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Output effects of inflation with fixed price- and quantity-adjustment costs.

I. INTRODUCTION

There is convincing evidence that fixed costs of price adjustment may be large. Levy et al. (1997) found such costs to be 0.7% of a firm's revenue, and Zbaracki et al. (2004) found them to be 1.22%. In the presence of fixed costs of price adjustment, a monopolistic firm does not adjust its nominal price continuously, with the result that the real price and output generally deviate from their static monopoly level. At low inflation rates, the average output is higher than the static monopoly output if there is positive discounting (Danziger 1988) and depends on higher order derivatives of the profit and demand functions if there is no discounting (Benabou and Konieczny 1994). (1)

Most of the literature assumes that only price adjustments are costly, while output can be continuously adjusted without cost. However, Bresnahan and Ramey (1994) document that there may be large fixed costs of quantity adjustments. Relatedly, many publications show that adjusting labor and capital inputs involves significant fixed costs. (2) Such costs may derive from loss of organizational capital (Baily et al. 2001; Jovanovic and Rousseau 2001), as well as from job protection rules, severance pay, and legal and administrative complications.

In a framework with both price--and quantity-adjustment costs, Andersen (1995) and Andersen and Toulemonde (2004) demonstrate that only intermediate-size shock--but not large or small shocks--may affect output. For a constant inflation rate, Danziger (2001) shows that a firm's permanent production decreases with inflation at low inflation rates if discounting is positive, and Danziger and Kreiner (2002) that output capacity decreases with inflation if the elasticity of demand is constant and there is no discounting (see also Danziger 2003).

The purpose of this article is to provide a simple general characterization of the output effects of a constant inflation rate if both price and quantity adjustments involve fixed costs. It is assumed that quantity adjustments are at least as costly as price adjustments, which implies that a firm's production is held constant at a permanent level. (3) Each firm then keeps its nominal price unchanged in periods of equal length, and the nominal price is adjusted so that the initial real price is the same in each period. Thus, a firm's optimal strategy consists of the initial real price, the duration of the periods with unchanged nominal price, and the permanent level of production.

The article shows that in the absence of discounting, the effect of inflation on the aggregate production depends on how fast firms' marginal real revenue decreases with demand and that this effect is fully determined by the elasticity of the marginal real revenue with respect to demand. Thus, aggregate production decreases with inflation if the elasticity of the marginal real revenue always exceeds minus unity, increases with inflation if the elasticity of the marginal real revenue is always less than minus unity, and is invariant to inflation if the elasticity of the marginal real revenue always equals minus unity.

Several studies have found that the comovement between output and prices is typically negative in the long run (Cooley and Ohanian 1991; Den Haan 2000; Den Haan and Sumner 2004; Fiorito and Kollintzas 1994). Within the framework of the present model, the negative comovement indicates that the empirically relevant situation is when the elasticity of the marginal real revenue always exceeds minus unity. This will be the case, among others, if demand functions exhibit a constant price elasticity less than minus unity (as assumed in Danziger and Kreiner 2002).

II. THE MODEL

Consider an economy with a unit continuum of identical consumers and a unit continuum of monopolistic firms producing differentiated perishable products. Each firm and its product are indexed by i, which is uniformly distributed on (0, 1]. Labor is used as the numeraire good. The real wage is therefore always unity, and the real prices and the real profits are the nominal prices and the nominal profits divided by the competitively determined nominal wage rate. The rate of inflation, which is given by the rate of change in the nominal wage rate, is a constant [mu] > 0.

Consumers

A consumer's instantaneous utility at time t is

[[integral].sup.1.sub.0] u([x.sub.it])di + [l.sub.t],

where [x.sub.it] [greater than or equal to] 0 is the instantaneous consumption of the ith product and [l.sub.t] [greater than or equal to] 0 is the instantaneous leisure. It is assumed that u'([x.sub.it]) > 0, and u"([x.sub.it]) < 0, and for demand to be elastic, that u'([x.sub.it])/[[x.sub.it]u" ([x.sub.it])] < -1.

A consumer's income stems from his work and the distribution of the firms' profits. His instantaneous budget constraint at time t is

I + [A.sub.t] = [[integral].sup.1.sub.0][z.sub.it][x.sub.it]di + [l.sub.t],

where I > 0 is the consumer's endowment of time, [A.sub.t] is his share of the distributed instantaneous real profits, and [z.sub.it] is the real price of the ith product.

Each consumer chooses consumption of the different products and of leisure to maximize utility, given the budget and rationing constraints. The latter constraints express that if the aggregate demand for a product exceeds the available supply, then consumers are rationed with each being allocated the same quantity of the product. Assuming that the firms' real prices are not so high that there is no demand for their products and that the consumption of leisure is positive, a consumer's purchase of good i at time t is determined by u'([x.sub.it]) = [z.sub.it] if there is no rationing of that good and by the rationing constraint otherwise. A consumer's (unrationed) instantaneous demand for good i therefore depends only on the real price of that good and is given by D([z.sub.it]) [equivalent to] [(u').sup.-1]([z.sub.it]). Because the set of consumers has a unit measure, D([z.sub.it]) is also the aggregate demand for good i.

Firms

Each firm sets its output and its nominal price, and adjusting either involves a fixed cost. Quantity adjustments are at least as expensive as price adjustments, so a firm keeps its output constant at a permanent level, while adjusting the nominal price at equally spaced intervals. The initial real price is the same in all periods with unchanged nominal prices.

Let Y denote a firm's permanent production, and C(Y), C'(Y) > 0, denote the real cost of production. The real price at which demand equals production is defined by [z.sub.Y] [equivalent to] [D.sup.-1](Y). At higher real prices, a firm sells less than it produces, while at lower real prices, the firm could sell more than it produces. Hence, the ith firm's instantaneous real profit from production at time t is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If S denotes the initial real price in a period with a constant nominal price, then [z.sub.i[tau]] = [Se.sup.-[mu][tau]] is the real price after [tau] of a period has elapsed, and [T.sub.Y] [equivalent to] (1/[mu]) ln(S/[z.sub.Y]) is the time it takes for the real price to reach [z.sub.Y]. It is clear that at the optimum 0 < [T.sub.Y] < T, where T is the duration of the period. If c is the fixed real cost of a price adjustment incurred at the beginning of the period, a firm's average real profit in a period is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first integral is the total real revenue when the firm sells less than it produces, and the second integral is the total real revenue when the firm sells everything it produces.

There is no discounting, and a firm chooses S, T, and Y to maximize the average real profit. The first-order conditions for a maximum are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where s [equivalent to] [Se.sup.-[mu]T] is the terminal real price. These can be simplified to (4)

(1) SD(S) - sY = 0,

(2) sY - C(Y) - V = 0,

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Condition (1) shows that the optimal initial real price is determined so that the initial and terminal real revenues are equal, or equivalently, that the initial and terminal real profits from production are equal. Condition (2) expresses that the optimal length of a period with a constant nominal price is set to make the terminal real profit from production equal to the average real profit (which takes the fixed cost of price adjustment into account). Condition (3) indicates that the optimal production makes the average marginal real revenue equal to the marginal real cost. It is assumed that conditions (1)-(3) yield a unique maximum and that the average real profit is positive.

III. INFLATION AND AGGREGATE PRODUCTION

If there were no inflation or no fixed cost of price adjustment, each firm would always charge the static monopoly real price and produce the static monopoly output determined by the firm's marginal real revenue being equal to its marginal real cost. In the presence of inflation and a fixed cost of price adjustment, however, a firm keeps its nominal price unchanged in periods during which the real price decreases and the demand for the firm's product increases. If the firm would produce the static monopoly output, its average marginal real revenue might therefore differ from the static monopoly marginal real revenue, with the consequence that the firm may choose a production level that deviates from that of the static monopoly.

Because there is a unit measure of firms, the aggregate production in the economy is also given by Y. It follows that inflation may cause the aggregate production to be different from that of a frictionless economy. To determine how the aggregate production varies with the inflation rate, differentiate conditions (1)-(3) totally with respect to la, which yields

dY/d[mu] = [[phi]D(S) - C'(Y)Y]B,

where

[phi] [equivalent to] D(S)/D' (S) + S

is the marginal real revenue at the initial real price, and

B [equivalent to] cS/(Y[mu]T{[phi]s[Y/D' ([z.sub.Y]) + s - 2C'(Y) + YC"(Y)[mu]T] + SC'[(Y).sup.2])})

is positive from the second-order condition for a maximum. Hence, dY/d[mu] has the same sign as

(4) [phi]D(S) - C'(Y) Y.

To understand this result, note that as the inflation rate increases, the initial real price increases and the terminal real price decreases. A firm sells less than it produces when real prices are high, but all it produces when real prices are low, so a higher S tends to reduce its output, and a lower s tends to increase its output. Hence, the effect of inflation on the aggregate production depends on whether firms' output response of the higher S or the lower s dominates.

The smaller [phi] is, the less the real revenue at the beginning of a period with a constant nominal price increases in D(S), and the lower is the loss of initial real revenue from an increase in S relative to the loss of terminal real revenue from a decrease in s. Accordingly, the smaller [phi] is, the less s decreases for a given increase in S. Because the average marginal real revenue decreases when S increases, and increases when s decreases, (5) a smaller [phi] is associated with a smaller average marginal real revenue for a given output. It follows that to make the average marginal real revenue equal to the marginal real cost, as required by condition (3), a smaller [phi] tends to make Y an decreasing function of inflation.

Specifically, expression (4) shows that Y decreases with inflation if the marginal real revenue decreases sufficiently slowly with demand that [phi]D(S) is an increasing function of D(S) (and hence a decreasing function of S). Put differently, Y decreases with inflation if the marginal real revenue always changes less than inversely proportional with demand, or equivalently, if the elasticity of the marginal real revenue with respect to demand always exceeds minus unity. The aggregate production is therefore always below what it would be if there were no inflation. Conversely, Y increases with inflation if the marginal real revenue decreases sufficiently fast with demand that [phi]D(S) is a decreasing function of D(S) (and hence an increasing function of S); that is, if the marginal real revenue always changes more than inversely proportional with demand, or equivalently, if the elasticity of the real revenue with respect to demand is always less than minus unity. The aggregate production is then always above what it would be if there were no inflation. Finally, Y is independent of inflation if the marginal real revenue is inversely proportional to demand, making [phi]D(S) constant. This occurs if the elasticity of the real revenue with respect to demand always equals minus unity.

IV. AN EXAMPLE

In this example, depending on the value of a parameter, the aggregate production either decreases with inflation, increases with inflation, or is independent of inflation.

Let a consumer's instantaneous utility at time t be given by

[[integral].sup.1.sub.0] [([alpha]/2)[ln.sup.2][x.sub.it][dx.sub.it] + [beta]ln[x.sub.it] + [gamma][x.sub.it]]di + [l.sub.t],

where [alpha] > 0, ([alpha]ln[x.sub.it] + [beta])/[x.sub.it] + [gamma] > 0, [x.sub.it] [greater than or equal to] [e.sup.1 - [beta]/[alpha]], and 0 < 1 + ([gamma]/[alpha])[x.sub.it]. Because [x.sub.it] = D([z.sub.it]), the inverse demand for [x.sub.it], is

[z.sub.it] = [[alpha]lnD([z.sub.it]) + [beta]]/D([z.sub.it]) + [gamma].

It follows that the marginal real revenue is [alpha]/D([z.sub.it]) + [gamma], and its elasticity with respect to demand is -1/[1 + [gamma]D([z.sub.it])/[alpha]]. Thus, the marginal real revenue is inversely proportional to demand if [gamma] = 0. Furthermore, because the demand increases with [gamma] without the demand curve changing its slope, the marginal real revenue and hence also the elasticity of the marginal real revenue increase with [gamma] for a given real price.

If [gamma] > 0, the elasticity of the marginal real revenue always exceeds minus unity. Hence each firm's production decreases with inflation and is always less than the static monopoly output. (6) The aggregate production therefore also decreases with inflation and is less than what it would be in an inflationless environment. If [gamma] < 0, the opposite is the case. If [gamma] = 0, in which case consumers' utility functions are log-quadratic in the different products, the elasticity of the marginal real revenue is always minus unity. Accordingly, each firm's production is independent of inflation and always equals the static monopoly output. The aggregate production is then always the same as in an inflationless environment.

V. CONCLUSION

This article has examined how the aggregate production varies with inflation when there are fixed price- and quantity-adjustment costs. It has been shown that such variation is determined by the elasticity of the firms' marginal real revenue with respect to demand. The aggregate production decreases with inflation if this elasticity always exceeds minus unity, whereas the aggregate production increases with inflation if the elasticity is always less than minus unity. The aggregate production is independent of inflation in the special case that the elasticity always equals minus unity. The latter occurs if demand is derived from a log-quadratic utility function.

doi: 10.1093/ei/cb1006

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(1.) See also Rotemberg (1983), Kuran (1986), Naish (1986), Benabou (1988), and Konieczny (1990).

(2.) For the cost of adjusting labor, see Davis and Haltiwanger (1992), Hamermesh (1992), Caballero et al. (1997), and Abowd and Kramarz (2003) . For the cost of adjusting capital, see Doms and Dunne (1998), Cooper et al. (1999). and Nilsen and Schiantarelli (2003).

(3.) See Danziger (2001) for a proof. Production clearly reacts to shocks and may even vary more than prices over the business cycle. However, the present model does not include shocks, the focus being on a fully anticipated, constant rate of inflation. See Danziger (1999). Chari el al. (2000), Golosov and Lucas (2004), and Gertler and Leahy (2006) for studies of the output consequences of monetary shocks in models with staggered price settings.

(4.) To simplify the expression for [partial derivative]V/[partial derivative]S, partially integrate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(5.) The average marginal real revenue can be written as ([z.sub.Y] - s)/ln(S/s).

(6.) As mentioned in the introduction, the condition that the elasticity of the marginal real revenue always exceeds minus unity is also satisfied by demand functions that have a constant price elasticity less than minus unity.

LEIF DANZIGER *

* I thank the anonymous referees for helpful comments.

Danziger: Professor, Department of Economics, Ben-Gurion University, Beer-Sheva 84105, Israel, and Central European University, 1051 Budapest, Hungary. Phone 972-8-647-2295, Fax 972-8-647-2941, E-mail danziger@bgu.ac.il
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Date:Jan 1, 2007
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