The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Discretionary Monetary Policy in the Calvo Model WP 11-03R Willem Van Zandweghe Federal Reserve Bank of Kansas City Alexander L. Wolman Federal Reserve Bank of Richmond Discretionary Monetary Policy in the Calvo Model∗ Willem Van Zandweghe† Alexander L. Wolman‡ March 14, 2017 Working Paper No. 11-03R Abstract We study discretionary equilibrium in the Calvo pricing model for a monetary authority that chooses the money supply, producing three main contributions. First, the model delivers a unique private-sector equilibrium for a broad range of parameterizations, in contrast to earlier results for the Taylor pricing model. Second, a generalized Euler equation shows how the monetary authority affects future welfare through its influence on the future state of the economy. Third, we provide exact solutions, including welfare analysis, for the transitional dynamics that occur if the monetary authority loses or gains the ability to commit. JEL Classification: E31; E52 Keywords: Time-consistent optimal monetary policy; Discretion; Markov-perfect equilibrium; Sticky prices; Relative price distortion ∗ We have benefited from discussions and comments from Gary Anderson, Roberto Billi, Andreas Horn- stein, Jinill Kim, Bob King, Per Krusell, Takushi Kurozumi, Stéphane Moyen, Víctor Ríos-Rull, Pierre Sarte, Russell Wong, Raf Wouters, Tack Yun, and two anonymous referees, and from the feedback of seminar participants at the Bundesbank, Carlos III University in Madrid, European Central Bank, Federal Reserve Banks of Kansas City and Richmond, Humboldt University in Berlin, Ludwig-Maximilians University in Munich, National Bank of Belgium, and Norges Bank. Jonathan Tompkins and Allen Sirolly provided outstanding research assistance. The views expressed in this paper are those of the authors alone. They are not the views of the Federal Reserve Bank of Kansas City, the Federal Reserve Bank of Richmond or the Federal Reserve System. † Corresponding author. Research Department, Federal Reserve Bank of Kansas City, 1 Memorial Drive, Kansas City, MO 64198. Tel: 816-881-2766. E-mail: willem.vanzandweghe@kc.frb.org. ‡ Research Department, Federal Reserve Bank of Richmond. E-mail: alexander.wolman@rich.frb.org. 1 1 Introduction Over the last two decades New Keynesian models have become the dominant framework for applied monetary policy analysis. This framework is characterized by optimizing privatesector behavior in the presence of nominal rigidities, typically Calvo (1983) pricing as described by Yun (1996). The fact that some prices are predetermined in these models leads to a time-inconsistency problem for monetary policy, and there is a vast literature studying aspects of discretionary, i.e. time-consistent, policy in New Keynesian models with Calvo pricing. While the typical practice, exempli…ed by Clarida, Gali and Gertler (1999) and Woodford (2003), has been to work with models approximated around a zero-in‡ation steady state, a growing literature studies the discretionary policy problem with global methods. This paper contributes to that literature in three ways. First, it shows that for a broad range of parameterizations, the Calvo model delivers a unique private-sector equilibrium when monetary policy is conducted without commitment, in contrast to earlier results for the Taylor model. Second, it derives a generalized Euler equation (GEE), as in Krusell, Kuruscu and Smith (2002) and Klein, Krusell and Rios-Rull (2008), and uses the GEE to decompose the dynamic policy tradeo¤s facing a discretionary policymaker. Third, it conducts welfare analysis (without approximation) of the transitional dynamics that occur when a policymaker loses or gains the ability to commit. The …rst contribution relates to an existing literature which has identi…ed discretionary policy as a source of multiple equilibria. Private agents make decisions, such as saving or price setting, based on expectations of future policy. Those decisions in turn are transmitted to the future through state variables, creating the potential for a form of complementarity between future policy and expected future policy when policy is chosen under discretion. Viewed from another angle, the fact that policy will react to endogenous state variables can be a source of complementarity among private agents’ actions. The link between discretionary policy and multiple equilibria has been especially prominent in the monetary policy literature. In particular, Khan, King and Wolman (2001) and King and Wolman (2004) show that in Taylor-style models with prices set for three and two periods respectively, multiple privatesector equilibria are pervasive under discretion.1 Calvo and Taylor models are similar in many 1 Albanesi, Chari and Christiano (2003) show that multiple equilibria arise under discretionary policy in a 2 ways, yet we …nd no evidence that discretionary policy generates equilibrium multiplicity in the Calvo model. Although we do not prove the uniqueness of discretionary equilibrium, we show that a policy analogous to the optimal policy in the Taylor model guarantees a unique equilibrium in the Calvo model, despite its greater potential for complementarity than the optimal discretionary policy. We trace the contrasting behavior of the two models to di¤erences in how current pricing decisions a¤ect the distribution of future predetermined prices, and how the future policymaker responds to that distribution. Uniqueness of private-sector equilibrium opens up the possibility of deriving a GEE, which represents the dynamic tradeo¤ facing a discretionary policymaker in equilibrium. While the GEE has been extensively studied in …scal policy applications, and recently extended to the Rotemberg sticky-price model by Leeper, Leith and Liu (2016), to the best of our knowledge it has not previously been derived for the Calvo model.2 Under discretion, the policy problem is dynamic only to the extent that endogenous state variables a¤ect future welfare, either directly or by shifting the future policymaker’s problem. With Calvo price setting, an index of predetermined relative prices is an endogenous state variable. The GEE reveals three distinct channels through which this state variable links current policy to future welfare. First, the state variable a¤ects welfare directly because it acts as an aggregate productivity shifter. Second, the state variable in‡uences the future stance of policy, which a¤ects consumption and leisure. Third, the state variable a¤ects price setting in the future, model in which a fraction of …rms have predetermined prices. Siu (2008) extends King and Wolman’s (2004) analysis and Barseghyan and DiCecio (2007) extend Albanesi, Chari and Christiano’s (2003) analysis, by incorporating elements of state-dependent pricing and showing that Markov-perfect discretionary equilibrium is unique. Those papers assume that monetary policy is conducted with a money supply instrument. Dotsey and Hornstein (2011) show that with an interest rate instrument there is a unique Markov-perfect discretionary equilibrium in a Taylor model with two-period pricing. 2 Our paper is closely related to Anderson, Kim and Yun (2010). They study optimal allocations without commitment in the Calvo model. Their approach cannot be used to investigate the possibility of multiple private-sector equilibria for a given policy action, or to derive a GEE. Their solution method, like ours, is based on Chebyshev collocation. While they study a slightly di¤erent region of the parameter space, the nature of their solutions is consistent with our …ndings. Ngo (2014) extends their analysis to a stochastic environment with the zero bound on nominal interest rates, and Leith and Liu (2016) use their approach to compare the Calvo and Rotemberg models. 3 which has an independent e¤ect on welfare. When any aspect of public policy su¤ers from a time-consistency problem, it is important to know the value of commitment. Thus, the third contribution of the paper is to provide exact solutions to the transitional dynamics that occur (i) when an economy that had been operating with optimal policy under commitment unexpectedly …nds itself with a policymaker who cannot commit to future policy, and (ii) when an economy that had converged to a discretionary steady state unexpectedly …nds itself with a policymaker that can commit to future policy. The …rst of these transitions is a straightforward analysis of the discretionary equilibrium studied in the …rst part of the paper. The second transition is more involved, as it requires also solving for the dynamics of optimal policy under commitment. In both cases, we …nd that the welfare loss (or gain) from the transition is quite close to the steady-state welfare di¤erence between discretion and commitment. The paper proceeds as follows. The next section contains a description of the Calvo model. Section 3 de…nes a discretionary equilibrium. Section 4 contains the numerical results for the discretionary equilibrium, emphasizing the issue of multiplicity or lack thereof. Section 5 describes the GEE approach. Section 6 presents the results on transitional dynamics and welfare. Section 7 relates our analysis to the early literature on discretionary monetary policy and concludes. Secondary material is contained in appendixes. 2 The Calvo model The model is characterized by a representative household that values consumption and dislikes supplying labor, a money demand equation, a competitive labor market, a continuum of monopolistically competitive …rms producing di¤erentiated goods, and a monetary authority that chooses the money supply. Each …rm faces a constant probability of price adjustment. We assume that the model’s exogenous variables are constant, so there is no uncertainty about fundamentals. 2.1 Households There is a large number of identical, in…nitely-lived households. They act as price-takers in labor and product markets, and they own shares in the economy’s monopolistically compet4 itive goods-producing …rms. Households’preferences over consumption (ct ) and labor input (nt ) are given by 1 X j j=0 " # n1+ t+j 1+ ln(ct+j ) 2 (0; 1) ; ; 0; > 0; where consumption is taken to be the Dixit-Stiglitz aggregate of a continuum of di¤erentiated goods with elasticity of substitution " > 1, ct = Z 1 ct (z) " 1 " " " 1 dz (1) : 0 The consumer’s ‡ow budget constraint is Pt wt nt + Rt 1 Bt 1 + Z 1 dt (z) dz Pt ct + Bt ; 0 where wt is the real wage, Rt is the one-period gross nominal interest rate, Bt is the quantity of one-period nominal bonds purchased in period t, dt (z) is the dividend paid by …rm z, and Pt is the nominal price of a unit of consumption. The aggregator (1) implies the demand functions for each good, " Pt (z) Pt ct (z) = (2) ct ; where Pt (z) is the price of good z. The price index is given by Pt = Z 1 1 " 1 1 " Pt (z) dz : (3) 0 From the consumer’s intratemporal and intertemporal problems we have the e¢ ciency conditions: wt = ct nt Rt ct+1 = ct where t Pt =Pt 1 (4) ; t+1 denotes the gross in‡ation rate between periods t 1 and t. We assume there is a money demand equation such that the quantity of money is equal to the nominal value of consumption, Mt = Pt ct : (5) This constant-velocity money demand equation simpli…es the model by abstracting from any distortions arising from money demand, and enables a straightforward comparison with 5 the existing literature (e.g. King and Wolman, 2004). It will be convenient to write the money demand equation normalizing by the lagged price level, which serves as an index of the predetermined nominal prices in period t: Mt = Pt 1 mt (6) t ct : We will refer to mt as the normalized money supply. 2.2 Firms Each …rm z 2 [0; 1] produces output yt (z) using a technology that is linear in labor nt (z), the only input, with a constant level of productivity that is normalized to unity: yt (z) = nt (z) : A …rm adjusts its price with constant probability 1 each period, as in Calvo (1983).3 As …rms are owned by households, the value of a …rm upon adjustment is given by " #) (1 " " X ct Xt Xt Pt j Xt ct+j Pt+j wt+j ct+j : max ( ) Xt Pt+j ct+j Pt+j Pt+j j=0 The factor j is the probability that a price set in period t will remain in e¤ect in period t + j. The optimal price is determined by di¤erentiating with respect to Xt . We will denote the pro…t-maximizing value of Xt by P0;t and we will denote by p0;t the nominal price P0;t normalized again by the previous period’s price level, p0;t P0;t =Pt 1 . Thus, we write the …rst-order condition as P0;t p0;t = = Pt t " " 1 P1 j=0 ( P1 j=0 ( )j (Pt+j =Pt )" wt+j )j (Pt+j =Pt )" 1 : (7) The real wage is equal to real marginal cost because …rm-level productivity is assumed constant and equal to one. With the constant elasticity aggregator (1) a …rm’s desired markup of price over marginal cost is constant and equal to "=(" 1). Because the …rm cannot adjust its price each period, if the real wage or the in‡ation rate are not constant then the …rm’s markup will vary over time. The optimal pricing equation (7) indicates 3 In Yun’s (1996) version of the Calvo model there is price indexation, whereas the version in King and Wolman (1996) has no indexation. We analyze the Calvo model without indexation. 6 that the …rm chooses a constant markup over an appropriately de…ned weighted average of current and future marginal costs. Note that the economy-wide average markup is simply the inverse of the real wage. The optimal pricing condition can be written recursively by de…ning two new variables, St and Ft , that are related to the numerator and denominator of (7), respectively: St = " t Ft = " 1 t (wt + (8) St+1 ) ; (1 + Ft+1 ) ; (9) then, p0;t = " " St : Ft 1 (10) Because of Calvo pricing, the price index (3) is an in…nite sum, "1 #11" X 1 " ; Pt = (1 ) j P0;t j (11) j=0 but it can be simpli…ed, …rst writing it recursively and then dividing by the lagged price level: 1 1 " 1 " ) p0;t + = (1 t : (12) Without loss of generality, we have normalized all period-t nominal variables by the period t 2.3 1 price level. Market clearing Goods market clearing requires that the consumption demand for each individual good is equal to the output of that good: (13) ct (z) = yt (z) ; and labor market clearing requires that the labor input into the production of all goods equal the supply of labor by households: Z 1 nt (z)dz = nt : (14) 0 The labor market clearing condition is nt = 1 X (1 j=0 7 ) j nj;t ; (15) where nj;t is the labor input employed in period t by a …rm that set its price in period t j. Combining this expression with the goods market clearing condition (13), then using the demand curves (2) for each good, and dividing the expression by the consumption aggregator yields nt X = (1 ct j=0 1 ) j P0;t Pt " j ; (16) ; (17) which can be written recursively as t = " t ) p0;t" + (1 t 1 where t and we call 2.4 t 1 nt =ct ; (18) the inherited relative price distortion. Monetary authority and timing The monetary authority chooses the money supply, Mt . In a discretionary equilibrium the money supply will be chosen each period to maximize present-value welfare. We assume the sequence of actions within a period is as follows: 1. Predetermined prices (P0;t j , j > 0) are known at the beginning of the period. 2. The monetary authority chooses the money supply. 3. Firms that adjust in the current period set their prices, and simultaneously all other period-t variables are determined. Timing assumptions are important in models with staggered price setting. Transposing items 2 and 3 or assuming that …rms and the monetary authority act simultaneously would change the nature of the policy problem and the properties of equilibrium. 3 Discretionary equilibrium in the Calvo model We are interested in studying Markov-perfect equilibrium (MPE) with discretionary monetary policy. In an MPE, outcomes depend only on payo¤-relevant state variables; trigger strategies and any role for reputation are ruled out. Hence, it is important to establish what 8 the relevant state variables are. Although there are an in…nite number of predetermined nominal prices (P0;t j ; j = 1; 2; :::), for an MPE a state variable is relevant only if it affects the monetary authority’s set of feasible real outcomes. It follows that in an MPE the normalized money supply and all other equilibrium objects are functions of the single state variable t 1. A discretionary policymaker chooses the money supply as a function of the state, taking as given the behavior of future policymakers. In equilibrium, the future policy that is taken as given is also the policy chosen by the current policymaker. 3.1 Equilibrium for arbitrary monetary policy As a preliminary to studying discretionary equilibrium, it is useful to consider stationary equilibria for arbitrary monetary policy— that is, for arbitrary functions m = ( ). To de- scribe equilibrium for arbitrary policy we use recursive notation, eliminating time subscripts and using a prime to denote a variable in the next period. The nine variables that need to 0 be determined in equilibrium are S, F , p0 , , , c, n, w, and m, and the nine equations are the laws of motion for S (8) and for F (9); the optimal pricing condition (10); the price index (12); the law of motion for the relative price distortion (17); the labor supply equation (4); money demand (6); the de…nition of the relative price distortion (18); and the monetary policy rule m = ( ). A stationary equilibrium can be expressed as two functions of the endogenous state variable. The two functions S ( ) and F ( ) must satisfy the two functional equations S( ) = " F( ) = " 1 [w + 0 S( [1 + F( )] ; (19) 0 (20) )] ; where the other variables are given recursively by the following functions of p0 = 0 " " = (1 = " c = n = w = S( ) ; F( ) 1 ) p10 + ) p0 " + (1 m 0 " : (21) 1=(1 ") ; ; (22) (23) (24) c; (25) cn : (26) 9 Given an arbitrary policy of the form m = ( ), functions S () and F () that satisfy (19) (26) represent a stationary equilibrium. 3.2 Discretionary equilibrium de…ned A discretionary equilibrium is a particular stationary equilibrium with policy given by a mapping from the state to the money supply, m = ( ), in which the following property holds: If the current-period policymaker and current-period private agents take as given that all future periods will be described by a stationary equilibrium associated with the current-period monetary authority maximizes welfare by choosing m = ( ), then ( ) for every . More formally, a discretionary equilibrium is a policy function ( ) and a value function v ( ) that satisfy n1+ v ( ) = max ln (c) + v ( 0) m 1+ n1+ ( ) = arg max ln (c) + v( m 1+ (27) 0 ) when v () = v (). The maximand in (27) can be seen to be a function of m by combining (22) (26) with optimal pricing by adjusting …rms, p0 = " " " 1 " [w + 1 [1 + S ( 0 )] ; F ( 0 )] (28) where the functions S () and F () satisfy (19) and (20) in the stationary equilibrium associated with ( ). Note the subtle di¤erence between (28) and (19) (21): in (28), which applies in the current period, we have not imposed a stationary equilibrium. The monetary authority takes as given that the future will be described by a stationary equilibrium. It is an equilibrium outcome, not a constraint, that current-period policy is identical to that which generates the stationary equilibrium in the future. 4 Properties of discretionary equilibrium We use a projection method to compute numerical solutions for discretionary equilibrium, restricting attention to equilibria that are limits of …nite-horizon equilibria. This restriction may further reduce the number of discretionary equilibria (Krusell, Kuruscu and Smith, 10 2002), and allows us to derive a useful analytical result for the case of a (suboptimal) monetary policy that holds m constant. Computational details are provided in Appendix A. The quarterly baseline calibration is common in the applied monetary policy literature: " = 10, = 0:99, = 0:5, = 0, = 4:5. Prices remain …xed with probability = 0:5, which means that the expected duration of a price is two quarters. The demand elasticity " = 10 implies a desired markup of approximately 11 percent. With elastic. Given the values for " and , = 0 labor supply is perfectly = 4:5 is chosen to target a steady-state level of labor in the ‡exible-price economy of n = 0:2. The baseline calibration is chosen to facilitate comparison with King and Wolman (2004), but many other examples were computed that cover a wide range of structural parameter values. There are two levels to a complete description of a discretionary equilibrium. First, the equilibrium is characterized by the value function, v ( ) and the associated monetary policy function, m = ( ), along with the transition function for the state variable and policy functions for the other endogenous variables. Second, for given values of the state variable and the normalized money supply, private-sector equilibrium involves the …xed point of a pricing best-response function. We use the best-response function to study uniqueness. 4.1 Equilibrium functions Figure 1, Panel A plots the transition function for the state variable as well as the function mapping from the state to the in‡ation rate in a discretionary equilibrium.4 The …rst thing to note is that there is a unique steady-state in‡ation rate of 5:5 percent annually.5 Two natural benchmarks against which to compare the steady state of the discretionary equilibrium are the in‡ation rate with highest steady-state welfare and the in‡ation rate in the long run 4 Note that in the model is a gross quarterly in‡ation rate, but the …gures and the text refer to annualized net in‡ation rates obtained as 100( 5 4 1) percent. The steady-state in‡ation rate reaches the range of 9-10 percent in the Calvo model when " is lower than the baseline value, or when is larger but not too large. For instance, reducing the demand elasticity to " = 8, which implies a desired markup of 14 percent, raises the steady-state in‡ation rate to 9:6 percent. Increasing the probability of no price adjustment to percent, and it declines for even larger values of = 0:71 raises the steady-state in‡ation rate to 9:8 . Anderson, Kim and Yun (2010) point out similar relationships between the model’s structural parameters and the steady-state in‡ation rate. 11 under optimal policy with commitment. Following King and Wolman (1999), we refer to these benchmarks as the golden rule and the modi…ed golden rule respectively. For our baseline parameterization, the golden-rule in‡ation rate is just barely positive (less than one tenth of a percent) and the modi…ed golden-rule in‡ation rate is zero. The latter result is parameter-independent; we return to it in Section 6.1. In addition to showing the steady state, Panel A illustrates the dynamics of the state variable, which exhibit monotonic convergence to the steady state. This means that a policymaker inheriting a relative price distortion that is large relative to steady state …nds it optimal to bequeath a smaller relative price distortion to her successor. Together with the monotone downward-sloping equilibrium function for in‡ation, it follows that the in‡ation dynamics in the transition from a large relative price distortion (as would be implied by a high in‡ation rate) involve an initial discrete fall in in‡ation and a subsequent gradual increase to the steady state.6 Panel B of Figure 1 displays the policy variable (m) and welfare (v) as functions of the state variable in the discretionary equilibrium (m is plotted on the left scale and welfare on the right scale).7 Both functions are downward sloping. Intuition for the welfare function’s downward slope is straightforward. By de…nition, the current relative price distortion represents the inverse of average productivity. But the current relative price distortion is also a summary statistic for the dispersion in relative prices. The higher is the inherited relative price distortion, the higher is the inherited dispersion in relative prices, and through (23) this contributes to a higher dispersion in current relative prices. Higher dispersion in current relative prices in turn reduces current productivity, reducing welfare. It is less straightforward to understand the downward sloping policy function, m = ( ). At …rst glance, it seems consistent with the state transition function for m to be decreasing in : if equilibrium involves the relative price distortion declining from a high level, then a large inherited relative price distortion ought to be met with a relatively 6 Yun’s (2005) analysis of the Calvo model with a subsidy to o¤set the markup distortion displays similar transition dynamics of in‡ation. But in his model, the steady-state in‡ation rate under optimal policy is zero, so the transition from a steady state with positive in‡ation inevitably involves a period of de‡ation. 7 In Panel B of Figure 1 we have not converted welfare into more meaningful consumption-equivalent units. We defer a quantitative discussion of welfare to Section 6. 12 A. State transition and inflation 1.02 1.015 0 8 State transition (left) 45-degree line (left) Inflation (right) Steady-state inflation rate (right axis) 6 1.01 4 1.005 2 0 Steady-state " (left axis) 1 1 % 1.005 1.01 1.015 0 1.02 State variable ( " ) B. Policy instrument and welfare 0.204 -251.16 Policy instrument (left) Welfare (right) -251.17 0.201 ! *(" ) v*( " ) -251.18 0.198 0.195 1 1.005 1.01 1.015 State variable ( " ) Figure 1: Equilibrium as a function of the state 13 -251.19 1.02 low normalized money supply, so that newly adjusting …rms do not exacerbate the relative price distortion. Looking in more detail, the essential short-run policy trade-o¤ is that the policymaker has an incentive to raise the money supply in order to bring down the markup, but this incentive is checked by the cost of increasing the relative price distortion. It appears that the short-run trade-o¤ shifts in favor of the relative price distortion as the state variable increases. That is, in equilibrium the policymaker chooses lower m at larger values of because the value of the decrease in the markup that would come from holding m …xed at higher 4.2 is more than o¤set by welfare costs of a higher relative price distortion.8 Private-sector equilibrium Our computational approach has led to …nding a single discretionary equilibrium. The preceding discussion highlighted some of the properties of the equilibrium for the baseline calibration. Although we have not proved that the equilibrium is unique, in the many other examples described in Appendix A we have found no evidence of multiple equilibria. This is in stark contrast to the Taylor model with two-period price setting, in which King and Wolman (2004) proved the existence of multiple discretionary equilibria, which they traced to multiple private-sector equilibria. To help explain why multiplicity of private-sector equilibrium does not appear in any of our numerical solutions for the Calvo model, we turn to the best-response function for price-setting …rms. The best-response function describes an individual …rm’s optimal price as a function of the price set by other adjusting …rms. Figure 2 plots a typical best-response function in a discretionary equilibrium of the Calvo model, using the baseline calibration. It has a unique …xed point, and is concave in a neighborhood of the …xed point. In contrast, the best-response function in the two-period Taylor pricing model is upward sloping, strictly convex and generically has either two …xed points or no …xed points (see King and Wolman, 2004, Figure I).9 8 In the two-period Taylor model, the policymaker also faces a short-run trade-o¤ between reducing the markup and increasing the relative price distortion, but does not face a cost of leaving a higher inherited price distortion to future policymakers. Appendix B provides a description of the Taylor model and a quantitative comparison with the Calvo model. 9 Our computations have not revealed multiple …xed points in equilibrium. However, we have encountered 14 1.1 0.9 0.7 Pricing best-response function 45-degree line 0.5 0 0.5 1 1.5 2 p0 Figure 2: Pricing best-response function: State = 1:004, m = 0:202 15 The starkly di¤erent best-response functions in the two models re‡ect di¤erences in how future monetary policy reacts to the price …rms set in the current period. This relationship is linear in the Taylor model: the price set in the current period (P0 ) is precisely the index of predetermined nominal prices that normalizes the future money supply, and the normalized money supply is constant in discretionary equilibrium. The relationship is nonlinear in the Calvo model, for two reasons. First, the relationship between P0 and the future index of predetermined prices (P ) is nonlinear. Second, P0 a¤ects the real state variable to which future policy responds ( 0 ). We consider in turn how both these factors weaken the complementarity in price setting. First, suppose the future policymaker were to set a constant m, raising the nominal money supply in proportion to the index of predetermined prices. In the Taylor model, where such a policy is optimal, the price set by adjusting …rms is the index of predetermined prices, so the future nominal money supply rises linearly with the price set by adjusting …rms. Understanding that this future policy response will occur, and that the price it sets today will also be in e¤ect in the future, an individual …rm’s best response is to choose a higher price when all other adjusting …rms choose a higher price. In the Calvo model, in contrast, next period’s index of predetermined prices comprises an in…nite number of lagged prices, of which the price set by adjusting …rms today is just one element. Under a constant-m policy, the e¤ect of an increase in prices set today on next period’s nominal money supply depends on the e¤ect of such an increase on next period’s index of preset prices. That index of preset prices— today’s price index— is highly sensitive to low levels of the price set by …rms today and relatively insensitive to high levels of the price set by …rms today, because goods with higher prices have a lower expenditure share and thus receive a smaller weight in the price index. As the price set by …rms goes to in…nity, it has no e¤ect on the index of preset prices and no e¤ect on tomorrow’s nominal money supply. Thus, in the Calvo model a constant-m policy would lead to a nominal money supply instances of multiple …xed points for sub-optimal values of m. In Figure 2 there is a convex region of the best-response function to the left of the …xed point. In the case of multiple …xed points, the convex region of the best-response function intersects the 45-degree line twice, with a third …xed point located on the concave portion. 16 that is increasing and concave in the price set by adjusting …rms. Because a higher future money supply leads …rms to set a higher price today, concavity of the future money supply corresponds to decreasing complementarity of the prices set by adjusters. This intuition is con…rmed by the following result, which applies to our baseline calibration. Proposition 1 Suppose the money supply is always set according to a constant-m policy, regardless of the state, and let = 0. Then the Calvo model has a unique private-sector equilibrium. Proof. See Appendix C. The second reason for weaker complementarity in the Calvo model is that the relationship between the price set by adjusting …rms and the future nominal money supply depends on the future state variable. Indeed, the policy maker does not hold m constant, instead lowering it with the state (see Figure 1.B). The response of next period’s normalized money supply to the price set by adjusting …rms today therefore depends on the relationship between p0 and 0 . Combining the market clearing condition (23) with the transformed price index (22) yields 0 )p0 " + (1 = )p10 + (1 " "=(" 1) ; which implies that for high (low) values of p0 the future state is increasing (decreasing) in p0 , holding …xed the current state: @ 0 = @p0 " (1 + (1 Given that equilibrium m is decreasing in ) p0 " ) p01 1 " 1+["=(" 1)] ( p0 1) : (29) , future m is decreasing in p0 for high values of p0 and increasing in p0 for low values of p0 . That is, a higher price set by adjusting …rms— if it is greater than 1= — translates into a higher value of the future state, and thus a lower value of the future normalized money supply.10 10 This relationship is reversed at low values of p0 : increases in p0 reduce the future state, and the poli- cymaker would respond by raising future m. Such low values of p0 are not relevant for understanding the properties of equilibrium however, because they are associated with suboptimally low values of m. Indeed, if m were low enough that raising m would reduce both the markup and the relative price distortion, there would be no policy trade-o¤ and the policymaker would choose a higher m. 17 Summarizing the argument: in the Taylor model the normalized money supply is constant in equilibrium, and this results in an increasing convex best-response function with multiple …xed points. In the Calvo model, if policy kept the normalized money supply constant there would be a unique equilibrium: complementarity would be weaker at high p0 than in the Taylor model, because next period’s index of predetermined prices responds only weakly to p0 at high levels of p0 . Because the normalized money supply is not constant in the Calvo model, the complementarity is weakened even further; m is decreasing in the state, and future m is decreasing in p0 for high p0 . As both parts of this argument rely on the fact that there are many cohorts of …rms with predetermined prices, this feature appears key to explaining why the Calvo model does not have the same tendency toward multiple discretionary equilibria as the Taylor model with two-period pricing.11 Although we have not proved uniqueness of equilibrium, our computations have found only one equilibrium in every case, and Proposition 1 gives us con…dence that the numerical results do generalize: the constant-m policy, which is key to proving that there are multiple private sector equilibria in the Taylor model, implies a unique private sector equilibrium in the Calvo model. If, as we suppose, MPE is unique, the nature of the equilibrium ought to be invariant to (i) the policy instrument and (ii) whether we use an alternative approach to solving the policy problem, either by solving the GEE or solving the planner’s problem as in Anderson, Kim and Yun (2010). For our baseline parameterization we have con…rmed that the same steady-state in‡ation rate obtains whether the policy instrument is the money supply or the nominal interest rate. In addition, we have replicated the steady-state in‡ation rate of 2:2 percent for Anderson, Kim and Yun’s baseline case with = 0:75, " = 11, and = 1, for both interest rate and money supply instruments. Finally, we have computed equilibrium for our benchmark example using the GEE approach, which we discuss in the next section. 11 This reasoning suggests, however, that a Taylor model with longer duration pricing might not have multiplicity, because the same opportunities to substitute would be present. Khan, King and Wolman (2001) …nd multiplicity is still present with three-period pricing. Unfortunately, it is computationally infeasible to study discretionary equilibrium in a Taylor model with long-duration pricing. 18 5 Generalized Euler equation approach Until this point, we have been careful to allow for the possibility of multiple private-sector equilibria. This has meant eschewing a …rst-order approach to the policy problem, as we needed to check for uniqueness of private-sector equilibria for all feasible values of m. For the broad range of parameter values that we have studied, however, we have found that privatesector equilibrium is always unique at the optimal choice of m. Therefore, the …rst-order approach described by Krusell, Kuruscu and Smith (2002) and Klein, Krusell and Rios-Rull (2008, henceforth KKR) is appropriate for our problem, ought to yield equivalent results to those described above, and may provide additional insight into the nature of equilibrium. In this section we follow the approach of KKR for deriving the policymaker’s GEE. 5.1 Discretionary equilibrium restated To derive the GEE, we continue to view the policymaker as choosing the normalized money supply as a function of the state, taking as given the private sector’s equilibrium response. We now assume the monetary policy function is di¤erentiable and private-sector equilibrium is unique. We reformulate the de…nition of discretionary equilibrium to make it more convenient for deriving the GEE. Some of the functions used in the derivation are known functions of p0 , m and . In particular, the functional forms for consumption, labor input, and the relative price distortion are readily obtained from combining Eqs. (22) (25), so we use the shorthand notation c = C (p0 ; m), n = N ( ; p0 ; m), and Current utility is given by u(c; n) = ln (c) 0 = D ( ; p0 ) for these functions. n1+ =(1 + ). Discretionary equilibrium consists of a value function v, a monetary policy function , and a pricing function h such that for all ,m= ( ) solves12 max fu (C (p0 ; m) ; N ( ; p0 ; m)) + v (D ( ; p0 ))g ; m p0 = h ( ) satis…es the optimality condition for the price chosen by adjusting …rms p0 [1 + 12 F (D ( ; p0 ))] = " " 1 (1 ) p01 " + 1 1 " un + uc S (D ( ; p0 )) ; (30) We denote the value function and policy function in discretionary equilibrium by v and , dropping for simplicity the asterisk notation used in Section 3. 19 and v () is given by v( ) (31) u (C (h ( ) ; ( )) ; N ( ; h ( ) ; ( ))) + v (D ( ; h ( ))) : In (30), S () and F () are the same functions that were de…ned in (19) and (20). Under our assumptions of di¤erentiability and uniqueness, this description of equilibrium is equivalent to that in Section 3. 5.2 The GEE We derive a simpli…ed representation of the policymaker’s …rst-order condition by using the envelope condition to eliminate the derivative of the value function, as in KKR. To that end, we …rst de…ne the …rm’s “pricing wedge” ( ; m; p0 ), which is the (out-of-equilibrium) deviation from the optimal price-setting condition: ( ; m; p0 ) = p0 [1 + " " 1 F (D ( ; p0 ))] (1 ) p10 " + 1=(1 ") un + uc S (D ( ; p0 )) : Following KKR, given an equilibrium (v; ; h) and under some regularity conditions, the implicit function theorem guarantees that there exists a unique function H ( ; m), de…ned on some neighborhood of the steady state, satisfying ( ; m; H ( ; m)) borhood. The function H gives the price if the current state is 0 in that neigh- , current money is m, and price-setting …rms expect that future money will be determined by the equilibrium policy function . Thus, H describes the private sector’s response to a one-time deviation of monetary policy from the equilibrium policy. Continuing as in KKR, using this de…nition, which implies that Hm = m = p0 and H = = p0 , the …rst-order condition for the monetary authority is uc [Cp0 Hm + Cm ] + un [Np0 Hm + Nm ] + v 0 Dp0 Hm = 0: (32) Note that prime always denotes the next period, never derivative. The next step is to get an expression for v 0 . Begin by di¤erentiating (31) with respect to , replacing h ( ) with H ( ; m), using the fact that in equilibrium h ( ) = H ( ; ( )): v = uc (Cp0 (H + Hm ) + Cm + un (N + Np0 (H + Hm ) ) + Nm 20 ) + v 0 (D + Dp0 (H + Hm )) : From the …rst-order condition (32) we have uc [Cp0 Hm + Cm ] un [Np0 Hm + Nm ] : Dp0 Hm v0 = So the derivative of the value function can be written as v = un N (uc Cm + un Nm ) m D uc Cp0 + un Np0 Dp0 p0 (uc Cm + un Nm ) : (33) m This derivative, the change in equilibrium welfare with respect to a change in the inherited relative price dispersion, consists of three terms. The …rst term represents the direct e¤ect of a change in the state on current utility. The second term, in parentheses, re‡ects the fact that even if current pricing behavior does not change, a change in the state variable requires a change in the current money supply in order for the …rm’s optimality condition to hold. The factor ( = m) represents the change in the money supply with respect to a change in the state along the …rst-order condition for pricing. Of course, in equilibrium pricing decisions do respond to the state variable; the third term, in brackets, takes care of this e¤ect. When p0 changes there is a direct e¤ect on utility. There is also an indirect e¤ect because the change in pricing must correspond to a change in the money supply in order for the …rms’optimality condition to hold, which explains the factor p0 = m . Pushing (33) one period forward, we use it to eliminate the value function derivative from the monetary authority’s …rst-order condition (32), therefore writing that …rst-order condition as a GEE: + Hm Dp0 u0n N 0 0 0 m 0 0 (u0c Cm + u0n Nm )+ D0 Dp0 0 0 p0 0 m 0 = 0; (34) where uc Cm + un Nm + Hm (uc Cp0 + un Np0 ) : The GEE states that in equilibrium, a marginal change in the current money supply leaves welfare unchanged. The variable represents the change in current utility with respect to a change in the current money supply. The term in brackets, v 0 from (33), consists of the three e¤ects on future welfare discussed above: the direct e¤ect from a change in the future state variable, the e¤ect of a change in the future money supply associated with the future state, and the e¤ect of a change in the optimal price associated with the future state. The 21 coe¢ cient on future marginal value, Hm Dp0 represents discounting and the mapping from a change in current m to a change in the future state. The GEE highlights the lack of commitment. The optimality condition (34) for the current policymaker incorporates the response of future policy to the endogenous state variable, 0 0 0 captured by the terms in Cm , Nm , and Hm of which some are contained in 0 . This contrasts with the case of commitment, where future policy would not respond to the state variable. The e¤ect of current policy on future welfare directly through the state variable would also be present with commitment, although it would be quantitatively di¤erent because the state variable embeds expectations about future policy. In addition to its analytical value, the GEE can be used as the basis for an alternative approach to computing equilibrium. In a reassuring check on our work above, using the GEE approach we computed an identical steady-state in‡ation rate of 5.5 percent to that reported in Section 4, although away from steady state the equilibrium di¤ered slightly.13 6 Transitions to and from discretion The relatively high steady-state in‡ation rate under discretion raises the questions of the cost and bene…t, respectively, of losing and gaining the ability to commit. One way to measure these objects is by comparing the steady-state levels of welfare under commitment and discretion. However, both empirical and theoretical considerations suggest that the steady state comparison is inappropriate. Empirically, large changes in the in‡ation rate rarely occur instantaneously. For example, the famous Volcker disin‡ation played out over a period of at least three years. Theoretically, we have emphasized the presence of a state variable in the discretionary equilibrium, and commitment induces additional policy inertia, as emphasized by Woodford (2003). Answering these questions, then, means studying the transitional dynamics. Loss of ability to commit involves the transitional dynamics under discretion, starting from the steady state under commitment. Acquisition of ability to commit involves the transitional dynamics under commitment, starting from the steady state under discretion. 13 In an application to …scal policy, Azzimonti, Sarte and Soares (2009) also …nd that di¤erent com- putational approaches produce identical steady states under discretionary policy, but somewhat di¤erent dynamics. 22 Thus, we need to know the steady states and transitional dynamics under both commitment and discretion. 6.1 Optimal allocations with commitment In considering optimal policy under commitment, we set aside the issue of implementation and solve a social planner’s problem. That is, we consider the problem of a planner who can choose current and future prices and quantities, subject to the conditions that characterize optimal behavior by households and …rms, and subject to markets clearing. The planner’s problem can be written as max fct ;nt ;wt ; ~ ~ t ;St ;Ft ;p0;t ; t 1 gt=0 1 X t n1+ t 1+ ln (ct ) t=0 ; subject to the following constraints, for t = 0; 1; :::: p0;t = " " t 1 S~t = wt + F~t = 1 + t = (1 t = " t S~t F~t S~t+1 (35) " t+1 (36) " 1~ t+1 Ft+1 : ) p10;t " + ) p0;t" + (1 (37) 1 (38) 1 " (39) t 1 nt = t ct (40) wt = c t nt (41) These constraints are each familiar from the description of private-sector equilibrium above. However, the optimal pricing condition is transformed slightly, with S~t Ft = " 1 , t St = " t and Fet to reduce the number of state variables in the planning problem. Recall that without commitment, there was a single state variable, t 1. With commitment, the presence of future realizations of variables in the constraints means that there are two additional state variables, the lagged multipliers on the constraints (36) and (37). The …rst-order conditions for this problem can be simpli…ed to a system of nine nonlinear di¤erence equations in n o ~ ~ the nine variables ct ; St ; Ft ; t ; t ; t ; t ; t ; t , where t , t , t , and t are the Lagrange multipliers on (36), (37), (38), and (39). The nine-equation system is derived in Appendix D. 23 Solving for the steady state under discretion required us to solve for the functions describing equilibrium dynamics. Under commitment the steady state is simply the time-invariant solution to the nine-equation system implied by the planner’s …rst-order conditions. As shown in Appendix D, the steady-state in‡ation rate is zero; that is, = = p0 = 1.14 Comparing the steady states under commitment and discretion gives a rough estimate of the cost (bene…t) of losing (gaining) the ability to commit to future policies. Using the baseline calibration, both consumption and leisure are slightly higher in the commitment steady state, with zero in‡ation, than in the discretionary steady state, with 5.5 percent in‡ation. The welfare di¤erence between the two steady states is equivalent to 0.228 percent of consumption every quarter.15 In present value terms, the consumption increment represents 5.70 percent of annual consumption. Of course, this calculation is arti…cial in the sense that the economy cannot simply jump from one steady state to the other. Next we analyze the transitions between steady states. 6.2 Losing the ability to commit If an optimizing policymaker loses the ability to commit, then the economy behaves according to the transitional dynamics under discretion with an initial condition of = 1, the steady state under commitment. These dynamics can be inferred from Figure 1, but we plot them explicitly in Figure 3. The solid lines display the paths of in‡ation, the relative price distortion, the markup, and the money growth rate along the transition to the discretionary steady state. Whereas the relative price distortion monotonically increases along the transition, the in‡ation rate jumps in the initial period (labeled zero) and then smoothly declines to the discretionary steady state. Panel D shows that the money growth rate essentially 14 We use the term “steady state” informally in the case of a policymaker with commitment. It is more accurate to refer to this allocation as the limit point in the long run under commitment. A planner who inherited only the state variable to zero in‡ation and 15 = 1 would choose some initial in‡ation before converging in the long run = 1; this is the nature of the time-consistency problem. The welfare calculation involves comparing the discretionary steady state to an allocation on the same indi¤erence curve as the commitment steady state, but with the same wage as the discretionary steady state. The number 0.228 percent represents the parallel rightward shift in the budget constraint (consumption on the horizontal axis). 24 6 A. Inflation 1.003 B. Relative price distortion Commitment to discretion Discretion to commitment 4 1.002 % 2 1.001 0 -2 -1 0 1 2 3 4 5 6 7 8 1.12 C. Markup 1 -2 -1 0 1 2 3 4 5 6 7 8 D. Nominal money growth rate 12 1.11 6 % 1.1 0 1.09 -2 -1 0 1 2 3 4 5 6 7 8 -6 -2 -1 0 1 2 3 4 5 6 7 8 Quarters Quarters Figure 3: Transitions to and from discretion 25 mimics the in‡ation rate. In present value terms, the welfare decline associated with loss of commitment is well-approximated by the steady-state welfare comparison: the representative household would be willing to forego 0.225 percent of consumption each quarter in order to avoid this transition (5.62 percent in present-value terms). It is notable that the transition to the discretionary steady state does not involve any transitory bene…ts: along the entire transition both the markup and the relative price distortion are increasing. One might have expected that in the initial period of the transition, the policymaker could e¤ectively exploit preset prices and reduce the markup. It is indeed the case that the markup for non-adjusting …rms falls substantially in the initial period. However, this decline is more than o¤set by an increase in the in‡ation rate that results from the behavior of adjusting …rms. This reasoning uses the identity Pt =M Ct = (Pt =Pt 1 ) (Pt 1 =M Ct ). Although the equilibrium path involves the markup rising, it is nonetheless the case that at each point in time the policymaker perceives a tradeo¤ between reducing the markup and increasing the relative price distortion. 6.3 Gaining the ability to commit If a policymaker previously operating with discretion gains the ability to commit, the economy behaves according to the dynamics under commitment, beginning in the discretionary steady state and— presumably— ending in the commitment steady state. The dynamics under commitment are represented by the aforementioned nine-variable system of nonlinear di¤erence equations, shown in Appendix D. To compute the transition path, we conjecture that convergence to the zero-in‡ation steady state is complete after T = 40 quarters. We then have a system of 9 T equations in n oT 1 n o ~ ~ ~ ~ the 9 T + 5 variables ct ; St ; Ft ; t ; t ; t ; t ; t ; t and 1 ; ST ; FT ; T ; T . If we t=0 assume there is a unique transition path, then to solve the system of 9 T equations we need n o ~ ~ to specify values for the initial condition 1 and the terminal conditions ST ; FT ; T ; T . Under the conjecture, the terminal conditions are given by the steady state under commitment (zero in‡ation). The initial condition is given by the steady-state value of under discretion. From the properties of the Jacobian matrix at the steady state, we know that locally there is a unique stable solution that converges to the steady state. Indeed, we are 26 able to compute a global solution satisfying the conjecture. The dashed lines in Figure 3 represent the resulting transition path from the steady state with discretion to the steady state with commitment. The transition contains an element of discretionary behavior: in the initial period, since the policy change is unexpected, the policymaker has an incentive to exploit the …xed prices of non-adjusters by increasing the money supply. This was also the case in the transition to discretion, but here the long-run policy involves lower in‡ation, so the temporary stimulus is not o¤set by frontloading of larger price increases. Instead, adjusting …rms frontload smaller future price increases, more than o¤setting any in‡ationary e¤ects of the temporary monetary stimulus. The welfare bene…t of the transition from discretion to commitment is again well approximated by the steady-state welfare comparison: the representative household would require a 0.222 percent increase in consumption each quarter in order to willingly choose the transition from commitment to discretion (5.54 percent in present-value terms). Initial-period policy under commitment and discretion illustrates the di¢ culty of exploiting initial conditions. The discretionary monetary authority would like to exploit initial conditions, but in equilibrium even in the short run it is unable to do so because of …rms’ forward-looking behavior. Conversely, that same forward-looking behavior means that a policymaker who can commit is able to exploit initial conditions (once!) by combining short-run expansionary policy with lower money growth and in‡ation in the long run. 7 Concluding remarks The vast literature on time-consistency problems for monetary policy is comprised of two seemingly disparate branches. Much of the profession’s intuition is derived from the seminal work by Barro and Gordon (1983), hereafter BG, which in turn built on Kydland and Prescott (1977). They studied reduced-form macroeconomic models in which the frictions giving leverage to monetary policy were not precisely spelled out. In contrast, the staggered pricing models popularized in the last two decades are precise about those frictions. We conclude here by summarizing the paper’s three main contributions and then explaining how the analysis relates to BG’s early work on time-consistency problems for monetary policy. The direct motivation for this paper comes from King and Wolman (2004), who showed 27 that in a model with Taylor-style staggered price setting, discretionary policy induces complementarity among …rms su¢ cient to generate multiple private-sector equilibrium. In the Calvo model, which shares many features with the Taylor model but has been much more in‡uential for applied monetary policy analysis (often in its linearized form), we …nd that the presence of many predetermined variables and a real state variable works against complementarity, and multiple private-sector equilibria do not arise. The state variable measures the dispersion in predetermined relative prices; in contrast, the Taylor model has only one predetermined nominal price and thus no predetermined relative prices. The Calvo model’s combination of a real state variable and unique private-sector equilibrium leads us to an analysis of discretionary equilibrium using the GEE approach. The GEE is a simpli…ed representation of the policymaker’s …rst-order condition, which highlights the various channels through which current policy can a¤ect future welfare. Without commitment, each of these channels works through the state variable; the state variable a¤ects future outcomes directly and it shifts the future policymaker’s problem. While the GEE approach has been applied extensively to study …scal policy, the Calvo application is new to this paper. Finally, we use the steady-state and dynamic properties of the discretionary equilibrium together with the solution under commitment to study the processes of gaining and losing the ability to commit. The present-value welfare gain or loss is approximately equivalent to 0.2 percent of consumption each period, or a one-time gain or loss of 5.5 5.6 percent of annual consumption. While we compute these welfare numbers using the full transition paths, they di¤er little from the steady-state welfare comparisons. At the heart of the time-consistency problem for monetary policy is the notion that a discretionary policymaker takes as given private agents’ expectations, but in equilibrium those expectations accurately incorporate the policymaker’s optimal behavior. This property holds in modern staggered pricing models such as the Calvo model and in the BG model. In BG, however, the expectations just referred to are current expectations about current policy; dynamics only arise through serial correlation of exogenous shocks. Without other intertemporal links, the policy problem is a static one in BG: treating expectations as …xed, higher in‡ation is costly in its own right but brings about a bene…cial reduction in unemployment. In equilibrium, private expectations are validated, and the policymaker 28 balances the static marginal cost and marginal bene…t of additional in‡ation. In contrast, in staggered pricing models prices set in the past incorporate expectations about current policy. Whereas in BG equilibrium requires that current policy actions be consistent with current-period expectations, in staggered pricing models equilibrium requires that current policy actions be consistent with expectations formed in the past.16 The intertemporal nature of price setting also means that unlike BG, staggered pricing models generally contain one or more state variables that can be a¤ected by a policymaker, even under discretion. Thus, the discretionary policymaker does not face a purely static tradeo¤ between in‡ation and real activity. That tradeo¤ is present, but it is complicated by the fact that the current policy action a¤ects tomorrow’s state, and thus tomorrow’s value function. One of the main points of this paper is that the details of this intertemporal element di¤er across staggered pricing models, leading to di¤erent implications for the nature of equilibrium under discretionary monetary policy. While staggered pricing models generate a static output-in‡ation tradeo¤ super…cially similar to the one in BG, these models are explicit about the source of that tradeo¤. Because of the forward-looking elements in these models, the details of the policy tradeo¤— and whether it even appears in equilibrium— depend critically on the entire path of expected future policy. For example, Section 6 has shown that an unexpected reduction in in‡ation can be stimulative, if it signals the transition to a permanently lower in‡ation rate (gaining commitment). Likewise, an unexpected increase in in‡ation can be contractionary if it signals the transition to a permanently higher in‡ation rate (losing commitment). Although these transitions may suggest that no output-in‡ation tradeo¤ is present, in the discretionary equilibrium the policymaker perceives such a tradeo¤: a one-period deviation toward more expansionary policy would raise output and in‡ation, as it reduced the markup and raised the relative price distortion. The e¤ects on welfare would be o¤setting, and thus the policymaker does not deviate. Of course, the properties of discretionary equilibrium are determined by the speci…cs of the model. The de…ning feature of the Calvo model is the assumption that a fraction 16 With di¤erent timing assumptions in staggered pricing models the BG version of expectational consis- tency would also be required to hold. 29 of …rms are prohibited from adjusting their price. This assumption makes for a relatively tractable framework, undoubtedly the main reason the Calvo model has come to serve as the basis for so much applied work on monetary policy. It has recently become feasible to conduct some forms of policy analysis in models which relax this assumption, allowing …rms to adjust their price by incurring a cost. Those models typically have a large number of state variables, currently rendering it infeasible to perform the kind of analysis conducted here (see for example Nakov and Thomas, 2014).17 Nonetheless, we hope that our work can serve as a useful input for future research on discretionary policy in quantitative state-dependent pricing models. 17 While Barseghyan and DiCecio (2007) and Siu (2008) study discretionary policy in models with state- dependent pricing, both papers limit the state space by allowing …rms to adjust costlessly after one and two periods, respectively. 30 A Computational details This appendix describes how numerical solutions for the discretionary equilibrium are computed and how uniqueness of the equilibrium is veri…ed in a large number of examples. Computing a discretionary equilibrium The value function and the expressions for S () and F () are approximated with Chebyshev polynomials. This computational method involves selecting a degree of approximation I, and then searching for values of vi and for the state variable i i, for i = 1 : : : I, that solve (27) at the grid points de…ned by the Chebyshev nodes. The optimization problem (27) is solved using the following algorithm. 1. Grids and initial values. The example of the baseline calibration (i.e., 0:99, = 10, = 0, and = 0:5, = = 4:5) uses a degree of approximation I = 12 on the interval [1; 1:1] for the state variable. As an initial guess for v (), S () and F () the discretionary equilibrium for the static model is used, which is the …nal period of a …nite horizon model. To compute the private-sector response to an arbitrary policy, grids fm1 ; : : : ; mIm g and fp0;1 ; : : : ; p0;Ip g are speci…ed for the money supply and the optimal price. In the case of the baseline calibration, the grid for m consists of Im = 600 evenly spaced points between 0.01 and 0.25 and the grid for p0 consists of Ip = 600 evenly spaced points between 0 and 2. 2. Private-sector responses. For each possible value of and m, compute the private- sector responses by solving (28) as a …xed-point problem. Speci…cally, compute the right hand side of (28) and call it p^0 , then use linear interpolation to …nd the …xed points p0 = p^0 . 3. Policy function and value function. On each grid point i, select the value of m that maximizes the value function. If the value function and policy function that solve the optimization problem are identical to the guess, then they form a discretionary equilibrium. Speci…cally, iteration j is the …nal iteration if jjv j+1 are smaller than the tolerance level 1:49 10 8 v j jj1 and jj j+1 j jj1 (the square-root of machine precision). If not, the starting values are updated by pushing out the initial guess one period into 31 the future, and assuming the one-period-ahead policy and value functions are the ones that solved the optimization problem. To assess the accuracy of a solution, the di¤erence between the left hand side and the right hand side of (27) is calculated using that solution on a grid of 100,000 points that do not include the Chebyshev nodes. With the baseline calibration, this residual function has a maximum absolute approximation error of order 10 6 . Many other examples were computed that cover a wide range of values for , " and . These include the values for = 0:1, 0:15, 0:2, 0:25, 0:3, 0:35, 0:4, 0:45, 0:55, 0:6, 0:61, 0:62, 0:63, 0:64, 0:65, 0:66, 0:67, 0:68, 0:69, 0:70, 0:71, 0:72, 0:73, 0:74, 0:75, 0:76, 0:77, 0:78, 0:79, 0:8, 0:85, 0:9, and 0:95, values for " = 6, 7, 8, 9, and 11, and values for = 0:25, 0:5, 0:75, 1, 1:25, 1:5, 1:75, and 2. In addition to these 46 solutions, which consider alternative values of one parameter at a time, solutions were computed for eight combinations of extreme parameter values: ( ; "; ) = (0:1; 4; 0), (0:1; 4; 2), (0:1; 11; 0), (0:1; 11; 2), (0:9; 4; 0), (0:9; 4; 2), (0:9; 11; 0), (0:9; 11; 2). Uniqueness of the solutions All the examples described above yield a unique solution. Step 2 in the model solution algorithm allows for the possibility of multiple …xed points at an arbitrary monetary policy at each value of the state. Suppose the money supply that maximizes the value function in iteration j 1 induces multiple private-sector responses. Then the inherited relative price dispersion in iteration j is not uniquely determined and neither is the money supply in iteration j. Therefore, if j is the …nal iteration there are multiple discretionary equilibria. However, this hypothetical sequence of outcomes does not arise in any of our examples, where discretionary equilibrium is always unique. Multiple …xed points were only encountered for sub-optimal values of m, in which case the largest …xed point was arbitrarily selected (the same solutions were found when selecting the smallest …xed point). Speci…cally, for the alternative calibrations with values of between 0:63 and 0:69, or with the value " = 6, the …nal iteration of the solution algorithm exhibited multiple …xed points for values of m in a range that is positive but smaller than the optimal value of m. 32 B Comparison with the Taylor model This appendix describes the two-period Taylor model and provides a quantitative comparison with the Calvo model. B.1 Model In the Taylor model each …rm sets its price for two periods. The description of the representative household remains unchanged, but the value of a …rm upon adjustment is given by max Xt ( 1 X j=0 j Pt Pt+1 j j ct " Xt ct+1 Xt Pt+j " ct+j Pt+j wt+j Xt Pt+j " ct+j #) ; and the optimal price satis…es the …rst-order condition, P0;t = Pt " " 1 wt + (Pt+1 =Pt )" wt+1 : 1 + (Pt+1 =Pt )" 1 This condition indicates the …rm chooses a constant markup over a weighted average of current and future marginal costs. Whereas in the Calvo model the index of predetermined prices was given by Pt 1 , in the Taylor model there is just one predetermined price, P0;t 1 . Normalizing the optimal price and the price index by P0;t P0;t =P0;t 1 and pt and using the de…nitions p~0;t 1 Pt =P0;t 1 , we have p~0;t = pt wt + (~ p0;t pt+1 =pt )" wt+1 : 1 + (~ p0;t pt+1 =pt )" 1 " " 1 (42) Money demand (5) is normalized by the lagged optimal price instead of the lagged price level m ~t Mt = p t ct : P0;t 1 (43) We eliminate the predetermined variable from the price index, Pt = 1 1 " 1 1 " + P0;t 1 P 2 0;t 2 1 1 " ; by dividing by the lagged optimal price: pt = 1 1 " 1 p~ + 2 0;t 2 33 1 1 " : (44) The labor market clearing condition yields 1X = nj;t 2 j=0 1 nt nt 1 " p0;t ) = p (~ ct 2 t " (45) +1 : There is one predetermined nominal price (P0;t 1 ), but there are no state variables in the labor market clearing condition. The equations (42) (45) and the labor supply equation (4), together with the behavior of future policymakers, implicitly de…ne the set of feasible values for wt , ct , nt , pt and p~0;t attainable by the current monetary policymaker in the Taylor model. The current policymaker chooses the money supply, or equivalently m ~ t , the money supply normalized by the predetermined price. Unlike the Calvo model, no state variables constrain the monetary authority in an MPE. The lagged optimal price P0;t 1 matters for the levels of nominal variables, but is irrelevant for the determination of real allocations. B.2 Results Do the seemingly small di¤erences between the Taylor and Calvo models generate similar predictions for in‡ation under discretion? The remainder of this appendix provides a quantitative comparison of the steady-state in‡ation rates in the two models. For this comparison, we solve the social planner’s problem associated with the Taylor model to circumvent the issue of equilibrium multiplicity. That is, instead of the policymaker choosing the money supply, we assume a planner chooses consumption allocations. The calibration is the same as for the Calvo model: " = 10, = 0:99, = 0, = 4:5. Since = 0:5 in the Calvo model, the average frequency of price adjustment is two quarters in both models. Under this calibration, the steady-state in‡ation rate in the Taylor model is 9:1 percent, exceeding the steady-state in‡ation rate of 5:5 percent in the Calvo model. In the Taylor model, the policymaker faces only the short-run trade-o¤ between the markup and the relative price distortion. In the Calvo model, when weighing the bene…t of a lower markup against the cost of a higher relative price distortion, the policymaker also takes into account the cost to future policymakers of inheriting a higher relative price distortion. By choosing a higher in‡ation rate today, the policymaker would pass on a higher relative price distortion to the next policymaker, which is an additional cost of high in‡ation relative to the Taylor 34 model. Therefore, the steady-state in‡ation rate is lower in the Calvo model. C Proof of Proposition 1 This appendix presents the proof of Proposition 1. Recall from equation (12) that in‡ation is the following function of the optimal reset price 1 1 " 1 " )p0;t + (p0;t ) = (1 ; which is increasing and strictly concave: 0 (p0;t ) = (1 00 (p0;t ) = " " 1 ) + p"0;t 1 ) (1 3(" 1) )" (p0;t )2" 1 p0;t (1 = (1 ) (p0;t ) p0;t " >0 (46) (47) < 0; and has a …nite limit lim 1 1 " (p0;t ) = p0;t !1 (48) : Consistent with the computation of discretionary equilibrium as the stationary limit of the …nite-horizon economy, we compute equilibrium with a constant-m policy as the limit of the …nite-horizon economy. Let T denote the …nal period, so ST +1 = FT +1 = 0. Then: " ST = " 1 (p0;T )" FT = (p0;T )" 1 [1 + 1 [ mT + ST +1 (p0;T )] = " " 1 (p0;T )" 1 mT ; FT +1 ] = (p0;T )" 1 ; (49) (50) and the pricing best-response function is p^0;T = ST = FT " " 1 (51) mT : The outcomes p0;T , ST , and FT do not depend on the state because monetary policy does not depend on the state. Moreover, there can be no complementarity in price setting in period T , because the pricing best-response function (51) of any given …rm does not depend on other …rms’price decisions. Note from (49) that ST = ST (mT ; p0;T ) and from (50) that FT = FT (p0;T ). We can now analyze the period T 1 pricing best-response function to determine whether there is a unique …xed point. We have: ST 1 = FT 1 = " " 1 (p0;T 1 )" (p0;T 1 [1 + 1) " 1 [ mT 1 + FT (p0;T )] ; 35 ST (mT ; p0;T ) (p0;T 1 )] so the period T 1 best-response function is p^0;T 1 " = " mT 1 + 1 ST (mT ; p0;T ) (p0;T 1 + FT (p0;T ) 1) (52) The optimal price does not depend on the state because the monetary policy function and the functions ST and FT do not depend on the state. To see that the best-response function has a unique …xed point, …rst write (52) as p^0;T where AT 1 (p0;T ) 1 = AT 1 (p0;T )mT 1 > 0 and BT + BT 1 (mT ; p0;T ) (p0;T 1 ); > 0 because mT ; p0;T > 0. It follows from 1 (mT ; p0;T ) (46) (48) that @ p^0;T @p0;T @ 2 p^0;T @p20;T lim p0;T 1 !1 1 = BT 1 0 = BT 1 00 (p0;T 1) >0 1) <0 1 1 1 p^0;T 1 " = " 1 (p0;T " mT 1+ 1+ # ST (mT ; p0;T ) : FT (p0;T ) 2 " 1 " Because the best-response function is always positive and concave and has a …nite limit, it has a unique …xed point. Therefore, there exists a unique private-sector equilibrium in period T 1. Write ST 1 = ST 1 (mT 1 ; mT ; p0;T 1 ; p0;T ) and FT 1 = FT 1 (p0;T 1 ; p0;T ). In period T 2 we obtain ST 2 = FT 2 = " " 1 (p0;T 2 )" Hence the period T p^0;T where AT 2 2 = AT (p0;T 1 " 1 2) [1 + [ mT FT 2 + ST 1 (mT 1 ; mT ; p0;T 1 ; p0;T ) (p0;T 2 )] 1 (p0;T 1 ; p0;T )] : 2 best-response function can be written as 2 (p0;T 1 ; p0;T )mT 2 > 0 and BT 2 + BT > 0 because mT ; mT 2 (mT 1 ; mT ; p0;T 1 ; p0;T ) 1 ; p0;T ; p0;T 1 as above there is a unique …xed point in period T (p0;T 2 ); > 0. By the same arguments 2. Repeating the same steps, we can show that for period t, St = Ft = " " 1 (p0;t )" (p0;t )" 1 [1 + 1 [ mt + S(mt+1 ; mt+2 ; : : : ; p0;t+1 ; p0;t+2 ; : : :) (p0;t )] F (p0;t+1 ; p0;t+2 ; : : :)] : 36 The period-t best-response function can therefore be written as p^0;t = At (p0;t+1 ; p0;t+2 ; : : :)mt + Bt (mt+1 ; mt+2 ; : : : ; p0;t+1 ; p0;t+2 ; : : :) (p0;t ); where At > 0 and Bt > 0 because mt+j ; p0;t+j > 0 for j = 1; 2; : : :. Therefore, by backward induction, there is a unique private-sector equilibrium associated with the arbitrary constantm policy. D Derivation of commitment solution Here we derive the equations characterizing optimal policy with commitment using the following Lagrangian: L = + + + 1 X t=0 1 X t=0 1 X t t ln (ct ) h ~ t St t t t=0 1 X t n1+ t 1+ n t + i + t=0 1 X t t " + p0;t 1 X o [wt + S~t t F~t " t t t=0 1 1 " ) p10;t " + t ct ) t " ~ t+1 St+1 (1 (nt t t=0 wt + t 1 X 1 X " h 1 F~t # " 1~ t+1 Ft+1 1+ t " t t t i ) p0;t" + (1 t 1 t=0 c t nt ] : t=0 The …rst order conditions are as follows: ) p10;t " + 1 " t " 1 p0 : t t S~ : t t 1 " t F~ : t t 1 " 1 t " 1 t 1" t : + t + c: 1 + ct t w: t t t ) p0;t" (1 " t+1 1 1) + t ct " 2 t t 1 F~t =0 ) " " 1 t p0;t =0 (53) (54) (55) " t " 1 S~t F~t (56) (57) =0 (58) =0 nt + t " (1 " (" t 1 nt t 2 ct " (1 ) p0;t" + 1 =0 t F~t S~t " t 2 = 0 " 1 F~t t t+1 nt ct nt t + S~t " 1 t t" t : n: (1 (59) =0 (60) = 0; 37 as well as the constraints (35) (41) in the body of the text. This is a system of 15 equations, which can be reduced to nine equations as follows. First, eliminate p0;t directly using (35). Second, eliminate t directly using (60). Third, eliminate nt as nt = ct 1+ t ct eliminate wt as wt = using (41). Fifth, eliminate t 1 = + c1+ t t t Finally, eliminate t t = 1 t t t using (40). Fourth, using (58) and (40), as : using (53), noting that we would also substitute out for p0;t using (35): t " 1 " ) p10;t " + (1 ) p0;t" (1 t " (1 ) " " 1 : t p0;t The nine remaining equations are as follows, where the variables (nt ; wt ; p0;t ; t ; t; t) should be understood to be substituted out as described above: S~t = wt + " ~ t+1 St+1 F~t = 1 + " 1~ t+1 Ft+1 1 1 " ) p10;t " + t = (1 t = " t 0 = t 0 = t t 1 " t 0 = t t 1 " 1 t 0 = 0 = 1 ct nt t + ) p0;t" + (1 1" t + t t nt + " 1 t t" t t 1 t + S~t " 1 t " t+1 t 1 1 F~t " t t (1 ct " " 1 (" ) p0;t" 1 1) + t S~t F~t2 " 2 t F~t " t " 1 S~t F~t t 1 " t+1 This is the system of nonlinear di¤erence equations that we must solve to compute the transition path in Section 6.3. It is straightforward to show that zero in‡ation ( = 1) solves the steady-state system of equations. 38 References [1] Albanesi, S., V.V. Chari, and L.J. Christiano (2003), “Expectation Traps and Monetary Policy,”Review of Economic Studies 70 (4), 715-741. [2] Anderson, G., J. Kim and T. Yun (2010), “Using a Projection Method to Analyze In‡ation Bias in a Micro-Founded Model,”Journal of Economic Dynamics and Control 34 (9), 1572-1581. [3] Azzimonti, M., P.D. Sarte and J. Soares (2006), “Distortionary Taxes and Public Investment when Government Promises Are Not Enforceable,”Journal of Economic Dynamics and Control 33 (9), 1662-1681. [4] Barseghyan, L. and R. DiCecio (2007), “Optimal Monetary Policy, Endogenous Sticky Prices, and Multiple Equilibria,”B.E. Journal of Macroeconomics 7 (1), Article 8. [5] Barro, R.J. and D. Gordon (1983), “Rules, Discretion and Reputation in a Model of Monetary Policy,”Journal of Monetary Economics 12 (1), 101-121. [6] Calvo, G.A. (1983), “Staggered Prices in a Utility-Maximizing Framework,”Journal of Monetary Economics 12 (3), 383-398. [7] Clarida, R., J. Gali and M. Gertler (1999), “The Science of Monetary Policy: A New Keynesian Perspective,”Journal of Economic Literature 37 (4), 1661-1707. [8] Dotsey, M. and A. Hornstein (2011), “On the Implementation of Markov-Perfect Monetary Policy,”Working Paper 11-29, Federal Reserve Bank of Philadelphia. [9] Khan, A., R.G. King and A.L. Wolman (2001), “The Pitfalls of Discretionary Monetary Policy,”Working Paper 01-16, Federal Reserve Bank of Philadelphia. [10] King, R.G. and A.L. Wolman (2004), “Monetary Discretion, Pricing Complementarity, and Dynamic Multiple Equilibria,”Quarterly Journal of Economics 119 (4), 1513-1553. [11] King, R.G. and A.L. Wolman (1999), “What Should the Monetary Authority Do When Prices Are Sticky?” in Monetary Policy Rules, J.B. Taylor ed., University of Chicago Press, 1999, 349-398. [12] King, R.G. and A.L. Wolman (1996), “In‡ation Targeting in a St. Louis Model of the 21st Century,”Proceedings, Federal Reserve Bank of St. Louis, 83-107. 39 [13] Klein, P. P. Krusell and J.-V. Rios-Rull (2008), “Time-Consistent Public Policy,”Review of Economic Studies 75 (3), 789-808. [14] Kydland, F.E. and E.C. Prescott (1977), “Rules Rather than Discretion: the Inconsistency of Optimal Plans,”Journal of Political Economy 85 (3), 473-491. [15] Krusell, P., B. Kuruscu, and A.A. Smith (2002), “Equilibrium Welfare and Government Policy with Quasi-geometric Discounting,”Journal of Economic Theory 105 (1), 42-72. [16] Leeper, Monetary, E.M., C. Fiscal Leith and and Debt D. Liu. Maturity (2016), “Optimal Policy,” Time-Consistent Available at SSRN: http://dx.doi.org/10.2139/ssrn.2719457. [17] Leith and D. Liu. (2016), “The In‡ation Bias under Calvo and Rotemberg Pricing,” Journal of Economic Dynamics and Control 73, 283-297. [18] Nakov, A. and C. Thomas (2014), “Optimal Monetary Policy with State-Dependent Pricing,”International Journal of Central Banking, September, 49-94. [19] Ngo, P.V. (2014), “Optimal Discretionary Monetary Policy in a Micro-Founded Model with a Zero Lower Bound on Nominal Interest Rate,” Journal of Economic Dynamics and Control 45, 44-65. [20] Siu, H.E. (2008), “Time Consistent Monetary Policy with Endogenous Price Rigidity,” Journal of Economic Theory 138 (1), 184-210. [21] Woodford, M., Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press, 2003. [22] Woodford, M. (2003), “Optimal Interest-Rate Smoothing,”Review of Economic Studies 70 (4), 861-886. [23] Yun, T. (2005), “Optimal Monetary Policy with Relative Price Distortions,”American Economic Review 95 (1), 89-108 [24] Yun, T. (1996), “Nominal Price Rigidity, Money Supply Endogeneity, and Business Cycles,”Journal of Monetary Economics 37 (2), 345-370. 40
@FedFRASER