 can choose optimally the new price pt. The price index must satisfy P 1t  θ ¼ λð1  δÞP 1t 1θ þ ½nt  λð1  δÞnt  1 p1t  θ 21 Remarkably, this is a new case in which Calvo and Rotemberg pricing do not deliver NKPCs that are observationally equivalent as in the baseline model (see Ascari and Rossi, 2012, for the case of trend inflation). 22 In particular, their calibration implicitly assumes η ¼ 1, which leads to a steady state number of firms high enough to limit the competition effect. Moreover, in the simulations they confine the range of values for θ under θ ¼ 4, which again limits the strength of the competition effect.

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which leads to the following modified NKPC23: 

ctþ π t ¼ βEt π t þ 1 þ κ  mc



βEt n^ t þ 1 þ n^ t  1 1 þ β½λ 1  δ 2 ^  n θ1 λð1  δÞðθ 1Þ t

ð32Þ

The slope remains the same as in (28), but both past and future firms active in the market are pushing up current inflation. In such a model, the impact of expansionary shocks generates additional business creation, due to the advantage that new firms have (relative to the incumbents) in choosing optimal prices.24 3. Normative analysis Strategic interactions and endogenous business creation affect price dispersion and the gains from variety, which in turn affects welfare. In this section we characterize the optimal monetary policy when different instruments are available for the policymaker. The issue has been recently analyzed in related models with entry. Bilbiie et al. (2008) have shown that in case of monopolistic competition and Rotemberg pricing the optimal (unconstrained) allocation of resources can be replicated in the standard way, with the traditional constant labor subsidy and zero (producer price) inflation, which insures markup synchronization and minimizes price adjustment costs. In the same model, Bilbiie et al. (2014) have analyzed the Ramsey allocation when fiscal tools are not available but the monetary authority can choose as an instrument only the inflation rate: they have found that zero inflation remains optimal in steady state with CES preferences (but positive long run inflation can be desirable with other homothetic preferences). Since Bertrand competition and Calvo pricing add an additional inefficiency due to strategic interactions and a different inefficiency due to price dispersion, the optimal policy problem in our framework is more complex. 3.1. The social planner problem and the optimal fiscal and monetary policy We solve for the social planner problem and show that it can be replicated with zero producer price inflation at all times under more demanding conditions on the fiscal instruments than in standard models. First of all, the first best can be reached with two fiscal instruments rather than one: a labor subsidy and a dividend tax. Second, both instruments must be time-dependent, namely functions of the number of firms that are active in each period, which may create practical problems of implementation. The first best problem can be obtained assuming fully flexible prices and choosing directly consumption, labor supply and number of firms in each period to maximize the discounted sum of utilities. Using the resource constraint we have 1=ð1  θÞ C t þ net V t ¼ At Lt =Δt , where under flexible prices Δt ¼ nt . Using this, the endogenous entry condition and the evolution 1=ðθ  1Þ of the number of firms we can express consumption as C t ¼ At Lt nt  η½nt þ 1 =ð1  δÞ  nt . Accordingly, the unconstrained social planner problem becomes ! 1 υL1 þ ϕ t max E0 ∑ β log C t  t C t ;Lt ;nt þ 1 1þϕ t¼0

n  1=ðθ  1Þ tþ1 s:t: C t ¼ At Lt nt η nt 1δ After substituting the constraint in the objective function, the FOCs are ϕ

1=ðθ  1Þ

Lt : υLt C t ¼ At nt

ð33Þ

( " #) ð2  θ Þ=ðθ  1Þ   At þ 1 Lt þ 1 nt þ 1 nt þ 1 : C t 1 ¼ β 1  δ Et C tþ11 1 þ ηðθ  1Þ

ð34Þ

which are clearly different from the equilibrium conditions of the decentralized economy (11) and (12) in the absence of taxes and subsidies. The social planner equalizes the marginal disutility of labor to its social marginal productivity, which differs from the wage because it is positively affected by the number of goods provided in the market, leading to more labor supply and production. In turn, the social planner chooses how many goods to provide comparing the social marginal benefit from variety with the marginal cost of business creation, while the number of goods provided in the decentralized equilibrium depends on the comparison between gross profits and entry costs. It is easy to verify that, the social planner allocation cannot be replicated adopting the usual policies of a constant labor subsidy financed with lump sum taxation and a zero (producer price) inflation. However, we can show that the first best allocation can be replicated under zero (producer price) inflation by adopting two time-varying fiscal tools, namely an 23

Further details on the derivation of the NKPC and simulations are available from the authors. Notice that the mechanics of such a model in general equilibrium is complicated by the different values of new firms (price-setters by assumption) and incumbent firms. However, the advantage in price-setting for the new born firms does not appear to be an entirely realistic source of business creation. 24

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appropriate labor subsidy τt and an appropriate dividend tax τD t , always with lump sum taxation guaranteeing budget balance. To characterize the optimal fiscal and monetary policy that replicates the first best, notice that producer price stability, 1=ðθ  1Þ namely p ¼ nt P t constant, involves the following consumer price inflation rate:  1=ð1  θÞ nt π nt ¼ 1 ð35Þ nt  1 which is decreasing when the number of firms is increasing: a boom associated with entry of firms requires a reduction in the consumer price inflation. This leads all firms to adopt a constant price corresponding to the equilibrium price under flexible prices: p¼

θ nt  θ þ 1 W t ðθ  1Þðnt  1Þ At

ð36Þ

Diving by Pt both sides allows one to derive the real marginal cost as mct ¼

ðθ 1Þðnt 1Þ 1=ðθ  1Þ n θnt  θ þ 1 t

ð37Þ

Since optimality requires that the right hand sides of (33) and (11) with this marginal cost are equal, setting 1=ðθ  1Þ ð1 þ τt Þmct At ¼ At nt , we obtain the optimal labor subsidy as

τnt ¼

nt ðθ 1Þðnt 1Þ

ð38Þ 1=ðθ  1Þ

which is always positive and allows one to verify that the optimal producer price p ¼ nt P t is indeed constant. The optimal labor subsidy is decreasing in the elasticity of substitution and becomes negligible when goods are highly substitutable (since the market power distortion tends to disappear). Moreover, it is decreasing in the number of firms: a boom associated with entry strengthens competition and allows the policymaker to reduce the subsidy. 1=ðθ  1Þ To determine the optimal tax on dividends, let us use wt Lt ¼ mct At Lt ¼ At Lt nt =ð1 þ τnt Þ to express dividends as dt ¼

Y t  wt Lt At Lt ntð2  θÞðθ  1Þ τnt ¼ nt 1 þ τnt

Replacing in (12) we can rewrite the decentralized equilibrium condition as " !# ð2  θ Þ=ðθ  1Þ n   ð1  τD τt þ 1 t þ 1 ÞAt þ 1 Lt þ 1 nt þ 1 C t 1 ¼ β 1  δ Et C tþ11 1 þ ηð1 þ τnt þ 1 Þ

ð39Þ

ð40Þ

Optimality requires that also the right hand sides of (34) and (40) should be equal in each period, therefore we need n n ð1  τD t Þτ t ðθ  1Þ ¼ 1 þ τ t for any t, or

τnt D ¼

θ  1  nt nt ðθ 1Þ

ð41Þ

which is positive if the number of firms is small enough and goods are substitutable enough, namely if nt o θ 1, which is the case on which we will focus.25 In this case the market generates excess entry and it is optimal to limit entry through profit taxation (which here corresponds to capital income taxation). Remarkably, the optimal tax remains positive when goods are close to homogenous, limθ-1 τnt D ¼ 1=nt . The cyclical variation of the optimal dividend tax is also simple: τnt D decreases in the number of firms, therefore a boom associated with entry requires policymakers to reduce both the labor subsidy and the dividend tax. The main results are summarized as follows : Proposition 3. Under Bertrand competition with endogenous entry and Calvo pricing, the unconstrained social optimum allocation can be obtained with (1) a labor subsidy τnt ¼ nt =ðθ 1Þðnt 1Þ decreasing with entry, (2) a dividend tax τnt D ¼ ðθ  1  nt Þ=nt ðθ  1Þ decreasing with entry, and (3) zero producer price inflation. It is important to keep in mind the distortions that characterize our model: the sticky price distortion which induces inefficient price dispersion and the Bertrand competition distortion which generates inefficient production and inefficient entry.26 Three instruments are needed to correct these three inefficiencies. Price dispersion is eliminated adjusting 25 Notice that τnt D can be negative only if the number of firms is large enough. Remarkably, we can verify that in the limit case of monopolistic competition we need only one constant fiscal tool, since





τn ¼ lim  τtnD ¼

lim nt -1 t

nt -1

1

θ1

and a constant subsidy on labor income and dividends is enough to achieve the first best. 26 Notice that Bilbiie et al. (2008), consider a model with monopolistic competition and Rotemberg (1982) pricing and thus price dispersion is absent in their model. However, the Rotemberg framework is characterized by the presence of convex price adjustment costs. Price dispersion and price adjustment

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consumer price inflation in such a way that each firm finds it always convenient to maintain constant its own producer price: this eliminates price dispersion and thus the variability of output between firms. The labor subsidy fixes the static distortion associated with market power: however, since markups are variable and countercyclical, the subsidy can be reduced when the number of firms increases and competition is strengthened reducing markups, and increased when the number of firms decreases. Finally, the dividend tax is chosen to fix the dynamic distortion associated with inefficient entry. Of course one could replicate the first best with other instruments, for instance a sale subsidy instead of a dividend tax (alternative optimal taxation systems are discussed in Etro, 2009, Chapter 3). However, an important corollary of our result is that the traditional instrument of a labor subsidy financed with lump sum taxation is not sufficient to guarantee the general optimality of zero (producer price) inflation at all times when firms compete à la Bertrand.

3.2. The Ramsey problem with inflation as the only instrument Given the complexity of the optimal fiscal tools needed to replicate the first best allocation, we now move to analyze the optimal steady state allocation obtained in the absence of any fiscal instruments, namely with the inflation rate as the only instrument available for the monetary authority.27 This is important because, in the absence of appropriate subsidies and taxes (and lump sum taxation), inflation can be used to improve the allocation of resources, in particular to control markups and entry. Nevertheless, we are able to show analytically that the optimal inflation rate is still zero in the steady state (as in the standard NK model of Schmitt-Grohé and Uribe, 2011). The optimal plan is determined by the monetary authority to maximize the discounted sum of utilities given the constraints of the competitive economy, assuming that an ex ante commitment is feasible. Using the resource constraint we have C t þ net V t ¼ At Lt =Δt where the price dispersion aggregator depends on inflation and the number of firms as in (18). The endogenous entry condition and the evolution of the number of firms allow us to express consumption as C t ¼ At Lt =Δt  η½nt þ 1 =ð1  δÞ nt . Accordingly, the Ramsey problem can be stated as follows: ! 1 υL1t þ ϕ t max E0 ∑ β log C t  1þϕ fπ t ;Δt ;C t ;Lt ;nt þ 1 ;mct ;pnt g t¼0

n  At Lt tþ1 η nt s:t: C t ¼ Δt 1δ ð6Þ; ð8Þ; ð9Þ; ð10Þ; ð11Þ; ð13Þ; ð14Þ; ð16Þ; ð18Þ; ð22Þ;

τt ¼ τDt ¼ T t ¼ 0 Its general solution is provided in the Appendix. Looking at the optimal steady state inflation rate amounts to computing the modified golden rule steady state inflation, i.e. the steady state inflation rate obtained imposing steady state conditions ex post on the first order conditions of the Ramsey plan.28 One may expect that in the absence of fiscal instruments, some producer price inflation could be used to reduce the steady state distortions, that is to reduce the average markups and the rate of business creation, that are both above the efficient steady state level under Bertrand competition.29 However, Faia (2012) has already shown that this is not the case under Rotemberg pricing. This result was justified by the fact that the Rotemberg convex costs of price adjustment are the dominant distortions cancelled only under zero steady state inflation. Similarly, we show that zero inflation remains optimal in steady state also in the presence of Calvo pricing: the costs in terms of price dispersions associated with Calvo pricing under a positive inflation are too high to justify a systematic use of the inflationary tool to address the two mentioned inefficiencies. Thus, as in the standard NK model (King and Wolman, 1999; Schmitt-Grohé and Uribe, 2011) and as in the case of Rotemberg pricing with Bertrand competition and endogenous entry (Faia, 2012), also our case with staggered prices à la Calvo delivers the optimality of zero inflation in the long run. A long derivation provided in the Appendix allows us to prove: Proposition 4. Under Bertrand competition with endogenous entry and Calvo pricing, the optimal monetary policy without fiscal tools requires zero inflation rate in steady state. Two remarks can be useful to complete the picture on the topic of optimal policy in models with strategic interactions and endogenous entry. First, augmenting the model with more general homothetic preferences, as the translog used in (footnote continued) costs affect differently the slope of the NKPC and both the steady state and the model dynamics up to a second order approximation. Nevertheless, the two sticky price distortions move in the same direction of reducing the level of output as inflation increases. 27 The recent literature on optimal monetary policy often refers to the instrument used by the monetary authority, i.e. the inflation rate, as to the inflation tax. For this reason we keep the label Ramsey problem, even if there are not fiscal instruments in the toolkit available. For a classic analysis see Lucas and Stockey (1983) and for a modern treatment Ljungqvist and Sargent (2004). 28 To have a confirmation of the steady state results, together with our analytical proof we also compute the numerical solution of the modified golden rule steady-state equilibrium. The numerical solution is calculated by using an OLS approach, as described by Schmitt-Grohé and Uribe (2012). The code is available upon request. 29 See also King and Wolman (1999), Yun (2005), and in particular Khan et al. (2003), Schmitt-Grohé and Uribe (2007), Faia (2009) and Faia and Rossi (2013) for a discussion on Ramsey optimal policy under a distorted steady state.

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Bilbiie et al. (2014),30 would generate non-zero long run inflation also under Bertrand competition, but with differences due to the strategic interactions. Second, one may reintroduce distorsive fiscal instruments and characterize the Ramsey allocation in this case: without lump sum taxes and under budget balance a dividend tax would still be useful to reduce the entry distortion and finance public spending and the labor subsidy (or a reduction of the labor tax). Remarkably, during recessions one would have to increase the labor subsidy, or reduce the labor tax, and increase the dividend tax.31 4. Conclusion We have reconsidered the New-Keynesian framework under time-dependent staggered pricing à la Calvo in markets characterized by a small number of competitors engaged in Bertrand competition. Such a description of the relevant form of competition can be quite realistic from an industrial organization point of view. Most local markets for traditional goods and services do involve a small number of competitors and represent a big portion of our economies. However, also many global markets tend to be highly concentrated because of a process of escalation of R&D costs. In these conditions, strategic interactions cannot be ignored and create important real rigidities that affect the propagation of shocks. Price adjusters do change their prices less when there are more firms that do not adjust: this strengthens the impact of nominal rigidities, which is the mechanism at the heart of New-Keynesian economics. Further, endogenous market structures affect the fiscal policy prescriptions to restore the social planner allocation, but confirm the optimality of zero producer price inflation in the long run.

Acknowledgments The authors are grateful to Pierpaolo Benigno, Paolo Bertoletti, Fabio Ghironi, EsterFaia and seminar participants at Yonsei University, Hitotsubashi University and University of Milan for comments. Appendix A. The modified NKPC ðθ  1Þ=θ θ=ðθ  1Þ t Let us consider the representative sector with the usual CES consumption index C t ¼ ½∑nj ¼  , which 1 C t ðjÞ implies that demand at time t of good i is C t ðiÞ ¼ ½pt ðiÞ=P t   θ C t , where pt(i) is the price of the firm and the average price index is " #1=ð1  θÞ

Pt ¼

nt

∑ pt ðjÞ1  θ

j¼1

A.1. Calvo pricing In each period, only a fraction 1  λ of firms can reoptimize the nominal price to the optimal level pt, which maximizes the discounted value of future profits. Therefore, the average price index can be expressed as (15), whose log-linearization around a zero inflation steady state is   1λ b t ¼ λP bt  1 þ 1  λ p bt  P n^ θ1 t

ð42Þ

which depends on the change of the number of firms. In every period t, each optimizing firm i chooses the same new price pt to maximize the expected profits until the next adjustment, taking into account the exogenous probability of exit at each period δ and the probability that there will be a new adjustment λ: ( )  θ 1  pt k Ct þ k maxEt ∑ ð1  δÞk λ Q t þ k pt MC t þ k pt Pt þ k k¼0 where MCt is nominal marginal cost, Q t þ k ¼ β P t C t =P t þ k C t þ k is the stochastic discount factor, and we used the fact that demand at time t þ k is C t þ k ðiÞ ¼ ½pt ðiÞ=P t þ k   θ C t þ k . The problem can be simplified as k

1

maxEt ∑ γ k pt

k¼0

½pt MC t þ k pt θ I t 1θ tþk ∑j ¼ 1 pt þ k ðjÞ n

where γ  λβð1  δÞ and expenditure I t ¼ P t C t is taken as given. Notice that at each time t þ k the denominator nt þ k 1θ ∑j ¼ includes a fraction λ of the nt firms that cannot adjust the nominal price, maintaining the pre1 pt þ k ðjÞ 30 Endogenous market structures with demand functions derived from the general class of homothetic preferences are analyzed in Bertoletti and Etro (2014). 31 Therefore in case of endogenous market structures the classic optimality of zero taxes on capital income in the long run (Chamley, 1986) disappears.

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determined price level and a fraction 1  λ that can reoptimize. The FOC of this problem provides, after imposing symmetry:

γk

1

pt Et ∑

1θ k ¼ 0 Pt þ k

γ k ðpt  MC t þ k Þ

1

 θEt ∑

P 1t þkθ

k¼0

1 γ k ðp  MC   t þ kÞ t ¼ 1  θ p1t  θ Et ∑  θÞ P t2ð1 k¼0 þk

1θ tþk where we used P 1t þkθ ¼ ∑j ¼ . The FOC can be rearranged as 1 pt þ k ðjÞ    2  θ  1  θ pt p p Gt  t Ht Ft ¼ Kt þ t Pt Pt Pt n

where 1

F t  Et ∑ ðλβÞk ðP t þ k =P t Þθ  1 k¼0

and Kt 

1 θ E ∑ ðλβÞk mct þ k ðP t þ k =P t Þθ θ1 t k ¼ 0

which correspond to the terms emerging under monopolistic competition (see Benigno and Woodford, 2005), plus: 1

Gt  Et ∑ ðλβÞk ðP t þ k =P t Þ2ðθ  1Þ k¼0

and 1

H t  Et ∑ ðλβÞk mct þ k ðP t þ k =P t Þ2θ  1 k¼0

A closed form solution for pt =P t is not available as it was for monopolistic competition. However, the steady state implies ðp=PÞ1  θ ¼ 1=n and p  MC n ¼ p θn þ1  θ

ð43Þ

θ1 k We split the FOC in four parts. The log-linearization of pt ∑1 is k ¼ 0 γ ðP t þ k Þ 1

pð1 þ p^ t ÞEt ∑ γ k P θ  1 ð1 þ ðθ 1ÞP^ t þ k Þ k¼0

¼ pP

θ1

1

1

1

k¼0

k¼0

k¼0

∑ γ k þ pP θ  1 ðθ  1ÞEt ∑ γ k P^ t þ k þpP θ  1 p^ t ∑ γ k

θ1 k The log-linearization of θEt ∑1 is k ¼ 0 γ ðpt MC t þ k ÞðP t þ k Þ h 

i

 1    d θEt ∑ γ k p 1 þ p^ t  MC 1 þ MC P θ  1 1 þ θ  1 P^ t þ k tþk k¼0

1 1   ¼ θ ∑ γ k P θ  1 ðp  MC Þ þ θ θ  1 P θ  1 ðp  MC ÞEt ∑ γ k P^ t þ k k¼0

k¼0

1

þ θP θ  1 pp^ t ∑

k¼0

γ k  θpP

θ  1 MC

p

1

d Et ∑ γ k MC t þk k¼0

2ðθ  1Þ k The log-linearization of ð1  θÞp2t  θ Et ∑1 is k ¼ 0 γ ðP t þ k Þ 1

ð1  θÞ½p2  θ ð1 þ ð2  θÞp^ t ÞEt ∑ γ k P 2ðθ  1Þ ½1 þ 2ðθ  1ÞP^ t þ k  k¼0

1

¼ ð1  θÞp2  θ P 2ðθ  1Þ  ð1  θÞp2  θ P 2ðθ  1Þ ðθ 2Þp^ t ∑ γ k k¼0

2 2  θ 2ðθ  1Þ

2ð1  θÞ p

P

1

Et ∑ γ P^ t þ k k

k¼0

2ðθ  1Þ k Finally, the log-linearization of ð1  θÞp1t  θ Et ∑1 is k ¼ 0 γ MC t þ k ðP t þ k Þ 1

2ðθ  1Þ d bt þ k Þ ð1  θÞp1  θ ½1 þð1  θÞp^ t Et ∑ γ k MCð1 þ MC ð1 þ 2ðθ 1ÞP t þ k ÞP

¼ ð1  θÞMCp

k¼0 1

1  θ 2ðθ  1Þ

P

1

bt ∑ γ k ∑ γ k þ ð1  θÞ2 MCp1  θ P 2ðθ  1Þ p

k¼0

k¼0

332

F. Etro, L. Rossi / Journal of Economic Dynamics & Control 51 (2015) 318–340 1

1

k¼0

k¼0

1  θ 2ðθ  1Þ 2 d bt þ k þ ð1  θÞMCp1  θ P 2ðθ  1Þ Et ∑ γ k MC P Et ∑ γ k P t þ k  2ð1  θ Þ MCp

Simplifying for the constant terms through the steady state relation we get 1 1   0 ¼ pP θ  1 θ 1 Et ∑ γ k P^ t þ k þpP θ  1 p^ t ∑ γ k k¼0 k¼0   1   MC Et ∑ γ k P^ t þ k  θ θ  1 pP θ  1 1  p k¼0 1

 θP θ  1 pp^ t ∑ γ k þ θpP θ  1 k¼0





2  θ 2ðθ  1Þ

 1θ p

P

1 MC d Et ∑ γ k MC t þk p k¼0

1   2  θ p^ t ∑ γ k k¼0

1

þ2ð1  θÞ2 p2  θ P 2ðθ  1Þ Et ∑ γ k P^ t þ k k¼0

2 1  θ MC

þð1  θÞ p 



p

1  θ MC

þ 1θ p

p

pP

pP

2 1  θ MC

2ð1  θÞ p

p

2ðθ  1Þ b

2ðθ  1Þ

pP

1

pt ∑ γk k¼0 1

d Et ∑ γ k MC tþk

2ðθ  1Þ

k¼0 1

bt þ k Et ∑ γ k P k¼0

d b t þ k , and MC bt ; P Notice that pP θ  1 multiplies everything. Cancelling it and collecting terms in p t þ k we have 1

1

1

k¼0

k¼0

k¼0

d b t þ k þ a2 Et ∑ γ k MC a0 p^ t ∑ γ k þ a1 Et ∑ γ k P t þk ¼ 0

ð44Þ

where, using the steady state conditions ðp=PÞ1  θ ¼ 1=n and (43) we obtain " # θ1 ðθ 1Þ2 ðn 1Þ nþθ2 a0 ¼  n θn þ1  θ a1 ¼ a2 ¼

ðθ  1Þ2 θn þ 1  θ ðθ  1Þðn  1Þ n

kb To rewrite the expression above in terms of the real marginal cost we add and subtract a2 ∑1 k ¼ 0 γ P t þ k , so that we define d b c mc t þ k ¼ MC t þ k  P t þ k and (44) becomes 1

1

1

k¼0

k¼0

k¼0

b t þ k þa2 Et ∑ γ k mc c t þk ¼ 0 a0 p^ t ∑ γ k þ ða1 þ a2 ÞEt ∑ γ k P

ð45Þ

One can verify that a1 þ a2 ¼ a0 . Therefore we can rewrite 1

1

1

k¼0

k¼0

k¼0

b t þ k þ a2 Et ∑ γ k mc c tþk ¼ 0 a0 p^ t ∑ γ k  a0 Et ∑ γ k P and solve for p^ t in recursive form:         b t  1  λβ 1  δ a2 mc c t þ λβ 1  δ Et p^ t þ 1 p^ t ¼ 1  λβ 1  δ P a0

ð46Þ

ð47Þ

In order to substitute for p^ t and p^ t þ 1 , consider the log-linearization of the price index (42). Solving it for p^ t we have b t  λP bt  1 P n^ t þ 1λ θ1 b t þ 1  λP b t Þ=ð1  λÞ þEt n^ t þ 1 =ðθ  1Þ. Substituting both into (47) we have so that Et p^ t þ 1 ¼ ðEt P   b t  λP bt  1    P 1 b t  a2 mc ct n^ t ¼ 1  λβ 1  δ þ P a 1λ θ1 !0 b t þ 1  λP bt   Et P 1 E n^ þ þ λβ 1  δ 1λ θ 1 t t þ 1 p^ t ¼

ð48Þ

F. Etro, L. Rossi / Journal of Economic Dynamics & Control 51 (2015) 318–340

333

b t and simplifying, we obtain Multiplying each side by 1  λ, adding to both sides λP

    

   a ct λ Pb t  Pb t  1 ¼ λβ 1  δ Et Pb t þ 1  Pb t  1  λ 1  λβ 1  δ 2 mc a0  1  λ 1λ E n^ n^ þ λβ 1  δ  θ 1 t t þ 1 θ  1 t bt  P b t  1 and π e ¼ P bt þ 1  P b t . Then, dividing by We know that π t ¼ P tþ1 expression for inflation:

λ and substituting for πt and π etþ 1 , we reach an

ð1  λÞ½1  λβð1  δÞ a2 ct mc a0 λ βð1  δÞð1  λÞ ^ ð1  λÞ Et n t þ 1  n^ þ θ1 λðθ  1Þ t 



π t ¼ β 1  δ Et π t þ 1 

Using a2 =a0 ¼ ðn  1Þ½θðn  1Þ þ 1=½θ  1 þ θnðn  1Þ and substituting we finally get the NKPC. When n is exogenous, we impose n^ t þ 1 ¼ n^ t ¼ 0 and δ ¼0 which delivers (25), and when n is endogenous we obtain (27). A.2. Rotemberg pricing In every period, each firm i chooses the price pt to maximize its own value: ( )  2 1  χ pt þ k ðiÞ k  1 pt þ k ðiÞC t þ k ðiÞ V t ¼ Et ∑ ð1  δÞ Q t þ k pt þ k ðiÞ  MC t þ k C t þ k ðiÞ  2 pt þ k  1 ðiÞ k¼0 Its problem can be simplified as follows: ( " )  2 #  2 χ pt ðiÞ   p ðiÞ 1 pt ðiÞ 1  MC t pt ðiÞ  θ 1  δ βχ t þ 1  1 pt þ 1 ðiÞ1  θ 2 pt  1 ðiÞ pt ðiÞ max  nt 1θ 1θ t pt ðiÞ ∑nj ¼ p ðjÞ 2∑ 1 t j ¼ 1 pt þ 1 ðjÞ The FOC is (

"

) 2 # pt ðiÞ 1 pt ðiÞ 1   MC t   2 pt  1 ðiÞ   pt ðiÞ  θ  1θ Pt P 2t  θ "    2 #  1  θ χ pt ðiÞ 1θ MC t pt ðiÞ  θ 1 þ þθ  pt  1 ðiÞ Pt Pt P t pt 2P t     θ    p χ pt ðiÞ pt ðiÞ ðiÞ pt þ 1 ðiÞ 1  θ pt þ 1 ðiÞ 1 ¼0  þβ 1δ χ t þ1 1 Pt pt ðiÞ Pt þ 1 P t pt  1 ðiÞ p2t ðiÞ

χ



Imposing symmetry, defining the real marginal cost as mct ¼ MC t =P t and the (gross) producer price inflation as Π t ¼ pt =pt  1 , and finally using nt ¼ pt =P t θ  1 , we have       1θ χ  θ=ðθ  1Þ 1=ð1  θÞ 2  1  θ nt 1 þ 1  θ mct nt þ ðΠ t 1Þ nt 1 þ1  θ þ θmct nt 2  

  1θ χ nt 2  ðΠ t  1Þ  χ ðΠ t  1ÞΠ t þ β 1  δ χ ðEt Π t þ 1  1ÞEt Π t þ 1 ¼0 2 nt þ 1 Log-linearizing the FOC provides the producer price inflation rate π~ t as     θ  1 ðn  1Þ θ 1 þ θnðn  1Þ ^ ct  n mc π~ t ¼ β 1  δ Et π~ t þ 1 þ χn χ n θn  θ þ1 t which goes back to the result of Bilbiie et al. (2008) under monopolistic competition (for n-1) and is a particular case of  θ  1 the result in Lewis and Poilly (2012). To recover the consumer price inflation πt, we use nt ¼ pt =P t to derive   π~ t ¼ π t þ nb t  nb t  1 =ðθ  1Þ from which we obtain the final NKPC in the rate of consumer price inflation (31) with        θ  1 þ θnðn  1Þ θ  1  χ n θn  θ þ 1 1  β 1  δ    ϱðnÞ ¼ χ n θn  θ þ 1 θ  1

334

F. Etro, L. Rossi / Journal of Economic Dynamics & Control 51 (2015) 318–340

Appendix B. Equilibrium conditions of the model The equilibrium conditions in the absence of taxation are summarized as follows: Consumption Euler equation:

1 þ it C tþ11 1: C t 1 ¼ βEt 1 þ πt þ 1 Asset pricing condition: ( )    Ct þ 1  1 V t þ 1 þ dt þ 1 2: V t ¼ β 1  δ Et Ct Entry condition: 3: V t ¼ η Aggregate resource constraint: 4: C t þnet V t ¼ nt dt þ wt Lt Dividends: 5: dt ¼

Y t  wt Lt nt

Law of motion of the number of firms:    6: nt ¼ 1  δ nt  1 þ net Labor market equilibrium: ϕ

7: At mct ¼ υLt C t Production function: 8: Y t ¼

At Lt

Δt

Relative price: 9: pnt ¼

1  λð1 þ π t Þθ  1 ð1  λÞnt

!1=ð1  θÞ

Price setting equation:  2  θ  1  θ 10: pnt F t ¼ K t þ pnt Gt  pnt Ht where Ft, Kt, Gt and Ht can be written in recursive form as n o 11: F t ¼ 1 þ λβEt ð1 þ π t þ 1 Þθ  1 F t þ 1 12: K t ¼

n o θ mc þ λβEt ð1 þ π t þ 1 Þθ K t þ 1 ; θ1 t

n o 13: Gt ¼ 1 þ λβEt ð1 þ π t þ 1 Þ2ðθ  1Þ Gt þ 1 n o 14: H t ¼ mct þ λβEt ð1 þ π t þ 1 Þ2θ  1 H t þ 1 Price dispersion:

  nt 1 ð1 þ π t Þθ Δt  1 þ 1  λð1 þ π t Þθ  1 pnt 15: Δt ¼ λ nt  1 plus a monetary policy rule. To solve for the steady state, notice that:   θ  1 ðn  1Þ 1=ðθ  1Þ mc ¼ n θn  θ þ 1

     Fixing A and η, we have V ¼ η and consequently d ¼ η 1  β 1  δ =β 1  δ . Moreover, we have output y ¼ ALn1=ðθ  1Þ , wage ϕ w¼ mcA, and consumption C ¼ mcA=υL . Using these in the steady state system we obtain a relation between number of

F. Etro, L. Rossi / Journal of Economic Dynamics & Control 51 (2015) 318–340

335

firms and labor demand: LðnÞ ¼ nη

    1 ð1  βð1  δÞÞ 1=ðθ  1Þ ðθ 1Þðn 1Þ n A 1 βð1  δÞ θn  θ þ 1

and a single equation for n, that is        θ  1 ðn  1Þ ð2  θÞ=ðθ  1Þ A θ  1 ðn 1Þ ð2  θÞ=ðθ  1Þ δη η 1  β 1  δ   þ n n LðnÞ  ϕ þ ¼ A¼0 υ 1δ θn  θ þ1 θn  θ þ 1 β 1δ From the latter we derive n and, accordingly, L(n) and all the other steady state variables. The equilibrium can be log-linearized around the steady state in the following equations:   C^ t ¼ Et C^ t þ 1  ^ı t  Et π t þ 1

    d b b b V^ t ¼ β 1  δ Et V^ t þ 1 þ β 1  δ Et d t þ 1  Et C t þ 1  C t V bt ¼ 0 V e ne V b nd ndb wL wL bt þ n Vn be þ bt þ d btþ b w C Vt ¼ n Lt tþ C t C C C C C

Y b wL wL bt þ n bt b bt ¼ Y d w Lt t dn dn dn   e n^ t þ 1 ¼ 1  δ n^ t þ δn^ t b t þ ϕb bt ¼C Lt w b t þ mc ct bt ¼A w bt n bt þ b bt ¼ A yt  Y Lt þ θ 1 plus the monetary rule (30) and the NKPC (27). Notice that ı^t  log it  logð1=βÞ is the log-deviation of the nominal interest rate from its steady state value. Appendix C. The Ramsey problem The Ramsey optimal plan is determined by maximizing the discounted sum of utilities given the constraints of the competitive economy and assuming that ex-ante commitment is feasible. Since it is not possible to combine all equations of the competitive equilibrium in a single implementability constraint, we follow an hybrid approach in which the competitive equilibrium conditions are summarized with a minimal set of equations, as reported below. In this context, the monetary authority chooses the policy instrument, namely the inflation rate, to implement the optimal allocation obtained as solution to the Ramsey problem. We now reduce the 15 equilibrium conditions to a smaller number. First of all, notice that the equation of the instrument it , i.e. the Euler equation for bonds is not necessary. This comes from the fact that the nominal interest rate enters only into the Euler equation for bonds. Then, once the monetary authority has chosen the optimal path for Ct and π t (using all the other constraints) the nominal interest rate is determined ex post. As for the standard NK model, the Lagrangean multiplier associated with the Euler equation for bonds is always zero. This means that we are left with 14 equations. We then combine all the static equations with the eight dynamic ones, i.e the law of motion of nt , the equation for the optimal price, the price dispersion equation and the Euler equation for firms shares, plus the 4 auxiliary variables Ft, K t , Gt and Ht. In particular, using the resource constraint and substituting for the equation of firms profits, we have C t þ net V t ¼ nt dt þwt Lt ¼ At Lt =Δt . Combing now the endogenous entry condition and the equation of evolution of the number of firms we get

n  AL t t t þ1 Ct þ η  nt ¼ 1δ Δt ϕ

Substituting into the previous equation the labor market equilibrium equation solved for C t ¼ Lt constraint of the Ramsey problem, which is ð1RÞ:

At mct

υ

1þϕ

n  At Lt ϕ tþ1 þη  nt Lt ¼ : 1δ Δt

At mct =υ, we find the first

336

F. Etro, L. Rossi / Journal of Economic Dynamics & Control 51 (2015) 318–340

Then, considering      dt þ 1 C t 1 ¼ β 1  δ Et C tþ11 1 þ

η

ϕ

ϕ

and substituting for C t ¼ ðLt At mct Þ=υ and C t þ 1 ¼ ðLt þ 1 At þ 1 mct þ 1 Þ=υ, we have 2 3 !1 !  1  ϕ ϕ   Lt þ 1 At þ 1 mct þ 1 Lt At mct dt þ 1 5 4 ¼ β 1  δ Et 1þ

υ

υ

η

Substituting profits dt ¼

At Lt At mct Lt  n t Δt nt

we find the second constraint of the Ramsey problem: 2 3 !1 ϕ Lt þ 1 At þ 1 mct þ 1 6 7 !1 ϕ 7   6 Lt At mct υ 6 7 ¼ β 1  δ Et 6  ð2RÞ: 7    6 7 υ At þ 1 Lt þ 1 4 1þ 1 5  At þ 1 mct þ 1 Lt þ 1 ηnt þ 1 Δt þ 1 The third and the fourth constraints come from the equations of price dispersion and of the optimal price:

  nt 1 ð1 þ π t Þθ Δt  1 þ 1  λð1 þ π t Þθ  1 pnt ð3RÞ: Δt ¼ λ nt  1 and  2  θ Gt  p1t  θ H t ð4RÞ: pnt F t ¼ K t þ pnt We then rewrite the four auxiliary variables Ft, Kt, Gt and Ht in recursive form, that is n o ð5RÞ: F t ¼ 1 þ λβEt ð1 þ π t þ 1 Þθ  1 F t þ 1 ð6RÞ: K t ¼

n o θ mc þ λβEt ð1 þ π t þ 1 Þθ K t þ 1 θ1 t

n o ð7RÞ: Gt ¼ 1 þ λβEt ð1 þ π t þ 1 Þ2ðθ  1Þ Gt þ 1 and n o ð8RÞ: H t ¼ mct þ λβEt ð1 þ π t þ 1 Þ2θ  1 H t þ 1 Finally, we have n

ð9RÞ: pt ¼

1  λð1 þ π t Þθ  1 ð1  λÞnt

!1=ð1  θÞ

The Ramsey problem follows: 1

max

fLt ;π t ;Δt ;nt þ 1 ;mct ;F t ;K t ;Gt ;Ht ;pnt g s:t:

E0 ∑ β

t

log

t¼0

 At Lt

n  υL1 þ ϕ tþ1 η nt  t Δt 1δ 1þϕ

ð1RÞ  ð9RÞ

!

ð49Þ

where we have already substituted Ct into the utility function. We define the Lagrangean multipliers for each constraint as μj;t for j ¼ 1; 2; …; 9. The FOCs that solve the Ramsey problem are !

n  At Lϕ 1 At ϕ tþ1 ϕ1  t Lt : 0 ¼  υLt þ μ1;t ϕη nt Lt  1þϕ C t Δt 1δ Δt !2 ϕ ϕ1 L At mct Lt At mct þ μ2;t ϕ t 0

υ

υ

!2 ϕ1   ϕ Lt At mct B  ϕβ 1  δ Lt At mct B υ υ 1B þ μ2;t  1 B    βB 1 At Lt @  1þ  At mct Lt ηnt Δt

1 C C C C C A

F. Etro, L. Rossi / Journal of Economic Dynamics & Control 51 (2015) 318–340

337

0 1 !  1    Lt ϕ At mct 1@  1 At A  μ2;t  1 A mc β 1δ υ ηnt Δt t t β

    nt θð1 þ π t Þθ  1 Δt  1 þ μ3;t θ  1 λð1 þ π t Þθ  2 pnt  1 nt  1   1  μ5;t  1 β θ  1 λβð1 þ π t Þθ  2 F t  μ6;t  1 β  1 θλβð1 þ π t Þθ  1 K t    1  1  μ7;t  1 β 2 θ  1 λβð1 þ π t Þ2ðθ  1Þ  1 Gt  μ8;t β λβ 2θ  1 ð1 þ π t Þ2θ  2 Ht !θ=ð1  θÞ ! 1  λð1 þ π t Þθ  1 λð1 þ π t Þθ  2  μ9;t ð1  λÞnt ð1  λÞnt

π t : 0 ¼  μ3;t λ

Δt : 0 ¼ 

1þϕ

1 At Lt At Lt þ μ1;t C t Δ2 Δ2

nt þ 1 þ μ3;t  μ3;t þ 1 βλ ð1 þ π t þ 1 Þθ nt t !1 !  Lt ϕ At mct 1 At Lt

t

 μ2;t  1 β

nt þ 1 : 0 ¼ 

1



β 1δ

υ

η nt Δ2t

η 1 1 η ϕ þβ η  μ1;t þ 1 βηLϕt þ 1 þ μ1;t L Ct þ 1 1  δ Ct 1δ t 2 3   1

ϕ

6 6  μ2;t β 1  δ 6 4

Lt þ 1 At þ 1 mct þ 1



ηn12

7 7 7 5

υ



A t þ 1 Lt þ 1

At þ 1 mct þ 1 Lt þ 1  1 2 nt þ 2 þ μ3;t þ 1 β  λ ð1 þ π t þ 1 Þθ Δt þ μ3;t þ 2 β λ 2 ð1 þ π t þ 2 Þθ Δt þ 2 nt nt þ 1 

mct : 0 ¼ μ1;t

At

υ

 μ2;t

Δt þ 1

tþ1

ϕ

Lt

At mct

υ

!2

2



ϕ

Lt

ϕ

Lt

υ

At

At mct

2

ϕ

Lt

3 At

7   υ υ  16 þ β 1  δ μ2;t  1 β 6

 7 4

5 1 At Lt  1 þ ηnt Δt  At mct Lt 



þ β 1  δ μ2;t  1 β

1

ϕ

Lt

At mct

!1

υ

1

ηnt

At Lt  μ6;t

θ μ θ  1 8;t

λβð1 þ π t Þθ  1 ¼ 0

F t : μ4;t pnt þ μ5;t  μ5;t  1 β

1

K t :  μ4;t þ μ6;t  μ6;t  1 β

1

βλð1 þ π t Þθ ¼ 0

 2  θ 1 Gt :  μ4;t pnt þ μ7;t  μ7;t  1 β λβð1 þ π t Þ2ðθ  1Þ ¼ 0  1  θ 1 H t : μ4;t pnt þ μ8;t  μ8;t  1 β λβð1 þ π t Þ2θ  1 ¼ 0

  2 þ μ9;t pnt : 0 ¼ μ3;t 1  λð1 þ π t Þθ  1 pnt

  n 1  θ     θ  þ μ4;t F t  2  θ pt Gt þ 1  θ pnt Ht The 10 FOCs together with the 9 constraints associated to the 9 Lagrangean multipliers define a system of 19 equations in 19 unknowns characterizing the optimal inflation. Out of the steady state one can only characterize its fluctuations with simulations. However, following Schmitt-Grohé and Uribe (2011), we can determine the optimal inflation rate in the long run. We first characterize the steady state version of the Ramsey plan and we then guess π ¼ 0. Given this, we verify whether our guess

338

F. Etro, L. Rossi / Journal of Economic Dynamics & Control 51 (2015) 318–340

belongs to the solution of the system, that is if zero inflation steady state is the Ramsey optimal steady state inflation rate. Imposing π ¼0, the steady state of the Ramsey allocation implies: !   ALϕ 1A δn ϕ  1   υLϕ þ μ1 ϕη  1þϕ L 0¼ CΔ 1δ Δ !2 L  ϕ Amc L  ϕ  1 Amc þ μ2 ϕ

υ

0

υ

1    ϕ1 L Amc 1 AL A þ μ2 @  ϕ 1  δ 1þ  AmcL υ υ ηn Δ 0 1 !  1     L  ϕ Amc 1 1 A  μ2 @ 1  δ A mc υ ηn Δ  L  ϕ Amc



!2

   1   0 ¼  μ3 λθΔ þ μ3 θ  1 λ pn  μ5 θ  1 λF      μ6 θλK  μ7 2 θ  1 λG  μ8 λ 2θ  1 H  μ9

0¼ 

λ

n1=ðθ  1Þ ð1  λÞ

!  1      L  ϕ Amc 1 AL AL1 þ ϕ 1 AL þ μ þ μ 1  βλ μ 1  δ  1 3 2 C Δ2 υ η nΔ2 Δ2

η 1 1 η ϕ 2 1 þ β η  μ1 βηLϕ þ μ1 L þ μ3 β λ Δ C n 1δ C 1δ !1       L  ϕ Amc AL 1 1  μ2 β 1  δ  mc  μ3 β λ Δ 2 n υ ηn Δ

ð50Þ

ð51Þ

ð52Þ

0¼ 

!2 !1 ϕ  A L  ϕ Amc L  ϕA  L At mct AL þ 1  δ μ2 t 0 ¼ μ1  μ2 υ υ υ υ ηn !2      L  ϕ Amc L  ϕA AL 1 θ μ 1þ  mc  μ6 þ 1  δ μ2 υ υ ηn Δ θ 1 8

μ4 ¼  μ5

  1λ pn



μ4 ¼ μ6 1  λ





μ8 1  λ





At mc

υ

 ð58Þ

2

  1  θ     θ  þ μ 4 F  2  θ pn G þ 1  θ pn H þ μ9 ¼ 0

δn ϕ AL1 þ ϕ þη L  ¼0 1δ Δ

     AL 1  mc ¼0 1β 1δ 1þ ηn Δ 

Δ ¼ pn

1

 2  θ  1  θ pn F  K  pn G þ pn H¼0 F¼

1 1  λβ

ð55Þ

ð57Þ

pn1  θ

μ3 1  λ pnt

ð54Þ

ð56Þ

  μ 1λ μ4 ¼ 7 n2  θ p

μ4 ¼ 

ð53Þ

ð59Þ

ð60Þ ð61Þ ð62Þ ð63Þ ð64Þ

F. Etro, L. Rossi / Journal of Economic Dynamics & Control 51 (2015) 318–340

K¼

 mc θ ¼ mcF θ 1 1  λβ θ1

θ



G¼F H¼

339

mc ¼ Fmc 1  λβ

pn ¼ n1=ðθ  1Þ

ð65Þ ð66Þ ð67Þ ð68Þ

Once we have guessed π ¼0 the steady state system (50)–(68) becomes a system of 19 equations and 18 unknowns: pn ; F; G; H; K; mc; Δ; L; n; μ1 ; μ2 ; μ3 ; μ4 ; μ5 ; μ6 ; μ7 ; μ8 ; μ9 . As shown by Schmitt-Grohé and Uribe (2011), in order to verify that zero inflation is indeed a solution, we should find that one equation is redundant. Indeed, the FOCs with respect to π and pn , that is Eqs. (51) and (59) can be shown to be identical. Using (55)–(58) Eq. (51) can be rewritten as follows:

        1   2  θ   1  θ  1  λ þ μ4 pn θ  1 F  θK  pn 2 θ 1 F þ pn 2θ  1 H  μ 9 pn ¼ 0 ð69Þ  μ3 pn Further, using (63) together with (66), rearranging and simplifying, this becomes

   1   2  θ  2  θ      1  λ  μ4 pn F þ pn 2F þ pn θF þ pn 1  θ 1  θ H  μ9 pn ¼ 0  μ3 pn Finally, multiplying Eq. (59) by ð pnt Þ, it coincides with (70). Thus, one of the two equations is redundant and solution of the steady state system.

ð70Þ

π ¼ 0 is a

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