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Variable price adjustment costs.

I. INTRODUCTION

Costly price adjustment models (often called menu cost models--see Mankiw |1985~), are based on the assumption that nominal price changes in product markets are costly. With aggregate inflation the optimal pricing policies are state-contingent: the firm lets its real price drift until it reaches a threshold; the nominal price is then changed so as to make the new real price equal to a target value. The simplest case is the one-sided (s, S) policy studied by Sheshinski and Weiss |1977~: the real price is increased to S after it has been eroded by inflation to s.(1) Nominal prices are held constant for extended periods of time, as is "...apparent to anyone with eyesight" (Rotemberg |1987~).

As the timing of price changes is chosen optimally, the optimal policies dominate time-contingent policies.(2) Not surprisingly the models have been adopted in the New-Keynesian literature as a microfoundation for nominal rigidities. Along with monopolistic competition and coordination failures they have entered the mainstream of that literature.(3)

Given their importance for the New-Keynesian approach, costly price adjustment models require closer scrutiny. There are two basic problems: first, it is not clear what the costs of price adjustment are; second, there is empirical evidence inconsistent with the standard, fixed cost model.(4)

1. More complex policies are studied by Barro |1972~, Tsiddon |1988~, Caplin and Leahy |1991~ and Dixit |1991~.

2. See discussions in Blanchard and Fischer |1990~ and Caplin and Leahy |1991~. A notable exception are time-inconsistent problems which may arise in long-term contracts.

3. Rotemberg |1987~, Ball, Mankiw and Romer |1988~, Blanchard |1990~ and Gordon |1990~ review macroeconomic applications. Textbook expositions are in Blanchard and Fischer |1990~, McCafferty |1990~, Abel and Bernanke |1992~ and Mankiw |1992~.

4. Another issue is how a smooth aggregate price level results from lumpy individual adjustments. On this see Caplin and Spulber |1987~ and Caballero and Engel |1991~.

The costs of price changes are of two basic types. First are the costs of market response: customers switching, competitors pricing more aggressively, etc. Second are the costs of changing the menu: deciding on the new price, changing labels, informing salesmen and customers, etc. In existing literature the costs are assumed to be fixed, which suggests that they are of the latter type.(5) The main criticism of this approach is that, as menu costs are small, the results are unlikely to be empirically relevant or economically significant. McCallum |1989~ summarizes this argument. In response Akerlof and Yellen |1985~ and Mankiw |1985~ argued that, as the profit function is flat around its maximum, profits are not responsive to changes in the real price. Thus even small costs may result in infrequent price changes.

Empirical evidence supports neither the criticism nor the response: the observed frequency of price changes is often too low, and real price variations are too large, to be generated by a small menu cost. For example Cecchetti |1986~ finds that a typical U.S. weekly magazine would change its nominal price once every four to six years, allowing the real price to deteriorate by about 25 percent. Results for monopolistic newspapers in Canada, studied by Fisher and Konieczny |1992~, are similar. The average price increase for the twenty-six products in Israel studied by Lach and Tsiddon |1992~ was 12 percent when the monthly inflation rate was 7.3 percent.

Also, in Kashyap |1990~, price changes vary greatly in size across products sold in the same catalog even though, presumably, the menu cost is similar.

With respect to the second problem, fixed cost models imply that the size of adjustment rises with inflation (the real price is allowed to be eroded more and is reset to a higher value). Yet in Lach and Tsiddon |1992~ this holds for only sixteen out of twenty-six products; adjustments become smaller for eight and there is no effect in two cases. Cecchetti |1986~ finds no effect of inflation on adjustment size.

This discussion suggests that it is the first type of costs, related to market response, which is responsible for infrequent individual price adjustment. The ideal approach would be to develop explicit models of market reaction. However, the problem of an oligopoly with costly price adjustment is very difficult and remains to be solved.(6)

Rather than modelling the second type of costs formally, therefore, I take a simpler approach and proxy market reaction with costs which depend on the size or on the frequency of nominal price changes. Several environments in which such costs may arise are described below. Also, for simplicity, I concentrate on the case of constant inflation.

This approach, suggested by Cecchetti |1986~, has powerful implications. First, the optimal policy is similar to that in fixed cost models. Thus costs of adjustment can be interpreted as consisting not only of (small) menu costs but also of costs of market response, which may be large. Also, the fixed cost specification can be usefully thought of as a simple, benchmark case. Second, as shown both theoretically and with simulations, empirical results are easily replicated with simple cost specifications. When costs vary with adjustment frequency, price changes may become smaller as inflation rises, as evidenced by Lach and Tsiddon |1992~. A simple quadratic specification replicates Cecchetti's |1986~ data. If costs depend on market response, the size of price changes is likely to be product, rather than seller, specific, as in Kashyap |1990~. Finally, while the frequency of price changes may fall as inflation rises, a restriction on the cost function is sufficient for a positive correlation which is found, without exception, in empirical studies.

The plan of the paper is as follows. In the next two sections I study the optimal pricing policy when costs vary with adjustment size (section II) and frequency (section III). The analysis of the relationship between inflation and adjustment frequency is in section IV. Numerical examples in section V show how the variable cost approach can account for empirical findings which imply rejection of fixed cost models. Section VI considers a tradeoff between adjustment costs and costs of operating price adjustment technology. Conclusions are in the last section.

II. THE OPTIMAL PRICING POLICY WITH COSTS VARYING WITH THE SIZE OF ADJUSTMENT

In this section I consider costs of changing prices which depend on adjustment size. This assumption can be motivated in two ways. First, consider an oligopolistic market in which each consumer purchases repeatedly from one seller and, as long as the nominal price remains constant, does not monitor competitors' prices. An increase in the nominal price signals a possibility that the relative price has changed (if other firms do not follow) and induces customers to check out the competition. A nominal price increase may result in a loss of customers; the larger is the price change, the more likely are they to switch. This leads firms to play a supergame of chicken: each attempts to maintain a constant nominal price for a longer period than opponents; hence a low frequency of price changes.(7)

Second, consider a monopolistic firm threatened by entry. Total profits in the market depend on a random state of the world (for example technology, demand or costs). Entry is profitable only when the state is good enough. By virtue of being in the market the incumbent has better information about the current state. In the presence of aggregate/relative misperceptions of the Lucas |1973~ type the entrant may misinterpret a price increase as signalling a good state. Since, unlike price, output is private information, it may be optimal for the incumbent to engage in limit pricing: he would react to good states by changing output and keeping price fixed. This generates a real, rather than nominal, rigidity. As shown in Konieczny |1993~, however, even a small fixed cost (which prevents keeping the real price fixed) may ensure a large loss from a large price change.

The model is as follows. Consider a monopolistic firm which produces a single, perishable good and expects the inflation rate to remain constant. Its real profit function, F(|center dot~), which depends on the real price only, is strictly quasiconcave and continuously differentiable. The real interest rate is zero.(8) The firm is competitive in factor markets and buys factors at constant real prices. It is convenient to set the model in logs: all prices are logs of nominal or real prices. The costs of changing the nominal price are

(1) c = c(|Delta~|p.sub.|Tau~~)

where |p.sub.t~ is the real price at time t;

|Mathematical Expression Omitted~

and c(|Delta~|p.sub.|Tau~~) is defined as

|Mathematical Expression Omitted~

The total cost of price changing consists of two parts: (i) the cost of market response, |c.sub.r~(|center dot~), which is assumed to depend on the size of the nominal price change, |Delta~|p.sub.|Tau~~, and (ii) the fixed menu cost, |c.sub.m~. It is incurred whenever the real price jumps discontinuously, which coincides with nominal price increases. In contrast, there are no costs as the real price is continuously eroded by inflation since then |Delta~|p.sub.|Tau~~ = 0. The menu cost is introduced to rule out continuous adjustment. It is assumed throughout the paper that |c.sub.m~ is small but positive. This is intuitive since price changes are never costless. The results do not depend on the size of the fixed cost; it can be very small. To obtain analytical solutions |c.sub.r~(|center dot~) is assumed to be a smooth function.

The firm's problem, at the time of the first price change (set, without the loss of generality, equal to zero), is to choose the timing of price changes and the initial prices at each change so as to maximize V

|Mathematical Expression Omitted~

where

|Mathematical Expression Omitted~

are the sequences of real prices set at each adjustment and the times of adjustments, respectively; |t.sub.T~ is the time of the last adjustment before T, |p.sub.T~ is the price set at |t.sub.T~ and g |is greater than~ 0 is the inflation rate. It is assumed that F(|center dot~), c(|center dot~) and g are such that an optimal (non-negative profits) pricing policy exists.

When adjustment costs are fixed the optimal pricing policy is of the (s,S) type: the real price is allowed to vary between two bounds, s and S; it is reset to S whenever it is eroded by inflation to s. For the present case the sufficient and locally necessary conditions for the optimality of an (s,S) policy are given in Proposition 1 below. Proofs are in the appendix, available from the author on request.

PROPOSITION 1: Let |Alpha~(|center dot~) denote the elasticity of the cost function: |Alpha~(x) |is Equivalent to~ xc|prime~(x)/c(x).

(a) If (s*,S*) is the optimal policy, then

(4) |Alpha~(S* - s*) |is less than~ 1.

(b) Assume c|prime~(center dot) |is greater than~ 0; c|double prime~(center dot) |is greater than~ 0. If, for every g |is greater than~ 0, there exists 0 |is less than~ |Beta~ |is less than~ 1 such that |Alpha~(x) |is less than~ |Beta~ for every x |is greater than~ 0 in the relevant range (non-negative profits) then there exists an optimal, unique, recursive (s*, S*) policy.(9)

The second-order conditions, which assure that the optimal policy is unique, are:

F|prime~(S) - gc|double prime~ |is less than~ 0; F|prime~(s) + gc|double prime~ |is greater than~ 0;

det |is Equivalent to~ F|prime~(S)F|prime~(s)-g|F|prime~(s) - F|prime~(S)~c|double prime~ |is less than~ 0.

If c|double prime~ |is greater than~ 0 they are met by quasiconcavity. They are also met for concave cost functions as long as c|double prime~ is not too small.

The interpretation is straightforward. The benefit to the firm from smaller price changes is that it charges prices closer to the profit maximizing value and its momentary profits are higher. This benefit becomes neglible for small changes. (s,S) pricing is optimal if the cost function is inelastic enough so that higher adjustment expenses more than offset the improvement in momentary profits. Otherwise profits increase as changes become more frequent. The impact on adjustment expenses of reducing the size of changes falls as the cost function elasticity, |Alpha~(|center dot~), rises. The expenses fall for |Alpha~(|center dot~) |is greater than~ 1, are unaffected for |Alpha~(|center dot~) = 1 and increase for |Alpha~(|center dot~) |is less than~ 1. If |c.sub.m~ = 0 and (b) is violated throughout (which is the case, for example, if c(S-s) = a |center dot~ |(S-s).sup.2~), then it is optimal to change prices continuously. |c.sub.m~ |is greater than~ 0, however small, assures that (b) is met for some (S-s) (as long as c(|center dot~) is differentiable and c|prime~(|center dot~) is finite).(10)

The similarity to the fixed cost case is easily established. If Proposition 1(b) holds, then s and S maximize

|Mathematical Expression Omitted~

and so

(5) F(S) = F(s) = V(s,S) - gc|prime~(S-s).

Differentiating (5) totally, using (4) and the second-order conditions:

(6) dS/dg = |F|prime~(s)/F|prime~(S)~ds/dg

= ||Alpha~(S-s) - 1~F|prime~(s)c(S-s)/|det |center dot~ (S-s)~ |is greater than~ 0.

It can be seen from (5)-(6) that the optimal policy is very similar to that with the fixed cost assumption, which is a special case of the current specification (the fixed cost results are obtained by substituting c(S - s) = c = constant, hence |Alpha~ = c|prime~ = c|double prime~ = 0, into (5)-(6) and the definition of det). The fixed cost approach is, therefore, a useful, simple benchmark case.

The effect of inflation on the size of price changes is, by (6), decreasing in |Alpha~(|center dot~). It is smaller than in the fixed cost case when price adjustment costs rise with the size of adjustment; as a consequence, real prices vary less over the pricing cycle. This means, for example, that the effect of inflation on the economy's responsiveness to real shocks, analyzed by Ball, Mankiw and Romer |1988~, is stronger while the effects of inflation on output considered in Benabou and Konieczny |1993~ and the output-productivity relationships in Konieczny |1990~ are weaker.

III. COSTS OF PRICE CHANGES VARYING WITH THE FREQUENCY OF ADJUSTMENT

In this section I consider costs of price changes which depend on the frequency of adjustment. This can be motivated as follows. Assume that customers take time to think about the purchase after sampling a price, as suggested by Rotemberg |1982~. Similarly, consider a market in which customers purchase repeatedly but infrequently (say, a market for consumer durables). Assume that search is time consuming and, as the price distribution changes over time, it is not known at the beginning of each search. In those environments price recall is valuable. Finally, consider search with limited memory as in Dow |1991~. Agents cannot remember the exact value of price offers but remember a partition element into which the offers fall. The decision to purchase is based on Bayesian inference about the actual offers. The lower is the frequency of price changes, the higher is the value of memory. In all environments it is optimal to start search with firms which are known to change prices infrequently. Those firms have a higher probability of being sampled and so would have, ceteris paribus, higher demand than sellers who change prices more often. This is precisely what Barro |1972~ suggested in his original contribution.

The adjustment cost, incurred whenever the nominal price is changed, is now:

|Mathematical Expression Omitted~

where f |is Equivalent to~ g/(S-s) is the frequency of price changes.

The problem is very similar to that analyzed in section II.(11) Proposition 1 continues to hold, except that (4) is replaced by

(4') |Alpha~(f) |is greater than~ - 1

where |Alpha~(f) |is Equivalent to~ fc|prime~(f)/c(f) is, as before, the elasticity of the cost function.

In contrast to the previous case, however, price changes may become smaller at higher inflation rates.

PROPOSITION 2:

(a) df/dg |is greater than~ 0;

b) dS/dg |is greater than~ 0 |is greater than~ ds/dg if and only if

(7) 1 + |Alpha~(f) + f|Alpha~|prime~(f)/|1 + |Alpha~(f)~ |is greater than~ 0.

The last result has no equivalent in fixed cost models where price changes always become larger with inflation. Empirical evidence in Lach and Tsiddon |1992~, which implies rejection of the fixed cost formulation, is consistent with the ambiguous result in Proposition 2: of the twenty-six products studied the adjustment size increases for sixteen, falls for eight and remains unaffected for two.

If (7) is not met, then the effect of reducing the time between changes on the average adjustment costs falls as the adjustment frequency rises. The firm, therefore, responds to higher inflation by rapidly increasing the frequency of price changes. This reduces price variability and allows the firm to charge prices closer to the profit maximizing level, raising momentary profits.(12)

IV. INFLATION AND THE FREQUENCY OF PRICE CHANGES

A central issue in the costly price adjustment literature is the relationship between the rate of inflation and the frequency of price changes. Intuition suggests that the correlation is positive. Fisher and Modigliani |1978~ argued, on this basis, that inflation reduces welfare. Ball, Mankiw and Romer |1988~ used the assumed positive correlation to discriminate between menu cost and misperceptions of the Lucas |1973~ type explanations of the short-term effects of money on output.

While all empirical studies find a positive correlation between the frequency of adjustment and inflation, this result need not hold in fixed cost models. As shown in Sheshinski and Weiss |1977~, a sufficient condition for positive correlation requires that F(|center dot~) be concave. With variable costs the correlation is positive when costs depend on frequency; when they depend on the size of adjustment it is sufficient to restrict only the cost function.

PROPOSITION 3: Let the cost function be given by (1)-(2) and denote x |is Equivalent to~ S - s. Assume an (s,S) policy is optimal and unique. Then df/dg |is greater than~ 0 if

(8) c(x) - xc|prime~(x) - |x.sup.2~c|double prime~(x) |is less than or equal to~ 0.

Proposition 3 provides a sufficient, but not necessary, condition. For (8) to hold the cost function must be convex since, by (4), c(x) - c|prime~(x)x |is greater than~ 0. This implies that the marginal cost of making bigger changes is increasing. If it is increasing fast enough the firm responds to higher inflation by making more frequent adjustments so as to avoid the high cost of each big change. If, on the other hand, the cost function is concave, increases in x have decreasing effect on the size of adjustment costs. The firm may therefore react to higher inflation by increasing adjustment size rapidly so as to make less frequent changes.

V. EXAMPLES

Numerical examples in this section show how the variable cost approach can account for empirical findings which contradict fixed cost models. The specification is very simple: linear demand, constant marginal production costs and adjustment cost functions which are low order polynomials. Parameters are chosen so as to meet the following conditions: (i) an (s,S) policy is optimal; (ii) the fixed menu cost is an insignificant portion of total adjustment cost (at most 1.2 percent of the total for inflation exceeding 1 percent per year); (iii) except for the last example, prices are changed yearly when the inflation rate is 5 percent.

With costs given by (1)-(2) momentary profits are F(z) = (|a.sub.1~ - |a.sub.2~z)(z - |a.sub.3~) and the average profits per unit of time are V = F(|p.sup.m~) + |x.sup.2~F|double prime~(|p.sup.m~)/24 - gc(x)/x where x |is Equivalent to~ S-s and |p.sup.m~ |is Equivalent to~ (|a.sub.1~ + |a.sub.2~|a.sub.3~)/2|a.sub.2~ is the profit maximizing price. The loss from charging a suboptimal price depends on the concavity of the profit function at |p.sup.m~. The last term is the average expense on price adjustment; with costs given by (2') it becomes |Mathematical Expression Omitted~.

In Figure 1 the adjustment cost function violates (7) for inflation rates above 7.26 percent. As a result, at higher inflation rates the frequency of price changes increases so rapidly that the size of adjustment falls. The model can, therefore, account for Lach and Tsiddon's |1992~ findings.(13)

Figure 2 shows an example of an adjustment cost function such that the correlation between frequency of price changes and inflation is negative. This requires that (8) be violated. Equation (8) is, however, a sufficient but not necessary condition: the cost function c(x) = ||Beta~.sub.0~ + ||Beta~.sub.1~|x.sup.|Gamma~~ does not meet (8), but the correlation is positive.

Finally, I consider a more precise calibration which replicates Cecchetti's |1986~ data. He was the first to provide evidence inconsistent with the fixed cost model and suggested the generalization used here. It is clear from (6) that his intuition is correct: the effect of inflation on adjustment size decreases, ceteris paribus, in |Alpha~(|center dot~). His data are: in 1953-65, average inflation = 1.68 percent, x = 0.2071; in 1974-79, average inflation = 7.63 percent and x = 0.2223. Using the inflation numbers with c(x) = 0.1 + 100x + 1.9275|x.sup.2~; F|double prime~(|p.sup.m~) = -0.7858, the values of x are replicated with an error of the order |10.sup.-4~.

VI. THE TECHNOLOGICAL TRADEOFF

In this section the assumption that the menu cost is exogenous to the firm is relaxed. This is motivated by the fact that firms usually have some control over administrative type costs of price adjustment. I assume that the firm can choose among a continuum of price changing technologies. For the illustration of the problem assume that technologies with lower price adjustment costs (say, barcode readers) have higher maintenance costs, which may include costs imposed on consumers. For simplicity the adjustment cost is assumed to be fixed, independent of the size or frequency of adjustment. The firm's momentary profit function is of the following form:

(9) F(z,c) = F(z) - H(c)

where H(c) is the maintenance cost, and H|prime~(|center dot~) |is less than~ 0. The firm, in addition to the sequences of adjustment times and prices, chooses also the size of the (fixed) adjustment cost so as to maximize (3) with the momentary profit function (9). Assuming that the second-order conditions are met it is straightforward to show that

dc/dg |is less than~ 0 if and only if df/dg |is greater than~ 0.

With this specification a higher rate of inflation induces investments to lower price adjustment costs, provided the frequency of price changes increases. This is a special result, though. If, for example, the increase in maintenance costs were proportional to output or revenue, the effect of inflation will, in general, be ambiguous. Contrary to intuition, higher inflation need not induce firms to invest in technologies with lower adjustment costs.

VII. CONCLUSIONS

If price changing were costless, prices would adjust in continuous (or near-continuous) fashion. They do not, which implies that price changes are costly. Small, fixed menu costs cannot generate the large changes observed in many markets; the costs must be due to unfavourable market reaction to price changes.(14) Such reaction is proxied in the paper by variable cost specifications.

I argued in the introduction that fixed cost models have two problems: it is not clear what the costs are and they are inconsistent with evidence. The generalization to variable cost specification deals successfully with the second problem: price changing patterns found in empirical studies can be generated with simple functional forms even in the simple, nonstochastic case considered here.

There are several conclusions. First, the fixed cost specification can be treated as a benchmark case. Second, the relationship between the rate of inflation and adjustment frequency or size should be inferred from empirical data; they support the intuition in Fischer and Modigliani |1978~ and Ball, Mankiw and Romer |1988~ that price changes become more frequent as inflation rises. Last, and most important, the analysis strongly suggests that the costs of price changes are due to unfavourable market reaction to price changes. The arguments in the literature about the existence of such costs have not been elaborated much beyond Barro's |1972~ original assertion that "... a firm with a more variable price history is likely to experience a lower demand ... for any current price level". Further development of the costly price adjustment approach requires developing explicit environments in which, with aggregate inflation, firms' optimal policies imply infrequent price changes. Some examples were suggested here. Blinder's |1991~ study provides further ideas. Such models would allow us to analyze the nature of price changing costs, which may be implicit, as suggested by Rotemberg |1982~, and resulting monetary non-neutralities.

5. There are two exceptions. Rotemberg |1982~ studies, in discrete time, the dynamic behaviour of aggregate variables when firms face costs which are quadratic in the size of price changes. In continuous time this specification of costs results in continuous adjustment, as shown below. In Tsiddon |1991~ the costs are linear in adjustment size.

6. A monopoly would not do. A popular informal story is that customers dislike price gouging (as experimental evidence suggests: see Kahneman, Knetsch and Thaler |1987~) and react negatively to price increases. This, however, implies a real, not nominal, rigidity since customers care only about real prices. The argument fails when, under general inflation, nominal prices are increased just to restore the real price to the original level.

Some progress in the oligopoly case has recently been made by Halperin |1990~ and Benabou and Gertner |1991~.

7. Even though customers are interested only in relative prices, without perfect information they must make inferences from changes in nominal prices. It may be noted that the model implies, unlike in Rotemberg and Saloner |1987~, that price changes are more frequent in monopolistic markets.

See Benabou |1992~ and Diamond |1990~ for a similar approach with monopolistically competitive firms.

8. This assumption simplifies the analysis greatly without affecting results.

9. Given the form of c(|center dot~) the optimal policy depends on the starting point. The results can be thought of as applying in the steady state, after S has reached its new equilibrium level, S*.

10. If (b) is met for some x only, there may not exist an optimal (s,S) policy and if it does exist, it may not be unique. If c|prime~(|center dot~) |is less than~ 0, an optimal (s,S) policy may exist; additional assumptions are needed to ensure a unique, internal (positive profits) solution.

11. As before, the results should be thought of as holding in the steady state.

12. As long as c|prime~, c|double prime~ are both finite, (7) is met at small frequencies (and so at small inflation rates) since it is equivalent to c + 3fc|prime~ + |f.sup.2~c|double prime~ |is greater than~ 0. This is quite intuitive: the size of the price adjustment must rise at small inflation rates. By (4') 1+|Alpha~(f)|is greater than~0 and so a necessary condition for (7) to be violated is that |Alpha~|prime~ |is less than~ 0.

13. A similar result was obtained numerically by Diamond |1990~ in a sticker cost model.

14. Evidence in Lach and Tsiddon |1992~ is, perhaps, the most striking. In 1982 the Israeli economy had suffered high or very high inflation for fifteen years and so its institutions were, by all accounts, attuned to it. The remaining menu costs must have been small (changing a price list rather than repricing individual goods, etc.). Yet the average adjustment size was at least 10 percent for twenty-five out of twenty-six goods.

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