Customer markets and the welfare effects of monetary policy

Journal of Monetary Economics 58 (2011) 206–219

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Customer markets and the welfare effects of monetary policy  ¨ Johan Soderberg Department of Economics, Uppsala University, Box 513, SE-751 20 Uppsala, Sweden

a r t i c l e in f o

abstract

Article history: Received 17 February 2010 Received in revised form 26 May 2011 Accepted 31 May 2011 Available online 30 June 2011

A customer market model in which firms and customers form long-term relations is developed and integrated into the canonical New Keynesian framework. This leads to two important differences compared to the standard model. First, the purely forwardlooking Phillips curve is replaced by a hybrid variant where current inflation also depends on past inflation. Second, the welfare cost of inflation is much lower, which leads to an optimal monetary policy where relatively more weight is put on output gap stabilization than previously found in the literature. & 2011 Elsevier B.V. All rights reserved.

1. Introduction Survey evidence consistently ranks implicit contracts and other forms of customer relations as important factors that firms consider when setting prices. For instance, price raises are often refrained from for fear of adverse customer reactions that may damage long-term relations. The importance of these factors for pricing has been documented in numerous studies for different countries; for recent studies, see Apel et al. (2005), Amirault et al. (2005), and Fabiani et al. (2007). In recent years, narrative evidence that documents the importance of implicit contracts has also emerged. Young and Levy (2006) provide evidence for implicit contracts in the marketing of Coca-Cola, while Nakamura and Steinsson (in press) survey the media and find numerous examples of firms communicating their intentions not to raise their prices. The customer market model, first proposed by Phelps and Winter (1970), formalizes the idea that firms and customers form long-term relations. In their model, a firm’s customer base is a valuable asset that only gradually adjusts to price changes. It is now well established in the literature that the dynamic interaction between prices and demand arising in the customer market model has important implications for price-setting behavior. Early contributions include Bils (1989) and Gottfries (1991), who use customer market models to explain why short-run variations in demand have weak effects on prices. In order to expand their customer base, firms refrain from increasing their prices in times of high demand. More recent examples on this theme are Ravn et al. (2006) and Kleshchelski and Vincent (2009), who construct general equilibrium customer market models that predict countercyclical markups, providing a source of real rigidity. The idea that firms fear adverse customer reactions is formalized in the customer anger model in Rotemberg (2005), where it is assumed that consumers react negatively to prices they perceive as unfair. This fear of antagonizing consumers makes firms scrupulous about price changes, which has the potential of generating nominal price rigidity. Nakamura and Steinsson (in press) analyze a model with forward-looking customer markets and find that nominal price rigidity is sustainable as an equilibrium outcome. This paper investigates how customer markets affect inflation dynamics and the optimal conduct of monetary policy in the context of the New Keynesian framework. The economic environment I have in mind is one where there are costs associated with the acquisition and processing of information about prices, so that households only occasionally  Tel.: þ46 18 471 11 00.

E-mail address: [email protected] 0304-3932/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.jmoneco.2011.05.012

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reoptimize their allocation of consumption among different goods. For instance, if different goods are sold at different physical locations, households may choose to only infrequently compare price quotes across stores. This means that information about better shopping opportunities diffuses slowly through the economy. This interpretation is in the spirit of the original customer market model proposed by Phelps and Winter (1970). But even if households are fully informed about prices, there may be costs associated with the time and cognitive effort required to optimally allocate consumption between goods. Such frictions, as well as uncertainty about other product attributes, are factors known to give rise to repeat purchase behavior; see Solomon et al. (2006, Chapter 8) and references therein. Explicit modeling of information frictions is technically complicated, and I therefore resort to a variant of the signaling mechanism proposed by Calvo (1983). It is assumed that a household allocates consumption among different goods by choosing the relative consumption of each good, i.e., the quantity consumed of the good relative to the total basket consumed. But in each period, the household is only allowed to reoptimize for a randomly chosen subset of all goods. By applying the restriction of infrequent reoptimization across goods, as opposed to across households, the representative household construct is preserved, which fundamentally simplifies aggregation and solution of the model. The result is a model where a firm’s market share depends on its lagged market share as well as on current and expected future prices. Because demand is a function of expected future prices, there is a problem of time inconsistency in price setting. Firms would like to promise low future prices, but renege on these promises when the future arrives. The time inconsistency problem is resolved by assuming that firms commit to state contingent price plans. This assumption is unrealistic if taken literately, but it captures the idea that firms can, to a considerable extent, make promises to their customers. The ability of firms to commit should be interpreted as a stylized way of modeling implicit contracts between firms and their customers. Nakamura and Steinsson (in press) obtain a similar demand formulation by assuming internal deep habits, i.e., households form habits in the consumption of individual goods. They analyze a partial equilibrium model where firms are unable to commit to price policies and show that nominal price rigidity is sustainable in a reputational equilibrium under imperfect information. In their model, a firm compensates for the lack of commitment by setting a price cap above which it will not raise its price. In this paper, in contrast, it is assumed that firms can commit to a price policy. Taking price stickiness as given, the general equilibrium implications for aggregate dynamics and monetary policy are analyzed. A central difference compared to Ravn et al. (2006) and Nakamura and Steinsson (in press) is that customer markets in this paper are the result of frictions that do not alter households’ preferences for different goods. As a consequence, the consumption Euler equation is unaffected by the introduction of customer markets. This has the advantage of allowing me to study the implications of customer markets without having to account for simultaneous changes in preferences and aggregate demand. The customer market framework is integrated in an otherwise standard New Keynesian staggered price-setting model. The resulting model differs from the standard one in two important respects. First, the Phillips curve is no longer purely forward-looking but also depends on lagged inflation, leading to endogenous inflation persistence. This is a consequence of the forward-looking nature of demand. A firm that desires a higher price, but is constrained by price-setting frictions, will, as a second best option, commit to raising the price in the future. This results in firms continuing to raise their prices even after the factor that initially led them to desire higher prices has dissipated. Second, the welfare criterion places a much lower weight on inflation stabilization. The main lesson emerging from the utility-based analysis in the canonical New Keynesian model is the importance of inflation stabilization over output gap stabilization. Inflation leads to price dispersion, which distorts the allocation of consumption among the different goods in the economy. Customer markets reduce this distortion by slowing down the reallocation of consumption when relative prices are dispersed. This makes price dispersion less distortionary, which reduces the welfare cost of inflation and leads to an optimal monetary policy that involves a substantially higher volatility of inflation and a lower volatility of the output gap. The remainder of this paper is organized as follows. Section 2 describes the model and Section 3 presents the welfare criterion. Section 4 describes the calibration and Section 5 shows the results from the numerical simulations. Section 6 concludes. 2. The model In this section, the decisions of households and firms are analyzed, and a log-linear approximation to the model is derived. 2.1. Households The economy is populated by a large number of households, indexed by h 2 ½0,1. A household h derives utility from the consumption of a large number of different goods, indexed i 2 ½0,1, according to the aggregator: "Z #Z=ðZ1Þ Cth ¼

1

0

ðCith ÞðZ1Þ=Z di

,

ð1Þ

208

J. S¨ oderberg / Journal of Monetary Economics 58 (2011) 206–219

where Cith denotes the household’s consumption of good i. The household’s utility is   1 X 1 1 E0 bt ðCth Þ1sC  ðNth Þ1 þ sN , 1sC 1þ sN t¼0

ð2Þ

where b 2 ð0,1Þ is the subjective discount factor, and Nth is the number of working hours supplied. The household’s budget constraint is Z 1 Bht þ Pit Cith di ¼ Rt1 Bht1 þ Wt Nth þ Ft , ð3Þ 0

where Bht denotes bond holdings from t to t þ1, Pit is the price of good i, Rt is the gross nominal interest rate paid off in t þ1, Wt is the nominal wage, and Ft is dividends from firm ownership. The household solves the intertemporal problem of maximizing (2), subject to (1) and (3). However, I impose the additional restriction that the allocation of consumption among different goods is subject to Calvo (1983) style frictions. In each period, the household draws a random subset of measure 1y of all goods. For each of these goods, the household is allowed to reoptimize the relative consumption of the good, i.e., the quantity consumed of the good relative to the total basket consumed. For the remaining goods, the household is not allowed to adjust its relative consumption.1 The draw is assumed to be uncorrelated both in time and across households. R1 h h Let Pt  0 Pit C~ it di be an aggregate ‘‘price index’’, where C~ it ¼ Cith =Cth denotes the relative consumption of good i. Because the subset of goods for which a household reoptimizes is chosen at random, the law of large numbers implies that Pt is identical across households. This fundamentally simplifies aggregation and ensures that all households choose the same allocations of aggregate consumption, labor supply, and bond holdings. The first-order conditions for consumption and bond holdings yield the familiar consumption Euler equation: CtsC ¼ bEt Rt

Pt C s C , Pt þ 1 t þ 1

ð4Þ

and the first-order condition for labor supply yields NtsN Wt ¼ : Pt CtsC

ð5Þ

The optimal relative consumption of good i is given by  Z n Git , C~ it ¼

ð6Þ

Gt

P1

k

k

where Git ¼ Et k ¼ 0 ðybÞk Dt,t þ k Ct þ k Pit þ k is the expectation of a weighted sum of future prices for good i, b Dt,t þ k ¼ b R 1 1Z ðCt þ k =Ct ÞsC ðPt =Pt þ k Þ is the nominal stochastic discount factor between periods t and t þk, and Gt ¼ ½ 0 Git di1=ð1ZÞ is an 2 average across firms of expected future prices. When a household decides how much to consume of a particular good, it realizes that in each future period it will not be able to reoptimize with probability y. Therefore, the household takes expected future prices of the good into account, discounted at a rate that incorporates the probability that reoptimization has not occurred. Integrating consumption over households yields the market share of good i, Y~ it  Yit =Ct , given by  Z Git Y~ it ¼ yY~ it1 þ ð1yÞ : ð7Þ

Gt

A fraction y of the market share remains from the previous period because some of the households have not been given the opportunity to reoptimize their relative consumption of the good in this period. The remaining fraction 1y of the market share, consisting of demand from households who reoptimize, depends on the current price and expected future prices, relative to the current and expected future price levels in the economy. 2.2. Firms Good i is produced by a monopolist with technology: Yit ¼ Nit :

ð8Þ

The time inconsistency problem in price setting, arising as a result of demand being a function of future prices, is resolved by assuming that the firm commits to a state contingent price plan. It is also assumed that the plan was set up an infinitely 1 It is not possible for a household to end up in a situation where it cannot adjust its relative consumption for the subset of goods for which it is allowed to reoptimize without adjusting its relative consumption for the remaining goods. It follows from (1) and the law of large numbers that the exponential sum of relative consumption for the subset of goods for which the household is not allowed to reoptimize must be unity. But this also implies that the exponential sum of relative consumption for the subset of goods for which the household is allowed to reoptimize must be unity, i.e., the adjustments of relative consumption of different goods cancel each other out. 2 See web Appendix A for details of the household’s problem.

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long time ago, so as to prevent the firm from exploiting the fact that expectations are fixed when the plan is announced. As discussed in Introduction, this is as a stylized way of modeling the existence of implicit contracts. Price-setting frictions are introduced by assuming that prices are set according to the mechanism in Calvo (1983). In each period, the firm is allowed to reoptimize its price with probability 1a. Given this restriction, the firm’s problem is to maximize its discounted profit stream: E0

1 X

bt D0,t ½Pit Yit Wt Nit ,

ð9Þ

t¼0

subject to (7) and (8). The firm’s Lagrangian, written in terms of the market share, is L ¼ E0

1 X

(



bt D0,t ðPit Wt ÞY~ it Ct þ nit yY~ it1 þ ð1yÞ

t¼0



Git Gt

Z

" #)  1 X Y~ it þ rit Git  ðybÞk Dt,t þ k Ct þ k Pit þ k :

ð10Þ

k¼0

Differentiating (10) with respect to Y~ it and Git yields

nit ¼ ðPit Wt ÞCt þ ybEt Dt,t þ 1 nit þ 1

ð11Þ

and 

rit ¼ Zð1yÞ

Git Gt

Z1

1

Gt

nit ,

ð12Þ

The multiplier nit is the shadow value of a marginal increase in the firm’s market share. Eq. (11) says that the value of a marginal increase in the firm’s market share is the profits generated by the additional customers today plus the present value of future profits from these customers, given by ybEt Dt,t þ 1 nit þ 1 . The multiplier rit is the shadow cost of a marginal increase in Git , the discounted sum of expected future prices. Eq. (12) says that the cost of a marginal increase in Git is the resulting decrease in the firm’s market share, given by Zð1yÞðGit =Gt ÞZ1 =Gt , multiplied by the value of the lost customers nit . The firm’s price must satisfy the optimality condition: Et

1 X

ðabÞk Dt,t þ k Ct þ k ½Y~ it þ kjt Cit þ kjt  ¼ 0,

ð13Þ

k¼0

P j ~ where Cit ¼ 1 j ¼ 0 y ritj . Under flexible prices ða ¼ 0Þ, the additional revenue generated by a marginal price increase, Y it , must equal the shadow cost of the price increase, Cit , resulting from the corresponding reduction in the firm’s market share.3 When setting up its price plan, at the beginning of time, the firm considers how a price change affects its market share at time t, captured by rit . But since households that reoptimize take expectations about future prices into account, the firm must also consider how the price change affects its market share in periods before t, which is why Cit depends on rit1 , rit2 , . . . The notation t þ kjt denotes the value of a variable at time t þk, conditional on the firm’s price last being reoptimized at time t. When prices are sticky, the optimality condition under flexible prices holds only as an expected discounted average over the expected duration of the price. Traditionally, customer markets have been analyzed under the assumption that demand is independent of expected future prices.4 Under that formulation, the price at time t has no effect on demand before t, so the last first-order condition would, ignoring price-setting frictions, correspond to Y~ it ¼ rit . In such a model, a firm faces a trade-off between investing in its customer base or capitalizing on it. By raising its price, the firm reaps a higher revenue from its existing customers, but at the cost of a smaller market share, which reduces revenue in future periods. This mechanism is also present in the model developed in this paper. But when firms commit to price plans, the insight that a price increase at time t also reduces demand before t adds another intertemporal aspect to price setting. This counteracts the incentive for firms to exploit their customers. The parameter y can be viewed as a measure of the degree of customer markets. Increasing the value of y has two opposing effects on the pricing decision. On the one hand, a price increase generates more revenue from existing customers because fewer households are able to reoptimize. On the other hand, a price increase in period t has a bigger negative effect on sales in the past and the loss of market shares will be more persistent. If prices are flexible, it turns out that the optimal price is set with a fixed markup of ðZ1Þ1 over marginal cost, which is identical to the optimal price obtained without customer markets. The two effects of increasing y cancel each other out when prices are flexible.5 3 The problem in (10) is not recursive, since it involves expected values of future prices, but can be transformed into a recursive saddle point problem, by applying the methods in Marcet and Marimon (1998), where Cit1 acts as an additional state variable. 4 Ravn et al. (2006) show that such a specification can be derived from microfoundations by assuming external deep habits, i.e., that households form habits in the consumption of individual goods, but the habitual component depends on aggregate consumption of that particular good. 5 This result also holds with internal deep habits under commitment, as shown by Nakamura and Steinsson (in press).

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2.3. Log-linearization and equilibrium The model is solved by taking a first-order log-linear approximation of the equilibrium conditions around a steady state with zero inflation. Note that a firm’s optimal price depends both on its market share and on the shadow cost of a price increase Cit . These, in turn, depend on the whole history of previous prices. This implies that different firms, which reoptimize their prices at the same time, will in general not set the same price, making the state space of the model infinite dimensional. Still, it is possible to derive a solution to the log-linearized model using the method of undetermined coefficients, applying the same logic as in Woodford (2005).6 Log-linearization of the Euler equation in (4) yields the IS curve: e xt ¼ Et xt þ 1 s1 C ðrt Et pt þ 1 rt Þ,

ð14Þ

7

where xt denotes the output gap, rt ¼ log Rt is the nominal interest rate, pt ¼ logðPt =Pt1 Þ is the inflation rate between t 1 and t, and rte is an exogenously introduced disturbance. The disturbance rte is the real interest rate consistent with the efficient level of output and will henceforth be referred to as the efficient interest rate. The IS curve derived here is identical to that obtained in standard New Keynesian model. Aggregate demand, up to a first-order approximation, is not affected by the presence of customer markets. Shocks emanating from preferences and technology are modeled by e assuming that the efficient interest rate follows the AR(1) process rte ¼ ð1rr Þr þ rr rt1 þ ert , where r ¼ log b is the steady r 2 state value of the efficient interest rate, and et is i.i.d. N ð0, sr Þ. As shown in web Appendix B, inflation dynamics in the model is determined by a modified Phillips curve of the form: ½ðpt ypt1 ÞbEt ðpt þ 1 yEt1 pt Þ ¼ os^ t þ ybEt ½ðpt þ 1 ypt Þbðpt þ 2 ypt þ 1 Þ,

ð15Þ

where o ¼ ðð1aÞð1abÞ=aÞz, and s^ t ¼ ðsC þ sN Þxt is the log deviation of aggregate real marginal cost from its steady state value. Inflation dynamics is more complex with customer markets. The Phillips curve includes various leads and lags of inflation and also the previous period’s expectation of current inflation. The Phillips curve can be written on a more familiar form by solving (15) forward to obtain:

pt ¼ ypt1 þ oEt

1 X

bk

k¼0

k X

yj s^ t þ k þ ybet ,

ð16Þ

j¼0

where et ¼ pt Et1 pt is the inflation forecast error between t  1 and t. This Phillips curve differs from that obtained in the standard New Keynesian model in several respects. First, inflation depends on past inflation—with the coefficient on the backward-looking component of inflation coinciding with the backward-looking component of the firms’ market share equation—and on the inflation forecast error. Second, the slope of the Phillips curve o is lower, as z is a decreasing function of y.8 Third, the rate at which future marginal costs are discounted is affected. To make the policy problem non-trivial, an inefficient time-varying disturbance to marginal cost is introduced in the form of a cost-push shock. This captures shocks unrelated to preferences and technology, e.g., markup shocks or timevarying tax wedges. Written in terms of the output gap, the relation in (15) then reads ½ðpt ypt1 ÞbEt ðpt þ 1 yEt1 pt Þ ¼ oðsC þ sN Þxt þ ybEt ½ðpt þ 1 ypt Þbðpt þ 2 ypt þ 1 Þ þ out , u t,

where the cost-push ut follows the AR(1) process ut ¼ ru ut1 þ e

u t

ð17Þ

2 u Þ.

and e is i.i.d. N ð0, s

2.4. Price setting and inflation dynamics To get some intuition for how customer markets affect inflation dynamics, it is instructive to log-linearize the pricesetting condition in (13), which yields Et

1 X

ðabÞk zit þ kjt ¼ 0,

ð18Þ

k¼0

where zit ¼ log Y~ it log Cit . One can think of zit as a measure of a firm’s incentive to change its price if unconstrained by price rigidities; if zit is positive, the firm desires a higher price, and vice versa. Eq. (18) thus implies that firms set their prices so that the incentive to adjust the price over the expected duration of the price is zero on average. Log-linearization of (11) and (12) yield that the law of motion for zit is given by ( ) 1 X k zit ¼ yzit1 þð1yÞ ðZ1Þð1ybÞEt ðybÞ ½logðPit þ k =Pt þ k Þs^ t þ k  : ð19Þ k¼0

If profit margins are expected to be high, the term inside the curly bracket is negative. Then, customers are more valuable and this gives the firm an incentive to lower its price in order to attract more customers. But (19) indicates that the incentive to 6 7 8

The details of this derivation can be found in web Appendix B. The output gap is defined as the log deviation of the actual level of output from the efficient level of output. The negative relation between y and z has not been proved analytically, but has been confirmed for all numerical values used in this paper.

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change the price also depends on what that incentive was in the previous period. This is without consequence for price setting when prices are flexible, as zit will be equalized to zero at all times, but has important implications when prices are sticky. Consider a firm that experiences a temporary one-time reduction in its marginal cost. It would like to temporarily lower its price in order to expand its customer base. But if it is unable to adjust the price, it will instead, as a second best option, commit to lower prices in the future, which, because of the forward-looking aspect of demand, is already something that attracts customers today. This commitment is encoded in the evolution of zit: when the firm fails to adjust its price, zit becomes negative, which gives the firm an incentive to lower its price in the next period. This implication is in contrast to the standard New Keynesian model, where a firm that fails to adjust its price in response to a temporary reduction in marginal cost will have no incentive to change its price in the future. By averaging (19) across firms, we obtain a measure for the aggregate ‘‘price pressure’’ in the economy, given by zt ¼ yzt1 þð1yÞðZ1Þð1ybÞEt

1 X

ðybÞk s^ t þ k :

ð20Þ

k¼0

If real marginal costs are expected to be high, this means that profit margins are depressed on average and that firms on average desire to raise prices. When constrained by infrequent price adjustment, firms commit to raising their prices in the future instead. As shown in web appendix B, zt is negatively correlated with the quasi-rate of inflation acceleration 1 Et pt þ 1 b pt .9 The dependence of zt1 in (20) implies that if the expected rate of inflation acceleration in the previous 1 period, Et1 pt b pt1 , was negative, as would be the case if inflation was expected to return to steady state, this puts upward pressure on prices in the current period. As seen in (16), this translates into inflation today depending positively on both inflation in the previous period and the inflation forecast error. Intuitively, inflation is persistent because price pressure in the economy accumulates over time and only gradually dissipates, leading firms on average to continue to raise their prices even after the rise in marginal cost has receded. An implication of (19) is that if a firm is allowed to adjust and raises its price, so that zit is lowered, this reduces the firm’s incentive to raise the price in the future because future values of zit will also be lower. The price increase erodes the firm’s customer base, which alleviates the desire to raise the price in coming periods. Because firms set their prices so that the incentive to adjust them over the expected duration of the price is on average zero, this counteracts the firm’s incentive to raise the price following an increase in marginal cost. This explains the lower value of o in the model with customer markets. Because zit depends on a sum of discounted future profit margins, this means that future marginal costs will, compared to the standard New Keynesian model, have a higher weight in the firm’s price-setting decision. This period’s marginal cost only enters in zit, but next period’s marginal cost enters in both zit and abzit þ 1 , and so on. As seen in (16), this affects the discounting of future marginal costs in the Phillips curve. This period’s marginal cost is discounted by 1, next period’s by bð1 þ yÞ, and so on. If y is sufficiently high, future marginal costs may be more important for inflation dynamics than the current marginal cost. 3. Welfare Welfare is evaluated by taking a second-order approximation to the ‘‘representative household’s’’ expected utility.10 Assuming that a subsidy is in place that neutralizes the distortion from monopolistic competition, so that the steady state is efficient, this yields an expression for welfare, expressed as a fraction of steady state consumption, given by 1 1 X bt fðsC þ sN Þx2t þ Z1 var i log Cith g,  E0 2 t¼0

ð21Þ

ignoring terms independent of policy and higher-order terms. This welfare criterion is identical to that obtained in the standard New Keynesian model. Optimality requires both that aggregate output is at its efficient level and that the same amount is consumed of all goods. The welfare criterion can be written in terms of the output gap and inflation: 1 1 X  E0 bt ½lx x2t þ lp p2t , 2 t¼0

ð22Þ

where lx ¼ sC þ sN and lp ¼ a=ðð1aÞð1abÞÞZe. Inflation reduces welfare because it leads to price dispersion, which distorts the allocation of consumption among the different goods in the economy. The parameter e, which takes on a value of one in the standard New Keynesian model, is a decreasing function of y.11 Customer markets reduce the welfare cost of inflation by slowing down the reallocation of consumption when relative prices are dispersed. 9 Price pressure is also negatively correlated with the quasi-rate of inflation acceleration in the standard New Keynesian model, but without customer markets, price pressure in the economy only arises from variations in current period real marginal cost. 10 The details of the derivation can be found in web Appendix C. 11 The same proviso as in footnote 8 applies for the negative relation between y and e. The relation between inflation and the dispersion of consumption across goods is significantly more complicated with customer markets. This is a result of the ‘‘mechanical’’ inertial behavior of market shares, but also of households taking both current and expected future prices into account when allocating consumption among goods.

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One might wonder how preventing households from reoptimizing their allocation of consumption for some goods can increase their welfare. Should not households be able to increase their utility, when relative prices are dispersed, by adjusting their consumption to the changes in relative prices? While this conjecture is correct in itself, observe that households prefer a balanced consumption basket, so the increase in consumption utility can only be achieved if the reallocation of consumption leads to an increase in the household’s total consumption of goods. For a given budget, households buy more of the goods that are cheap and less of those that are expensive, so total consumption increases. But in general equilibrium, this increase in consumption must be accompanied by a corresponding increase in economy-wide labor supply. Because different goods are imperfect substitutes, utility from consumption will increase less than the disutility from increased labor supply in the economy, and this is why there will be a decrease in welfare when consumption becomes more dispersed.

4. Calibration Table 1 summarizes the benchmark calibration, where one period corresponds to one quarter. The value of b implies a steady state annual interest rate of about 4%. Nakamura and Steinsson (2008) report a monthly median frequency of price adjustment for consumer goods—excluding sales and product substitutions—of about 9%.12 Converted to a quarterly frequency, this corresponds to a value of a of 0.75, implying that prices on average remain fixed for four quarters.13 The novelty is the calibration of y. Observing this parameter directly from households’ behavior is hardly a viable option. But it can be inferred from the model’s structural relations: the backward-looking component of the firms’ market share equation is equal to y, and so is the backward-looking component of inflation in (16). Unfortunately, empirical studies that estimate market share equations are scarce. An exception is Gottfries (2002), who uses time-series data to estimate a customer market equation in which a firm’s market share depends on its relative price and its lagged market share. In his full sample estimate, the backward-looking component of the market share equation is found to be 0.92. There is an abundance of empirical literature estimating the backward-looking behavior of inflation. While most studies find that lagged inflation is statistically significant in estimates of hybrid New Keynesian Phillips curves, there is little consensus regarding the importance of backward-looking behavior in explaining inflation dynamics. Galı´ and Gertler (1999) and Galı´ et al. (2001) argue that the purely forward-looking New Keynesian Phillips curve provides a good fit for inflation dynamics in both the US and Europe. In contrast, the estimates in Rudd and Whelan (2005) and Linde (2005) indicate that the backward-looking behavior of inflation is more important than the forward-looking. For instance, the baseline estimates in Rudd and Whelan (2005) of the backward-looking component of inflation are in the range of 0.79–0.91; the lower estimates are obtained when a measure of marginal cost is used instead of the output gap.14 I set y ¼ 0:875 as a benchmark value, indicating a substantial degree of inflation persistence, but I also consider lower values as a sensitivity check. This value implies that a household on average reoptimizes the relative consumption of a good every eight quarters. The elasticity of substitution between goods, Z, is calibrated to imply a 10% steady state markup, consistent with the estimates in Basu and Fernald (1995). The long-run price elasticity is Z; a permanent price change will over time have the same effect on demand as in an economy without customer markets. The short-run price elasticity depends on households’ expectations about future prices. To get a sense of the magnitude of this parameter, I calculate the implied price elasticity for two special cases: when a price change is perceived to be completely transitory and when it is expected to be permanent. In the former case, the short-run price elasticity is Zð1yÞð1ybÞ, and in the latter case it is Zð1yÞ. For the value of y assumed here, these two cases correspond to a short-run price elasticity of 0.18 and 1.38 respectively. As a comparison, Gottfries (2002) and Lundin et al. (2009) estimate a short-run (within-quarter) price elasticity of 0.27 and 0.13, respectively. The elasticity of intertemporal substitution is set following Rotemberg and Woodford (1998). The value of sC may appear to imply an implausibly high intertemporal elasticity of substitution but, as discussed by Woodford (2003), this parameter should not be calibrated, in a model that abstracts from capital, from estimates of consumer expenditure. Instead it should be interpreted as the intertemporal elasticity of overall private spending, including interest rate sensitive investment spending. The elasticity of the real wage with respect to output in the model is sC þ sN . The labor supply elasticity is set to match the estimate in Solon et al. (1994) of a real wage elasticity with respect to output of 0.62. The standard deviations of the shocks are calibrated so that the responses to shocks are of reasonable magnitudes (cf. Smets and Wouters, 2007).

12 The mean frequency is higher, but Nakamura and Steinsson (2010) recommend using the median frequency when calibrating a single-sector model, on grounds that this yields a similar degree of monetary non-neutrality as in a multi-sector model calibrated to account for the heterogeneity in frequency of price change between sectors. 13 The durations reported in Nakamura and Steinsson (2008) are somewhat lower because they calculate durations using the continuous time formula 1=ðlog aÞ, whereas I calculate the duration using the discrete time formula 1=ð1aÞ. 14 It should be noted, however, that these estimates are not directly transferable to the model in this paper because their specification is different from (16).

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Table 1 Calibration. Parameter

Value

Description

b

0.99 0.75 0.875 11 0.46 0.16 0.8 0.8 0.001 0.01

Households’ subjective discount factor Calvo parameter firms Calvo parameter households Elasticity of substitution between goods Inverse of (Frisch) labor supply elasticity Inverse of intertemporal elasticity of substitution Persistence of shock to the efficient interest rate Persistence of cost-push shock Standard deviation of shock to the efficient interest rate Standard deviation of cost-push shock

a y

Z sN sC rr ru sr su

Note: This table shows the benchmark calibration used in the numerical simulations.

Output gap

Annualized pp

1 0.5 0 −0.5

0.4 0.2 0

0.3 0.2 0.1

−0.2 0

6 Quarters Output gap

12

Annualized pp

0 pp

−0.2 −0.4 −0.6

0 0

0.2

6 Quarters Inflation

12

0.8

0.4

0.6

0.3

0.4 0.2

0

6 Quarters

12

0

6 Quarters Nominal rate

12

0

6 Quarters

12

0.2 0.1

0

−0.8

Nominal rate

0.4

Annualized pp

pp

Inflation

0.6

Annualized pp

1.5

0 0

6 Quarters Customer markets

12

Standard

Fig. 1. Impulse responses of the output gap, inflation, and the nominal interest rate under the Taylor rule. The top row shows the responses to a one standard deviation shock to the efficient interest rate. The bottom row shows the responses to a one standard deviation cost-push shock. The response of the output gap is expressed in percentage points (pp); the responses of inflation and the nominal interest rate are expressed in annualized percentage points.

5. Results In this section, aggregate dynamics is first analyzed by assuming an interest rule and then an analysis is made of how optimal monetary policy and welfare are affected by the introduction of customer markets. 5.1. Monetary policy under a Taylor rule Suppose that monetary policy follows a Taylor rule of the form: rt ¼ ð10:8Þðr þ1:5pt þ 0:1xt Þ þ 0:8rt1 ,

ð23Þ

These coefficients are roughly consistent with empirical evidence for the Greenspan era; see, e.g., Taylor (1999) and Clarida et al. (2000). The top row of Fig. 1 shows the effects of a shock to the efficient interest rate. The central bank could in principle completely offset the shock by letting the nominal interest rate perfectly track the rise in the efficient rate. However, the

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k=1

k=2 0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

%

0.4

%

%

Flexible prices 0.4

0

0

0

−0.1

−0.1

−0.1

−0.2

−0.2 0

6 Quarters

12

−0.2 0

12

0

k=8 0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

%

0.4

0

0

0

−0.1

−0.1

−0.1

−0.2

−0.2 0

6 Quarters

12

Price

6 Quarters

12

k=12

0.4

%

%

k=4

6 Quarters

−0.2 0

Market share

6 Quarters

12

Price New Keynesian

0

6 Quarters

12

Price Level

Fig. 2. Price and market share responses in the model economy with customer markets following a one standard deviation shock to the efficient interest rate. A firm reoptimizes its price for the first time k  1 periods after the shock and thereafter every four quarters. All responses are expressed in percentages.

equilibrium interest rate response implied by the Taylor rule is too small to attain this, so the shock will trigger a boom.15 The bottom row of Fig. 1 shows the effects of a cost-push shock. In this case, the central bank is unable to simultaneously stabilize the output gap and inflation. Since the Taylor rule prescribes a heavy weight on inflation stabilization, the central bank drives down the output gap to stabilize inflation, leaning strongly against the wind. As expected, the model economy with customer markets exhibits a greater degree of inflation persistence. After a shock to the efficient interest rate, inflation returns to steady state several quarters later than in the standard model. In a simulated business cycle, driven by both shocks to the efficient interest rate and cost-push shocks, the first-order autocorrelation of inflation increased from 0.61 to 0.78, while remaining virtually unchanged for the output gap and the nominal interest rate.

5.2. Price and market share dynamics To gain further insights into price and inflation dynamics, it is instructive to look at the evolution of prices and market shares at the firm level. Fig. 2 plots the price and market share, following a shock to the efficient interest rate in the model economy with customer markets, for firms with different ex post realizations of the Calvo signal. This includes firms that reoptimize their prices for the first time k 1 periods after the shock and thereafter, corresponding to the average duration of prices, every four quarters. Also plotted is the price that would have been set by a firm that follows the price-setting rule implied by the standard New Keynesian model, i.e., sets its price as a discounted average of expected future marginal costs. Consider first a firm that is allowed to adjust its price in the period that the shock occurs. The New Keynesian firm, anticipating that marginal costs will fall in the future, sets its price lower than the flexible price. Customer markets have no effect on prices when they are flexible, yet they call for a smaller initial price response when prices are sticky. As discussed in Section 2.4, a firm that is about to reoptimize realizes that a price increase will have a negative effect on its market share, which reduces the incentive to raise the price. Perhaps the most salient characteristic of Fig. 2 is the observation that price changes are typically larger in size and that price decreases are more common with customer markets. A firm that is unable to adjust its price for some time will typically see its market share either rise or fall, affecting both the direction and the size of the eventual price adjustment. 15

For all simulations, Dynare software, available at http://www.dynare.org/, was used.

J. S¨ oderberg / Journal of Monetary Economics 58 (2011) 206–219

Output gap

Inflation 0.6

0

0.4

0.4

−0.4 −0.6 −0.8

Annualized pp

0.6 Annualized pp

pp

Nominal rate

0.2

−0.2

0.2 0 −0.2 −0.4

−1 0

6 Quarters

12

0.2 0 −0.2 −0.4

0

Output gap

6 Quarters

12

0

Inflation

0

0.4

0.4

−0.6 −0.8

Annualized pp

0.6

Annualized pp

0.6

−0.4

0.2 0 −0.2 −0.4

−1 0

6 Quarters

12

6 Quarters

12

Nominal rate

0.2

−0.2 pp

215

0.2 0 −0.2 −0.4

0

6 Quarters

Customer markets

12

0

6 Quarters

12

Standard

Fig. 3. Impulses responses of the output gap, inflation, and the nominal interest rate to a one standard deviation cost-push shock under optimal policy. The top row is policy based on the value of l implied by the respective model. The bottom row is policy when the value of l obtained without customer markets is imposed in both models. The response of the output gap is expressed in percentage points (pp); the responses of inflation and the nominal interest rate are expressed in annualized percentage points.

As illustrated in the figure, firms that are late to adjust after the shock have gained market shares and therefore have big incentives to raise their prices. Still, the price level adjusts slowly because market share movements cancel each other out in the aggregate. Arguably, this pattern of price adjustment is more in line with empirical evidence of price-setting behavior that typically finds quite a lot of flexibility in prices at the microeconomic level, even though there is considerable inertia in the aggregate price level (see, e.g., Klenow and Kryvtsov, 2008 and Nakamura and Steinsson, 2008). A detailed analysis of price dynamics is beyond the scope of this paper. I note, however, that in order to match the size of price changes found in the data, one must reasonably add idiosyncratic shocks at the firm level, in order to amplify the size of price changes.16 5.3. Optimal monetary policy It was shown above that for a given monetary policy, the main consequence of customer markets pertains to the dynamic response of inflation. I now turn to the question of how customer markets affect the conduct of optimal monetary policy. The central bank’s problem is to choose a sequence fxt , pt g1 t ¼ 0 to minimize the discounted sum of normalized period loss functions: L ¼ p2t þ lx2t ,

ð24Þ

where l ¼ lx =lp is the relative weight on output gap stabilization, subject to the model’s equilibrium relations given by (14) and (17). The policy problem is solved under the assumption that the central bank is able to make state contingent commitments about future policy actions. The baseline calibration implies that the annualized value of l is 0.948, making output gap stabilization about as important as inflation stabilization.17 The corresponding value without customer markets is 0.0774, giving the central bank a much stronger motive to stabilize inflation. The top row of Fig. 3 shows the response of the model economy to a cost-push shock. The hump-shaped decline in the output gap engineered by the central bank is more gradual with customer markets, resulting in a less intense, but more 16 Klenow and Kryvtsov (2008) and Nakamura and Steinsson (2008) both find that for consumer prices excluding sales, the median absolute size of price changes is about 10%. 17 Since the model is calibrated to a quarterly periodicity, the value of l is annualized by multiplying it by 42.

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5 4.5 4

Variance output gap

3.5 3 2.5 2 1.5 1 0.5 0 0

0.1

0.2

0.3 0.4 Variance inflation

0.5

0.6

0.7

Fig. 4. Trade-off between inflation and output gap stabilization when the business cycle is driven by both shocks to the efficient interest rate and costpush shocks. The solid line shows the efficient policy frontier for the baseline calibration of y ¼ 0:875. The circles show optimal points for different degrees of customer markets. The variance of the output gap is expressed in percentage points; the variance of inflation is expressed in annualized percentage points.

prolonged, downturn. The central bank finds it optimal to accommodate some of the inflationary pressure arising due to the cost-push shock. It is well known from the literature that the central bank, in this class of models, anchors inflation expectations by committing to reverting the price level back to its pre-shock level. This result carries over to the customer market model. After the initial rise in inflation, the central bank therefore keeps inflation negative for some time. Both the degree of accommodation and the subsequent drop in inflation are larger and more persistent with customer markets. We see that the optimal monetary policy response is substantially affected by the introduction of customer markets. A question that follows is how much of the difference in policy is due to the new hybrid Phillips curve, and how much is due to the higher relative weight on output gap stabilization. To separate these two effects, I also consider the experiment of imposing the value of l obtained without customer markets in both models; this comparison is plotted in the bottom row of Fig. 3. The impulse responses are very similar, suggesting that the main difference in policy is due to the higher relative weight on output gap stabilization with customer markets. The central bank’s trade-off between inflation and output gap stabilization can be illustrated by means of Fig. 4, which shows the variance of inflation and the output gap for a simulated business cycle driven by both shocks to the efficient interest rate and cost-push shocks. The solid line is the efficient policy frontier for the baseline calibration; the filled circle indicates the optimal point based on the value of l implied by the welfare criterion. Instead, imposing the value of l obtained without customer markets leads to the point on the efficient frontier indicated by the asterisk. Comparing these two points, it is evident that the higher relative weight on output gap stabilization obtained with customer markets involves an optimal policy with substantially higher volatility of inflation and lower volatility of the output gap. The unfilled circles in Fig. 4 show optimal points for model economies with different degrees of customer markets (with l based on the value implied by the welfare criterion in each model). The values of y correspond to households reoptimizing every quarter and every two, four, and six quarters. In the baseline case, when households reoptimize every eight quarters, the introduction of customer markets leads to a decrease in the volatility of the output gap of almost 40%, but there is a more than sevenfold increase in the volatility of inflation. If, instead, households reoptimize every four or six quarters, the change in policy implied by the introduction of customer markets is smaller compared to the baseline calibration. Yet, it is not trivial; for instance, setting y ¼ 0:75 still leads to a reduction in the volatility of the output gap of 21%, but a more than two and half times increase in the volatility of inflation. Thus, even relatively modest degrees of customer markets imply an optimal policy that, compared to the standard New Keynesian model, involves substantially higher volatility of inflation.

J. S¨ oderberg / Journal of Monetary Economics 58 (2011) 206–219

217

0.12 Optimal policy Baseline rule Alternative rule I Alternative rule II

0.1

Welfare loss

0.08

0.06

0.04

0.02

0 0

0.2

0.4

0.6

0.8

1

θ Fig. 5. Average welfare loss per period, expressed as a percentage of steady state consumption, for different values of y under alternative policies, with the business cycle driven by both shocks to the efficient interest rate and cost-push shocks. The baseline rule corresponds to (23). Alternative rule I sets the reaction coefficient with respect to inflation to 5. Alternative rule II sets the reaction coefficient with respect to the output gap to 0.3.

5.4. Welfare In this section, the implications for the welfare of different policies are evaluated. Welfare is measured by calculating the average welfare loss per period, given by 1 2½lx varðxt Þ þ lp varð t Þ:

p

ð25Þ

Fig. 5 shows the welfare loss for different values of y under optimal policy, the baseline Taylor rule, and two alternative Taylor rules. Keeping the other coefficients at their baseline values, alternative rule I sets the reaction coefficient with respect to inflation to 5, while alternative rule II sets the reaction coefficient with respect to the output gap to 0.3. For all policies, the welfare loss is decreasing in y. When this parameter approaches unity, the welfare cost associated with inflation vanishes, as price dispersion no longer distorts the allocation of consumption among the different goods. In this case, optimal policy involves complete stabilization of the output gap. Of the Taylor rules, alternative rule I, with a strong reaction to inflation, performs the best without customer markets when the gain from inflation stabilization is large. Alternative rule II, which puts a high weight on output gap stabilization, leads to the largest welfare losses in this case. The more balanced baseline Taylor rule lies in-between the two alternative rules. Without customer markets, alternative rule I is relatively close to optimal policy, while the other rules lead to substantially higher welfare losses. For high values of y, when the loss associated with inflation volatility is small and the gain from stabilizing the output gap is large, the situation is reversed. Alternative rule II performs the best, while alternative rule I, as a result of its focus on inflation stabilization, performs the worst. In this case, however, the difference between the different rules are much smaller. 5.5. Commitment versus discretion An important question is how the results would change if firms were unable to commit to future prices. In a strict interpretation, commitment requires firms to write binding contracts specifying prices in all future states of the world. Even if firms are scrupulous about not antagonizing their customers, such contracts are, at least for consumer goods, inconceivable. Unfortunately, the fact that the firm’s market share and its price are state variables in the optimization makes even the Markov perfect equilibrium intractable when firms act with discretion. A full analysis is therefore beyond the scope of this paper. Suppose, however, that one is able to obtain a solution to the Markov perfect equilibrium under discretion. How would the results be likely to change? First, as discussed in Section 2.4, we should consider that the backward-looking nature of inflation is a result of firms’ ability to commit. Under discretion, one would therefore expect the return of a purely

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forward-looking Phillips curve. Markup dynamics would also be different. Under commitment, a firm’s incentive to raise its price when profit margins erode is counteracted by concerns of its customer base in past periods. Under discretion, the firm has no such concerns and will want to raise its price more than is necessary to restore its markup. Hence, optimal markups will plausibly be time-varying, as has traditionally been found in the customer market literature. Moreover, markups will be higher on average when firms cannot commit to low prices in order to improve their market share positions. The characterization of optimal monetary policy will also be different. For one thing, the time-varying markups introduce an inefficient disturbance to the natural level of output that generally makes it impossible for the central bank to simultaneously stabilize the output gap and inflation, even in the absence of exogenously imposed cost-push shocks. Preferences are not affected, so given a subsidy that makes the steady state efficient, the expression in (21) is still the relevant welfare criterion. The relation between inflation and the dispersion of consumption across goods will be different however, as prices are set differently. One may hypothesize that, when unconstrained by past commitments, firms will react more strongly to market share movements. This would likely increase the dispersion of prices in the economy and counteract the reduction in the welfare cost of inflation. Reasonably, customer markets reduce the welfare cost of inflation, and lead to an optimal monetary policy that assigns a higher relative weight to output gap stabilization, also under discretion, but the quantitative effects may be smaller. 6. Conclusion Two salient characteristics of the canonical New Keynesian model are the purely forward-looking nature of inflation and the utility-based welfare criterion’s emphasis on inflation stabilization over output gap stabilization. In this paper, a customer market model in which firms and customers form long-term relations is developed and integrated in the standard New Keynesian model. This leads to a Phillips curve of the hybrid variant, where current inflation depends on past inflation. In the literature, inflation persistence is usually generated by imposing generalized price indexation schemes. Besides the ad hoc nature of such schemes, they also have the counterfactual implication that all prices are adjusted at all times, albeit not optimally. The customer market model developed in this paper, in contrast, displays endogenous inflation persistence, arising as a consequence of the forward-looking nature of demand and firms’ ability to commit to future prices. The model has important implications for monetary policy. When customers respond sluggishly to relative price changes, price dispersion is a significantly less distortionary phenomena, leading to lower welfare costs of inflation and an optimal monetary policy that assigns relatively more weight to output gap stabilization. There are also implications of the model that merit further investigation. It is important to determine the exact role of firms’ ability to credibly commit to price plans, and how the result would change under discretion. Furthermore, it would be interesting to investigate how price adjustment at the microeconomic level, reasonably combined with some kind of firm-specific shocks, fits the empirical facts.

Acknowledgments ¨ ¨ I would like to thank Nils Gottfries, Mikael Carlsson, Ulf Soderstr om, Morten Ravn, and participants at various seminars for helpful comments and suggestions. Part of the research was conducted while visiting Sveriges Riksbank. Financial support from Handelsbanken forskningsstiftelser is gratefully acknowledged. Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jmoneco.2011.05.012.

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