Inflation and relative price variability

Economics Letters 76 (2002) 27–33 www.elsevier.com / locate / econbase

Inflation and relative price variability Hasan Bakhshi* Bank of England, Threadneedle Street, London EC2 R 8 AH, UK Received 15 October 2001; accepted 27 November 2001

Abstract Theoretical explanations of the inflation-relative price variability relationship are often observationally equivalent. This paper compares two theoretical models: menu costs and imperfect information. The microfoundations of the models are used to generate new testable differences between the theories.  2002 Elsevier Science B.V. All rights reserved. Keywords: Inflation; Information; Menu; Costs; Competition JEL classification: E31

1. Introduction There is a vast empirical literature analysing the relationship between inflation and relative price variability (RPV). Marquez and Vining (1984) and Danziger (1987) are classic references. One reading is that there is no robust relationship: a positive one is uncovered in some countries over certain periods, while in other cases no relationship is found. Some studies even document a negative relationship (Fielding and Mizen, 2000; Silver and Ioannidis, 2001). Some stress a distinction between anticipated and unanticipated changes in inflation, though again no robust empirical relationships are uncovered. Caucutt et al. (Caucutt et al., 1994, 1998) look at industry-level pricing data and conclude that the differences in the relationship between RPV and inflation across industries is ‘tremendous’. There are fewer well-specified theoretical explanations for a relationship between inflation and RPV. Exceptions are the menu cost models of Sheshinski and Weiss (1977), Rotemberg (Rotemberg, 1982, 1983) and Caplin and Spulber (1987); the search models of Benabou (Benabou, 1988, 1992), Benabou and Gertner (1993) and Diamond (1993) that combine (S,s) models with the assumption that consumers engage in costly search, and Hercowitz (1981) and Cukierman’s (Cukierman, 1984) imperfect information model with heterogenous supply elasticities. * Tel.: 144-207-601-5996; fax: 144-207-601-5018. E-mail address: [email protected] (H. Bakhshi). 0165-1765 / 02 / $ – see front matter PII: S0165-1765( 02 )00031-9

 2002 Elsevier Science B.V. All rights reserved.

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Menu cost models predict a positive relationship between both anticipated and unanticipated inflation and RPV. In dynamic versions, firms use (S,s) pricing rules: they adjust nominal prices whenever the real price of their goods fall to a lower bound, s, at which time they should raise nominal prices so that real prices equal the upper bound, S. Provided that firms do not adjust prices at the same time, the models predict that as inflation increases, the difference between the optimal s and S increases, and, other things being equal, relative price dispersion is greater. Danziger (1987) uses Rotemberg’s (1982) version of the menu cost model to show that the standard measure of RPV—the variance of individual inflation rates—can rise or fall with inflation depending on the sampling frequency of the price data relative to the frequency with which firms adjust. In the imperfect information models of Hercowitz (1981) and Cukierman (1984), firms with higher supply elasticity adjust their prices less in response to an unexpected demand shock than do firms with lower supply elasticity: there is a positive relationship between unanticipated inflation and RPV. The empirical literature has tried to discriminate between menu cost and imperfect information models by testing whether it is only unanticipated inflation that is related to RPV. But it is notoriously difficult to decompose inflation into that part truly anticipated by agents and that which is not. So empirical researchers have often complained that the theories are otherwise observationally equivalent (Hartman, 1991). I compare a micro-founded version of the imperfect information model and a menu cost model to demonstrate that the imperfect information model predicts the relationship between inflation and RPV is stronger in industries with greater competition, whereas the menu cost model predicts the opposite. This suggests using cross-industry or -country data to discriminate between theories.

2. A model of relative price variability and unanticipated inflation Firms observe the overall current level of demand for their goods but cannot distinguish between aggregate shocks M and idiosyncratic shocks e fi (they do not observe the aggregate price level, P). Firms are assumed to employ only their own labour and this is supplied competitively (they are ‘yeoman farmers’). There are no state variables in this model, so the firms’ optimisation problems collapse down into a series of static problems. Firm i sets prices to maximise expected utility subject to demand and production technology:

HS D

¯ P M d ]i Yi 2 ] L iß 1 ]i P ß P

Max EUi 5 E Pi

J

(1)

subject to e M D, S D S]] P

Pi Yi 5 ] P

2u

fi

S O D m

1 Yi 5 L ai i , P ; ] P 12u m i 51 i

1 ] 12u

and fi | Ns0,s 2fd.

Pi is firm i’s price; P is the Dixit and Stiglitz (1977) aggregate price index; Li is firm i’s labour input; ß measures the marginal disutility of working; ai , 1 denotes the returns to labour in firm i’s ¯ i is firm i’s desired money holdings; u is the constant elasticity of production technology; M

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substitution between goods (assumed.1 to ensure existence of equilibrium); and m is the number of goods.1 The first-order condition from Eq. (1), assuming that the aggregate price level is log-normally distributed, ln P 5 p | NsEp,s 2pd, can be written as: pi 5 cism 1 fid 1s1 2 cidEp 1 c i

(2)

where

sß 2 aids1 2 xd ci 5 ]]]]], ß 2 ai x u 21 0 , xi 5 ]] , 1 2 , and c i is a firm-specific constant 3 (dropped from this point on). Lower-case u lettering denotes logs. Assume further that

sß 2 aids1 2 xd ci 5 ]]]]] | Nsc,s c2 d ß 2 ai x 4

and that this distribution is known by firms. And the money supply follows a stochastic money growth rule with expected growth gt : 2

˜ t , where m ˜ t | Ns0,s m˜ d mgt 5 m t 2 m t 21¯ 5 gt 1 m

(3)

Following Hercowitz (1981) I solve for the aggregate price level by the method of undetermined coefficients and then can compute an expression for the variance of the rates of change in individual prices at time t, g 2t the usual measure of RPV:

O

1 m g ; ] fs p it 2 p it 21d 2s pt 2 pt 21dg 2 ;m large m i 51 2 t

2 t

2 c

2 f

g 5 2ss1 2 jd s 1s j 1 cs1 2 jdd ds 1s1 2 jd s 2

2

2

2 c

s 2m˜ 1ss 2f /cd 2 ]]]] spt 2 Eptd 2 s 2m˜ 1 s 2f

1

2

(4)

where 5

1

It is straightforward to introduce a total factor productivity shock into this model, but the results in this paper are unaffected. 2 x is an indicator of intensity of competition in the goods market (higher x means more competition). 3 The constant contains a term in the variance of aggregate prices, s 2p . 4 Eq. (2) is a micro-founded version of Hercowitz’s (Hercowitz, 1981) Eq. (3) on page 332. 5 The algebraic derivation, tedious but straightforward, makes use of the common first-order approximation that the log aggregate price level equals the arithmetic mean of log individual goods prices.

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s 2m˜ j 5 ]]]]]. 1 s 2m˜ 1 ] s 2f c

SD

Differentiation gives Hercowitz’s results that RPV is increasing in the unanticipated rate of inflation (result 1), and the rate at which it does so is increasing in the cross-sectional variance of technologies across the economy, s 2c (result 2). But the micro-foundations in this analysis generate novel results. First, the rate at which RPV increases with unanticipated inflation is decreasing in the marginal disutility of labour (result 3): ≠ 2g 2t ]]]]] , 0. ≠spt 2 Eptd≠ß So a more inelastic labour supply means a weaker relationship between unanticipated inflation and RPV. This is intuitive: the elasticity of labour supply is common across production units, so it is a common factor in the marginal cost of production across different firms. If the elasticity of labour supply is low, there is a common factor raising the marginal costs of production for all firms. This ‘drowns’ out the effects of unanticipated money shocks on relative prices arising from heterogeneous ai s. The low labour supply elasticity for individuals estimated in the labour economics literature points to models of imperfectly competitive labour markets if imperfect information is to be a significant explanation for the inflation–RPV relationship. 6 Second, greater competition implies a stronger relationship between unanticipated inflation and RPV (result 4): ≠ 2g 2t ]]]]] . 0. ≠spt 2 Eptd≠x Cooper and John (1988) define strategic complementarity as existing when an increase in the action of all agents except agent i increases the marginal return to agent i’s action. Eq. (2) shows that firm i’s profit-maximising price is increasing in the prices that other firms are perceived to be setting. So prices are strategic complements. The degree of strategic complementarity is increasing in x. So the more intense is competition, the more likely are firms’ pricing decisions to respond to perceived changes in the aggregate price level. And given that the inflation–RPV relationship arises from firms’ misperception of the aggregate price level, there is a stronger relationship between unanticipated inflation and RPV.

3. Menu cost models Models with perfect information but staggered price setting also predict a positive relationship between inflation and RPV. But the micro-foundations of these models have not previously been exploited to relate this to structural parameters. Here I simply take a first step in that direction. 6

That models with a competitive labour market require elastic labour supply to match the data is a familiar conclusion in the macro literature (see e.g. Ball and Romer, 1990; Chari et al., 2000; Kiley, 1997).

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In Rotemberg (1983), monopolistically competitive firms face the familiar demand curve in Eq. (1). There is no heterogeneity in production functions. Firms incur a common fixed cost when adjusting prices. The money supply grows at a deterministic rate, g and there is no uncertainty. Rotemberg further assumes that production costs are quadratic: 1 ] 2

UPt Y i2 ,

(5)

where U is a positive constant. In the absence of menu costs, firms would set prices, P *i to maximise expected profits. But in the face of fixed costs of adjusting prices, actual prices deviate from the instantaneous profit-maximising price. Rotemberg (Rotemberg, 1982, 1983) shows that the instantaneous loss of profit from charging pi instead of p *i is approximately Bs pi 2 p i*d 2

(6)

where

H

S

u B 5 u log ]] u 21

M 2u U DJH]] J S] u 21 PD 2u 2] 11u

t

2 ] 11u

t

When adjusting prices, agents choose both their new price and the period over which they keep prices fixed so as to maximise profits net of costs of price adjustment. Assuming that agents are uniformly distributed over the time of their last price adjustment, Rotemberg proves that g, the rate of money growth, is also the rate of inflation. In which case real money balances are constant. The symmetry of the model means that all agents adopt the same price adjustment policy. In particular, assuming that A.0 denotes the fixed cost of adjusting prices, Mussa (1981) shows in this model that agents keep their prices constant in periods of length:

S Dg

6A T5 ] B

1 ] 3

2 2] 3

(7)

Denoting the length of time between the researchers’ observations of prices as h, in the case where each agent is assumed to increase his price at most once between successive observations 7 , Danziger (1987) shows that the variance of rates of price change between successive observations is given by:

S D hg 2 g h

6A V5 ] B

1 ] 3

4 ] 3

2

2

(8)

This relationship is non-linear and depends on the value of h. In fact Danziger shows that the relationship is negative for some values of inflation. It is tedious but nonetheless straightforward to show numerically that: ≠ 2V ]] , 0 ≠g ≠u 7

(9)

This is a plausible assumption given firms are typically assumed to adjust their prices at a lower frequency than the data used in empirical studies (which is typically quarterly or even monthly).

32

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The relationship between RPV and inflation in the Rotemberg (1982) menu cost model is decreasing in the degree of goods market competition (result 5). Intuitively, greater competition means that firms opt for a shorter interval during which they keep their prices fixed, T. This quicker adjustment means there is a weaker relationship between inflation and RPV. Rotemberg (1982) also sets out a version of the menu cost model with competitive labour supply. He demonstrates that the simple relation in Eq. (6) is robust. It can be shown that the inflation-relative price variability relationship is decreasing in the elasticity of labour supply, as in the imperfect information model (result 6).

4. Empirical evidence There are surprisingly few studies of how the inflation–RPV relationship differs across industries. Exceptions have focused on how the relationship varies with industrial concentration. Again there is no robust finding. Domberger (1987), Slade (1991) and more recently Beaulieu and Mattey (1999) find evidence that the strength of the inflation–RPV relationship is negatively related to industry concentration. Caucutt et al. (1994) find the opposite. But it is difficult to interpret industrial concentration as a measure of competition. Both Caucutt et al. (1994) and Beaulieu and Mattey (1999) go further in testing how the inflation–RPV relationship varies with the industry advertising to sales ratio, a proxy for the degree of product differentiation. Product differentiation is the appropriate measure of competition in our context. Caucutt et al. (1994) find a large and significant negative effect on the inflation–RPV relationship, consistent with the imperfect information model, whereas Beaulieu and Mattey (1999) find the opposite, consistent with menu costs. These papers do not offer a theoretical framework for interpreting these particular results. Testing further the predictions of these models using industry-level data is an important area for research.

Acknowledgements I am grateful to Larry Ball, Shamik Dhar, Jens Larsen, Lavan Mahadeva, Steve Millard, Paul Mizen, Kalin Nikolov, Simon Price, Tony Yates and in particular Barbara Rudolf-Leuscher and Richard Harrison for their comments. The views expressed in this paper are those of the author and not necessarily those of the Bank of England.

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