Staggered Contracts and the Frequency of Price Adjustment...

Staggered Contracts and the Frequency of Price Adjustment

Stephen G. Cecchetti

The Quarterly Journal of Economics, Vol. 100, Supplement. (1985), pp. 935-959.

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STAGGERED CONTRACTS AND THE FREQUENCY OF PRICE ADJUSTMENT*

This paper describes a methodology for measuring the frequency of price change in order to test the relevance of assuming prices to be set for discrete periods of time a t overlapping intervals. Taylor [I9801 has related the frequency of adjustment to the rigidity of the economy in responding to unanticipated events. Estimates of the frequency of price change are computed from data on the component parts of the deflator for personal consumption expenditure. The results show a substantial decrease in the period between price changes during the middle 1960s, and marked fluctuations in the 1970s. The movements suggest changes in the rigidity associated with both changes in general price inflation and changes in the posture of the fiscal and monetary authorities.

Theoretical models of the inflation process that assume wages and prices to be fixed for discrete periods of time at overlapping intervals predict persistent effects on output and employment from unanticipated shocks. In a recent paper John Taylor [19801 simulates the dynamic properties of staggered contracts, showing that the serial correlation exhibited by unemployment in the United States is consistent with contract lengths of between three and four quarters. Contracts in this context are both explicit and implicit, defining periods of time for which prices or wages are held constant. This paper describes a method of measuring the frequency of price change in order to test empirically the validity of the staggered contract assumption. Such research bears both on the theoretical debate over the merits of the market-clearing as- sumptions of the new classical macroeconomics, and on the current discussion among policy makers over the size of the output losses necessary to lower inflation.

If prices change only infrequently and in an overlapping fashion, as long as the period for which prices are fixed is longer than the period of observation, some prices will be observed to change between observations, while others will not. This implies that an

*This paper is a revised version of the first essay of my Ph.D. Dissertation completed in August 1982 at the University of California, Berkeley. Thanks are due especially to George Akerlof and also to Jeffrey Frankel, Barry Ickes, Peter Rappoport, Peter Berck, Ronald Warren, Daniel Saks, Bill Greene, and the par- ticipants of the Macroeconomics Seminar a t U. C. Berkeley. Partial financial support was provided by the National Commission for Employment Policy.

@ 1985by the President and Fellows of Harvard College. Published byJohn h'iley & Sons, Inc. The Qua~ter~Journai ofEconomics, Vol. 100. Supplement, 1985

936 QUARTERLY JOURNAL OF ECONOMICS

estimate of the periodicity1 of price change in the Taylor model can be inferred from a measure of the cross-sectional dispersion in inflation across the economy. The following section discusses a simple model of inflation where prices adjust only periodically. An example is developed to show how the economy-wide dispersion in inflation increases with the inflation rate for a given frequency of price adjustment. Section I11 sets forth a more complex model and derives the relationship between the dispersion in inflation and the period for which prices are fixed. The model is then examined using data on the newsstand prices of magazines over the past thirty years. The results from individual price data help in testing and interpreting the properties of the model. In this case the period for which prices are fixed can be calculated directly as well as estimated from the cross-sectional dispersion in inflation across magazine prices. It is shown that the model tracks very accurately, indicating that the abstractions it makes are not critical. Estimates for the frequency of price change have been calculated from the dispersion in the component parts of the deflator for personal consumption expenditure and are presented in Section V. The measure for the average periodicity of price change is found to decrease first in the middle 1960s and again in the middle 1970s, probably as a result of increases in inflation, dropping by as much as one-half from the beginning to the end of the sample period considered. The concluding section discusses interpretations of empirical results.

The models considered by Akerlof [1969], Fischer [19771, and Taylor [I9801 assume that monopolistic competitors adjust prices with constant frequency, all other things equal, and that firms set prices in a staggered manner.' Sheshinski and Weiss [I9771 provide a basis for the first assumption. They show that under conditions of monopolistic price adjustment with nonzero adjust-

1.Periodicity refers to the interval of time that passes between changes in the price. The frequency is the inverse of the periodicity.

2. Several alternatives to models that resume staggered monopolistically competitive firms have arisen. Phelps and Faylor [I9771 and Iwai [I9811 work with a scheme where firms must precommit to product and pricing strategies in advance of sales. Blanchard [I9821 examines the consequences of desynchronized price setting a t different stages in the production process. All of these models lead to some price stickiness that gives rise to partial quantity adjustments in the face of unexpected events.

STAGGERED CONTRACTS AND PRICE ADJUSTMENT 937

ment costs and a continuously compounded inflation rate n, it is optimal to change prices with a constant periudicity k. The value of k is determined by inflation, the elasticity of demand with respect to price, the curvature of the real unit cost function, the rate of interest used to discount the profit stream, and the cost of adjusting the price.3 The assumption of staggering is easy to accept on the basis of casual empirical observation.

To understand best how staggered contract models behave and how one might measure the frequency of price adjustment from the dispersion in inflation across firms, consider an example with two firms. Assume that each firm changes its price every two quarters in alternate quarters. If all prices are sampled every period, and the length of time for which prices are fixed remains unchanged, the dispersion in inflation will rise with an increase in inflation. Simple algebraic calculations show exactly why this is so. Between observations, one firm's price will change, while the others will not. The dispersion in inflation can be calculated as the market share weighted sum of squared deviations in individual firm inflation from the inflation in the average price level. If the average inflation rate is n, the firm whose price changed between observations will exhibit inflation of 2n, while the other firm's price will remain constant. If d ( p )is the economy- wide dispersion in i n f l a t i~n ,~

With the periodicity of price change constant, the observed dispersion in inflation across firms will rise with the square of inflation. A simple numerical example has been constructed to aid in understanding exactly what is happening. Stay with the two- firmcase and see that the example assumes that each finnchanges price every two quarters. Table I shows the pricing sequence fol- lowed by each firm. From the beginning of Year One until the third quarter of Year One, the inflation rate is assumed to be steady at 10percent per year, or a continuously compounded rate

3. Rotemberg [I9821 models the behavior of monopolists who face rice adjustment costs. His model differs in that agents adjust every period, gut only gradually since small price adjustments are assumed to be less costly than large ones.

4. The two firms are assumed to split the market evenly.


TABLE I

PRICEAMUSTMENTS INFLATIONWITH CHANGING RATES (Adjustments are made every other quarter.)

Year Quarter Firm 1 Firm 2 Average Inflation rate*

Zero 1 2 3 4

One 1 2 3 4

Two 1 2 3 4

*Inflation is at a continuously compounded rate for one quarter.

of 0.024 per quarter. Beginning in the third quarter of Year One, the inflation rate jumps to 15 percent per year, representing a continuously compounded rate of 0.035 per q ~ a r t e r . ~ Prices are set by each firm to equal the price set in the previous quarter by the other firm times e", where n is the continuously compounded inflation rate. Inflation is the only reason for which prices change.

The dispersion in inflation across firms can be calculated once a period of observation is chosen. If the time between observations is an integer multiple of the periodicity of price setting (i.e., 2,4,6,8 . . . quarters), then the measured dispersion during a period of steady inflation will be zero. Calculation of the dispersion of relative price inflation over periods of observation that are not exactly an even number of quarters in length will yield information about the frequency of price change. For single-quarter sampling, the calculations in Table I1 illustrate the point.

The dispersion clearly rises with inflation. But the example assumes the frequency of price change to be constant. While a steady state would yield data with a constant price change frequency, the world we observe does not. Iwai [I9811 has developed a model where individual agents adopt price change rules. They monitor conditions and adjust their fixed nominal price when their

5. I t is assumed for convenience that the periodicity of price change is unchanged when inflation increases. This need not be the case.


TABLE I1

CHANGESIN THE DISPERSIONOF RELATIVE WITH CHANGESINFLATION IN THE RATEOF INFLATION

Third to fourth One quarter inflation Dispersion quarter of year continuously compounded in inflation

0 0.024 0.00058 1 0.030 0.00086 2 0.035 0.00123

Source.Calculated from Table I.

current price is far enough out of line. In the Iwai model the rule may change, but even if it does not, prices will not be observed to change in an orderly fashion. Sudden changes in the environment could prompt many agents to change price suddenly but retain the same rule. This means that a model must be constructed which allows for changes in the length of time between price changes. By noting what happens to the measured dispersion as the periodicity of price change approaches the periodicity of observation, it can be seen how this might be done. In the example, imagine adjustment occurring every quarter. The dispersion would fall to zero. It is suggested from these simple calculations that observation of the dispersion in inflation across the economy and the average inflation rate can be used to estimate the degree to which prices are rigid. The following section examines the relationship among these three quantities.

The most elementary model begins with the periodic pricing equation,

This represents the inflation in the price of a firm setting its price at time t for k periods. It is assumed that inflation is the only reason a firm will change its price. With no disturbances to the economy, k will be constant over time.6

6. There are several reasons k may change. In models of the type considered by Iwai [19811, where a firm's fixed nominal price is changed when it deviates sufficiently from a (stochastic) profit-maximizing price, k depends on numerous things including the level of general rice inflation. To the extent that changes in inflation bring about only small cganges in k in the short run, they can be ignored. The short run here is essentially the period between observations discussed below. Changes in k over longer time horizons, which come about as a result of movements in 7, can be measured using the techniques discussed here.


Unfortunately, continuous observation of a firm's price is not practical. Any attempt to infer the magnitude of the k's, or their distribution, must take into account that observation occurs at periodic intervals. There is a relationship between the length of the observation period (call this T), the frequency parameter k, and the observable dispersion. As was pointed out in Section 11, with reference to the example, if T = k, the observed dispersion in inflation across firms will be zero. This highlights the fact that there are two important cases for which the model must allow. These depend on whether k is larger or smaller than T.7

Fundamental to the model being considered is that members of a group setting prices for k periods act at nonsynchronized times. In the spirit of the models of Akerlof and Taylor, as well as the example above, take the firms setting prices for k periods to be uniformly distributed over the interval from zero to k. This assumption, which will be called "uniform nonsynchronization," implies that if prices are monitored first in period t and next in period (t + k)/2, one-half of the firms in the k group would have been observed to have changed their prices. Observation at t and t + k would find that all firms had changed their prices8

To proceed with the model, since the period of observation is smaller than the periodicity of price change, and since the firms are uniformly nonsynchronized, some firms will have changed their prices between observations, and some will not. This creates a dispersion in inflation across the group that can be calculated. The mean price change for the group can be calculated by observing that a fraction Tlk percent of the firms will have changed their price between the observations. The amount of each firm's price change is ~k from (3), SO the mean is TT. The dispersion is simply

Simplifying (4) yields

7. The case where k < T is discussed in detail in the appendix to the first essay of Cecchetti [19821. The analysis there shows how k cannot be uniquely estimated when the dispersion becomes small relative to the level of inflation. It is shown that when the estimate fork exceeds 1.125T, the problem does not occur.

8. This assumption is not that important. The model allows consideration of groups with different k's. That is all that seems really necessary. The uniform nonsynchronization assumption simplifies the algebra.


This is just the weighted sum of squared deviations from the mean in individual firm inflation. Equation (5) exhibits the property required by the example-that the dispersion increases with the inflation rate, holding k constant. It is important to note (and fundamental to the analysis) that increases in k for a given IT

increase the dispersion as well.9 Equation (5) is the dispersion that will be observed assuming

that every firm in the economy sets its price for the same length of time and that firms are uniformly nonsynchronized. The model allows for a distribution of k7s across the economy. To find an expression for the dispersion in inflation across the economy, assume that f(k) describes the frequency function for k such that J; f(k)dk = 1and the fraction of the economy with k = a is f(a). By simply adding up the appropriate values from (5) for each group, an expression for the economy-wide relative inflation dispersion can be derived. Since it is assumed that k > T, the appropriate expression is

d(b) = S(k)f(k) dk. T

The assumption of uniform nonsynchronization makes S(k) linear in k. This, together with the observation that ITT is the mean of both the economy-wide inflation and the inflation in a single group, allows computation of d(p) as a function of IT and the first moment of f(k). To see this, define to be the expectation of k conditional on k > T. Substituting (5) into (6) and evaluating the integral yields

the final expression for the dispersion in inflation across the economy.lo

Before turning to some preliminary empirical results, several remarks can be made about estimates from such a simple model. The first is that the difficulty in dealing with solutions where the time between observations is smaller than k emphasizes the need for a short observation period. A further problem arises when one makes the trivial observation that prices change for a variety of

9. In the example, T = 1 quarter, and k = 2, so S ( k ) = a2. 10. A n expression similar to (7) can be derived from the (s,S)-type models

employed by Sheshinski and Weiss [I9771 and Iwai [19811. Such an analysis leads to a relationship between d ( p ) ,a ,and the probability o f observing a price change.


reasons, some of them not simply the result of infrequent adjustment. Relative prices are constantly adjusting to accommo- date shifting demand, changing technologies, and the like. The model examined here attributes all dispersion in inflation across commodities to the phenomenon of periodic adjustment. This means that estimates of k from the simple model will be biased upward. But as will be argued below, available data mask much of the dispersion in inflation by presenting information in the form of averages from commodity groups, ignoring the differences in adjustment that occur within the groups, and biasing the estimates of the periodicity downward.

IV. A TEST: THE NEWSSTAND PRICESOF MAGAZINES

Before attempting to estimate the average period for which prices are fixed in the entire economy, a simple microeconomic example is useful to help understand the characteristics of the model presented in the previous section. The newsstand prices of magazines have the desired property of infrequent and periodic adjustment.ll Individual price changes can be observed directly, and the proposition that knowledge of the dispersion in inflation provides information about the periodicity of price change can be tested. In addition, the findings concerning magazine prices are interesting in and of themselves. The adjustment pattern of this individual set of prices yields information about what one can expect when looking a t the economy as a whole.

Data were collected on a monthly basis for thirty-six magazines over the past thirty years. But in order to obtain sufficient dispersion in the inflation of the magazine prices, the shortest observation window used was one year. Since no magazine changed price more than once per year only the January data were used.'' To obtain estimates from the model, equation (7) was solved for -k. Set T = 1to define the units of z,which yields

11.It is worth noting that even competing magazines change their prices in different years. The fact that sharply differing prices for essentially the same y d u c t are actually observed at the same time poses a uzzle for which no solution

as yet been offered. Some progress has been made by Pethke and Policano [19831, who are able to show that in a market with two identical competing producers, welfare improvements may result from staggered adjustment. But, unfortunately, their analysis does not readily generalize to the case with more than two firms.

12. A list of the magazines appears in the Appendix.


Once estimates for 7~ and d ( p )have been obtained, the estimate for % follows. Inflation and dispersion can be estimated as the average inflation in magazine prices over a period and the average squared deviation of individual magazine price inflation from this mean:

where pit is the change in the log of the ith magazine's price from period ( t - 1) to t, n is the number of magazines in the sample, and a caret denotes an estimator.13 The estimator for the dispersion follows as

Substituting (9)and (10)into (8) yields an estimator for z,which will be labeled k .

In the case of the magazine price data, there is more information than just the dispersion in inflation. One can observe directly whether or not a particular price changed during a given period. This suggests an alternative way to measure the periodicity of price change, one based directly on the number of prices observed to change. Take the period of time between changes of an individual magazine price to be a random variable with a waiting time distributed as Poisson, each magazine having a Pois- son parameter drawn from a Beta distribution; then the data set containing all magazines taken together can be treated as a sample drawn from a negative binomial distribution. The mean of this distribution, and consequently the average Poisson parameter, can be estimated by dividing the number of magazines whose price changed by the total number of magazines (36).The inverse of this average is the expected waiting time between price changes and can be compared with k . 1 4 For example, between January

13. The weights in the dispersion measure should represent the importance of an individual price in the economy. A natural measure would be the market share of the product sold for that price. In this context, weighting each magazine price equally assumes that each magazine has equal importance in the industry.

14. This technique allows calculation of the expected waiting time during every period. The estimate is unconditional on any previous activity. It is rea- sonable to believe that the probability of observing a price change in a articular period should be dependent on the history of a firmprior to that periocf Such an approach is not taken here but is the subject of the second essay in Cecchetti [1982]. There a logistic model of magazine price change is derived and estimated. But for the purposes of this paper, the unconditional waiting time is all that is required.


1964 and January 1965, seven magazines changed price. Con- sequently, the expected waiting time in that year is calculated as (7136)-l, which equals 5.143. This can be compared with a k of 5.7.

Table I11 presents estimates of % obtained from equation (8) as well as the expected waiting time W. The estimated valu:s are extremely close to the true values, the correlation between k and W is 0.98 for each of the observation windows reported. It is important to note that k is always greater than or equal to the expected waiting time. This is a consequence of ignoring underlying trends or relative price changes and ascribing all dispersion in inflation to nonsynchronized adjustment.

What is most striking about the numbers is that as the observation window is allowed to grow, the correlation between k and W does not deteriorate. But the upward bias does increase. Simple univariate regressions can be used to tell this story. When k is regressed on W, the estimated slope coefficient is not statis- tically significantly different from one at the 5 percent level for an observation window of either one or two years. This is not true when the observation window is allowed to grow to five years. -Driving this result is the consistent and sizable overestimate of k during the 1970s when W is close to the length of the observation window.

The intent of this section thus far has been to bring up both the positive and negative attributes of the estimator for the periodicity of price change derived from the simple model. The example shows the importance ofhavingfrequent observations, as well as the upward bias introduced by failing to account for true relative price changes. While this upward bias in the estimates is important when examining the individual price data, when aggregate data are examined in the following section, there is a large downward bias that is almost certainly much more important.

The results in Table I11 not only are useful as a test of the model, but also are interesting as predictors of what one expects to find when examining data on the entire economy. The pattern of the estimated k's for magazines is not surprising. Price changes are found to occur more frequently in the latter half of the sample than in the earlier half, with the notable exception of the incomes policy period from 1971 to 1974. In the case of yearly observations, breaks occur around 1966 and again in 1975. Examination of the average length of time between price changes, a number calcu-


TABLE I11

ESTIMATES OF ADJUSTMENT PRICESOF THE PERIODICITY IN THE NEWSSTAND OF MAGAZINES*

(in sears)

Length of observation period ( T )

One year Two years Five years Year A W A W k W

~~~~~

*Calculations are all January to January k 1s estimated uslng equat~on (8) The expected waitlng time W is calculated by dividing the number of total magazines (36) by the number of magazines that changed over the observation perlod

lated by averaging completed spells during a period, shows a drop from seven years in the 1950s to three years in the 1970s. Cec- chetti [I9821 presents a model of price change at the micro level. By examining the determinants of the probability of observing a magazine price change, that paper concludes that increases in


general price inflation bear primary responsibility for this fall in periodicity.

Estimates of the periodicity of price setting in the economy as a whole can be obtained in a way analogous to that used in the case of magazines. Equation (8) can be used to generate estimates for k once estimates for the dispersion and inflation have been obtained. Recall from the earlier discussion that estimates from the simple model of Section I11 have three interpretations. They can be viewed as estimates of the mean of f(k), %. Alter-natively, the estimates represent the period for which prices are fixed if every firm in the economy sets its prices with the same frequency. Finally, there is the interpretation from the last section, that the estimates are the expected waiting time between price changes.

Parks's [I9781 method has been used to estimate the dispersion has been estimated from the component parts of the deflator for personal consumption expenditure (PCE) of the National In- come and Product Accounts. The estimates for d(p) were calculated using the following technique. For the ith group pit = log Pit - log Pi(,- l,. An estimate of the average inflation rate can be calculated as

and the estimator for d(p) as

The measure expressed in (12) is thus a Divisia index (see Theil [19671).15

Estimates have been obtained for one-year, one-quarter, and one-month observation periods. They were calculated using equations (11)and (12), and the results were substituted into equation

15. The dispersion calculation is a weighted average of the deviation of individual price inflation from the mean. The weights are intended to represent the proportion of economic activity bridging two time periods. Difficulties occur when the weights change over time. This problem is similar to the one encountered in choosing an index from which to measure inflation where the question of income and substitution effects arises. Blinder [19801 discusses the price index number problem a t length. The choice here is to use a weighting scheme similar to a chain weight index. Theil [I9671 provides an explanation of the technique.


(8).16 Estimates for the period from 1960 to 1981 are presented in three graphs. Figure I presents the estimates from annual data. Figures I1and 111present the results from quarterly and monthly data.17 The estimates in the second and third graphs are moving averages of the year previous to the date at which the point is plotted.''

The general character of the results is that the estimates fall during 1967, and fluctuate during the 1970s with some notable rises in late 1971 and again in early 1976. The results using quarterly data show ff falling in late 1967 from between 0.8 and 1to about 0.4 years. In 1971, probably due to the incomes policy initiated in August of that year, the estimates rise above 0.5, falling back to 0.4 in late 1974 and rising to 0.5 after 1976. This is exactly the result expected during a period when government monitoring increased the cost of changing prices. The estimated k's for the last four months of 1971 are substantially higher than any others after 1967. This is not surprising, since it is the only month during which there was an effective wage and price freeze. The monthly estimates show the same pattern, with differences only in the absolute magnitude of the estimates and a significant rise in 1976."

As was mentioned at the end of the third section, there are several important biases in the estimates of k. The first, an upward bias, is the result of failing to remove the true relative price fluctuations before calculating the frequency parameter. By ac-counting for all of the relative price dispersion with a model based

16. In estimating k, it is assumed that the inflation rate used by each firm setting prices is in fact the one that obtains. The relative price dispersion that is attributable to variation in expectations is ignored.

17. Results from annual and quarterly data are reported because of the well- known deficiencies in the actual information content of monthly price data. The frequency of the price survey and the method used to compute the index ensure that adjacent monthly observations of the components of the index do not all contain new information.

18. Smoothing was necessary for two reasons. First, as inflation approaches zero during a time period, the model breaks down and the estimates can jump unrealistically. Smoothing also eliminates some of the problems caused by the static nature of the model. Taken literally, the k's are estimated under the un- realistic assumption that prices are set based only on current inflation. An alternative model would require exact specification of the way in which price changes depend both on past inflation and on future expected inflation, both of which are likely to be functions of k. At a minimum this suggests smoothing inflation before computing k. This partially addresses the problem created b the static inflation assumption, but when tried it made no difference to the resugs. The decision was made to avoid another level of complexity and sacrifice what is likely tq be a small improvement in accuracy for tremendous simplicity. As a result, the k's are best interpreted as index levels; their actual values should not be taken literally.

19. These results are not affected by exclusion of items such as food, energy, and housing services from the inflation and dispersion calculations.

Periodicity of Price Change, from Monthly Data Source. Same as Figure I with monthly data. Plot is of a moving average of the

previous twelve months.


solely on staggered contracting, the estimates of the length of the implicit contracting period are biased upward. The Appendix discusses a generalization to the model in Section 111. The more complex model allows explicit consideration of relative price movements. The size of the upward bias is estimated using a technique that exploits information contained in the dispersion of inflation computed over a two-year observation frequency; T is set equal to twenty-four months. The method derived allows computation of k adjusted for the relative price changes. Figure IV plots the results and shows the same pattern a? in Figures I, 11, and 111. The absolute level of the estimates of k is substantially lower (the vertical scale is now in months, not years), but the fall during the late 1960s is still evident.20 The one disturbing thing about the results in Figure IV is that the adjusted k s do not fall during the middle 1970s, but instead show a slight tendency to increase. The fact that the estimates of the two-year dispersion used to estimate the relative price changes are stable from 1972 through 1981 explains this result.

It is also extremely important to note a downward bias in the estimated k's. To the extent that staggering within a commodity group creates dispersion in intragroup inflation, the technique used here is not capable of measuring it. Under the assumption of uniform nonsynchronization within a group, the average group inflation rate during an observation period will be equal to the economy-wide average. However, if a group functions similarly to the automobile industry, changing prices in synchronized fashion, this will not be a problem. Subsection 3 of the Appendix shows that the estimate of d ( j )is biased downward by the average of the within-commodity-group dispersion. This translates directly into a bias in k . It seems likely that within- group dispersion should fall when between-groyp dispersion falls. If this is the case. then the downward bias in k will shrink with k. This implies that the downward bias causes a compression in the vertical scale in all four figures; changes in k are artificially deempha~ized.'~

20. As is noted a t the end of subsection 2 of the Appendix, it is likely that computing the relative price variation from the dispersion measured over a period as short as two years biases the estimates of the adjusted k's downward.

21. As a consequence of the problems mentioned in both the text and the Appendix, the estimates of the periodicity of price adjustment should be treated as indexes and not as literal measures of the average time between changes. It is therefore difficult to conclude that the estimates are large or small in any absolute sense.

QUARTERLY JOURNAL OF ECONOMICS

STAGGERED CONTRACTS A N D PRICE ADJUSTMENT 953

It might be asked: if the individual groups exhibit synchronized price-setting behavior, why not simply observe each individual k directly? This suggests a simple regression of current- period group inflation on a series of past lagged price changes. Such a technique would yield little information. There is no reason to believe the lag pattern would be stable over any sample period long enough to actually make precise calculations. Further, as it turns out, the lag pattern cannot be estimated independently of the expectation formation process. What this means is that the k's cannot be defined well enough to use straightforward techniques. Instead, the fact that they change, as do the inflation process and the inflation rate, means that only proxies can be found for the frequency of price change. The measure based on the dispersion divided by the squared average inflation rate can be justified as one that will exhibit the same tendencies under a variety of different models. The nonsynchronization assumption itself is not crucial. So long as the macroeconomy is not a lock- step process and as long as prices are changed by individual actors on different days and fixed for different periods into the future, there will be some dispersion across inflation that can be used to infer the degree of rigidity.

One final point should be mentioned. The data examined here could have been generated by a completely different model that has very different implications. Instead of a staggered contract model, consider a model where each firm changes price every period. The price change may be the consequence of precommit- ment, as in Phelps and Taylor [I9771 and Iwai [19811, or based on current, but imperfect, information. All of the dispersion in inflation, as measured by d ( p ) , would represent relative price fluctuations. Some of it would be unintentional. Such a scheme could be specified using a purely stochastic model, and there would be no way of identifying whether the staggered contract model or the random process model generated the observed data. But k retains an interpretation in this alternative setting. The estimated period would be the average length of time it takes an individual price to adjust to the current rate of inflation, main- taining relative prices unchanged. While this second view is based solely on relative price change, it does give rise to general inflation in all prices. The speed at which prices are adjusting to the current general inflation is measured by k. This too, is a measure of inertia in response to unanticipated changes.


VI. CONCLUSION

This paper has developed a method for estimating the length of the period for which prices are set from an aggregate measure of the dispersion of inflation across the economy. It has been argued that the results based on inflation in the component parts on the PCE deflator can properly be interpreted as measures of the rigidity in the economy under the hypothesis that the changing of prices is not synchronized. The fact that the estimates move over time is quite important and not at all surprising. The findings give the impression that changes in rigidity are associated with both changes in the level of inflation and changes in the posture of the fiscal and monetary authoritie~.~' The significance of this is at least twofold. The first concerns empirical macroeconomic research in general. Changes in the frequency of price change represent shifts in the underlying economic structure that most macroeconometric studies assume to be stable. This means that adjustment parameters estimated in models that assume struc- tural constancy over the period from 1960 to 1980 will not be reliable.23

The second significant implication of this study concerns the market-clearing assumption of the new classical macroeconomics. While the hypothesis that markets clear cannot be rejected for every observed period over the past twenty years, the fluctuation in the estimated periodicity of price change implies that for a t least some years the economy adjusted slowly. Consequently, the effectiveness of policy, as well as the persistence of the effects on output and employment of unanticipated shocks, changes with the economic environment. There may be times when adjustments are rapid and policy ineffective, but this cannot be true for the entire period under consideration. Careful examination of both the estimates and the history of the period will be needed to make these statements more precise.

22. The fact that changes in policy affect the price determination mechanism is support for the claim made by Lucas [I9761 that the effectiveness of policy depends on how the institutions in the economy respond to the policy itself.

23. Regressions of inflation on a distributed lag of its past values, such as those in Gordon [1982], necessarily presume that the lag pattern is fixed from one year to the next. This implies that the speed of price adjustment is invariant to changes in the environment. In light of the results presented here, it is difficult to accept such a premise.


1.Magazines

The single copy prices. of the following magazines were used in the calculations in Section IV: Americas, American City and County, American Rifleman, Architectural Review, Atlantic, Bet- ter Homes and Gardens, Business Week, Commentary, Current History, Ebony, Esquire, Films in Review, Foreign Affairs, Good Housekeeping, Gourmet, Harper's Magazine, Hi h Fidelity, House and Garden, House Beautiful, Interiors, Mo c fern Photography, Musical Quarterly, The Nation, National Geographic, Natural History, The New Republic, The New Yorker, Newsweek, Parents', Popular Science, Reader's Digest, Scientific American, Sunset, Time, U. S . News and World Report, and Yachting.

2. The Upward Bias: Desired Relative Price Changes This Appendix describes the technique used to construct the

estimates reported in Figure IV of Section V. As was noted in the body of the paper, the estimates of the length of the implicit contracting period reported in Figures I, 11, and I11 are biased upward. This is a consequence of the fact that the simple model assumes that the only motivation for a price change is general price inflation, and as a result all of the relative price dispersion is presumed to be a result of staggered adjustment. The model presented in Section 111can be extended to include the possibility of relative price changes, in addition to responses to general price inflation. A generalization of equation (3) states that the price of a firm set a t time t for k periods is

where Xk( t )represents the change in the relative price. The ad- ditional term has expectation zero and variance dependent on both k and T. Assume that k is greater than the observation frequency, the dispersion in inflation across uniformally nonsynchronized firms with the same I t , and see that the generalization to S ( k ) is

(A21 S 1 ( k )= n2T(k - T)+ (TIk) E(Xk( t I2) .

In order to proceed, a model for X k ( t )is required. I t is most natural to assume that X is the sum of individual period disturbances, so

k -1

(A3) Xk(t) = 2 ~ t - i , i = O

where the e,'s represent single-period relative price movements, each with expectation zero. Unfortunately, this formulation of


Xk(t) makes S1(k) a nonlinear function of k, so the analysis can proceed only under the assumption that all prices change with the same frequency.

Information about the variance of Xdt ) can be obtained by measuring the dispersion in inflation over longer and longer observation periods using equations (11)and (12). Call these $(TI'S.As T gets very large relative to k, the measured dispersion will reflect only relative price changes. It will contain no information about stag ered adjustment. The techni ues developed in the appendix to 8ecchetti [I9821 for the consi 3eration of cases where k is smaller than T can be used to show that

(-44) lim S' (k,T) = E(XT(tI2),( k l T H O

where S1(k,T) is the dispersion measured over T period# and XT(t) measures relative price movements over T periods. d (T) is an estimate of the right-hand side of (A4).

The next step is to impose a stochastic structure on the e,'s. Several possibilities come to mind. First, the relative price shocks could be independent, implying that the log of the firm's price level is a random walk. Alternatively, the e,'s might be autocor- related. Movements in relative prices can be serially correlated for numerous reasons. The spot price of a storable agricultural commodity in the face of an anticipated future supply shock would exhibit substantial serial correlation.

Each of the stochastic models for e, has direct implications for the pattern of the estimated d(T)'s. If the e,'s are identically independently distributed with variance u:, then

(A51 E(XT(tI2)= T u ~ .

This implies that the $(T)'s should grow linearly with T. In fact, they grow quite a bit faster. A F-test can be constructed to test the null hypothesis that relative price movemeqts are a random walk. The test involves taking the ratio of two d(T)'s measured over two adjacent nonoverlapping periods of different lengths, each divided by the length of the period T. Under the assumption that the e,'s are stationary from the beginning of the earlier period to the end of the later one, the resulting statistic will be distributed as F(59,59), since sixty commodity groups are involved in computation of each d(T). The test has very low power, but still rejects that the E'S are i.i.d. a t the 1percent level for numerous subperiods of the sample.

The rejection of the hypothesis that the e,'s are i.i.d. leads to the more complex first-order autoregressive formulation where

(-46) et = pet-l + ut. The v,'s are assumed to be i.i.d. with expectation zero and variance u: Substituting (A6) into (A31 and using a result from the econ-


ometric examinations of autoregressive processes (see Theil [1971, p. 2561) shows that

Estimates of the average valye of p and u,2 can bepbtained using equations (A6), (A7), and the d (T)'s. For large T, d (T) equals the right-hand side of (A7), with T substituted for k.

Because of the large amount of information available, there are many ways to construct estimates of p and u,2 to use in computing estimates of k. In order to allow the distribution of e, to change as often as possible, it is assumed that for T = 24 months, the bulk of d (T) represents true relative price changes. This re- quires the assumption that the e,'s are stationary over each two- year period. Once the time period is chosen, the following procedure is used to compute the frequency of adjustment k. First the relative price inflation during each of the two years is computed for each commodity group. These are sums of twelve adjacent E ~ S .Then the relative price inflation in the second year of each two-year period is regressed on the change in the first year. The regression is run once for each of the eleven two-year periods ending with 1980-1981, using fifty-nine of the sixty commodity groups as observations for each regression. The slope coefficient from one of the regressions is a complex function of p during that two-year period. The relationship between the slo e and p can be computed and used to construct an estimate of p. ft is interesting to note that the estimates of p range from 0.56 to 0.96 with an average of 0.84. The estimate of p is then combined with the d (T) measured over the entire two-year period and substituted into (A7) to yield an estimate of a:. Then the adjusted k's are computed using (A71 substituted into (A21 with p and u: set to the same value for each month during the two-year period over which they are computed. Figure I V ~ l o t s the twelve-month lagged moving averages of the adjusted k's.

The procedure used to adjust the estimates of the frequency parameter almost certainly overcompensates for the bias i t at- tempts to remove. Because of the length of the period over which d(T) is computed, twenty-four months, u? is overestimated, and k is underestimated. This is especially true when k is high. The reason is that if the perbd between adjustments is long, staggered adjustment causes the d(T)'s to be contaminated measures of the true relative inflation variance. It will systematically overestimate E(XT(t)2). The importance of the contamination can be examined by comparing the dispersion induced by staggered adjustment with the observed d(T)'s, for T = 24. Using the results in the appendix to Cecchetti [I9821 makes i t possible to show that during the 1960s, when inflation was between 1 and 4 percent


per year and k was six months or higher, up to 50 percent of d(24) may be accounted for by staggered adjustment. But after 1967, when k fell below two months and inflation rose well over 4 percent per year, it is unlikely that even 10 percent of d(24) is contamination. Since the upward bias i n k is more of a problem in the 1970s when k is small, using d(24) seems appropriate.

3. An Examination of the Downward, Aggregation Bias The downward bias in the measure dispersion in inflation in

Section V is a consequence of the fact that data on commqdity groups, as opposed to individual prices, are used to construct d(p). The desired measure of dispersion d(p) is the sum of two components-the dispersion in inflation within comrpodity groups and the dispersion among commodity groups. Since d(p) includes only the second of these two, the bias is the average dispersion in inflation within the commodity groups. To see this, take P..to be the inflation in the price of the j th commodity in group i. ff wi is the weight given the grour, i in the economy as a whole, and w: - . is the wiighYt given the j th commodity in grokp i, then thedesired measure d(p) can be expressed as

(A8) d(p) = z x w i w j ( ~ ,- =I2, i j

where both the wi's and the wj's sum to one and .rr is the economy- wide average level of inflation. The dispersion estimated d (p) can be written as

The difference between the desired and the actual measures of dispersion is simply

The expression in braces is the dispersion in inflation within the j th group. Consequently, the estimate d(p)is low by the average of the within-group dispersion. When a group exhibits uniform nonsynchronization, the bias is large. If a group changes price in a synchronized fashion, the bias is small.

The bias in d(p) translates into a bias in &. If within-group dispersion falls as between-grour, dispersion falls, then the downward bias in k will shrink with &.A comparison of the dispersion computed over sixty commodity groups with the dispersion computed using the eleven major components of personal consumption expenditure suggests that the bias may be proportional to the level of the dispersion. The two measures of dispersion have a


simple correlation slightly above 0.9 and the more disaggregated one is on average 60 percent higher than the more aggregated one. The consequence of this is that k is underestimated by more when k is high. This implies that the vertical scale in the figures is compressed and that changes in k are artificially deemphasized.

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