Endogenous Markups in the New Keynesian Model: Implications for In ation-output Trade-O and Welfare

Transcription

1 Endogenous Markups in the New Keynesian Model: Implications for In ation-output Trade-O and Welfare Ozan Eksi TOBB University of Economics and Technology March 203 Abstract The standard new Keynesian (NK) model with nominal price rigidities is unable to produce the in ation-output trade-o that Central Banks have to deal with, unless the model is modi ed by exogenous shocks to price or wage markups. Blanchard and Gali (2007) proves that a supply side real rigidity (rigidity in real wages) can modify the standard NK model to produce the in ation-output trade-o in response to more conventional shocks. My paper has a similar set-up with this study but analyzes the demand side real rigidity: the endogenous markup setting. It documents that unlike the real wage rigidity setting and also exogenous markup shocks, the endogenous markup setting cannot improve the NK models to produce the in ation and output trade-o. The paper also investigates the optimal monetary policy under endogenous markup setting. It shows that the exible price markup is una ected; hence, the optimal policy is to target the exible price equilibrium. JEL Codes: E32, E52 Keywords: Monetary Policy, Endogenous Markups, In ation Output Trade-o Introduction Let y n t represents the equilibrium level of output obtained with the rms having exible prices, y t represents the realization of output when there are also rms with xed prices, and ~y = y t yt n is the output gap. Using these de nitions, the new Keynesian Phillips Curve (NKPC) equation in () t = E t f t+ g + ~y t () indicates that stabilizing the output gap also results in the stabilization of the in ation, called the divine coincidence. Therefore, the standard NK model is unable to create the in ation In the AS-AD framework, the Divine Coincidence can be desribed as the ability of monetary authority for keeping the in ation constant and the output equal to its natural level by counteracting upon the changes in AD and LRAS curves.

2 output trade-o that central banks face. The common way of obtaining this trade-o from NK models is to modify equation () by adding it a cost push shock (exogenous changes in price or wage markups; Galí, Gertler and Clarida (999)) as follows t = E t f t+ g + ~y t + u t : (2) Equation (2) indicates that a positive cost push shock leads either to a rise in in ation or a decline in the output gap; hence, the divine coincidence disappears. 2 The weak side of this approach is that the markup shocks in (2) are exogenous. In an attempt to obtain in- ation output policy trade-o in response to more conventional such as the technology or preference shocks, Blanchard and Galí (2007) uses a rigidity in real wages. 3 The current paper uses another type of real rigidity, the endogenous markup setting, and asks whether this setting can result in similar implications with real (wage) rigidity setting in generating in ation and output trade-o from the basic NK model. In general, a real rigidity is a source that makes rms unwilling to change their relative prices. For the rst time Ball and Romer (990) emphasizes the need for real rigidities. Kimball (995) and Rotemberg (996) are some other early examples. The need for real rigidities in monetarist models stem from the ndings that the sluggishness in the real price data cannot be fully explained by nominal rigidities. 4 In the application, these rigidities are either on the supply side (one way of obtaining a supply side real rigidity is to assume a rm speci c capital. Then rms would be reluctant to change their prices and their sales because each rm s short run marginal cost curve is increasing in its own output), or on the demand side (if the elasticity of consumer demand is increasing in rms prices, then again rms would be reluctant to change their relative prices). 5 In this paper I apply this demand side real rigidity, called the endogenous markup setting, into the standard NK framework. The endogenous markup setting is rst applied to a monetarist model by Kimball (995), and followed by many others such as Kimball (995), Eichenbaum and Fisher (2004) and Klenow and Willis (2006). In this world the rms having exible prices also consider that their decisions a ect the aggregate price index in the economy, so they diverge from the constant elasticity of demand assumption of Dixit and Stiglitz (977). For instance when a positive supply shock hits the economy, the rms that are able to lower their prices will obtain higher 2 In the AS-AD framework, cost push shocks are shifs in the SRAS curve. In response, monetary authority can stabilize either the price level or output. 3 They show that under the real wage rigidity i.e. the sluggish adjustment of real wages, the di erence between natural rate of output, and rst best output that occurs when there is no market imperfection or distortion such as nonzero markups is not constant. Therefore, stabilizing the output gap (y t yt n ) is no longer equal to stabilizing the welfare relevant output gap (y t y fb t ) and is no longer desirable from the welfare point of view. 4 See Christiano, Eichenbaum and Evans (999), Romer and Romer (2003), Bernanke, Boivin and Eliasz (2004) for a recent evidence in favor of price sluggishness in the real data. Klenow and Kryvtsov (2005), Dhyne et al. (2005) supply micro empirical ndings that indicate nominal rigidities based on the frequency of rms price changes are not enough to explain price sluggishness in the data. 5 For rigidities on the supply side see Gali and Gertler (999), Woodford (2003), Altig et al. (2005). 2

3 market share and in result will be confronted with lower elasticity of demand, leaving them less incentive to reduce their prices. Likewise, at the time of a negative supply shock, the rms that are able to increase their prices will end up with lower market share and in result will be confronted with higher elasticity of demand, leaving them less incentive to increase their prices. Consequently, the sluggishness of prices increases in the Kimball s model. Since the change in prices is lower, and in result the change in output gap is higher under the variable elasticity of demand assumption when compared with the constant elasticity of demand assumption, my motivation can be stated as to analyze in ation-output trade-o and optimal monetary policy implications of this countercyclically between in ation and output-gap. The method I follow is to incorporate the endogenous markup setting into the standard New Keynesian framework a convenient contribution, I believe and then use it together with the same economic model with Blanchard and Galí (2007). I show that the endogenous markup setting does not lead to the in ation output trade-o, also that the exible price markup is una ected in the endogenous markup setting; hence, the optimal monetary policy is to target the exible price equilibrium. The rest of the paper is organized as follows. Section 2 explains the model and search for its implications for the in ation-output trade-o of central banks. Section 3 analyzes the optimal monetary policy in this environment. Section 4 concludes. 2 The Model The baseline economic model is taken from Blanchard and Galí (2007). 2. Consumers The utility function for the representative consumer is given by U(C; N) = log(c) exp fg N + + ; where is preference parameter. Time and rm-speci c subscripts are suppressed until they are necessary. This utility function is used to obtain the following intratemporal marginal rate of substitution, which is equal to the real wages MRS = U n U c = exp fg N =C = W (in logs: mrs = w = c + n + ): (3) The Endogenous Markup Setup. The consumption aggregate C t is de ned as Z 0 ( c(i) )di = ; C 3

4 where () = and (x) is a strictly increasing and concave function for all x(i) = c(i)=c 0, and x(i) is the share of the goods in the consumption basket. When (x) = x ( )=, this consumption function results in Dixit-Stiglitz type consumption aggregator that assumes constant elasticity of substitution among goods. Kimball s (995) study, on the other hand, uses elasticity of substitution between goods is not constant but is increasing in rm s price level. As a result, the rms that are able to increase their prices have less incentive to do that. The cost minimization problem of the consumer is as follows Z min 0 p(i)y(i)di s:t: = Z 0 ( c(i) C )di: The goods are non-storable and consumed in the same period they are produced. In a closed economy environment, this assumption implies c = y. Using this equality, the solution of the above problem leads to the following implicit demand curve (the derivations for this section are provided in appendix A) 0 ( y(i) Y ) = 0 () p(i) P : (4) Finally, the elasticity of the demand for equation (4) is obtained as (x(i)) = 0 (x(i)) x(i) 00 (x(i)) : (5) Unlike Dixit and Stiglitz (977) approach, the elasticity formula in (5) is not constant but depends on the market share of a rm, x(i), i.e. on the price level of a rm. 2.2 Firms Monopolistically competitive rms use a Cobb-Douglas type of production function Y = M N (in logs: y = m + ( )n); (6) where N is labor input and M is non-produced input with exogenous supply. The use of M allows for supply (technology) shocks in the model, otherwise it is constant. The marginal product of labor can be found as MP N = ( )Y=N (in logs: mpn = (y n) + log( )): (7) The optimal markups can be pin down by the Lerner formula by using the elasticity of demand de ned in (5) (x) = (x(i)) (x(i)) : (8) 4

5 This markup has an elasticity (x(i)). approximate it around x = ; and denote it by : Following Kimball (995) and Woodford (2003), I 2.3 Baseline Scenario: The Equilibrium with Flexible Prices When prices are exible, the symmetry of rms implies that the market shares, and as a result markups are constant along the rms. Hence, the need for endogenous changes in markups does not arise with the exible prices. Since labor is the only input of production with a positive cost, the markups can be calculated as: = mpn w. Taking w and mpn from equations (3) and (7), I nd = ( + )n 2 + log( ) ; (9) where the subscript 2 denotes the second best (natural) level of labor. It is second best because even though the prices are exible there is still an imperfect competition in the goods market ( > 0). Combining (9) with the production function in (6) yields the following equation = ( + ) y 2 m ( ) + log( ) : (0) The real marginal cost of production is equal to the minus markups (mc = = w mpn) mc = ( + ) y 2 m ( ) log( ) + : () Finally, rearranging (0) results in the second best level of output y 2 = m The Equilibrium with Sticky Prices ( ) (log( ) ): (2) ( + ) Nominal price stickiness in the model is created by using the Calvo (983) approach, which yields the following NKPC equation (appendix B. provides the derivations) t = Et f t+ g + where is the discount factor and ( at a given period. 6 ( ")( ") " [ + + =( )] (mc t mc); (3) ") is the probability that the rm may reset its price (3) indicates that in ation is positively correlated with the deviations of real marginal cost from its exible price equilibrium. 6 If (x) is not approximated around x = (tha is ), the solution for NKPC is obtained as t = Et f t+g + [ (x)= + + (mct + ) =( )] 5

6 The mc t term in (3) can be written by using (), which holds for any time t mc t = ( + ) y t m ( ) log( ) + : (4) Using () and (4), (3) can also be written in terms of the deviations in the output rather than the deviations in the real marginal cost t = Et f t+ g + ( ")( ") " 2.5 Results for the In ation Output Trade-o ( + ) [ + + =( )] ( ) (y t y t;2 ); (5) Equation (5) shows that with the endogenous markup setting the stabilization of the output gap is equal to stabilization of in ation, which points out that no trade-o is generated from the model. On the other hand, Blanchard and Galí (2007) indicate that...in contrast with the baseline NKPC model, (with the model modi ed 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 rst-best level. In contrast to the baseline model, the distance between the rst- and the secondbest levels of output is no longer constant, but is instead a ected by the shocks... The authors indicate that the distance between the rst and second best levels of output is not constant in their models but it is a complicated function of model parameters; hence, stabilizing the output gap to its second best level is no longer desirable. To check if this possibility arises with our model, notice that equation (2) that de nes the second best level of output with no (real or nominal) rigidities. Using = 0 in that equation gives the rst best allocation y = m + The nal equation together with (2) results in ( ) (log( ) ): ( + ) y y 2 = ( ) : (6) ( + ) Equation (6) shows that under the endogenous markup setting the distance between the rst best and the second best levels of output equals to a constant term, i.e. is una ected from technology (m) or preference () shocks. Therefore, stabilizing the output gap at its second best level ( y y 2 ) is equal to stabilizing the welfare relevant output gap ( y y ), and no tradeo arises between in ation and output in a model with endogenous markup setting. Since the term + + in (5) is greater than the zero (this is because is approximated around, and and are positive terms that, respectively, de ne the elasticity of substitution between di erent goods and the labor share in production), according to the equations (3) and (5) the 6

7 endogenous markup setting only causes a real rigidity, i.e. a strategic complementarity between (relative) pricing decisions of the rms. Hence, even though the change in in ation is lower and the change in output gap is higher under the endogenous markup case when compared with the constant markup case, which could be de ned as a relatively countercyclical movement with respect to the case with constant elasticity of demand assumption, output gap still increases in the endogenous markup case along with the in ation. As a result, both of these measures can be both stabilized at a time unless the model is modi ed with exogenous shocks, which is shown below. 2.6 Exogenous Changes in Markups If we assume that markups are not endogenous but subject to exogenous cost push shocks as suggested by Galí, Gertler and Clarida (999), then the NKPC equation can be derived as (shown in appendix B.2) t = Et f t+ g + [ + =( )] (mc t mc) + [ + =( )] ( t ) (7) (7) shows that the term ( t ) that represents an exogenous change in the markup is a ective on the in ation in addition to the term (mc t mc) in (3). As a result, a shock to markup can either be confronted with an increase in t, or a decrease in mc t (i.e. a decrease in y t ), and the model obtains the in ation output trade-o. 3 Optimal Monetary Policy (6) shows that the distance between the rst best and the second best levels of output is constant under the endogenous markup setting. This result implies that the utility losses arise from deviations from e cient allocation (y ) remain parallel to those arise from deviations from the exible price allocation (y 2 ), which further requires the optimality of the monetary rules for the standard NKPC equation (see Galí, 2008) t = Et f t+ g + (y t y t;2 ) to remain valid. The only di erence is that now = ( ")( ") " ( + ) [ + + =( )] ( ) ; instead of the one obtained with constant elasticity of demand assumption = ( ")( ") " ( + ) ( ) : 7

8 The optimal policy should target exible price level of output and zero in ation. In general, monetary policies bene t from less variance in in ation and su er more from the variance in the output gap. 4 Conclusions This paper questions whether the central banks dilemma for stabilizing output gap or stabilizing in ation can be obtained with a model using interaction between aggregate shocks and rms markups, i.e. the endogenous markup setting. The use of this setting in the literature is to create price stickiness, and its in ation output trade-o and policy implications are not investigated yet. Moreover, even though constant elasticity of demand assumption is a convenient assumption for economic modelling, the strategic complementarity in pricing decisions that arises with the endogenous markup setting proves to be a more realistic representation of the real world markets. I modify the standard New Keynesian framework with the endogenous markup setting and use it within the model studied in Blanchard and Galí (2007). The ndings show that the tradeo between output gap stabilization and in ation stabilization that is confronted by the central banks is not produced by this set-up. The results follow the fact the strategic complementarity in pricing decisions only creates countercyclical movements in in ation and output conditional on the e ect of the shock that deviates the markups, and unlike with exogenous markup shocks, we still observe procyclical changes in in ation and output. Finally, it is brie y indicated that the exible price markup remains una ected from the endogenous markup setting; hence, targeting the level of output obtained with exible prices is equivalent to targeting the rst best level of output and is the optimal monetary policy. APPENDIX A-Derivation of the Demand Side Equations The consumer problem is de ned as follows Z min 0 p(i)y(i)di s:t: = Z 0 ( y(i) Y )di Using this problem in a Lagrangian and taking its derivative with respect to y t (i) yields p(i) = 0 ( y(i) Y ) Y (8) 8

9 where is the Lagrange Multiplier. By calculating (8) at p(i) = P and y t (i) = Y, we nd = P Y 0 () ; Using this equality in (8) results in the inverse demand equation given in the text p(i) = P t 0 ( y(i) Y ) 0 () : (4) By de nition, the elasticity of demand is = p(i) : (9) Calculating p t (i) t (i)=@y t (i) from (4) and using them together with (9) result in (x(i)) = y(i) P t 0 ( y(i) Y ) 0 () P 00 ( y(i) Y ) Y 0 () = y(i)=y 0 ( y(i) Y t ) 00 ( y(i) Y ) = x t (i) 0 (x(i)) 00 (x(i)) : (5) When we use the elasticity of demand equation in (5) to log-linearize the inverse demand equation in (4) around the steady state at x(i) =, we obtain the familiar demand equation for monopolistically competitive markets with constant elasticity of demand assumption ln( y(i) Y ) = ln( p(i) P ) where = (); (20) which can also be written as y(i) Y = (p(i) P ) B-Firms Pro t Maximization Problem The maximization problem of rms follows Galí (2008) and is given as max P t " k E t Qt;t+k (Pt Y t+k=t t+k(y t+k=t )) (2) where Q t;t+k = k (U 0 (C t+k )=U 0 (C t ))(P t =P t+k ) is the stochastic discount factor, is the nominal cost function, Y t+k=t denotes output in period t + k for a rm resetting its price last time in period t, P is the optimal price set by a rm at time t, and ( ") is the probability that 9

10 the rm may reset its price: The problem is subject to the following demand conditions Y t+k=t = ( P t P t+k ) Y t+k (22) Inserting this constraint into the rm maximization problem in (2) and taking derivative with respect to the optimal price (P t ) yields " k E t Qt;t+k Y t+k=t (Pt (x t (i)) 0 t+k (Y t+k=t)) = 0 The last equation, when divided by P t, can be written as " k E t Q t;t+k Y t+k=t ( P t P t+k (x t (i))mc P t+k=t = 0 (23) t P t where MC is the real cost of marginal production and equal to the derivative of the nominal cost function, 0 t+k (Y t+k=t ), with respect to P t+k. At the zero in ation steady state the following conditions must hold: P t = P t = P t+k, Y t+k=t = Y, MC t+k=t = MC = = and Q t;t+k = k. Applying log linearization to (23) around this steady state nds p t p t = ( ") B.: Endogenous Markups (") k E t ^(xt+k (i)) + m^c t+k=t + (p t+k p t ) (24) Equation (24) is used to derive NKPC equation in (3). To ^(x t+k (i)) term in (24) is the deviation of the rm i s markup from its steady state, and it can be written in terms of the demand elasticity of markups that is de ned under equation (8) and approximated around x = ^(x t+k (i)) = (y t+k (i) y t+k ) combining the previous equation with the demand equation in (22) yields ^(x t+k (i)) = (p t p t+k ) (25) To derive the m^c t+k=t term in (24), the real marginal cost can be written as mc t (i) = w t mpn t (i) = w t [y t (i) n t (i) + log( )] ; where w denotes the real wages that depends on the economy wide labor market and the mpn denotes the marginal product of labor that is taken from (7). When we also take n t from the 0

11 production function in (6), the previous equation becomes so that mc t (i) = w t + mc t+k=t (i) = w t+k + Combining the previous two equations we obtain using (20) with this equation yields mc t+k=t = mc t+k + (y t(i) m t ) log( ); (y t+k=t(i) m t+k ) log( ): (y t+k=t y t+k ) mc t+k=t = mc t+k (p p t+k ) (26) Inserting equations (25) and (26) into equation (24) nds p t p t = ( ") (") k E t m^ct+k [ + + =( )](p p t+k ) + (p t+k p t ) adding ()[ + =( )]p t to the RHS (right hand side) of this equation we obtain p t p t = ( ") (") k E t m^ct+k ( + =( ))(p p t ) + [ + + =( )](p t+k p t Now collecting all the p t p t terms at the LHS of the equation and then divide the equation by [ + + =( )] gives p t p t = ( ") (") k E t [ + + =( )] m^c t+k + (p t+k p t ) : Similarly, E t (p t+ p t ) can be written as E t (p t+ p t ) = ( ") (") l E t [ + + =( )] m^c t++l + (p t++l p t ) : l=0 Combining the last two equations p t p t = ( ") [ + + =( )] m^c t + "E t (p t+ p t ) + ( ")(p t p t ): Using t = ( ")(p t p t ) (this equality indicates that in ation is determined by the number

12 of rms that are able to set their prices to the optimal level), the last equation becomes t = Et f t+ g + ( ")( ") " [ + + =( )] m^c t: (3) B.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 taken as a constant term. Calculating (26) at the constant elasticity of demand and using the result in (24) yields p t p t = ( ") (") k E t ^t+k + m^c t+k [=( )](p p t+k ) + (p t+k p t ) adding ()[=( )]p t to RHS of the equation, it becomes p t p t = ( ") (") k E t ^t+k + m^c t+k [=( )](p p t ) + ( + =( ))(p t+k p t ) collecting the p t p t terms at the LHS of the equation and dividing the equation by [ + =( )], we get p t p t = ( ") (") k E t [ + =( )] ^ t+k + [ + =( )] m^c t+k + (p t+k p t ) : Similarly, E t (p t+ p t ) can be written as E t (p t+ p t ) = ( ") (") l E t [ + =( )] ^ t++l + [ + =( )] m^c t++l + (p t++l p t ) : l=0 Combining the last two equations p t p t = ( ") [ + =( )] ^ t + ( ") [ + =( )] m^c t + "E t (p t+ p t ) + ( ")(p t p t ) Finally, using t = ( ")(p t p t ) in the last equation results in (7) t = E t f t+ g + ( ")( ") "[ + =( )] m^c ( ")( ") t + "[ + =( )] ^ t (7) 2

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