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Endogenous markups in the new Keynesian model: Implications for inflation–output trade-off and welfare
Economic Modelling 51 (2015) 626–634
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Endogenous markups in the new Keynesian model: Implications for inflation–output trade-off and welfare Ozan Eksi ⁎ TOBB University of Economics and Technology, Department of Economics, Sogutozu Street, No: 43, Ankara 06560, Turkey
a r t i c l e
i n f o
Article history: Accepted 3 September 2015 Available online 29 September 2015 Keywords: Endogenous markups Monetary policy Inflation–output trade-off
a b s t r a c t This paper extends the standard new Keynesian (NK) model by using the endogenous markup setting a la Kimball (1995). In this setting, consumers' price elasticity of demand for a good is increasing in the good's relative price level, which affects the desired price markup of firms. In the literature, this setting is mainly used to improve the NK models in matching sluggishness of prices in the data. Our paper analyzes the monetary policy implications of the model. It is shown that unlike the cases of real wage rigidity and exogenous markup shocks, the endogenous markup setting does not improve the NK models in generating the inflation–output trade-off. It is also discussed that the optimal monetary policy in this environment is to target the flexible price equilibrium. © 2015 Elsevier B.V. All rights reserved.
1. Introduction A well-known result in the literature is that the standard NK model with nominal rigidities is unable to create the inflation–output trade-off faced by the central banks. As an illustration of this, let ynt represent the potential output under flexible prices, yt represent the actual output in the presence of a fraction of firms with fixed prices, and ỹt = yt − ynt is the output gap. Using these definitions, the following new Keynesian Phillips Curve (NKPC) equation ~t πt ¼ βEt fπtþ1 g þ κ y
ð1Þ
indicates that stabilizing the output gap leads to the stabilization of inflation, called the divine coincidence by Blanchard and Galí (2007). 1 A common way of obtaining inflation–output trade-off from NK models is to modify Eq. (1) by adding a cost push shock (exogenous changes in price or wage markups; Clarida et al., 1999) as follows: ~t þ ut : πt ¼ βEt fπtþ1 g þ κ y
ð2Þ
According to (2), a positive cost push shock raises current inflation, unless the level of output is reduced via contractionary monetary
⁎ Tel.: +90 312 290 4542; fax: +90 312 290 4104. E-mail address: [email protected]. URL: http://oeksi.etu.edu.tr. 1 In the AS–AD (aggregate demand–aggregate supply) framework, the divine coincidence can be described as the ability of the monetary authority to keep inflation constant and output equal to its natural level by counteracting to the changes in AD and LRAS (longrun aggregate supply) curves.
http://dx.doi.org/10.1016/j.econmod.2015.09.005 0264-9993/© 2015 Elsevier B.V. All rights reserved.
policies. As a result, the divine coincidence no longer holds.2 The weak side of this approach is that the cost push shocks are exogenous. In an attempt to obtain inflation–output policy trade-off in response to more conventional shocks, Blanchard and Galí (2007) use a model with real wage rigidity, and Divino (2009) uses an open economy model.3 Our paper, on the other hand, uses a real price rigidity, the endogenous markup setting, and asks whether this setting can generate inflation–output trade-off from NK models. The paper then investigates the optimal monetary policy under this setting. A real price rigidity is a source that makes firms unwilling to change their relative prices. The need for such rigidities stems from the fact that the monetary non-neutralities cannot be fully explained by nominal rigidities of simple NK models.4 With regard to the theoretical ways of obtaining real price rigidities, Rotemberg and Woodford (1992) use strategic pricing decisions between colluding firms, Galí (1994) uses a 2 In the AS–AD framework, cost push shock is a shift in the SRAS (short-run aggregate supply) curve. Or, put it differently, it is a shock to the Phillips curve that does not change the potential output. In response, monetary authority can stabilize either the price or the output level. 3 Blanchard and Galí (2007) show that under the real wage rigidity—i.e. the sluggish adjustment of real wages for a fraction of workers at each point in time—the difference between natural rate of output and first best output—that occurs when there is no market imperfection or distortion such as nonzero markups—is not constant. Therefore, stabilizing the output gap (yt − ynt ) is no longer equal to stabilizing the welfare relevant output gap (yt − ytfb) and is no longer desirable from the welfare point of view. Divino (2009) offers a model in which both foreign and domestic demand shocks increase the real marginal cost of production, which makes the same effect with the exogenous cost push shocks in (2). 4 Ball and Romer (1990) is the first study emphasizing that real rigidities have a crucial role in explaining nominal rigidities and non-neutrality of money. Christiano et al. (1999) and Romer and Romer (2004) provide a more recent discussion on monetary nonneutralities. Klenow and Kryvtsov (2008) supply micro empirical findings indicating that nominal rigidities based on the frequency of firms' price changes are not enough to explain price sluggishness in the data.
O. Eksi / Economic Modelling 51 (2015) 626–634
setting in which the elasticity of substitution across goods in consumption to differ from that in production, Altig et al. (2011) use short run marginal cost curve of firms that is increasing in its own output in the presence of firm-specific capital. Importance of these studies arises from the fact that all of them improve the fit between a theoretical model and the real data. Yet, probably the most common way of obtaining real price rigidity in the monetarist models is the endogenous markup setting of Kimball (1995). This model, instead of using the constant elasticity of demand assumption of Dixit and Stiglitz (1977), assumes increasing price elasticity of demand so that the desired markups of firms are decreasing in their relative prices. For instance, when a negative supply shock hits the economy, the firms that are able to increase their prices will be confronted with higher elasticity of demand, leaving them with less incentive to increase their prices. Likewise, at the time of a positive supply shock, the firms that are able to lower their prices will be confronted with lower elasticity of demand, leaving them with less incentive to reduce their prices. Consequently, the change in prices is lower, and as a result, the change in output gap is higher in this setting compared to the case with constant elasticity of demand assumption. In this paper we analyze if this countercyclicality between prices and output gap changes the implications of the standard NK model for the inflation–output policy trade-off. Since strategic interactions between the pricing decisions of the firms are used to enhance NK models towards a more realistic picture of the world markets, we also study the optimal monetary policy implications of the model. In this regard, Blanchard and Galí (2007) note that “ Rotemberg and Woodford (1992, 1996) restricted their analysis to the real implications of endogenous markups, without exploring their consequences for inflation and monetary policy in the presence of nominal frictions”. Hence, the aim of our paper is to fill this gap in the literature. The method we follow in this study is to incorporate the endogenous markup setting into the standard new Keynesian framework and then use it together with the same economic model of Blanchard and Galí (2007). The results show that the endogenous markup setting does not lead to the inflation output trade-off but only creates a real rigidity in prices that results in lower volatility in inflation and higher volatility in output gap when compared to the case with constant markups. With regard to the optimal monetary policy in this environment, the results show that stabilizing inflation ensures not only stabilization of the output gap but also stabilization of the welfare relevant output gap, which is the discrepancy between the actual output and the first best level of output with zero markups. Therefore, the optimal monetary policy is to target the flexible price level of output and zero inflation. The rest of the paper is organized as follows. Section 2 explains the model and analyzes its implications for the inflation–output trade-off of central banks. Section 3 investigates the optimal monetary policy in this environment. Section 4 discusses the results and Section 5 concludes.
2. The model We borrow the baseline model from Blanchard and Galí (2007) and extend it with endogenous markup setting.
2.1. Consumers An infinitely living representative consumer seeks to maximize the expected value of his/her discounted future utilities
E0
∞ X t¼0
βt U ðC t ; Nt Þ
627
where the utility function is given by U ðC t ; Nt Þ ¼ logðC t Þ− exp fξg
N1þϕ t : 1þϕ
Here, β is the discount factor for future utility, C is composite consumption—defined below—,N is employment, ϕ is the inverse of the elasticity of labor supply with respect, ξ is a preference parameter between consumption and leisure. This utility function is subject to the following sequence of budget constraints P t C t þ Q t Bt ≤ Bt−1 þ W t Nt þ T t ; where P is the aggregate price index for consumption goods, B represent purchases of one-period bonds, Q is the bond price, W is the nominal wages and T is a lump-sum component of income. Under the assumption of perfect competition in the labor market, the above-defined consumer problem can be used to obtain the following intratemporal optimality condition between the real wages and the marginal rate of substitution of leisure for consumption
MRS ¼ −
Un expfξgNϕ W ðin logs : mrs ¼ w ¼ c þ ϕn þ ξÞ; ð3Þ ¼ ¼ P Uc 1=C
and also the following intertemporal optimality condition between saving and consumption Q t ¼ βEt
U c;tþ1 P t : U c;t P tþ1
2.1.1. The consumption aggregate and the endogenous markup setup The consumption aggregate, C, is defined as (time and firm specific subscripts are suppressed unless they are necessary) Z
1 0
C ψ i di ¼ 1; C
ð4Þ
where ψ(ρi) satisfies ψ(1) = 1, ψ′(ρi) N 0, ψ″(ρi) b 0 for all ρi = Ci/C ≥ 0, where ρi is the share of good i in the consumption basket (this share, in the case of a representative consumer model of ours, is also equal to the market share of the firm producing that good). Note that when ψ(ρ) = (ρ)( − 1)/, so that ψ(Ci/C) = (Ci/C)( − 1)/, the consumption function in (4) results in Z C¼
1 0
!=ð−1Þ ð−1Þ=
Ci
di
;
which is the Dixit–Stiglitz type consumption aggregator that assumes constant elasticity of substitution () between different goods. The specification in (4), on the other hand, follows Kimball's (1995) approach and uses consumers' price elasticity of demand for a good that is increasing in its price level. To derive the formula for this elasticity, we start with the cost minimization problem of the consumer, which can be written as follows Z 1 Z 1 C min P i Y i di s:t: 1 ¼ ψ i di: C 0 0 In this setup, the goods are non-storable and are consumed within their production period. In a closed economy environment, this assumption implies C = Y. Using this equality, the solution of the above
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O. Eksi / Economic Modelling 51 (2015) 626–634
problem leads to the following implicit demand equation (shown in Appendix A.1) ψ0
Yi P ¼ ψ0 ð1Þ i ; Y P
ð5Þ P ðY i Þ ¼ M ðρi Þ
where P is the aggregate price index in the economy. Since ψ′(ρi) is a strictly decreasing function (as ψ″(ρi) b 0), higher values of Pi are associated with lower Yi/Y ratios. That is, Eq. (5) implies that, as usual, there is an inverse relationship with the relative price of good i and its relative demand. The actual price elasticity of the demand of consumers is then obtained as (in Appendix A.2) ðρi Þ ¼ −
∂Y i =Y i ψ0 ðρi Þ : ¼− ∂P i =P i ρi ψ″ ðρi Þ
this case, the term (ρi)/[(ρi) − 1] is the markup (the ratio of price to the marginal cost of production), which we denoted as M(ρi). Therefore, Eq. (9) can be rewritten as 7
ð6Þ
∂ΨðY i Þ : ∂Y i
ð10Þ
The elasticity of this markup, M(ρi), with respect to the firm's market share is denoted by ημ(ρi). Following Kimball (1995) and Woodford (2003), we approximate it around ρ = 1. We then denote it by ημ⁎. In the equilibrium within the flexible price framework, the symmetry of firms implies that market shares, and as a result markups are constant along the firms. Hence, Eq. (10) can be written as P = M ⋅ (∂Ψ(Y)/∂Y). Next, by dividing both sides of this equation by the price level P, we find
Notice that unlike Dixit and Stiglitz (1977) approach, the elasticity in (6) is not constant but depends on the share of good i in the consumption basket of consumer, ρi. Moreover, as in Kimball (1995), we assume (ρi) is decreasing in ρi. 5 Intuitively speaking, this assumption implies that when the share of good i in the consumption basket of consumer increases (or when, according to (5), the relative price of good i compared to aggregate price index decreases), consumers' price elasticity of demand for good i is decreasing.
where MC is the real marginal cost of production that is equal to nominal marginal cost of production divided by the price level. Hence, markup is the reciprocal of the real marginal cost of production, that is, percentage changes in this term are reflected as opposite changes in markups. The markups can also be calculated from the following equation
2.2. Firms
MPN ¼ M W=P ðin logs : mpn ¼ μ þ wÞ
Monopolistically competitive firms use a Cobb-Douglas type of production function
where MPN is the real marginal product of labor and W/P is the real wage. Equality of Eqs. (11) and (12) is given in Appendix B.2. Intuitively speaking, Eq. (11) relates real unit price of a good to its production cost in terms of labor, and Eq. (12) relates the real price of goods produced by one labor to the cost of employing that labor. Taking w and mpn from Eqs. (3) and (8), and using the equilibrium condition in a closed economy: y = c, Eq. (12) can be written as
Y ¼ Sα N1−α ðin logs : y ¼ αs þ ð1−α ÞnÞ;
ð7Þ
where S is non-produced input with exogenous endowment. In this case, shocks to S are interpreted as supply shocks.6 N is the labor input. Using (7), the real marginal product of labor can be found as MPN ¼ ð1−α ÞY=N ðin logs : mpn ¼ ðy−nÞ þ logð1−α ÞÞ:
ð8Þ
2.3. The equilibrium with flexible prices In case of flexible prices, with no capital in the model, firms solve a static profit maximization problem max P ðY i ÞY i − ΨðY i Þ;
μ ¼ −ð1 þ ϕÞn2 þ logð1−α Þ − ξ;
ð11Þ
ð12Þ
ð13Þ
where the subscript ‘2’ denotes the second best (natural) level of labor. It is second best because even though the prices are flexible there is still an imperfect competition in the goods market (μ N 0). Combining (13) with the production function in (7) yields the following equation μ ¼ −ð1 þ ϕÞ
y2 −αs þ logð1−α Þ − ξ: ð1−α Þ
ð14Þ
Using the equality in (11), 0 = μ + mc, the final equation becomes
Yi
where Ψ(Yi) is the nominal cost function. Taking the first order condition with respect to Yi, and rearranging the equation, we find (see Appendix B.1) ðρi Þ ∂ΨðY i Þ : P ðY i Þ ¼ ðρi Þ−1 ∂Y i
1 ¼ M MC ðin logs : 0 ¼ μ þ mcÞ
ð9Þ
Eq. (9) relates nominal prices to the nominal marginal cost of production and to the consumers' price elasticity of demand for goods. In
5 Kimball (1995), notes that it is easy to find a variety aggregator ψ to match any desired dependence of the elasticity of demand (ρi) on ρi. Dotsey and King (2005) further specify an aggregator for which (ρi) is decreasing in ρi. 6 In their original model, Blanchard and Galí (2007) explain why they do not use technology shocks but S (they denote S by M) as a source of supply shocks. They indicate that technology shocks in business cycle models are defined as a public, non-rival good, whose use by firms entail no cost, and they are not directly observable in the data. Yet, the use of S allows clear mapping with observable supply shocks, like oil price disturbances.
mc ¼ ð1 þ ϕÞ
y2 −αs − logð1−α Þ þ ξ: ð1−α Þ
ð15Þ
Finally, rearranging (14) results in the second best level of output, y2, below y2 ¼ αs þ
ð1−α Þ ð logð1−α Þ−μ − ξÞ: ð1 þ ϕÞ
ð16Þ
Eq. (16) indicates that the natural level of output depends on the level of technology (s), the consumers' preferences (ϕ and ξ), the parameter of the production function (a) and the markup (μ), which in turn depends on consumer elasticity of demand over goods (). 7 Note that Eq. (10) defines the markup as M = P/Ψ′(Yi), the ratio of price to the marginal cost of production, which is a standard approach in the New Keynesian literature (see Blanchard and Galí, 2007; Kimball, 1995; Woodford, 2003). The alternative approach would be to define markups as in P = (1 + M)Ψ′(Yi), which uses the difference between price and marginal cost of production to define markup. The latter results in M = (P − Ψ′)/Ψ′. This term and the one we use in (10) are monotonic transformations of each other.
O. Eksi / Economic Modelling 51 (2015) 626–634
2.4. The equilibrium with sticky prices Nominal price stickiness is incorporated into the model by using the Calvo (1983) approach. In this setup, while resetting their prices at time t, firms take into account that their prices will be effective in each of the subsequent periods with some probability θ so that the probability that their prices will be effective in period t + k is given by θk. Hence, the profit maximization of firm i, which uses the discounted value of future profits of the firm while its price remains effective, can be written as max P t;i
∞ n X o θk Et Q t;tþk P t;i Y tþk=t;i − Ψtþk Y tþk=t;i
ð17Þ
k¼0
where Q t,t + k = βk(Uc,t + k/Uc,t)(Pt/Pt + k) is the stochastic discount factor coming from the utility maximization problem of consumers, Ψ ⁎ is the optimal price set by the firm at is the nominal cost function, Pt,i time t, and Yt + k/t,i denotes the output of the firm in period t + k given that it charges the price which is set in period t. The problem in (17) is subject to the following sequence of demand conditions Y tþk=t;i ¼
P t;i P tþk
no trade-off is generated from the model. Next, we check whether stabilizing the output gap to its second best level is desirable. In this regard, Blanchard and Galí (2007) indicate that “ … in contrast with the baseline NKPC model, (with the model modified by the real wage rigidity) the divine coincidence no longer holds, since stabilizing the output gap (y − y 2) is no longer desirable. This is because what matters for welfare is the distance of output not from its second-best level, but from its first-best level. In contrast to the baseline model, the distance between the first- and the second-best levels of output is no longer constant, but is instead affected by the shocks… ”. To check if this possibility arises with our model, we use μ = 0 in (16). This exercise, by combining zero markups with no (real or nominal) rigidities, leads to the first best allocation
y1 ¼ αs þ
ð1−α Þ ð logð1− α Þ − ξÞ: ð1 þ ϕÞ
The final equation together with (16) results in
−ðρi Þ ð18Þ
Y tþk
where the elasticity of demand (ε) depends on the market share of a firm, ρi. The solution of the above problem, when aggregated at the country level, yields the following NKPC equation (see Appendixes C.1 and C.1.1 for the details)8 πt ¼ βEt fπtþ1 g þ
ð1−θÞð1−βθÞ 1 h i ðmct −mcÞ: θ 1 þ η þ α =ð1−α Þ μ
ð19Þ Eq. (19) indicates that inflation is positively correlated with the deviations of real marginal cost from its flexible price level. It also indicates that increase in ε results in lower inflation. This result follows the fact that ε measures the elasticity of substitution of consumers between different types of goods. An increase in this elasticity reduces firms' incentive to deviate from the equilibrium price of the economy, which is the source of real rigidity in this paper. Eq. (19) can be rewritten in terms of the deviations in output rather than the deviations in real marginal cost. For this purpose, we take mc from Eq. (15) and, for time t, write it as mct ¼ ð1 þ ϕÞ
629
yt −αs − logð1−α Þ þ ξ: ð1−α Þ
ð20Þ
Combining (15), (19) and (20) results in
πt ¼ βEt fπtþ1 g þ
ð1−θÞð1−βθÞ 1 ð 1 þ ϕÞ h i yt −yt;2 : θ 1 þ η þ α =ð1−α Þ ð1−α Þ μ
ð21Þ
y1 −y2 ¼ μ
ð1−α Þ : ð1 þ ϕÞ
ð22Þ
Eq. (22) shows that under the endogenous markup setting the distance between the first best and the second best levels of output is equal to a constant term, i.e. remains unaffected from supply (s) or preference (ξ) shocks. Therefore, stabilizing the output gap at its second best level (y − y2) is equal to stabilizing the welfare relevant output gap (y − y 1) as well. This result is different than that of Blanchard and Galí (2007) mentioned above, which discusses that under real wage rigidities the distance between the first best and the second best levels of output is fluctuating in response to both supply and preference shocks. Eqs. (19) and (21) indicate that in a model with endogenous markup setting and nominal price rigidities no trade-off arises between inflation and output. As the term ημ⁎* + α*/(1 − α), which appears in these equations, is greater than zero,9 the results indicate that the endogenous markup setting only causes a real price rigidity, i.e. a strategic complementarity between (relative) pricing decisions of the firms. As a result, inflation volatility is lower and output gap volatility is higher compared to the constant markup case in response to technology or preference shocks. Even though these movements of variables could be defined as relatively countercyclical compared to the constant markup case, output gap still increases along with inflation. As a result, both of these variables can be both stabilized at a time, unless the model is modified with exogenous shocks, which is shown below. 2.6. Exogenous changes in markups If we assume that markups are not endogenous but are subject to exogenous cost push shocks as in Clarida et al. (1999), then the NKPC equation can be derived as (Appendix C.1.2)
2.5. Results for the inflation output trade-off Eq. (21) shows that with the endogenous markup setting the stabilization of the output gap is equal to stabilization of inflation; hence,
πt ¼ βEt fπ tþ1 g þ þ
λ ðmct −mcÞ ½1 þ α=ð1−α Þ
λ ðμ −μ Þ ½1 þ α=ð1−α Þ t
ð23Þ
8 If we do not use the approximation of ημ(ρ) around ρ = 1 (that is η⁎μ), the solution for NKPC is obtained as
π t ¼ βEt fπ tþ1 g þ h
λ ημ ðρÞ=ημ þ ημ þ α =ð1−α Þ
i ðmct þ μ Þ:
9 This is because η⁎μ is approximated around 1, ε is a positive term that defines the elasticity of substitution between different goods, and a ranges in 0 b α b 1 as 1 − α is the labor share in production.
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O. Eksi / Economic Modelling 51 (2015) 626–634
where is the constant elasticity of demand over goods. Once combined with Eqs. (15) and (19), Eq. (23) can also be written in terms of the deviations in output rather than the deviations in the real marginal cost: πt ¼ βEt fπ tþ1 g þ þ
λ ð1 þ ϕÞ y −yt;2 ½1 þ α=ð1−α Þ ð1−α Þ t
λ ðμ −μ Þ: ½1 þ α=ð1−α Þ t
ð24Þ
Eqs. (23) and (24) show that the term μt − μ represents an exogenous change in the markup and is effective on inflation in addition to the changes in the real marginal cost of production. As a result, a positive shock to markup can either be confronted with an increase in πt, or a decrease in mct (i.e. a decrease in yt), so that the model generates the inflation–output trade-off. 3. Optimal monetary policy Eq. (22) shows that the distance between the first best and the second best levels of output is constant under the endogenous markup setting. This result implies that utility losses due to the deviations from efficient allocation (y1) remain parallel to those that arise from deviations from the flexible price allocation (y2). This further requires the optimality of the monetary rules for the standard NKPC equation, given below, to remain valid (see Galí, 2008) πt ¼ βEt fπ tþ1 g þ κ yt −yt;2 : The only difference is that now κ¼
ð1−θÞð1−βθÞ 1 ð1 þ ϕÞ h i ; θ 1 þ η þ α =ð1−α Þ ð1−α Þ μ
instead of the one obtained with constant elasticity of demand assumption ð1−θÞð1−βθÞ ð1 þ ϕÞ : κ¼ θ ð1−α Þ Thus, the optimal policy should target flexible price level of output and zero inflation. In summary, as stated above, monetary policies benefit from less volatility in inflation and suffer from more volatility in the output gap under the endogenous markup setting. 4. Discussion New Keynesian (NK) models find a role for the monetary policy on the real economy by using sluggish adjustment of nominal prices. Yet, as the literature discusses, this sluggish adjustment mechanism may not be enough to account for the whole price dynamics of the economy. The strategic interactions between the pricing decisions of the firms are also considered to improve NK models in matching the characteristics of the real data, and thus it is natural to consider that these interactions may have implications for the optimal monetary policy. Extending NK models with the endogenous markup setting is a way to answer these questions. This paper incorporates the endogenous markup setting into a standard NK model in a very introductory way. Then it analyzes the predictions of the model not only for the optimal monetary policy but also for the inflation–output trade-off of central banks. The latter objective concerns the weakest side of the basic NK models. These models cannot create trade-off between stabilizing inflation and stabilizing output gap in response to conventional demand or technology shocks but can only do that in response to shocks that shift the Phillips curve, called markup shocks. These are supply side shocks. However, while a positive technology shock increases the potential output and reduces the prices, which
can be confronted with a monetary expansion that stabilizes both output and inflation, a positive markup shock increases prices without affecting the potential output. In this case the monetary authority faces a trade-off between stabilizing inflation at the expense of reducing the output. However, the source of these shocks is ambiguous. Mankiw (2009) indicates that10: “ It is common to tack an additional shock onto the Phillips curve equation, which automatically makes the divine coincidence disappear. But one might wonder where, at a deeper microeconomic level, this shock comes from.... Some new Keynesian modelers build in this shock as a markup shock… (I suspect that) these markup shocks are proxying for something funny going on within the price-adjustment process…”. 11 Hence, it seems that there is an ongoing debate on how to improve NK models to produce inflation output trade-off without relying on shocks that are difficult to interpret. Blanchard and Galí (2007) note that “ In our model, this interaction works through endogenous variations in wage markups, resulting from the sluggish adjustment of real wages. However, this is not the only possible mechanism: a similar interaction could work, for example, through the endogenous response of desired price markups to shocks, as in Rotemberg and Woodford (1996). Understanding these interactions should be high on macroeconomists' research agendas”. This is exactly the target of this study, that is, analyzing whether endogenous changes in desired price markups would lead to the inflation–output policy trade-off in response to conventional shocks. Our results, yet, indicate that the trade-off that arises from a model with exogenous changes in price markups does not arise in an environment with endogenous changes in price markups. The endogenous markup setting only causes a real price rigidity, i.e. a strategic complementarity between (relative) pricing decisions of the firms. As a result of the movements of output gap and inflation in the same direction, in response to supply or preference shocks, monetary authorities can stabilize both measures successfully. 5. Conclusions This paper questions whether the central banks dilemma for stabilizing output gap or stabilizing inflation can be obtained with a model using interaction between aggregate shocks and firms' markups. The main purpose of implementation of this setting in the literature is to create price stickiness, and its implications for inflation–output policy trade-off and welfare are not investigated yet. Moreover, even though constant elasticity of demand assumption is a convenient assumption for economic modeling, the strategic complementarity in pricing decisions that arises with the endogenous markup setting improves these models towards a more realistic representation of the real world markets. The paper extends the standard new Keynesian framework with the endogenous markup setting and incorporate it into the model studied in Blanchard and Galí (2007). The findings show that the trade-off between output gap stabilization and inflation stabilization that is confronted by the central banks is not produced in this setup. Finally, it is shown that the flexible price markup remains unaffected from the endogenous markup setting; hence, targeting the level of output under flexible prices is equivalent to targeting the first best level of output. As a result, the optimal monetary policy in this environment is to target the flexible price equilibrium. Acknowledgments I am grateful to Jordi Galí, Sushanta Mallick (the editor), and three anonymous reviewers for their constructive comments and very helpful suggestions. All remaining errors are my own. 10
Mankiw (2007) provides a similar discussion. The referenced studies in this quotation are Blanchard and Galí (2010) and Ball and Mankiw (1995). 11
O. Eksi / Economic Modelling 51 (2015) 626–634
631
which would be further simplified to
Appendix A. Consumers A.1. Solution to the consumer's problem
P i −P ψ″ ð1Þ Y i −Y ¼ 0 : P ψ ð1Þ Y
The consumer problem is defined as follows Z 1 Z 1 Y ψ i di: min P i Y i di s:t: 1 ¼ Y 0 0
Finally, when combined with (6), the equation above finds
Putting this problem into Lagrangian form and taking derivative with respect to Yi yields Y 1 P i ¼ λψ0 i Y Y
ðA:1Þ
where λ is the Lagrange multiplier. By calculating (A.1) at Pi = P and Yi = Y, we find
P i −P 1 Y −Y ¼− i P Yi which, given the following approximations for small changes: (ρi − ρ)/ρ ≈ ln(ρi/ρ) and (Yi − Y)/Yi ≈ (Yi − Y)/Y, can also be written as ln
Pi 1 Y ≈− ln i ; P Y
hence, PY λ¼ 0 ; ψ ð1Þ
− Yi P ≈ i where ¼ ð1Þ: Y P
using this equality with Eq. (A.1) results in the following inverse demand equation given in the text Y 1 : P i ¼ Pψ0 i Y ψ0 ð1Þ
ð5Þ
ðA:3Þ
Appendix B. Firm's problem in the flexible price setting B.1. Solution to the firm's problem Flexible price firms solve a static profit maximization problem max P ðY i ÞY i − ΨðY i Þ;
A.2. Derivation of the elasticity of demand
Yi
By definition, the elasticity of demand is ðρi Þ ¼ −
∂Y i =Y i P 1 ¼− i : Y i ∂P i =∂Y i ∂P i =P i
where Ψ(Yi) is the nominal cost function. Taking first order condition with respect to Yi finds ðA:2Þ 0¼
Calculating Pi and ∂Pi/∂Yi from (5) and using them together with (A.2) result in Eq. (6) in the text Y 1 Y Pψ0 i ψ0 i 0 Y ψ ð1Þ 1 1 1 ψ0 ðρi Þ Y ¼− ; ¼− ðρi Þ ¼ − Yi ″ Yi 1 1 Y i =Y ″ Y i ρi ψ″ ðρi Þ ψ Pψ 0 Y Y ψ ð1Þ Y
ð6Þ
∂P ðY i Þ Y i ∂ΨðY i Þ þ 1 P ðY i Þ− : ∂Y i P ðY i Þ ∂Y i
The final equation can be combined with the elasticity of demand equation which is, by definition, given as (ρi) = − [∂Yi/Yi]/[∂Pi/Pi]. This practice finds
1 ∂ΨðY i Þ þ 1 P ðY i Þ− : 0¼ − ðρi Þ ∂Y i which can be rearranged to find Eq. (9) in the text P ðY i Þ ¼
where we used the equilibrium condition in the goods market of a closed economy: Yi = Ci and Y = C, together with the definition of ρi which is the share of good in the consumption basket: ρi = Ci/C. One can obtain the familiar demand equation for monopolistically competitive markets by log-linearizing the inverse demand equation in (5) around the steady state at ρi = 1. To do so, we can take natural logarithm of the equation Y 1 þ ln 0 ln ðP i Þ ¼ ln ðP Þ þ ln ψ0 i Y ψ ð1Þ
ðρi Þ ∂ΨðY i Þ : ðρi Þ−1 ∂Y i
ð9Þ
B.2. Equality of Eqs. (11) and (12) Showing equality of Eqs. (11) and (12) only requires to show that MC = W/P ⋅ 1/MPN. To show this, notice that the nominal cost function of a firm that produces Y amount of good can be written as ΨðY Þ ¼ NðY ÞW subject to Y ¼ Sα N1−α :
and apply first order Taylor approximation The above equality and constraint can be combined into one equality 1 ψ ð1Þ 1 1 ln ðP Þ þ ðP i −P Þ ¼ ln ðP Þ þ ln ψ0 ð1Þ þ 0 Y ðY i −Y Þ þ ln 0 ; P ψ ð1Þ ψ ð1Þ
″
ΨðY Þ ¼
Y Sα
1=ð1−αÞ W:
632
O. Eksi / Economic Modelling 51 (2015) 626–634
Taking derivative of cost function with respect to Y and dividing by the price level finds the nominal marginal cost of production α=ð1−αÞ ∂ΨðY Þ 1 1 Y 1W ð¼ MC Þ ¼ ; 1−α Sα ∂Y P Sα P
∞ X Pt P θk Et Q t;tþk Y tþk=t −M ρtþk=t MC tþk=t tþk ¼0 P t−1 P t−1 k¼0
now inserting for Y MC ¼
α
1−α
1 S N 1−α Sα
!α=ð1−α Þ
1 W 1 1 W ¼ : 1−α N−α Sα P Sα P
Eq. (8) in the text finds the marginal product of labor as MPN = (1 − α)Y/N = (1 − α)SαN−α. When inserted into the above equation, it finds the necessary equality that ensures equalities of Eqs. (11) and (12) MC ¼
This equation states that, in expected terms, the discounted value of firm's future profits is equal to the discounted value of its future costs times the markups defined over with these costs. This equation, when divided by Pt − 1, and also when its second term is multiplied and divided by Pt + k, can be written as
W 1 : P MPN
where MC is the real cost of marginal production and is equal to the nominal marginal cost function, Ψ′t + k (Yt + k/t), divided by Pt + k. At the zero inflation steady state the following conditions must hold: Pt − 1 = Pt⁎ = Pt + k, Yt + k/t = Y, MCt + k/t = MC = 1/M by Eq. (11) and Q t,t + k = βk. Applying log linearization to (C.1) around this steady state ( ∞ X θk Et βk Q eq^t;tþk Yey^tþk=t k¼0
Appendix C. Firms' problem in the sticky price setting
The profit maximization problem of a typical firm is given as (for notational convenience, firm specific subscripts are suppressed)
Pt
∞ X θk Et Q t;tþk P t Y tþk=t −Ψtþk Y tþk=t
!) ¼ 0:
∞ n o X θk Et βk QYeq^t;tþk þy^tþk=t þp^t −p^t−1 k¼0
ð17Þ
k¼0
¼
∞ n o X ^ ^ ^ ^ ^ ^ θk Et βk Q Yeqt;tþk þytþk=t þμ ðρtþk=t Þþmctþk=t þptþk −pt−1
ð1Þ
k¼0
where Q t,t + k = βk(Uc,t + k/Uc,t)(Pt/Pt + k) is the stochastic discount factor coming from the utility maximization problem of consumers, Ψ is the nominal cost function, Pt⁎ is the optimal price set by a firm at time t, Yt + k/t denotes the output of the firm in period t + k given that it charges the price which is set in period t, and (1 − θ) is the probability that the firm may reset its price. Eq. (17) is subject to the following sequence of demand conditions Y tþk=t ¼
Pep^t Pep^tþk ^ −Meμ ðρtþk=t Þ MCem^ctþk=t p^ ^t−1 p Pe Pe t−1
where the deviation of a variable from its steady state value is denoted by a hat. Using (11), which states that M ⋅ MC = 1, and rearranging the equation, we find
C.1. Solution to the firm's problem
max
ðC:1Þ
P t P tþk
−εðρÞ
^t;tþk þ y ^tþk=t terms cancel out. Using the approximawhere QY and 1 þ q
^t and p ^t−1 are not affected from the tion eρ ≅ (1 + ρ), and the fact that p summation operator, we find ∞ n X o ^t −p ^t−1 ¼ ð1−βθÞ ðβθÞk Et μ^ ρtþk=t þ m^ctþk=t þ p ^tþk −p ^t−1 : p k¼0
Y tþk :
ðC:2Þ
ð18Þ
∂Y tþk=t Y tþk=t ¼ − ð ρ Þ : P t ∂P t
C.1.1. Endogenous markups This section uses Eq. (C.2) to derive NKPC equation in (19). First, we start with μ^ ðρtþk=t Þ term in (C.2), which is the deviation of firm i's markup from its steady state, and which can be written in terms of the demand elasticity of markups that is approximated around ρ=1
Hence, the derivative of the firm maximization problem in (17) with respect to the optimal price (Pt⁎) yields
^tþk=t −y ^tþk : μ^ ρtþk=t ¼ ημ y
where the elasticity of demand depends on the market share of a firm, (ρ). Note that the derivative of (18) with respect to Pt⁎ can be written as
∞ X Y tþk=t θk Et Q t;tþk Y tþk=t þ −ðρÞY tþk=t −Ψ0tþk Y tþk=t −ðρÞ ¼ 0; Pt k¼0
taking the Yt + k/t parentheses and multiplying both sides of the equation by Pt⁎, we obtain ∞ X θk Et Q t;tþk Y tþk=t P t ð1−ðρÞÞ þ ðρÞΨ0tþk Y tþk=t ¼ 0:
Combining the previous equation with the demand equation in (18) that is log-linearized around its steady state yields ^ t −p ^tþk : μ^ ρtþk=t ¼ −ημ p
ðC:3Þ
k¼0
To derive the mĉt + k/t term in (C.2), the real marginal cost can be written as
Dividing both sides of the equation by (1 − (ρ)) and using the definition of markup in (9), which is: M = (ρ)/((ρ) − 1), we find
mct ¼ wt −mpnt ¼ wt −½yt −nt þ logð1−α Þ;
∞ X θk Et Q t;tþk Y tþk=t P t −M ðρt ðiÞÞΨ0tþk Y tþk=t ¼ 0: k¼0
where w denotes the (log) real wages that depend on the economy wide labor market and the mpn denotes the (log) marginal product of
O. Eksi / Economic Modelling 51 (2015) 626–634
labor that is taken from (8). When nt is taken from the production function in (7), the previous equation becomes mct ¼ wt þ
α y −s − logð1−α Þ: 1−α tþk tþk
For the marginal cost of a firm at time t + k, which resets its price at time t, the preceding equation reads mctþk=t ¼ wtþk þ
α ytþk=t −stþk − logð1−α Þ: 1−α
α ytþk=t −ytþk ; 1−α
and using the log linearized version of (A.3), together with the following definitions mĉt + k/t = mct + k/t − mc, mĉt + k = mct + k − mc, ŷt + k/t = yt + k/t − y and ŷt + k = yt + k − y within the last equation above, we find α ^tþk : ^t −p p ¼ m^ctþk − 1−α
ðC:4Þ
Inserting Eqs. (C.3) and (C.4) into (C.2) finds
∞ X α ^t −p ^tþk −p ^t−1 ¼ ð1−βθÞ ðβθÞk Et m^ctþk − ημ þ ^tþk þ p ^t−1 : ^t −p p p 1−α k¼0
First, rearranging this equation as, ^t−1 ^t −p p
∞ X α α ^t−1 þ 1 þ ημ þ ^t −p ^tþk ¼ ð1−βθÞ ðβθÞk Et m^ctþk − ημ þ p p 1−α 1−α k¼0
^t−1 to the RHS (right hand side) then, by adding ðÞ½ημ þ α =ð1−αÞp of this equation we obtain ^t−1 ^t −p p
α ^t −p ^t−1 p ¼ ð1−βθÞ ðβθÞk Et m^ctþk − ημ þ 1−α k¼0
α ^tþk −p ^t−1 : p þ 1 þ ημ þ 1−α ∞ X
f
g
^t −p ^t−1 terms which are not affected from the Collecting all the p summation term at the LHS (left hand side) of the equation and dividing the equation by [1 + ημ⁎* + α*/(1 − α)] gives ^t−1 ^t −p p
8 9 ∞ < X = 1 k ^tþk −p ^t−1 : i m^ctþk þ p ¼ ð1−βθÞ ðβθÞ Et h : 1 þ η þ α =ð1−α Þ ; k¼0
μ
^tþ1 −p ^t Þ can be written as Similarly, Et ðp 8 9 < ∞ X = 1 k ^t ¼ ð1−βθÞ ðβθÞ Et h ^tþ1þk −p ^t : ^tþ1 −p i m^ctþ1þk þ p Et p : 1 þ η þ α =ð1−α Þ ; k¼0 μ
Combining the last two equations delivers ^t−1 ¼ βθEt p ^t þ h ^tþ1 −p ^t −p p ^t −p ^t−1 ; þ βθp
1þ
^t −p ^t−1 : i m^ct þ p þ α =ð1−α Þ
πt ¼ βEt fπtþ1 g þ
ð1−θÞð1−βθÞ 1 h i m^ct : θ 1 þ η þ α =ð1−α Þ
ð19Þ
μ
C.1.2. Exogenous changes in markups We no longer assume that the changes in the markups are endogenous; as a result, the elasticity of demand, , is a constant now. Calculating (C.4) at the constant elasticity of demand and using it together with (C.2) yields
Combining the previous two equations we obtain mctþk=t ¼ mctþk þ
ð1−βθÞ ημ
^t −p ^t−1 ¼ ð1−θÞðp ^t −p ^t−1 Þ,12 the last equation above Using πt ¼ p yields Eq. (19) in the text
which holds at time t + k as well
m^ctþk=t
which can be simplified as ^t −p ^t−1 ¼ βθEt p ^t þ h ^tþ1 −p p
α ðy −st Þ− logð1−α Þ; 1−α t
mctþk ¼ wtþk þ
633
∞ n X o α ^ t−1 ¼ ð1−βθÞ ðβθÞk Et μ^ tþk þ m^ctþk − ^ tþk þ p ^t−1 ^t −p ^ −p ^tþk −p p p 1−α k¼0
where μ^ tþk defines the deviations in markups in response to shocks to ^t−1 to RHS of the this variable at time t + k. Adding ðÞ½α=ð1−αÞp equation, this equation becomes ∞ X α ^ −p ^t−1 ¼ ð1−βθÞ ðβθÞk Et μ^ tþk þ m^ctþk − ^t−1 ^t −p p p 1−α k¼0 α ^tþk −p ^t−1 : p þ 1þ 1−α
f
g
^t−1 terms at the LHS of the equation and dividing ^t −p Collecting the p it by [1 + α/(1 − α)], we get ∞ X ^t−1 ¼ ð1−βθÞ ðβθÞk Et ^t −p p k¼0
þ
f ½1 þ α=1ð1−αÞ μ^
tþk
g
1 ^tþk −p ^t−1 : m^c þ p ½1 þ α=ð1−α Þ tþk
^tþ1 −p ^t Þ can be written as Similarly, Et ðp ∞ X ^tþ1 −p ^t ¼ ð1−βθÞ ðβθÞk Et Et p k¼0
þ
f ½1 þ α=1ð1−αÞ μ^
tþ1þk
g
1 ^tþ1þk −p ^t : m^c þ p ½1 þ α=ð1−α Þ tþ1þk
Combining the last two equations ^tþ1 −p ^t−1 ¼ βθEt p ^t þ ^t −p p
ð1−βθÞ ð1−βθÞ μ^ þ m^ct ½1 þ α=ð1−α Þ t ½1 þ α=ð1−α Þ
^t −p ^t−1 : þp ^ t −p ^t−1 ¼ ð1−θÞðp ^t −p ^t−1 Þ in the last equation Finally, using π t ¼ p results in Eq. (23) in the text πt ¼ βEt fπtþ1 g þ
ð1−θÞð1−βθÞ ð1−θÞð1−βθÞ m^ct þ μ^ : θ½1 þ α=ð1−α Þ θ½1 þ α=ð1−α Þ t
ð23Þ
ð1−βθÞ 1þ
ημ
^t i m^ct þ ð1−βθÞp þ α =ð1−α Þ
12 ^ t −p ^ t−1 ) is determined by the proportion of This equality indicates that inflation (p firms that are able to reset their prices. This proportion (1 − θ) is indeed equal to the probability that a firm may reset its price.
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References Altig, D., Christiano, L., Eichenbaum, M., Lindé, J., 2011. Firm-specific capital, nominal rigidities and the business cycle. Rev. Econ. Dyn. 14 (2), 225–247. Ball, L., Mankiw, N.G., 1995. Relative-price changes as aggregate supply shocks. Q. J. Econ. 110 (1), 161–193. Ball, L., Romer, D., 1990. Real rigidities and the non-neutrality of money. Rev. Econ. Stud. 57 (2), 183–204. Blanchard, O.J., Galí, J., 2007. Real wage rigidities and the new Keynesian model. J. Money Credit Bank. 39 (1), 35–65. Blanchard, O.J., Galí, J., 2010. Labor markets and monetary policy: a new Keynesian model with unemployment. Am. Econ. J. Macroecon. 2 (2), 1–30. Calvo, G., 1983. Staggered prices in a utility maximizing framework. J. Monet. Econ. 12, 383–398. Christiano, L.J., Eichenbaum, M., Evans, C., 1999. Monetary Policy Shocks: What Have We Learned and to What End? In: Taylor, John B., Woodford, Michael (Eds.), Handbook of Macroeconomics vol. 1A. Elsevier, New York Clarida, R., Galí, J., Gertler, M., 1999. The science of monetary policy: a new Keynesian perspective. J. Econ. Lit. 37, 1661–1707. Divino, J.A., 2009. Optimal monetary policy for a small open economy. Econ. Model. 26 (2), 352–358. Dixit, A.K., Stiglitz, J.E., 1977. Monopolistic competition and optimal product diversity. Am. Econ. Rev. 67, 297–308.
Dotsey, M., King, R.G., 2005. Implications of state-dependent pricing for dynamic macroeconomic models. J. Monet. Econ. 52 (1), 213–242. Galí, J., 1994. Monopolistic competition, business cycles, and the composition of aggregate demand. J. Econ. Theory 63 (1), 73–96. Galí, J., 2008. Monetary Policy, Inflation and the Business Cycle: An Introduction to the New Keynesian Framework. Princeton University Press, Monograph. Kimball, M.S., 1995. The Quantitative Analytics of the Basic Neomonetarist Model. J. Money Credit Bank. 27, 1241–1277. Klenow, P.J., Kryvtsov, O., 2008. State dependent or time dependent pricing: does it matter for recent U.S. Inflation? Q. J. Econ. 123 (3), 863–904. Mankiw, N.G., 2007. Comments presented at federal reserve conference price dynamics: three open questions. J. Money Credit Bank. 39 (s1), 187–192. Mankiw, N.G., 2009. What are supply shocks? Web log post. Greg Mankiw's Blog (Retrieved from http://gregmankiw.blogspot.pt/2009_04_01_archive.html. April 8, 2009. Web. February 27, 2015) Romer, D.H., Romer, C.D., 2004. A new measure of monetary shocks: derivation and implications. Am. Econ. Rev. 94 (5), 1055–1084. Rotemberg, J., Woodford, M., 1992. Oligopolistic pricing and the effects of aggregate demand on economic activity. J. Polit. Econ. 100 (6), 1153–1207. Rotemberg, J., Woodford, M., 1996. Imperfect competition and the effects of energy price increases on economic activity. J. Money Credit Bank. 28, 549–577. Woodford, M., 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton University Press.
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Endogenous markups in the new Keynesian model: Implications for inflation–output trade-off and welfare Ozan Eksi ⁎ TOBB University of Economics and Technology, Department of Economics, Sogutozu Street, No: 43, Ankara 06560, Turkey
a r t i c l e
i n f o
Article history: Accepted 3 September 2015 Available online 29 September 2015 Keywords: Endogenous markups Monetary policy Inflation–output trade-off
a b s t r a c t This paper extends the standard new Keynesian (NK) model by using the endogenous markup setting a la Kimball (1995). In this setting, consumers' price elasticity of demand for a good is increasing in the good's relative price level, which affects the desired price markup of firms. In the literature, this setting is mainly used to improve the NK models in matching sluggishness of prices in the data. Our paper analyzes the monetary policy implications of the model. It is shown that unlike the cases of real wage rigidity and exogenous markup shocks, the endogenous markup setting does not improve the NK models in generating the inflation–output trade-off. It is also discussed that the optimal monetary policy in this environment is to target the flexible price equilibrium. © 2015 Elsevier B.V. All rights reserved.
1. Introduction A well-known result in the literature is that the standard NK model with nominal rigidities is unable to create the inflation–output trade-off faced by the central banks. As an illustration of this, let ynt represent the potential output under flexible prices, yt represent the actual output in the presence of a fraction of firms with fixed prices, and ỹt = yt − ynt is the output gap. Using these definitions, the following new Keynesian Phillips Curve (NKPC) equation ~t πt ¼ βEt fπtþ1 g þ κ y
ð1Þ
indicates that stabilizing the output gap leads to the stabilization of inflation, called the divine coincidence by Blanchard and Galí (2007). 1 A common way of obtaining inflation–output trade-off from NK models is to modify Eq. (1) by adding a cost push shock (exogenous changes in price or wage markups; Clarida et al., 1999) as follows: ~t þ ut : πt ¼ βEt fπtþ1 g þ κ y
ð2Þ
According to (2), a positive cost push shock raises current inflation, unless the level of output is reduced via contractionary monetary
⁎ Tel.: +90 312 290 4542; fax: +90 312 290 4104. E-mail address: [email protected]. URL: http://oeksi.etu.edu.tr. 1 In the AS–AD (aggregate demand–aggregate supply) framework, the divine coincidence can be described as the ability of the monetary authority to keep inflation constant and output equal to its natural level by counteracting to the changes in AD and LRAS (longrun aggregate supply) curves.
http://dx.doi.org/10.1016/j.econmod.2015.09.005 0264-9993/© 2015 Elsevier B.V. All rights reserved.
policies. As a result, the divine coincidence no longer holds.2 The weak side of this approach is that the cost push shocks are exogenous. In an attempt to obtain inflation–output policy trade-off in response to more conventional shocks, Blanchard and Galí (2007) use a model with real wage rigidity, and Divino (2009) uses an open economy model.3 Our paper, on the other hand, uses a real price rigidity, the endogenous markup setting, and asks whether this setting can generate inflation–output trade-off from NK models. The paper then investigates the optimal monetary policy under this setting. A real price rigidity is a source that makes firms unwilling to change their relative prices. The need for such rigidities stems from the fact that the monetary non-neutralities cannot be fully explained by nominal rigidities of simple NK models.4 With regard to the theoretical ways of obtaining real price rigidities, Rotemberg and Woodford (1992) use strategic pricing decisions between colluding firms, Galí (1994) uses a 2 In the AS–AD framework, cost push shock is a shift in the SRAS (short-run aggregate supply) curve. Or, put it differently, it is a shock to the Phillips curve that does not change the potential output. In response, monetary authority can stabilize either the price or the output level. 3 Blanchard and Galí (2007) show that under the real wage rigidity—i.e. the sluggish adjustment of real wages for a fraction of workers at each point in time—the difference between natural rate of output and first best output—that occurs when there is no market imperfection or distortion such as nonzero markups—is not constant. Therefore, stabilizing the output gap (yt − ynt ) is no longer equal to stabilizing the welfare relevant output gap (yt − ytfb) and is no longer desirable from the welfare point of view. Divino (2009) offers a model in which both foreign and domestic demand shocks increase the real marginal cost of production, which makes the same effect with the exogenous cost push shocks in (2). 4 Ball and Romer (1990) is the first study emphasizing that real rigidities have a crucial role in explaining nominal rigidities and non-neutrality of money. Christiano et al. (1999) and Romer and Romer (2004) provide a more recent discussion on monetary nonneutralities. Klenow and Kryvtsov (2008) supply micro empirical findings indicating that nominal rigidities based on the frequency of firms' price changes are not enough to explain price sluggishness in the data.
O. Eksi / Economic Modelling 51 (2015) 626–634
setting in which the elasticity of substitution across goods in consumption to differ from that in production, Altig et al. (2011) use short run marginal cost curve of firms that is increasing in its own output in the presence of firm-specific capital. Importance of these studies arises from the fact that all of them improve the fit between a theoretical model and the real data. Yet, probably the most common way of obtaining real price rigidity in the monetarist models is the endogenous markup setting of Kimball (1995). This model, instead of using the constant elasticity of demand assumption of Dixit and Stiglitz (1977), assumes increasing price elasticity of demand so that the desired markups of firms are decreasing in their relative prices. For instance, when a negative supply shock hits the economy, the firms that are able to increase their prices will be confronted with higher elasticity of demand, leaving them with less incentive to increase their prices. Likewise, at the time of a positive supply shock, the firms that are able to lower their prices will be confronted with lower elasticity of demand, leaving them with less incentive to reduce their prices. Consequently, the change in prices is lower, and as a result, the change in output gap is higher in this setting compared to the case with constant elasticity of demand assumption. In this paper we analyze if this countercyclicality between prices and output gap changes the implications of the standard NK model for the inflation–output policy trade-off. Since strategic interactions between the pricing decisions of the firms are used to enhance NK models towards a more realistic picture of the world markets, we also study the optimal monetary policy implications of the model. In this regard, Blanchard and Galí (2007) note that “ Rotemberg and Woodford (1992, 1996) restricted their analysis to the real implications of endogenous markups, without exploring their consequences for inflation and monetary policy in the presence of nominal frictions”. Hence, the aim of our paper is to fill this gap in the literature. The method we follow in this study is to incorporate the endogenous markup setting into the standard new Keynesian framework and then use it together with the same economic model of Blanchard and Galí (2007). The results show that the endogenous markup setting does not lead to the inflation output trade-off but only creates a real rigidity in prices that results in lower volatility in inflation and higher volatility in output gap when compared to the case with constant markups. With regard to the optimal monetary policy in this environment, the results show that stabilizing inflation ensures not only stabilization of the output gap but also stabilization of the welfare relevant output gap, which is the discrepancy between the actual output and the first best level of output with zero markups. Therefore, the optimal monetary policy is to target the flexible price level of output and zero inflation. The rest of the paper is organized as follows. Section 2 explains the model and analyzes its implications for the inflation–output trade-off of central banks. Section 3 investigates the optimal monetary policy in this environment. Section 4 discusses the results and Section 5 concludes.
2. The model We borrow the baseline model from Blanchard and Galí (2007) and extend it with endogenous markup setting.
2.1. Consumers An infinitely living representative consumer seeks to maximize the expected value of his/her discounted future utilities
E0
∞ X t¼0
βt U ðC t ; Nt Þ
627
where the utility function is given by U ðC t ; Nt Þ ¼ logðC t Þ− exp fξg
N1þϕ t : 1þϕ
Here, β is the discount factor for future utility, C is composite consumption—defined below—,N is employment, ϕ is the inverse of the elasticity of labor supply with respect, ξ is a preference parameter between consumption and leisure. This utility function is subject to the following sequence of budget constraints P t C t þ Q t Bt ≤ Bt−1 þ W t Nt þ T t ; where P is the aggregate price index for consumption goods, B represent purchases of one-period bonds, Q is the bond price, W is the nominal wages and T is a lump-sum component of income. Under the assumption of perfect competition in the labor market, the above-defined consumer problem can be used to obtain the following intratemporal optimality condition between the real wages and the marginal rate of substitution of leisure for consumption
MRS ¼ −
Un expfξgNϕ W ðin logs : mrs ¼ w ¼ c þ ϕn þ ξÞ; ð3Þ ¼ ¼ P Uc 1=C
and also the following intertemporal optimality condition between saving and consumption Q t ¼ βEt
U c;tþ1 P t : U c;t P tþ1
2.1.1. The consumption aggregate and the endogenous markup setup The consumption aggregate, C, is defined as (time and firm specific subscripts are suppressed unless they are necessary) Z
1 0
C ψ i di ¼ 1; C
ð4Þ
where ψ(ρi) satisfies ψ(1) = 1, ψ′(ρi) N 0, ψ″(ρi) b 0 for all ρi = Ci/C ≥ 0, where ρi is the share of good i in the consumption basket (this share, in the case of a representative consumer model of ours, is also equal to the market share of the firm producing that good). Note that when ψ(ρ) = (ρ)( − 1)/, so that ψ(Ci/C) = (Ci/C)( − 1)/, the consumption function in (4) results in Z C¼
1 0
!=ð−1Þ ð−1Þ=
Ci
di
;
which is the Dixit–Stiglitz type consumption aggregator that assumes constant elasticity of substitution () between different goods. The specification in (4), on the other hand, follows Kimball's (1995) approach and uses consumers' price elasticity of demand for a good that is increasing in its price level. To derive the formula for this elasticity, we start with the cost minimization problem of the consumer, which can be written as follows Z 1 Z 1 C min P i Y i di s:t: 1 ¼ ψ i di: C 0 0 In this setup, the goods are non-storable and are consumed within their production period. In a closed economy environment, this assumption implies C = Y. Using this equality, the solution of the above
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problem leads to the following implicit demand equation (shown in Appendix A.1) ψ0
Yi P ¼ ψ0 ð1Þ i ; Y P
ð5Þ P ðY i Þ ¼ M ðρi Þ
where P is the aggregate price index in the economy. Since ψ′(ρi) is a strictly decreasing function (as ψ″(ρi) b 0), higher values of Pi are associated with lower Yi/Y ratios. That is, Eq. (5) implies that, as usual, there is an inverse relationship with the relative price of good i and its relative demand. The actual price elasticity of the demand of consumers is then obtained as (in Appendix A.2) ðρi Þ ¼ −
∂Y i =Y i ψ0 ðρi Þ : ¼− ∂P i =P i ρi ψ″ ðρi Þ
this case, the term (ρi)/[(ρi) − 1] is the markup (the ratio of price to the marginal cost of production), which we denoted as M(ρi). Therefore, Eq. (9) can be rewritten as 7
ð6Þ
∂ΨðY i Þ : ∂Y i
ð10Þ
The elasticity of this markup, M(ρi), with respect to the firm's market share is denoted by ημ(ρi). Following Kimball (1995) and Woodford (2003), we approximate it around ρ = 1. We then denote it by ημ⁎. In the equilibrium within the flexible price framework, the symmetry of firms implies that market shares, and as a result markups are constant along the firms. Hence, Eq. (10) can be written as P = M ⋅ (∂Ψ(Y)/∂Y). Next, by dividing both sides of this equation by the price level P, we find
Notice that unlike Dixit and Stiglitz (1977) approach, the elasticity in (6) is not constant but depends on the share of good i in the consumption basket of consumer, ρi. Moreover, as in Kimball (1995), we assume (ρi) is decreasing in ρi. 5 Intuitively speaking, this assumption implies that when the share of good i in the consumption basket of consumer increases (or when, according to (5), the relative price of good i compared to aggregate price index decreases), consumers' price elasticity of demand for good i is decreasing.
where MC is the real marginal cost of production that is equal to nominal marginal cost of production divided by the price level. Hence, markup is the reciprocal of the real marginal cost of production, that is, percentage changes in this term are reflected as opposite changes in markups. The markups can also be calculated from the following equation
2.2. Firms
MPN ¼ M W=P ðin logs : mpn ¼ μ þ wÞ
Monopolistically competitive firms use a Cobb-Douglas type of production function
where MPN is the real marginal product of labor and W/P is the real wage. Equality of Eqs. (11) and (12) is given in Appendix B.2. Intuitively speaking, Eq. (11) relates real unit price of a good to its production cost in terms of labor, and Eq. (12) relates the real price of goods produced by one labor to the cost of employing that labor. Taking w and mpn from Eqs. (3) and (8), and using the equilibrium condition in a closed economy: y = c, Eq. (12) can be written as
Y ¼ Sα N1−α ðin logs : y ¼ αs þ ð1−α ÞnÞ;
ð7Þ
where S is non-produced input with exogenous endowment. In this case, shocks to S are interpreted as supply shocks.6 N is the labor input. Using (7), the real marginal product of labor can be found as MPN ¼ ð1−α ÞY=N ðin logs : mpn ¼ ðy−nÞ þ logð1−α ÞÞ:
ð8Þ
2.3. The equilibrium with flexible prices In case of flexible prices, with no capital in the model, firms solve a static profit maximization problem max P ðY i ÞY i − ΨðY i Þ;
μ ¼ −ð1 þ ϕÞn2 þ logð1−α Þ − ξ;
ð11Þ
ð12Þ
ð13Þ
where the subscript ‘2’ denotes the second best (natural) level of labor. It is second best because even though the prices are flexible there is still an imperfect competition in the goods market (μ N 0). Combining (13) with the production function in (7) yields the following equation μ ¼ −ð1 þ ϕÞ
y2 −αs þ logð1−α Þ − ξ: ð1−α Þ
ð14Þ
Using the equality in (11), 0 = μ + mc, the final equation becomes
Yi
where Ψ(Yi) is the nominal cost function. Taking the first order condition with respect to Yi, and rearranging the equation, we find (see Appendix B.1) ðρi Þ ∂ΨðY i Þ : P ðY i Þ ¼ ðρi Þ−1 ∂Y i
1 ¼ M MC ðin logs : 0 ¼ μ þ mcÞ
ð9Þ
Eq. (9) relates nominal prices to the nominal marginal cost of production and to the consumers' price elasticity of demand for goods. In
5 Kimball (1995), notes that it is easy to find a variety aggregator ψ to match any desired dependence of the elasticity of demand (ρi) on ρi. Dotsey and King (2005) further specify an aggregator for which (ρi) is decreasing in ρi. 6 In their original model, Blanchard and Galí (2007) explain why they do not use technology shocks but S (they denote S by M) as a source of supply shocks. They indicate that technology shocks in business cycle models are defined as a public, non-rival good, whose use by firms entail no cost, and they are not directly observable in the data. Yet, the use of S allows clear mapping with observable supply shocks, like oil price disturbances.
mc ¼ ð1 þ ϕÞ
y2 −αs − logð1−α Þ þ ξ: ð1−α Þ
ð15Þ
Finally, rearranging (14) results in the second best level of output, y2, below y2 ¼ αs þ
ð1−α Þ ð logð1−α Þ−μ − ξÞ: ð1 þ ϕÞ
ð16Þ
Eq. (16) indicates that the natural level of output depends on the level of technology (s), the consumers' preferences (ϕ and ξ), the parameter of the production function (a) and the markup (μ), which in turn depends on consumer elasticity of demand over goods (). 7 Note that Eq. (10) defines the markup as M = P/Ψ′(Yi), the ratio of price to the marginal cost of production, which is a standard approach in the New Keynesian literature (see Blanchard and Galí, 2007; Kimball, 1995; Woodford, 2003). The alternative approach would be to define markups as in P = (1 + M)Ψ′(Yi), which uses the difference between price and marginal cost of production to define markup. The latter results in M = (P − Ψ′)/Ψ′. This term and the one we use in (10) are monotonic transformations of each other.
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2.4. The equilibrium with sticky prices Nominal price stickiness is incorporated into the model by using the Calvo (1983) approach. In this setup, while resetting their prices at time t, firms take into account that their prices will be effective in each of the subsequent periods with some probability θ so that the probability that their prices will be effective in period t + k is given by θk. Hence, the profit maximization of firm i, which uses the discounted value of future profits of the firm while its price remains effective, can be written as max P t;i
∞ n X o θk Et Q t;tþk P t;i Y tþk=t;i − Ψtþk Y tþk=t;i
ð17Þ
k¼0
where Q t,t + k = βk(Uc,t + k/Uc,t)(Pt/Pt + k) is the stochastic discount factor coming from the utility maximization problem of consumers, Ψ ⁎ is the optimal price set by the firm at is the nominal cost function, Pt,i time t, and Yt + k/t,i denotes the output of the firm in period t + k given that it charges the price which is set in period t. The problem in (17) is subject to the following sequence of demand conditions Y tþk=t;i ¼
P t;i P tþk
no trade-off is generated from the model. Next, we check whether stabilizing the output gap to its second best level is desirable. In this regard, Blanchard and Galí (2007) indicate that “ … in contrast with the baseline NKPC model, (with the model modified by the real wage rigidity) the divine coincidence no longer holds, since stabilizing the output gap (y − y 2) is no longer desirable. This is because what matters for welfare is the distance of output not from its second-best level, but from its first-best level. In contrast to the baseline model, the distance between the first- and the second-best levels of output is no longer constant, but is instead affected by the shocks… ”. To check if this possibility arises with our model, we use μ = 0 in (16). This exercise, by combining zero markups with no (real or nominal) rigidities, leads to the first best allocation
y1 ¼ αs þ
ð1−α Þ ð logð1− α Þ − ξÞ: ð1 þ ϕÞ
The final equation together with (16) results in
−ðρi Þ ð18Þ
Y tþk
where the elasticity of demand (ε) depends on the market share of a firm, ρi. The solution of the above problem, when aggregated at the country level, yields the following NKPC equation (see Appendixes C.1 and C.1.1 for the details)8 πt ¼ βEt fπtþ1 g þ
ð1−θÞð1−βθÞ 1 h i ðmct −mcÞ: θ 1 þ η þ α =ð1−α Þ μ
ð19Þ Eq. (19) indicates that inflation is positively correlated with the deviations of real marginal cost from its flexible price level. It also indicates that increase in ε results in lower inflation. This result follows the fact that ε measures the elasticity of substitution of consumers between different types of goods. An increase in this elasticity reduces firms' incentive to deviate from the equilibrium price of the economy, which is the source of real rigidity in this paper. Eq. (19) can be rewritten in terms of the deviations in output rather than the deviations in real marginal cost. For this purpose, we take mc from Eq. (15) and, for time t, write it as mct ¼ ð1 þ ϕÞ
629
yt −αs − logð1−α Þ þ ξ: ð1−α Þ
ð20Þ
Combining (15), (19) and (20) results in
πt ¼ βEt fπtþ1 g þ
ð1−θÞð1−βθÞ 1 ð 1 þ ϕÞ h i yt −yt;2 : θ 1 þ η þ α =ð1−α Þ ð1−α Þ μ
ð21Þ
y1 −y2 ¼ μ
ð1−α Þ : ð1 þ ϕÞ
ð22Þ
Eq. (22) shows that under the endogenous markup setting the distance between the first best and the second best levels of output is equal to a constant term, i.e. remains unaffected from supply (s) or preference (ξ) shocks. Therefore, stabilizing the output gap at its second best level (y − y2) is equal to stabilizing the welfare relevant output gap (y − y 1) as well. This result is different than that of Blanchard and Galí (2007) mentioned above, which discusses that under real wage rigidities the distance between the first best and the second best levels of output is fluctuating in response to both supply and preference shocks. Eqs. (19) and (21) indicate that in a model with endogenous markup setting and nominal price rigidities no trade-off arises between inflation and output. As the term ημ⁎* + α*/(1 − α), which appears in these equations, is greater than zero,9 the results indicate that the endogenous markup setting only causes a real price rigidity, i.e. a strategic complementarity between (relative) pricing decisions of the firms. As a result, inflation volatility is lower and output gap volatility is higher compared to the constant markup case in response to technology or preference shocks. Even though these movements of variables could be defined as relatively countercyclical compared to the constant markup case, output gap still increases along with inflation. As a result, both of these variables can be both stabilized at a time, unless the model is modified with exogenous shocks, which is shown below. 2.6. Exogenous changes in markups If we assume that markups are not endogenous but are subject to exogenous cost push shocks as in Clarida et al. (1999), then the NKPC equation can be derived as (Appendix C.1.2)
2.5. Results for the inflation output trade-off Eq. (21) shows that with the endogenous markup setting the stabilization of the output gap is equal to stabilization of inflation; hence,
πt ¼ βEt fπ tþ1 g þ þ
λ ðmct −mcÞ ½1 þ α=ð1−α Þ
λ ðμ −μ Þ ½1 þ α=ð1−α Þ t
ð23Þ
8 If we do not use the approximation of ημ(ρ) around ρ = 1 (that is η⁎μ), the solution for NKPC is obtained as
π t ¼ βEt fπ tþ1 g þ h
λ ημ ðρÞ=ημ þ ημ þ α =ð1−α Þ
i ðmct þ μ Þ:
9 This is because η⁎μ is approximated around 1, ε is a positive term that defines the elasticity of substitution between different goods, and a ranges in 0 b α b 1 as 1 − α is the labor share in production.
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where is the constant elasticity of demand over goods. Once combined with Eqs. (15) and (19), Eq. (23) can also be written in terms of the deviations in output rather than the deviations in the real marginal cost: πt ¼ βEt fπ tþ1 g þ þ
λ ð1 þ ϕÞ y −yt;2 ½1 þ α=ð1−α Þ ð1−α Þ t
λ ðμ −μ Þ: ½1 þ α=ð1−α Þ t
ð24Þ
Eqs. (23) and (24) show that the term μt − μ represents an exogenous change in the markup and is effective on inflation in addition to the changes in the real marginal cost of production. As a result, a positive shock to markup can either be confronted with an increase in πt, or a decrease in mct (i.e. a decrease in yt), so that the model generates the inflation–output trade-off. 3. Optimal monetary policy Eq. (22) shows that the distance between the first best and the second best levels of output is constant under the endogenous markup setting. This result implies that utility losses due to the deviations from efficient allocation (y1) remain parallel to those that arise from deviations from the flexible price allocation (y2). This further requires the optimality of the monetary rules for the standard NKPC equation, given below, to remain valid (see Galí, 2008) πt ¼ βEt fπ tþ1 g þ κ yt −yt;2 : The only difference is that now κ¼
ð1−θÞð1−βθÞ 1 ð1 þ ϕÞ h i ; θ 1 þ η þ α =ð1−α Þ ð1−α Þ μ
instead of the one obtained with constant elasticity of demand assumption ð1−θÞð1−βθÞ ð1 þ ϕÞ : κ¼ θ ð1−α Þ Thus, the optimal policy should target flexible price level of output and zero inflation. In summary, as stated above, monetary policies benefit from less volatility in inflation and suffer from more volatility in the output gap under the endogenous markup setting. 4. Discussion New Keynesian (NK) models find a role for the monetary policy on the real economy by using sluggish adjustment of nominal prices. Yet, as the literature discusses, this sluggish adjustment mechanism may not be enough to account for the whole price dynamics of the economy. The strategic interactions between the pricing decisions of the firms are also considered to improve NK models in matching the characteristics of the real data, and thus it is natural to consider that these interactions may have implications for the optimal monetary policy. Extending NK models with the endogenous markup setting is a way to answer these questions. This paper incorporates the endogenous markup setting into a standard NK model in a very introductory way. Then it analyzes the predictions of the model not only for the optimal monetary policy but also for the inflation–output trade-off of central banks. The latter objective concerns the weakest side of the basic NK models. These models cannot create trade-off between stabilizing inflation and stabilizing output gap in response to conventional demand or technology shocks but can only do that in response to shocks that shift the Phillips curve, called markup shocks. These are supply side shocks. However, while a positive technology shock increases the potential output and reduces the prices, which
can be confronted with a monetary expansion that stabilizes both output and inflation, a positive markup shock increases prices without affecting the potential output. In this case the monetary authority faces a trade-off between stabilizing inflation at the expense of reducing the output. However, the source of these shocks is ambiguous. Mankiw (2009) indicates that10: “ It is common to tack an additional shock onto the Phillips curve equation, which automatically makes the divine coincidence disappear. But one might wonder where, at a deeper microeconomic level, this shock comes from.... Some new Keynesian modelers build in this shock as a markup shock… (I suspect that) these markup shocks are proxying for something funny going on within the price-adjustment process…”. 11 Hence, it seems that there is an ongoing debate on how to improve NK models to produce inflation output trade-off without relying on shocks that are difficult to interpret. Blanchard and Galí (2007) note that “ In our model, this interaction works through endogenous variations in wage markups, resulting from the sluggish adjustment of real wages. However, this is not the only possible mechanism: a similar interaction could work, for example, through the endogenous response of desired price markups to shocks, as in Rotemberg and Woodford (1996). Understanding these interactions should be high on macroeconomists' research agendas”. This is exactly the target of this study, that is, analyzing whether endogenous changes in desired price markups would lead to the inflation–output policy trade-off in response to conventional shocks. Our results, yet, indicate that the trade-off that arises from a model with exogenous changes in price markups does not arise in an environment with endogenous changes in price markups. The endogenous markup setting only causes a real price rigidity, i.e. a strategic complementarity between (relative) pricing decisions of the firms. As a result of the movements of output gap and inflation in the same direction, in response to supply or preference shocks, monetary authorities can stabilize both measures successfully. 5. Conclusions This paper questions whether the central banks dilemma for stabilizing output gap or stabilizing inflation can be obtained with a model using interaction between aggregate shocks and firms' markups. The main purpose of implementation of this setting in the literature is to create price stickiness, and its implications for inflation–output policy trade-off and welfare are not investigated yet. Moreover, even though constant elasticity of demand assumption is a convenient assumption for economic modeling, the strategic complementarity in pricing decisions that arises with the endogenous markup setting improves these models towards a more realistic representation of the real world markets. The paper extends the standard new Keynesian framework with the endogenous markup setting and incorporate it into the model studied in Blanchard and Galí (2007). The findings show that the trade-off between output gap stabilization and inflation stabilization that is confronted by the central banks is not produced in this setup. Finally, it is shown that the flexible price markup remains unaffected from the endogenous markup setting; hence, targeting the level of output under flexible prices is equivalent to targeting the first best level of output. As a result, the optimal monetary policy in this environment is to target the flexible price equilibrium. Acknowledgments I am grateful to Jordi Galí, Sushanta Mallick (the editor), and three anonymous reviewers for their constructive comments and very helpful suggestions. All remaining errors are my own. 10
Mankiw (2007) provides a similar discussion. The referenced studies in this quotation are Blanchard and Galí (2010) and Ball and Mankiw (1995). 11
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631
which would be further simplified to
Appendix A. Consumers A.1. Solution to the consumer's problem
P i −P ψ″ ð1Þ Y i −Y ¼ 0 : P ψ ð1Þ Y
The consumer problem is defined as follows Z 1 Z 1 Y ψ i di: min P i Y i di s:t: 1 ¼ Y 0 0
Finally, when combined with (6), the equation above finds
Putting this problem into Lagrangian form and taking derivative with respect to Yi yields Y 1 P i ¼ λψ0 i Y Y
ðA:1Þ
where λ is the Lagrange multiplier. By calculating (A.1) at Pi = P and Yi = Y, we find
P i −P 1 Y −Y ¼− i P Yi which, given the following approximations for small changes: (ρi − ρ)/ρ ≈ ln(ρi/ρ) and (Yi − Y)/Yi ≈ (Yi − Y)/Y, can also be written as ln
Pi 1 Y ≈− ln i ; P Y
hence, PY λ¼ 0 ; ψ ð1Þ
− Yi P ≈ i where ¼ ð1Þ: Y P
using this equality with Eq. (A.1) results in the following inverse demand equation given in the text Y 1 : P i ¼ Pψ0 i Y ψ0 ð1Þ
ð5Þ
ðA:3Þ
Appendix B. Firm's problem in the flexible price setting B.1. Solution to the firm's problem Flexible price firms solve a static profit maximization problem max P ðY i ÞY i − ΨðY i Þ;
A.2. Derivation of the elasticity of demand
Yi
By definition, the elasticity of demand is ðρi Þ ¼ −
∂Y i =Y i P 1 ¼− i : Y i ∂P i =∂Y i ∂P i =P i
where Ψ(Yi) is the nominal cost function. Taking first order condition with respect to Yi finds ðA:2Þ 0¼
Calculating Pi and ∂Pi/∂Yi from (5) and using them together with (A.2) result in Eq. (6) in the text Y 1 Y Pψ0 i ψ0 i 0 Y ψ ð1Þ 1 1 1 ψ0 ðρi Þ Y ¼− ; ¼− ðρi Þ ¼ − Yi ″ Yi 1 1 Y i =Y ″ Y i ρi ψ″ ðρi Þ ψ Pψ 0 Y Y ψ ð1Þ Y
ð6Þ
∂P ðY i Þ Y i ∂ΨðY i Þ þ 1 P ðY i Þ− : ∂Y i P ðY i Þ ∂Y i
The final equation can be combined with the elasticity of demand equation which is, by definition, given as (ρi) = − [∂Yi/Yi]/[∂Pi/Pi]. This practice finds
1 ∂ΨðY i Þ þ 1 P ðY i Þ− : 0¼ − ðρi Þ ∂Y i which can be rearranged to find Eq. (9) in the text P ðY i Þ ¼
where we used the equilibrium condition in the goods market of a closed economy: Yi = Ci and Y = C, together with the definition of ρi which is the share of good in the consumption basket: ρi = Ci/C. One can obtain the familiar demand equation for monopolistically competitive markets by log-linearizing the inverse demand equation in (5) around the steady state at ρi = 1. To do so, we can take natural logarithm of the equation Y 1 þ ln 0 ln ðP i Þ ¼ ln ðP Þ þ ln ψ0 i Y ψ ð1Þ
ðρi Þ ∂ΨðY i Þ : ðρi Þ−1 ∂Y i
ð9Þ
B.2. Equality of Eqs. (11) and (12) Showing equality of Eqs. (11) and (12) only requires to show that MC = W/P ⋅ 1/MPN. To show this, notice that the nominal cost function of a firm that produces Y amount of good can be written as ΨðY Þ ¼ NðY ÞW subject to Y ¼ Sα N1−α :
and apply first order Taylor approximation The above equality and constraint can be combined into one equality 1 ψ ð1Þ 1 1 ln ðP Þ þ ðP i −P Þ ¼ ln ðP Þ þ ln ψ0 ð1Þ þ 0 Y ðY i −Y Þ þ ln 0 ; P ψ ð1Þ ψ ð1Þ
″
ΨðY Þ ¼
Y Sα
1=ð1−αÞ W:
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Taking derivative of cost function with respect to Y and dividing by the price level finds the nominal marginal cost of production α=ð1−αÞ ∂ΨðY Þ 1 1 Y 1W ð¼ MC Þ ¼ ; 1−α Sα ∂Y P Sα P
∞ X Pt P θk Et Q t;tþk Y tþk=t −M ρtþk=t MC tþk=t tþk ¼0 P t−1 P t−1 k¼0
now inserting for Y MC ¼
α
1−α
1 S N 1−α Sα
!α=ð1−α Þ
1 W 1 1 W ¼ : 1−α N−α Sα P Sα P
Eq. (8) in the text finds the marginal product of labor as MPN = (1 − α)Y/N = (1 − α)SαN−α. When inserted into the above equation, it finds the necessary equality that ensures equalities of Eqs. (11) and (12) MC ¼
This equation states that, in expected terms, the discounted value of firm's future profits is equal to the discounted value of its future costs times the markups defined over with these costs. This equation, when divided by Pt − 1, and also when its second term is multiplied and divided by Pt + k, can be written as
W 1 : P MPN
where MC is the real cost of marginal production and is equal to the nominal marginal cost function, Ψ′t + k (Yt + k/t), divided by Pt + k. At the zero inflation steady state the following conditions must hold: Pt − 1 = Pt⁎ = Pt + k, Yt + k/t = Y, MCt + k/t = MC = 1/M by Eq. (11) and Q t,t + k = βk. Applying log linearization to (C.1) around this steady state ( ∞ X θk Et βk Q eq^t;tþk Yey^tþk=t k¼0
Appendix C. Firms' problem in the sticky price setting
The profit maximization problem of a typical firm is given as (for notational convenience, firm specific subscripts are suppressed)
Pt
∞ X θk Et Q t;tþk P t Y tþk=t −Ψtþk Y tþk=t
!) ¼ 0:
∞ n o X θk Et βk QYeq^t;tþk þy^tþk=t þp^t −p^t−1 k¼0
ð17Þ
k¼0
¼
∞ n o X ^ ^ ^ ^ ^ ^ θk Et βk Q Yeqt;tþk þytþk=t þμ ðρtþk=t Þþmctþk=t þptþk −pt−1
ð1Þ
k¼0
where Q t,t + k = βk(Uc,t + k/Uc,t)(Pt/Pt + k) is the stochastic discount factor coming from the utility maximization problem of consumers, Ψ is the nominal cost function, Pt⁎ is the optimal price set by a firm at time t, Yt + k/t denotes the output of the firm in period t + k given that it charges the price which is set in period t, and (1 − θ) is the probability that the firm may reset its price. Eq. (17) is subject to the following sequence of demand conditions Y tþk=t ¼
Pep^t Pep^tþk ^ −Meμ ðρtþk=t Þ MCem^ctþk=t p^ ^t−1 p Pe Pe t−1
where the deviation of a variable from its steady state value is denoted by a hat. Using (11), which states that M ⋅ MC = 1, and rearranging the equation, we find
C.1. Solution to the firm's problem
max
ðC:1Þ
P t P tþk
−εðρÞ
^t;tþk þ y ^tþk=t terms cancel out. Using the approximawhere QY and 1 þ q
^t and p ^t−1 are not affected from the tion eρ ≅ (1 + ρ), and the fact that p summation operator, we find ∞ n X o ^t −p ^t−1 ¼ ð1−βθÞ ðβθÞk Et μ^ ρtþk=t þ m^ctþk=t þ p ^tþk −p ^t−1 : p k¼0
Y tþk :
ðC:2Þ
ð18Þ
∂Y tþk=t Y tþk=t ¼ − ð ρ Þ : P t ∂P t
C.1.1. Endogenous markups This section uses Eq. (C.2) to derive NKPC equation in (19). First, we start with μ^ ðρtþk=t Þ term in (C.2), which is the deviation of firm i's markup from its steady state, and which can be written in terms of the demand elasticity of markups that is approximated around ρ=1
Hence, the derivative of the firm maximization problem in (17) with respect to the optimal price (Pt⁎) yields
^tþk=t −y ^tþk : μ^ ρtþk=t ¼ ημ y
where the elasticity of demand depends on the market share of a firm, (ρ). Note that the derivative of (18) with respect to Pt⁎ can be written as
∞ X Y tþk=t θk Et Q t;tþk Y tþk=t þ −ðρÞY tþk=t −Ψ0tþk Y tþk=t −ðρÞ ¼ 0; Pt k¼0
taking the Yt + k/t parentheses and multiplying both sides of the equation by Pt⁎, we obtain ∞ X θk Et Q t;tþk Y tþk=t P t ð1−ðρÞÞ þ ðρÞΨ0tþk Y tþk=t ¼ 0:
Combining the previous equation with the demand equation in (18) that is log-linearized around its steady state yields ^ t −p ^tþk : μ^ ρtþk=t ¼ −ημ p
ðC:3Þ
k¼0
To derive the mĉt + k/t term in (C.2), the real marginal cost can be written as
Dividing both sides of the equation by (1 − (ρ)) and using the definition of markup in (9), which is: M = (ρ)/((ρ) − 1), we find
mct ¼ wt −mpnt ¼ wt −½yt −nt þ logð1−α Þ;
∞ X θk Et Q t;tþk Y tþk=t P t −M ðρt ðiÞÞΨ0tþk Y tþk=t ¼ 0: k¼0
where w denotes the (log) real wages that depend on the economy wide labor market and the mpn denotes the (log) marginal product of
O. Eksi / Economic Modelling 51 (2015) 626–634
labor that is taken from (8). When nt is taken from the production function in (7), the previous equation becomes mct ¼ wt þ
α y −s − logð1−α Þ: 1−α tþk tþk
For the marginal cost of a firm at time t + k, which resets its price at time t, the preceding equation reads mctþk=t ¼ wtþk þ
α ytþk=t −stþk − logð1−α Þ: 1−α
α ytþk=t −ytþk ; 1−α
and using the log linearized version of (A.3), together with the following definitions mĉt + k/t = mct + k/t − mc, mĉt + k = mct + k − mc, ŷt + k/t = yt + k/t − y and ŷt + k = yt + k − y within the last equation above, we find α ^tþk : ^t −p p ¼ m^ctþk − 1−α
ðC:4Þ
Inserting Eqs. (C.3) and (C.4) into (C.2) finds
∞ X α ^t −p ^tþk −p ^t−1 ¼ ð1−βθÞ ðβθÞk Et m^ctþk − ημ þ ^tþk þ p ^t−1 : ^t −p p p 1−α k¼0
First, rearranging this equation as, ^t−1 ^t −p p
∞ X α α ^t−1 þ 1 þ ημ þ ^t −p ^tþk ¼ ð1−βθÞ ðβθÞk Et m^ctþk − ημ þ p p 1−α 1−α k¼0
^t−1 to the RHS (right hand side) then, by adding ðÞ½ημ þ α =ð1−αÞp of this equation we obtain ^t−1 ^t −p p
α ^t −p ^t−1 p ¼ ð1−βθÞ ðβθÞk Et m^ctþk − ημ þ 1−α k¼0
α ^tþk −p ^t−1 : p þ 1 þ ημ þ 1−α ∞ X
f
g
^t −p ^t−1 terms which are not affected from the Collecting all the p summation term at the LHS (left hand side) of the equation and dividing the equation by [1 + ημ⁎* + α*/(1 − α)] gives ^t−1 ^t −p p
8 9 ∞ < X = 1 k ^tþk −p ^t−1 : i m^ctþk þ p ¼ ð1−βθÞ ðβθÞ Et h : 1 þ η þ α =ð1−α Þ ; k¼0
μ
^tþ1 −p ^t Þ can be written as Similarly, Et ðp 8 9 < ∞ X = 1 k ^t ¼ ð1−βθÞ ðβθÞ Et h ^tþ1þk −p ^t : ^tþ1 −p i m^ctþ1þk þ p Et p : 1 þ η þ α =ð1−α Þ ; k¼0 μ
Combining the last two equations delivers ^t−1 ¼ βθEt p ^t þ h ^tþ1 −p ^t −p p ^t −p ^t−1 ; þ βθp
1þ
^t −p ^t−1 : i m^ct þ p þ α =ð1−α Þ
πt ¼ βEt fπtþ1 g þ
ð1−θÞð1−βθÞ 1 h i m^ct : θ 1 þ η þ α =ð1−α Þ
ð19Þ
μ
C.1.2. Exogenous changes in markups We no longer assume that the changes in the markups are endogenous; as a result, the elasticity of demand, , is a constant now. Calculating (C.4) at the constant elasticity of demand and using it together with (C.2) yields
Combining the previous two equations we obtain mctþk=t ¼ mctþk þ
ð1−βθÞ ημ
^t −p ^t−1 ¼ ð1−θÞðp ^t −p ^t−1 Þ,12 the last equation above Using πt ¼ p yields Eq. (19) in the text
which holds at time t + k as well
m^ctþk=t
which can be simplified as ^t −p ^t−1 ¼ βθEt p ^t þ h ^tþ1 −p p
α ðy −st Þ− logð1−α Þ; 1−α t
mctþk ¼ wtþk þ
633
∞ n X o α ^ t−1 ¼ ð1−βθÞ ðβθÞk Et μ^ tþk þ m^ctþk − ^ tþk þ p ^t−1 ^t −p ^ −p ^tþk −p p p 1−α k¼0
where μ^ tþk defines the deviations in markups in response to shocks to ^t−1 to RHS of the this variable at time t + k. Adding ðÞ½α=ð1−αÞp equation, this equation becomes ∞ X α ^ −p ^t−1 ¼ ð1−βθÞ ðβθÞk Et μ^ tþk þ m^ctþk − ^t−1 ^t −p p p 1−α k¼0 α ^tþk −p ^t−1 : p þ 1þ 1−α
f
g
^t−1 terms at the LHS of the equation and dividing ^t −p Collecting the p it by [1 + α/(1 − α)], we get ∞ X ^t−1 ¼ ð1−βθÞ ðβθÞk Et ^t −p p k¼0
þ
f ½1 þ α=1ð1−αÞ μ^
tþk
g
1 ^tþk −p ^t−1 : m^c þ p ½1 þ α=ð1−α Þ tþk
^tþ1 −p ^t Þ can be written as Similarly, Et ðp ∞ X ^tþ1 −p ^t ¼ ð1−βθÞ ðβθÞk Et Et p k¼0
þ
f ½1 þ α=1ð1−αÞ μ^
tþ1þk
g
1 ^tþ1þk −p ^t : m^c þ p ½1 þ α=ð1−α Þ tþ1þk
Combining the last two equations ^tþ1 −p ^t−1 ¼ βθEt p ^t þ ^t −p p
ð1−βθÞ ð1−βθÞ μ^ þ m^ct ½1 þ α=ð1−α Þ t ½1 þ α=ð1−α Þ
^t −p ^t−1 : þp ^ t −p ^t−1 ¼ ð1−θÞðp ^t −p ^t−1 Þ in the last equation Finally, using π t ¼ p results in Eq. (23) in the text πt ¼ βEt fπtþ1 g þ
ð1−θÞð1−βθÞ ð1−θÞð1−βθÞ m^ct þ μ^ : θ½1 þ α=ð1−α Þ θ½1 þ α=ð1−α Þ t
ð23Þ
ð1−βθÞ 1þ
ημ
^t i m^ct þ ð1−βθÞp þ α =ð1−α Þ
12 ^ t −p ^ t−1 ) is determined by the proportion of This equality indicates that inflation (p firms that are able to reset their prices. This proportion (1 − θ) is indeed equal to the probability that a firm may reset its price.
634
O. Eksi / Economic Modelling 51 (2015) 626–634
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