Price Reset Hazard Functions and Macro Dynamics

Transcription

1 Price Reset Hazard Functions and Macro Dynamics Fang Yao University of Erlangen-Nuremberg July 8, 22 Abstract This paper investigates implications of the price reset hazard function for aggregate dynamics. I rst document some general analytical results that highlight the central role of the price (accumulative) distribution in linking the hazard function at the micro level and the impulse response function at the macro level. In addition, I investigate how increasing hazard functions interact with other important features of modern monetary models, such as the forward-looking aggregate demand curve, strategic complementarity and positive steadystate in ation. The central message of the study is that considering non-constant hazard function has important implications for in ation dynamics, but to a lesser extent for the output dynamics. JEL classi cation: E2; E3 Key words: Hazard function, Nominal rigidity, Strategic complementarity, In ation persistence I am grateful to Michael Burda, Carlos Carvalho, Christian Merkl, Alexander Meyer-Gohde, Kevin Sheedy, Lutz Weinke, Alexander Wolman and other seminar participants at the Deutsche Bundesbank and in Berlin for helpful comments. I acknowledge the support of University of Erlangen-Nuremberg and the Deutsche Forschungsgemeinschaft through the SFB 649 "Economic Risk". All errors are my sole responsibility. Address: Chair of macroeconomics, Universität Erlangen-Nürnberg, Lange Gasse. 2, 943 Nürnberg, Germany, fang.yao@wiso.uni-erlangen.de

2 Introduction Empirical studies based on micro-level consumer price data widely conclude that the price reset hazard function is not constant across price durations. This evidence raises doubts about predictions of sticky price models based on the Calvo assumption (Calvo, 983), which is commonly used in the applied monetary policy analysis. The objective of this paper is to assess the implications of non-constant hazard functions for macro dynamics and to understand how the shape of the hazard function a ects the propagation mechanism of monetary shocks. To tackle these questions, I use the generalized time-dependent pricing model, set forth by Wolman (999), in which the probability of price adjustment is allowed to vary with respect to the time elapsed since the last price revision. As a rst pass, I solve a simpli ed version of the model analytically and document some general results concerning the role of the price (accumulative) distribution in bridging the price reset hazard function at the micro level and equilibrium dynamics at the aggregate level. Then, by solving the complete model numerically, I investigate how increasing hazard functions interact with other important features of modern monetary models, such as the forward-looking aggregate demand curve, strategic complementarity and positive steady-state in ation. In solving for analytical results, I work with a simpli ed model, in which the only feature making monetary disturbances matter is the nominal rigidity under a exible hazard function. The result shows that, after a money level shock, both impulse responses of output and the aggregate price are determined by the accumulative distribution of price durations, whereas the in ation response is driven by the distribution of price durations. Consequently, changing the shape of hazard function generates di erent macro dynamics through its e ects on the price distributions. To illustrate this proposition, I compare three models based on increasing, decreasing and constant hazard functions, respectively. The increasing-hazard model predicts a slower rate of decrease in the initial stage of the in ation response in comparison to the decreasing-hazard model. In contrast, the decreasing-hazard model produces the more inertial output response than the increasing-hazard model. The impulse responses predicted by the constant-hazard model lie between these two cases. Similarly, after a persistent money growth shocks, the increasing-hazard model predicts a hump-shaped response of in ation and a less persistent output response compared to the constant-hazard model. Based on these results, I conclude that the increasing-hazard model predicts more empirically realistic dynamics in terms of in ation inertia (See: e.g. Christiano et al., 25), whereas it also implies less persistent real e ects of monetary disturbances than the Calvo sticky price model. Next, I check whether the conclusion based on the simpli ed model also holds in a more realistic setting. To this end, I calibrate and solve the complete model combining the nominal rigidity with strategic complementarity, the forward-looking IS curve and positive steady-state in ation. The simulation results show that, on the one hand, strategic complementarity and positive steady-state in ation strengthen the ability of the increasing hazard function in generating more persistent in ation. On the other hand, the real e ect of monetary shocks is less sensitive to the shape of the hazard function in this general setting, because the forward-looking IS curve isolates output dynamics to a large extent from the forces of the supply side economy. As a See: e.g. Campbell and Eden (25), Alvarez (27) and Nakamura and Steinsson (28a). 2

3 result, the increasing hazard function helps account for both observed persistence of in ation and output in the complete model. 2 The economic reasons why strategic complementarity and positive steady-state in ation should reinforce the e ect of the increasing hazard function on in ation are that, rst, the increasing hazard function postpones the timing of the price adjustment. Shortly after the impact of a monetary shock, there is only a small fraction of rms adjusting their prices. At this stage, because of strategic complementarity, even the adjusting rms opt for small price changes in order to avoid deviating too much from the majority. Over time, however, when more and more rms reset their prices, the price norm in the economy switches slowly to the new level. Consequently, strategic complementarity now drives rms to catch up to the higher price norm. In this way, strategic complementarity ampli es the delayed e ect of the increasing hazard function. Furthermore, the presence of positive in ation accelerates relative price dispersion, which in turn enhances the interaction between strategic complementarity and the increasing hazard function further. In the literature, the implications of non-constant hazard functions have been studied by a number of papers. Wolman (999) raises the issue that in ation dynamics should be sensitive to hazard functions underlying di erent pricing rules. He demonstrates this result in a simple partial equilibrium analysis. Kiley (22) compares the Calvo and Taylor staggered price-setting in a general equilibrium setup and shows that output dynamics resulting from monetary shocks are both quantitatively and qualitatively quite di erent across the two pricing speci cations unless one assumes a substantial level of real rigidity in the economy. Mash (24) also studies a sticky price model with the general hazard function. He derives the generalized New Keynesian Phillips curve log-linearized around a constant-in ation steady-state, and shows numerically that this model can reconcile the tension between evidence on aggregate inertia at the macro level and the frequent price-setting at the micro level. Whelan (27) and Sheedy (2) focus on the implication of the shape of hazard functions on generating intrinsic in ation persistence. They show that the shape of hazard function a ects the sign and magnitudes of the coe cients on lagged in ation in the theoretical Phillips curve. This paper makes a theoretical contribution that extends the analysis of previous papers in the literature. To my best knowledge, the analytical results are new and more general in the sense that its conclusion is not limited to a speci c sticky price assumption or a speci c shape of the hazard function. In addition, this paper is so far the rst to address the issue concerning interactions between the increasing hazard function, strategic complementarity and positive steady-state in ation. The remainder of the paper is organized as follows: in section 2, I present the theoretical model with the generalized time-dependent pricing scheme; section 3 shows analytical results regarding new insights gained from relaxing the constant hazard function underlying the Calvo assumption; in section 4, I simulate the complete DSGE model and present the simulation results regarding how the increasing hazard function interacts with strategic complementarity and steady-state in ation; section 5 contains some concluding remarks. 2 The same conculsion is also drawn by Mash (24) in a similar setting without the positive steady-state in ation. 3

4 2 The model In this section, I present a model that mainly departs from the standard New Keynesian model by its sticky price assumption. Following Wolman (999), I set up a staggered price adjustment process, which is characterized by the price reset hazard function. In contrast to the Calvo sticky price model, the probability of adjusting a price is allowed to vary across price durations. The representative household has one unit of labor endowment in all periods. She works N t hours, consumes composite good C t, buys one-period non-contingent real bonds B t+, and holds a quantity of money M t at the end of the period t. Her period utility function is: U t = log C t + log Mt P t N + t + ; () where > is the inverse of Frisch labor supply elasticity and is the weight on disutility of labor hours. The composite consumption good C t is a constant-elastisity-of-substitution aggregator (Dixit and Stiglitz, 977) over a measure-one continuum of the intermediate goods: Z C t C t (j) dj ; (2) where C t (j) is consumption of good variety j 2 [; ] and > is the elasticity of substitution parameter. It follows that the cost-minimizing demand for C t (j) and the corresponding price index P t are given by Pt (j) C t (j) = C t ; (3) P t Z P t = P t (j) dj : (4) The ow budget constraint of the household at the beginning of period t is C t + B t+ R t + M t P t W t N t + M t + B t + F t + T t ; (5) P t P t P t P t where R t is the gross rate of real return on one-period bonds held from t to t +, the nominal wage is W t per hour of labor and F t is a share of nominal pro ts transferred from rms to the household. T t is the lump-sum nominal transfer from the government. The in nitely-lived household maximizes the sum of expected discounted lifetime utility subject to a sequence of budget constraints (5). The optimality conditions for the household are: N t C t = W t (6) P t Rt Ct+ = E t (7) M t P t = C t C t R t ; (8) 4

5 There is a continuum of monopolistically competitive rms, indexed on the unit interval by j 2 [; ]. To produce intermediate goods of type j, the rm uses the technology: Y jt = Z t L a jt; (9) where L jt is the amount of labor input demanded by the rm j in time t, and Z t is an aggregate technology shock. Parameter a is strictly greater than zero, but smaller or equal to one. As shown in Sbordone (22), when a < ; it incorporates strategic complementarity into the rms price-setting. Similar to the Calvo mechanism, monopolistically competitive rms adjust their prices according to a stochastic timing scheme. At each period, rms draw a price-resetting lottery, whose winning probabilities depend on the age of the price. This price adjustment scheme is summarized in term of a hazard function, de ned as h i = P (adjust at i j survival to i ): The hazard function gives the probability of price adjustment i period since the last revision conditional on the price has been xed for i periods, 8i = ; ; I. I is the maximum possible price duration, where I = is allowed. In addition, I assume 6 h i 6 ; for all i. Furthermore, I de ne the survival function as S i = i n= ( h n ) ; () which gives the probability of a price being xed at least for i periods. Note that dynamics of the stochastic price adjustment process can be modeled as a Markov chain with the price durations as its states and hazard rates de ning its transition matrix. We can derive the invariant distribution of the states, by calculating the eigenvector associated with the unit eigenvalue: i = S i ; 8i = ; ; I: () I S n n= Once the invariant distribution is reached, there is no need to track the dynamics of price distribution any more. In the rest of the paper, I assume that the price vintage distribution has converged to the invariant distribution and thereby I will use Equation () to calculate the aggregate price index. In a given period, the optimal price chosen should re ect the possibility that it will not be revised in the near future. Consequently, adjusting rms choose the optimal price that maximizes the discounted sum of real pro ts in the time horizon over which the new price is expected to be xed. Based on the generalized sticky price assumption, discussed above, the maximization problem of a price setter can be expressed as P max t = E I t i= S Pjt iq t;t+i P jt P t+i T C j;t+i Y j;t+i ; P t+i 5

6 where Q t;t+i is the real stochastic discount factor appropriate for discounting real pro ts from t to t + i, given by Equation (7). The adjusting rm maximizes real pro ts subject to demand for intermediate goods in period t + j given that the rm resets the price in period t Pjt Y t+jjt = Y t+j: P t+j It yields the rst order necessary condition for the optimal price P h i P + t = E I t i= S i Q t;t+i Y t+i P t+i MC t+i P h i ; (2) E I t i= S i Q t;t+i Y t+i P t+i where Pt denotes the average optimal relative price chosen by adjusting rms at period t and MC t is the average level of nominal marginal costs of all resetting rms, which is given by t Y t MC t = W tz P + t. Intuitively, the optimal price is equal to the markup multiplied by a weighted sum of future marginal costs, where weights depend on the survival probabilities, discount factor and future market conditions. In the Calvo case, where S i = ( h) i, this equation reduces to the Calvo optimal pricing condition. Finally, given the invariant price distribution i, the aggregate price index (4) can be rewritten as a distributed sum of past reset prices. Let de ne Pt periods ago, then the aggregate price index is obtained by 2. In ationary steady state P t = PI i= ip t i i as the aggregate optimal price set i : (3) I assume that, in the steady state, the gross growth rate of nominal money stock is equal to. The steady state of the economy is characterized by a constant path of real variables and a growing path of nominal variables at the rate of. De ne X as the steady state value of a generic variable X, then the optimality condition (2) can be rewritten as " P t P = MC PI # i= i S i i P t I : (4) i= i S i ( )i As discussed in Ascari (24), the optimal relative price ratio is equal to markup over the real marginal cost multiplied by an extra term, which represents the e ect of steady-state in ation on the relative price. When steady-state in ation is equal to one, this term disappears. On the other hand, when steady-state in ation is greater than one, the extra term is also greater than one. It implies that the adjusting rms front-load their prices in order to hedge the risk that they may not adjust again in the near future. As a result, the rm sets its price higher than the case of zero in ation. 2.2 Log-linearized model To solve the model, I log-linearize the optimality conditions and market clearing conditions around the steady state with positive in ation. The log-linearized optimal price equation is 6

7 obtained by ^p t = E t " PI i= ( ) i S i ( cmc t+i + ^p t+i ) # ; (5) where = P I i= ( ) i S i and ^p t, ^p t are the percentage deviations of prices from the trend price level. Similarly, I log-linearize equation (3) and obtain where (i) = ^p t = P I i= (i)^p t i; (6) i ( )i P I i= ( )i i Finally, other log-linearized optimality conditions and market clearing conditions are ^y t = ^n t + (^p t ^p t ); (7) ^p t = P I i i i= P I i= i i ^p t i; (8) ^c t = ^y t ; (9) cmc t = a + + ( a) + ^y t a + a a + a ^z t; (2) E t [^c t+ ] ^c t = i t E t [^ t+ ] ; (2) ^ t = ^p t ^p t : (22) All variables here denote percentage deviations from the steady state. To explore the implications of the price reset hazard function for macro dynamics, I proceed in two steps. First, to gain some intuitions, I will focus on a simpli ed setup and derive analytical results of the model. Then, I will solve and simulate it numerically and present the implications of the complete model. 3 Analytical results To deliver analytical results, I simplify the model by adding the following assumptions. First, I assume that there is no in ation at the steady state, i.e. =. Second, without loss of generality, I set the subjective discount factor to one. Third, I choose marginal cost elasticity a++( a) of output to be one, i.e. a+a =. This implies that there is no strategic complementarity in the price-setting. 3 Finally, I assume that log-deviation of money supply stock follows a quantity-theory equation ^m t = ^p t + ^y t ; (23) where ^m t is the log-deviation of nominal money supply from the steady state. 3 This assumption not only simpli es the solution, but also disentangles the e ects of nominal rigidity from those of "real rigidity" caused by the strategic complementarity. 7

8 As for the monetary disturbances, I will consider two di erent monetary regimes. First, I assume that the level of ^m t follows an autoregressive process: ^m t = ^m t + t : (24) Second, I consider a money growth shock, which is subject to an autoregressive process: ^m t = ^m t + t : (25) In both stochastic processes, innovations t v i:i:d:(; 2 m) and is the persistence parameter, which lies between zero and one. Under these simpli cation assumptions, the log-linearized model presented in Section (2) can be represented by the three equations ^p t = P I i= ie t [ ^m t+i ] ; (26) ^p t = P I i= i ^p t i; (27) ^y t = ^m t ^p t ; (28) where only structural parameters left are the distribution of price durations i, which are functions of hazard rates according to Equation (). 3. E ects of the level shock Proposition Assume that the log deviations of nominal money supply follows an AR( ) process speci ed in (24). Under the sticky-price model summarized by equation (26), (27), and (28), impulse response functions to a monetary level shock are given by IR ^p t+n = F n (29) IR(^p t+n ) = F P n i= i n i ; (3) IR (^y t+n ) = n F P n i i= i (3) where F = P I i= i i : (32) Proof: See Appendix (A). Several interesting insights stand out from this proposition. First, the impulse response function of the optimal price (Equation 29) is not only a ected by the rate () at which the monetary disturbance decays over time, but also by a scaling factor F, which captures rm s forward-looking pricing behavior. As seen in Equation (32), F is smaller than one, when <. The optimal price adjustment is scaled down, because rms expect the money level shock will fall by itself over time at the rate of. When =, implying a permanent shock, the scaling factor is equal to one, the optimal price will jump immediately to the new level that is one-to-one to the size of the money level shock. 8

9 Second, the impulse response of the aggregate price is a ected by the accumulative distribution of price durations. This is intuitive, because the accumulative distribution of price durations represents the fraction of rms that are allowed to react to the shock over time. Together with the size of the response, they determine the response of the aggregate price to a money level shock. An instructive case emerges when the shock is permanent ( = ). In this case, the shape of the aggregate price response is solely determined by the accumulative distribution of price durations IR (^p t+n ) = P n i= i: (33) Furthermore, one can show that the impulse response of in ation to a permanent money level shock is equal to the distribution of price durations. IR(^ t+n ) = IR (^p t+n ) IR (^p t+n ) h Pn = i= P i n i i= i = n (34) Third, the real e ect of the monetary level shock is just the ipping side of its nominal e ect. Again, the shape of the accumulative price distribution plays a central role in determining the output response to a money level shock. Using the example of a permanent shock again, the real e ect of a money level shock can be expressed as IR (^y t+n ) = P n i= i: Through these analytical results, I shed lights on the central role played by the price distributions in linking the shape of the price reset hazard function and impulse responses of aggregate variables. To illustrate this mechanism, in Figure (), I plot price distributions and impulse response functions to a permanent money level shock as the result of three di erent hazard functions. The rst hazard function is constant at the level of 25%, implying that the average price duration is of 4 quarters (solid line). The second hazard function is linearly increasing in the time-since-last-adjustment (dash line), while the third is linearly decreasing in the timesince-last-adjustment (star line). 4 The top-left gure shows the distribution of price durations implied by di erent hazard functions. With the constant hazard function, the probability mass falls geometrically as the price duration gets longer. By contrast, the price distribution of the increasing hazard function drops slower at the shorter durations, but faster at the longer durations, whereas the price distribution of the decreasing hazard function follows the opposite pattern. As shown in Equation (34), the shape of the price distribution determines the in ation response to the money level 4 Here, I parameterize the hazard function in a parsimonious way. In particular, the functional form I apply is the hazard function of the Weibull distribution, which has two parameters: h(j) = j (35) is the scale parameter, which controls the average duration of the price adjustment, while is the shape parameter to determine the monotonic property of the hazard function. I choose, such that it implies an average price duration of 4 quarters. The shape parameter is set at values in f; 2; :5g to represent constant, increasing and decreasing hazard function. 9

10 Inflation Output Price Distribution of Price Durations Constant HZ Increasing HZ Decreasing HZ..9.8 Impulse Responses of Price Time since last adjustment Quarters.35 Impulse Responses of Inflation.8 Impulse Responses of Output Quarters Quarters Figure : E ects of a Permanent Money Level Shock shock. As a result, the increasing-hazard model predicts a more persistent in ation response than the constant- and decreasing-hazard model (the top-right panel). The bottom panels illustrate how the change in the slope of hazard functions a ects the macro dynamics through its e ects on the accumulative distribution of price durations. As seen in the bottom-left panel, responses of the aggregate price converges to the new level at fastest in the increasing-hazard model, and at slowest in the decreasing-hazard model. Furthermore, since the output response is also determined by the accumulative distribution, as seen in the bottom-right panel, the decreasing-hazard model predicts a more persistent response of output to a permanent money level shock than the other two models. 3.2 E ects of the growth shock Proposition 2 Assume that nominal money supply growth rates follow an AR( ) process speci ed in (25). Under the sticky-price model summarized by equation (26), (27), and (28), the impulse response function of in ation to a money growth disturbance is given by IR( t+n ) = P I j= P I j i= i n+i j ; 8n + i j Proof: See Appendix (B). This proposition is intended to show how the price distribution a ects in ation dynamics after a money growth shock. Again, it is instructive to study a permanent money growth shock rst. In this case, the impulse response of in ation is equal to a complex summation of the price distribution. However, in the Calvo model, when i = ( ) i, where < < denotes non-adjustment rate, the impulse response function of in ation becomes IR( t+n ) = ( ) 2 P j= j P i= i =

11 Inflation Output.25.2 Calvo Model Inflation Output.25.2 Increasing HZ Model Inflation Output Figure 2: E ects of a Permanent Money Growth Shock.25 Impulse Responses of Inflation Calvo Increasing HZ.35 Impulse Responses of Output Calvo Increasing HZ Quarters Quarters Figure 3: E ects of a Persistent Growth Shock It implies that in ation reacts immediately to the permanent money growth shock and the size of the response is equal to the size of the money growth shock. Figure (2) illustrates the transitional dynamics of a disin ationary scenario predicted by the increasing- and constant-hazard model, respectively. In this simulation, I assume that, before t =, output is at its steady state level, while in ation is equal to 2.5% per quarter. At t =, the central bank sets the growth rate of nominal money stock from 2.5% per quarter to zero and keep it constant permanently. In the left panel, the constant-hazard model predicts a costless disin ation, i.e. in ation drops immediately to the new level, while real output is not a ected at all. This result recapitulates the famous criticism to the Calvo sticky price model (e.g. Ball, 994 and Mankiw, 2). By contrast, the increasing-hazard model predicts a recession up to 8 periods during the slow disin ationary process. Figure (3) compares further the e ects of the constant and increasing hazard functions after a persistent money growth shock ( = :5). The left gure illustrates the impulse responses of in ation. While the Calvo model predicts a monotonically declining response to the money

12 growth shock (the dash line), the increasing-hazard model generates a hump-shaped response of in ation (the solid line). Even though this hump is not very eminent, peaking only at the second period after the impact of the shock, it represents a qualitative improvement towards the empirically plausible shape of in ation response to a monetary shock (See: e.g. Christiano et al., 25). In the following sections, I will show that adding other features to the model will help enhance the delayed e ect of the increasing hazard function. The right gure depicts the impulse response function of output to a money growth shock. The Calvo model predicts a more persistent real e ect of money growth shocks than the increasing-hazard model. The economic intuition is that the increasing-hazard function implies less nominal rigidity on average, in the sense that there are less rms stuck to an old price for a long spell of time. As a result, it leads to a lesser real e ect of the monetary disturbance. To summarize the analytical results, I nd that there is a close correspondence between the impulse responses of output and aggregate price, because both of them are mainly determined by the accumulative distribution of price durations. On the contrary, there is a trade-o between persistent responses of output and in ation as the slope of the hazard function changes. Increasing hazard functions enhance in ation inertia, but it weakens the persistent response of output. 4 Numerical results Next, I will check to what extent the analytical results still hold in a more realistic setting. To do that, I solve and simulate the full- edged DSGE model described in Section (2). Speci cally, additional to nominal rigidity studied in Section (3), the economy is now characterized by strategic complementarity in price-setting, the intertemporal IS curve and non-zero steady-state in ation. 4. Calibration As the main theme suggested, the most important parameters in the model are those a ecting the price reset hazard function. Although empirical studies provide useful information on the average level of hazard rates across price durations, only mixed evidence on the shape of the hazard function is available. For example, Cecchetti (986) uses newsstand prices of magazines in the U.S. and Goette et al. (25) study Swiss restaurant prices. Both studies nd strong support for increasing hazard functions. By contrast, recent studies using more comprehensive consumer price index data nd that hazard functions are rst downward sloping and then mostly at, interrupted periodically by spikes (See, e.g.: Campbell and Eden, 25, Alvarez, 27 and Nakamura and Steinsson, 28a). Despite the recent evidence, in the numerical exercises, I will focus on comparing the implications of the increasing hazard function with those from the Calvo model. I justify my choice by referring to two reasons. First, the increasing hazard function is advocated by most of micro-founded sticky price models (See: e.g. Dotsey et al., 999, Golosov and Lucas, 27 and Nakamura and Steinsson, 28b). Second, Alvarez et al. (25) argue that the estimated downward-sloping hazards result mainly from the aggregation bias when price stickiness is heterogeneous in the economy. As a result, the decreasing hazard function is rather an aggregate phenomenon and hence less relevant to the pricing behavior at the rm s level. 2

13 Following Mash (24), I calibrate the hazard function based on the survey evidence by Blinder et al. (998) for the U.S. economy. As reported in Table (), this empirical hazard function is increasing in price durations and the maximum price duration is truncated at the 7th quarter 5. In the numerical exercise, I compare this empirical hazard with the Calvo hazard function, which has a mean price duration of 3 quarters. This is consistent with the mean price duration of 7-9 months estimated by Nakamura and Steinsson (28a). Given hazard functions, the survival function S j and the distribution of price durations j can be calculated using formulae presented in Equation() and (). Hazard Function h h 2 h 3 h 4 h 5 h 6 Calvo (983) /3 /3 /3 /3 /3 /3 Blinder et. al (998) Table : Calibration of hazard functions The calibration of the remaining structural parameters is generally standard. Labor share is set to :64, and the discount factor implies a steady state real return on nancial assets of about four percent per annum ( = :992). I choose the Frisch elasticity of labor supply to be, implying a unit labor supply elasticity. The elasticity of substitution between intermediate goods is equal to, implying the desired markup over marginal cost is about %. Finally, I assume that the growth rate of nominal money supply and percentage deviations of the technology shock follow rst-order autoregressive processes. ^m t = ^m t + t ; (36) ^z t = z ^z t + u t : (37) In both stochastic processes, innovations have a zero mean and nite variances. I choose = :5 and the standard deviation of the innovation to the nominal money growth rate to be 25 basic points per quarter. For the aggregate technology shock, I choose z = :95 and the standard deviation of.7, which is commonly used values in the RBC literature (see: e.g. King and Rebelo, 2). 4.2 Interaction with strategic complementarity In the rst numerical simulation, I focus on the interaction between the increasing hazard function and strategic complementarity, while keeping the steady-state in ation to be one. Table (2) reports theoretical moments predicted by models with di erent hazard functions. For each model, the standard deviation and rst-order autocorrelation coe cient are reported as the measures for volatility and persistence of macro dynamics. 6 5 This empirical hazard function is calculated by Mash (24) in Table. 6 All moments are for a Hodrick-Prescott ltered time series. 3

14 = % Volatility (%) Persistence Models ^ ^y ^ cmc ^y Ampli cation Ratio Calvo Model :2 :68 :66 :6 :79 :8 Increasing HZ Model :2 :7 :79 :6 :78 :3 Table 2: Theoretical Moments of Models Several patterns can be observed in the table. First, volatility of in ation and output is hardly a ected by the change in the shape of the hazard function, neither is the persistence of output and the real marginal cost. Second, the model with the increasing hazard function generates higher persistence in in ation than in the Calvo model. The stronger propagation mechanism of the increasing-hazard model is more conspicuous when we compare persistence of in ation with that of the real marginal cost. In both models, the real marginal cost is the underlying driving force of in ation inertia and model s ability to generate intrinsic in ation persistence can be measured by the ratio between the AR() coe cient of in ation and the real marginal cost. In the Calvo model, as the new Keynesian Phillips curve is purely forwardlooking, in ation persistence is solely ampli ed by the strategic complementarity in the pricesetting. As a result, the ampli cation ratio is :8, indicating that real rigidity resulted from strategic complementarity can only amplify 8 percent more in ation inertia. By contrast, the ampli cation ratio of the increasing-hazard model is :3, i.e. 22% more in ation persistence is generated out of the same driving force due to the increasing hazard function. This result would suggest that, if increasing hazard function is considered, nominal rigidity provides a more powerful propagation mechanism in generating in ation persistence than the real rigidity resulted from strategic complementarity. Figure 4 plots the impulse responses of in ation and output to technology shocks and monetary shocks. As seen in the upper panels, the in ation response of the increasing-hazard model is hump-shaped, whereas it is monotonically declining in the Calvo model. This nding is interesting, because it shows that, under a plausible calibration, the increasing-hazard model not only generates more persistent in ation, but also accounts for the empirical shape of impulse responses reported by the structural VAR evidence (See:e.g. Christiano et al., 25). It is important to emphasize that this model generates intrinsic in ation inertia without the reliance on the ad. hoc. assumptions of backward-looking pricing (See:e.g. Gali and Gertler, 999 and Christiano et al., 25). Mankiw and Reis (22) argue that the sticky information model outperforms the Calvo sticky price model because the latter is unable to generate a hump-shaped in ation response to a monetary shock. In the light of the increasing-hazard model, the monotonic in- ation response is not an intrinsic feature of sticky price models, rather merely results from the restrictive hazard function underlying the Calvo assumption. The economic intuition why an increasing hazard function can generate a delayed response of in ation is explained in Sheedy (2). The increasing-hazard function postpones the timing of the price adjustment, so that there are more catch-up rms than the roll-back rms. In this numerical exercise, however, I want to highlight a new piece of insight that strategic complementarity helps amplify the delayed e ect caused by the increasing hazard function. Under strategic complementarity, the price decision of one rm also depends on other rms price decisions. The interaction between the strategic complementarity and the increasing hazard 4

15 4 x Inflation to technology shock Calvo Increasing HZ 3 x Inflation to monetary shock x Output to technology shock x Output to monetary shock Figure 4: Impulse Response Functions under Di erent Hazard Functions function plays out as follows: shortly after the impact of a monetary shock, there are only a few rms adjusting their prices due to the increasing hazard function. In this phase, the low average price is still the price norm in the economy. As a result, even the adjusting rms opt for a small price adjustment, because it is not optimal to deviate too much from other rms in the economy. Over time, as more and more rms reset their prices, the price norm slowly switches to the new level. Now, strategic complementarity gives rms incentives to catch up to the higher price norm. In this way, strategic complementarity ampli es the delayed e ect of the increasing hazard function, leading to a more salient hump-shaped impulse response of in ation to a monetary shock. In the lower panels, impulse responses of output are virtually identical. Both models predict hump-shaped responses to a technology shock, but monotonic declining responses to a money growth shock. Contrast to the analytical results, where I show that there is a trade-o between persistence of in ation and output responses to monetary shocks, the more complex model predicts that the change in the shape of hazard functions has much stronger e ects on in ation than output dynamics. I argue that this di erence is due to the fact that the forward-looking IS curve makes the dynamics of output to be determined by the real interest rate, which is largely independent of the shape of hazard functions. As a result, it isolates output dynamics to a large extent from the forces a ecting the sticky price-setting. This numerical result is consistent with the related theoretical ndings in the literature, in which some authors (e.g. Fuhrer and Moore, 995 and Kiley, 22) show that the Calvo sticky price model can account for output persistence, but not for in ation inertia, while others (e.g. Mash, 24) conclude that the increasing-hazard sticky price model can generate persistent in ation as well as output. In the light of my analysis, the increasing hazard function is important for accounting for in ation inertia, while the forward-looking IS curve is crucial for preserving output persistence under the increasing hazard function. 5

16 5 5 x 4 Inflation to technology shock x Inflation to monetary shock x Output to technology shock x 3 Output to monetary shock % inflaion 5% inflation % inflation Figure 5: Impulse Response Functions under Positive Steady-State In ation 4.3 Interaction with steady-state in ation Ascari (24) shows that trend in ation has important implications for the model s dynamics, when the Calvo pricing model is log-linearized around non-zero trend in ation. Here I study the e ects of non-zero steady-state in ation in the increasing-hazard model. In Table (3), I report theoretical moments predicted by the increasing-hazard model under steady-state in ation of 5% and % per year. Volatility (%) Persistence Steady-state In ation ^ ^y ^ cmc ^y Ampli cation Ratio = % :2 :7 :79 :6 :78 :3 = 5% :9 :72 :83 :63 :78 :32 = % :8 :73 :87 :65 :77 :33 Table 3: Theoretical Moments of Increasing-Hazard Models under Steady-State In ation In this table, we observe the tendency that higher steady-state in ation weakens volatility of in ation, but enhances volatility of output. Those e ects on the volatility of macro dynamics, however, are modest. On the other hand, positive steady-state in ation enhances persistence of in ation. It is achieved through two channels. First, under positive steady-state in ation, the real marginal cost becomes more persistent. Second, the combination of positive steady-state in ation and increasing hazards strengthens the inner propagation even further. The latter can be measured by the ampli cation ratio that is equal to :32 under 5% steady-state in ation and :33 under % steady-state in ation. Figure (5) plots the impulse responses of in ation and output under various levels of steadystate in ation. With positive steady-state in ation, the hump shape of the in ation responses becomes even more salient. The maximum impact of both shocks is delayed further. In the 6

17 lower panels, impulse responses of output drift outward, implying larger real e ects of monetary shocks under higher steady-state in ation. The reason why high trend in ation ampli es the delayed e ect of the increasing hazard function is that the presence of positive in ation accelerates relative prices dispersion, which in turn enhances the interaction between strategic complementarity and the increasing hazard function discussed in the previous section. 5 Conclusion The central theme of this study is to investigate e ects of the price reset hazard function on macro dynamics. I present new analytical and numerical results, showing that, in the sticky-price model, the shape of the price distribution greatly a ects persistence of output and in ation. The distribution of price durations lies in the center of the propagation mechanism. In addition, the interaction between the hazard function and other features, which are found to be important for propagating monetary shocks, is also studied. The central message of the study is that considering non-constant hazard function has important implications for in ation dynamics, but to a lesser extent for the output dynamics. Admittedly, the model I use has its weaknesses. Above all, the timing decision is not endogenous, so that important dynamics resulted from the extensive margin of price adjustments miss in my analysis. To study this e ect, one would need to work with a micro-founded statedependent pricing model. Although the state-dependent pricing models are superior in principle, due to their tractability challenges, one can only solve them at expense of realism on other aspects of the economy. Since I show in the paper that features, such as strategic complementarity and forward-looking aggregate demand, are as important as the detailed modeling of the pricesetting behavior for the propagation mechanism of the monetary shocks, I view my modelling choice as a reasonable compromise. For future research, an optimal price reset hazard function is clearly a promising extension for the further exploration of the topic. 7

18 References Alvarez, L. J. (27), What do micro price data tell us on the validity of the new keynesian phillips curve?, Kiel Working Papers 33, Kiel Institute for the World Economy. Alvarez, L. J., P. Burriel, and I. Hernando (25), Do decreasing hazard functions for price changes make any sense?, Working Paper Series 46, European Central Bank. Ascari, G. (24), Staggered prices and trend in ation: Some nuisances, Review of Economic Dynamics, 7(3), Ball, L. (994), Credible disin ation with staggered price-setting, American Economic Review, 84(), Blinder, A. S., E. D. Canetti, D. E. Lebow, and J. B. Rudd (998), Asking about prices - A new approach to understanding price stickiness, Russell Sage Foundation, New York. Calvo, G. A. (983), Staggered prices in a utility-maximizing framework, Journal of Monetary Economics, 2(3), Campbell, J. R. and B. Eden (25), Rigid prices: evidence from u.s. scanner data, Working Paper Series WP-5-8, Federal Reserve Bank of Chicago. URL Cecchetti, S. G. (986), The frequency of price adjustment : A study of the newsstand prices of magazines, Journal of Econometrics, 3(3), Christiano, L. J., M. Eichenbaum, and C. L. Evans (25), Nominal rigidities and the dynamic e ects of a shock to monetary policy, Journal of Political Economy, 3(), 45. Dixit, A. K. and J. E. Stiglitz (977), Monopolistic competition and optimum product diversity, American Economic Review, 67(3), Dotsey, M., R. G. King, and A. L. Wolman (999), State-dependent pricing and the general equilibrium dynamics of money and output, The Quarterly Journal of Economics, 4(2), Fuhrer, J. and G. Moore (995), In ation persistence, The Quarterly Journal of Economics, (), Gali, J. and M. Gertler (999), In ation dynamics: A structural econometric analysis, Journal of Monetary Economics, 44(2), Goette, L., R. Minsch, and J.-R. Tyran (25), Micro evidence on the adjustment of sticky-price goods: It s how often, not how much, Discussion Papers 5-2, University of Copenhagen. Department of Economics. Golosov, M. and R. E. Lucas (27), Menu costs and phillips curves, Journal of Political Economy, 5,

19 Kiley, M. T. (22), Partial adjustment and staggered price setting, Journal of Money, Credit and Banking, 34(2), King, R. G. and S. T. Rebelo (2), Resuscitating real business cycles, NBER Working Papers 7534, National Bureau of Economic Research, Inc. Mankiw, N. G. (2), The inexorable and mysterious tradeo between in ation and unemployment, Economic Journal, (47), C45 6. Mankiw, N. G. and R. Reis (22), Sticky information versus sticky prices: A proposal to replace the new keynesian phillips curve, The Quarterly Journal of Economics, 7(4), Mash, R. (24), Optimising microfoundations for in ation persistence, Economics Series Working Papers 83, University of Oxford, Department of Economics. Nakamura, E. and J. Steinsson (28a), Five facts about prices: A reevaluation of menu cost models, The Quarterly Journal of Economics, 23(4), Nakamura, E. and J. Steinsson (28b), Monetary non-neutrality in a multi-sector menu cost model, NBER Working Papers 4, National Bureau of Economic Research, Inc. Sbordone, A. M. (22), Prices and unit labor costs: a new test of price stickiness, Journal of Monetary Economics, 49(2), Sheedy, K. D. (2), Intrinsic in ation persistence, Journal of Monetary Economics, 57(8), Whelan, K. (27), Staggered price contracts and in ation persistence: Some general results, International Economic Review, 48(), 45. Wolman, A. L. (999), Sticky prices, marginal cost, and the behavior of in ation, Economic Quarterly, 3(Fall),

20 A Proof of Proposition First, based on the assumption that nominal money supply follows an AR() process ^m t = ^m t + t ; where t v i:i:d:(; 2 m); (38) The impulse response function of nominal money supply to an innovation t is IR ( ^m t+n ) = n : (39) Using the de nition of impulse response function, I derive the impulse response function of the optimal price to a monetary level shock as IR ^p t+n t = E t (^p t+n) E t (^p t+n) = P I i= i [E t ( ^m t+n+i ) E t ( ^m t+n+i )] = P I i= iir ( ^m t+n+i ) t IR ^p t+n = n P I i= i i : (4) Following the aggregate price equation (27), I derive the impulse response function of the aggregate price as follows IR (^p t+n ) t = E t (^p t+n ) E t (^p t+n ) = P n i= ie t (^p P n t+n i) i= ie t (^p t+n i) = P n i= i Et ^p t+n i E t (^p t+n i) = P n i= iir ^p t+n i t : IR (^p t+n ) = P n i= iir ^p t+n i : (4) Finally, plugging (4) into (4), it yields IR (^p t+n ) = P h n i= i n i P I i= i ii = P I i= i i P n i= {z } i n i F = F P n i= i n i (42) Finally, using equation (28), I derive the impulse response function of real output as follows IR (^y t+n ) = IR ( ^m t+n ) IR (^p t+n ) = n F P n i= i n i = n F P n i i= i (43) 2

21 B Proof of Proposition 2 First, in ation de ned as ^ t = ^p t ^p t can be expressed as Then, substituting equation (26) for ^p t, I obtain ^ t = ^p t ^p t (44) = P I j= j ^p t j ^p t j ^ t = P I j= j ^p t j ^p t j = P I j= P I j = P I j= j i= ie t ( ^m t+i j ^m t+i j ) P I i= ie t [ ^m t+i j ] (45) Based on the assumption that nominal money growth rate follows an AR() process ^m t = ^m t + t ; where t v i:i:d:(; 2 m); we can derive the impulse response function of money growth shock to an innovation t as It follows that IR (^ t+n ) = P I j= P I j = P I j= j IR ( ^m t+n ) = n : (46) i= iir ( ^m t+n+i j ) P I i= i n+i j ; 8n + i j (47) 2