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Output effects of disinflation with staggered price setting.

1. Introduction

Monetary authorities who choose to reduce their country's rate of inflation, that is, to disinflate, are usually faced with subsequent recessions. A pattern of slower growth in the money supply, followed by a lower inflation rate and lower output, can be seen in historical statistics, such as those examined by Friedman and Schwartz (1963).

When there was thought to be a permanent Phillips curve trade-off between inflation and unemployment, an important policy question was what combination of inflation and unemployment was most desirable. As economists came to accept the idea of a short-run but not a long-run trade-off between inflation and unemployment, the stage was set to ask about how much output is lost as a result of a disinflation. Okun (1978), using results from several econometric models, estimated that an extra 1% unemployment rate for one year would reduce inflation between one-sixth and one-half of 1%. To reduce inflation by 1%, these figures suggested that the unemployment rate needed to be higher by 2 to 6% of the labor force. If one uses an "Okun's law" of 2.5% of gross national product (GNP) lost for an unemployment rate higher by 1%, then the output lost to reduce inflation by 1% would be between 5 and 15% of GNP.

The term "sacrifice ratio" is now widely understood to mean the ratio of the cumulated percentage loss of output (at an annual rate) to the reduction in the trend rate of inflation. (1) For the disinflation in the early 1980s in the United States, Fischer (1986) figures the sacrifice ratio was about five. Other empirical studies, such as those in Ball (1994b) and Jordan (1997), find a range of estimates around an average sacrifice ratio of about two. (2)

The pattern of slower rates of growth in the money supply being followed by recessions has posed a challenge to theorists to develop plausible models that help us understand this phenomenon. The existence of staggered price setting is one candidate explanation, and Taylor (1979, 1980) deserves credit for introducing formal analyses of how overlapping wage contracts can lead to a persistence in wages and deviations in output from its natural rate during disinflations.

Sargent (1983), in discussing conditions for a successful disinflation without much output loss, characterizes the Taylor model as follows: "In this class of models, in terms of unemployment it is costly to end inflation because firms and workers are now locked into long-term wage contracts that were negotiated on the basis of wage and price expectations that prevailed in the past. ... In addition, the wage contracting mechanism contributes some momentum of its own to the process, so that the resulting sluggishness in inflation cannot be completely eliminated or overcome by appropriate changes in monetary and fiscal policies" (p. 55). Similarly, Vegh (1992), explaining why ending hyperinflations appears to involve less output loss than ending moderate inflations, writes, "The fact that in hyperinflations backward-looking contracts (so prevalent in industrial and chronic-inflation countries) disappears is probably at the heart of the difference in output costs" (p. 656).

In a widely used textbook, Romer (1996, p. 273) writes that "the Taylor model exhibits price level inertia: the price level adjusts fully to a monetary shock only after a sustained departure of output from its normal level. As a result, it is often claimed that the Taylor model accounts for inflation inertia." Romer then adds the provocative sentence: "Ball (1994a) demonstrates, however, that this claim is incorrect." The link between the discrete-time model presented by Romer and the contention attributed to Ball is not immediately evident and needs to be clarified.

Beginning with an environment of steady growth in money and prices, Ball (1994a) assumes that a new regime suddenly announces a fully credible continuous linear decline in the growth rate of money. He also assumes that each firm sets price for one year and that the timing of the changes is spread out evenly over a year. Instead of deriving the path for prices and comparing it with the path of money, Ball uses an indirect approach in which he shows that prices will be below the level necessary to keep output constant if the decline in the money supply does not proceed too rapidly. He reports that the borderline case for no change in output occurs when money growth reaches zero at 0.68 of a year, or after about eight months. Any fully-anticipated steady reduction in the future growth rate of money that will reach zero in more than 0.68 of a year generates increases in output. On the basis of this result, in a bit of an overstatement, Ball concludes, "With credible policy and a realistic specification of stagger ing, quick disinflations cause booms" (p. 289).

In what follows, I will show that the discrete-time model can be readily used to analyze an array of disinflation scenarios, including Ball's, with explicit solutions for the paths of prices and output. This makes it possible to sort out how the degree of credibility may or may not overcome the negative output effects of price inertia imparted by the nonsychronization of price setters and helps build intuition about these tricky issues. The model is essentially the one used by Taylor (1979) in which two groups of workers have their wages fixed for two overlapping periods. The primary difference is that the model will be formulated in terms of price setters rather than wage setters. The basic structure for this analysis can be found in Romer (1996, pp. 265-73).

Section 2 sets out the model, and section 3 characterizes the steady-state solution. Section 4 analyzes the impact of a cold-turkey disinflation. Sections 5 and 6 study policies of gradual disinflation under different assumptions about policy credibility. Section 7 looks at the effect of a credibly anticipated deflation, and section 8 concludes.

2. The Model

The first assumption is that output is determined by the real money supply. A frequently used and simplest form of this hypothesis is the following equation:

y = m - p, (1)

where y is the log of output, m is the log of the money supply, and p is an average of the log of prices set by all firms. We can treat y = 0 as the natural rate of output so that in long-run equilibrium p = m.

The next assumption is that a firm's desired price is an average of m and p:

[p.sup.*] = [Phi]m + (1 - [Phi])p 0 < [Phi] < 1 (2)

There are a variety of ways to justify this specification. In Blanchard and Fischer (1989), for example, a firm's demand depends on both the real money supply and the firm's price relative to the average price set by all firms. Within that framework, the parameter [Phi] is a decreasing function of the degree of substitutability between the goods produced by different firms. If goods are close substitutes, [Phi] is small, and, because of a high elasticity of demand, firms are reluctant to let their prices differ by much from prices set by other firms.

Now suppose that there are two groups of staggered price setters. Half the firms set prices at the beginning of periods t, t + 2, t + 4, . . ., and the other half set prices at the beginning of periods t - 1, t + 1, t + 3, . . . .

Once prices are set, they stay fixed for two periods. Let [x.sub.t-1] be the price set at the beginning of period t - 1 and [x.sub.t] the price set at the beginning of period t. Then the average price in effect in period t will be

[p.sub.t] = .5([x.sub.t-1] + [x.sub.t]). (3)

Those firms that set prices at the beginning of period t choose

[x.sub.t] = .5([p.sub.t.sup.*] + [E.sub.t][p.sub.t+1.sup.*]), (4)

where [E.sub.t] denotes an expected value with information available at time t. The convention used here will be that the concurrent money supply [m.sub.t] is included in that information set.

Substituting from Equations 2 and 3 into Equation 4 and collecting terms gives the following equation:

[E.sub.t][x.sub.t+1] - 2(1 + [Phi])/1 - [Phi][x.sub.t] + [x.sub.t-1] = - 2[Phi]/1 - [Phi][[m.sub.t] + [E.sub.t][m.sub.t+1]] (5)

Romer's solution, after substitution for a redundant parameter, (3) can be written as

[x.sub.t] = [lambda][x.sub.t-1] + [(1 - [lambda]).sup.2]/2 [[m.sub.t] + (1 + [lambda])([E.sub.t][m.sub.t+1] + [lambda][E.sub.t][m.sub.t+2] + [[lambda].sup.2][E.sub.t][m.sub.t+3] + ...)] (6)


[lambda] = (1 - [square root of ([Phi])])/(1 + [square root of ([Phi])]). (7)

This is entirely within the spirit of Taylor's analysis. Note that prices set at time t depend both on prices that were set at time t - 1 and on the current and expected future levels of the money supply. The parameter [lambda] is a decreasing function of [Phi]. A lower [Phi] means that price setters are more concerned about other firms' prices in setting their optimal price. Thus, it makes intuitive sense that there is a greater persistence in prices (higher [lambda]) when firms put more weight on the prices of other firms and less weight on nominal aggregate demand, represented by the money supply variable m.

Equation 6 is extremely flexible in that it can accommodate any scenario about expected future money supplies in the context of staggered prices and can be combined with Equation 3 for the average price level and with Equation 1 for output. It can also be used for dynamic analyses in which subsequent prices depend on lagged prices, on money supplies that may differ from what had been expected, and on revised expectations.

3. Steady Growth of the Money Supply

Suppose that the money supply is expected to grow at a steady rate g:

[E.sub.t][m.sub.t+n] = g(t + n) for n = 0, 1, 2, ... (8)

In other words, the log of the money supply at time t is gt, and it is expected to be gt + g at time t + 1, gt + 2g at time t + 2, and so on. When these projected future money supplies are substituted into Equation 6, the result can be written as (4)

[x.sub.t] = [lambda][x.sub.t-1] + (1 - [lambda])gt + [(1 + [lambda])/2]g. (9)

To find the steady-state solution, assume that the money supply has always grown at a rate g. If [m.sub.t] = gt for all prior t, we can solve Equation 9 by making successive backward substitutions. In lag operator notation, the resulting equation can be expressed as

[x.sub.t] = 1/1 - [lambda]L [(1 - [lambda])gt + 1 + [lambda]/2 g] (10)

where the operator L is such that [Lx.sub.t] = [x.sub.t-1]. After a bit of algebra, Equation 10 boils down to

[x.sub.t] = gt + g/2. (11)

The steady inflation path is then

[p.sub.t] = ([x.sub.t] + [x.sub.t-1])/2 = gt. (12)

Therefore, on the steady inflation path, output is at its natural rate

[y.sub.t] = [m.sub.t] - [p.sub.t] = 0.

no matter how high or low the steady rate of inflation.

4. Cold-Turkey Disinflation and the Sacrifice Ratio

Now consider a conceptual experiment. Suppose that the economy is on a steady inflation path at a rate g and, between the beginning of period t and the beginning of period t + 1, a new monetary regime institutes a lower money growth rate. The algebra is simpler without changing the results if the new lower growth rate is zero. What happens to the average price relative to the money supply, and hence what is the effect on output?

To obtain a formal answer, use Equation 11 for the price set in period t and assume that from period t + 1 onward the money supply is expected to stay at gt. Making these substitutions in Equation 6 for period t + 1 instead of t,

[x.sub.t+1] = [lambda](gt + g/2) + [(1 - [lambda]).sup.2]/2 [1 + (1 + [lambda])(1 + [lambda] + [[lambda].sup.2] + ...)]gt

This reduces to

[x.sub.t+1] = gt + [lambda]g/2. (13)

Successive calculations yield

[x.sub.t+n] = gt + [[lambda].sup.n] g/2. (14)

With 0 < [lambda] < 1, there is a partial adjustment toward the new steady-state price level of gt. Because of the overlapping nature of the price setting, firms have backward- as well as forward-looking influences on their current price, so that the price set at time t exerts a persistent effect on subsequent prices.

To find the average price in period t + 1, substitute for from Equation 11 and for [x.sub.t+1] from Equation 13 into [p.sub.t+1] = .5([x.sub.t] + [x.sub.t+1]) and collect terms to get

[p.sub.t+1] = gt + (1 + [lambda])g/4. (15)

Subsequent average prices are

[p.sub.t+n] = gt + [[lambda].sup.n-1](1 + [lambda])g/4 n = 2, 3,... (16)

Thus, the average price also gradually fails toward the new steady inflation path.

Finally, one can assess the effects on output. The initial effect is

[y.sub.t+1] = [m.sub.t+1] - [p.sub.t+1] = -(1 + [lambda])g/4 < 0. (17)

A cold-turkey policy of disinflation unambiguously depresses output. Parenthetically, the model also implies an output gain when there is an unexpected but permanently higher rate of money growth. (5) This is the traditional view of how overlapping prices impart a real effect when money growth rates are changed.

One can readily use this stylized model to obtain an analytical expression for the sacrifice ratio. Suppose, after the cold-turkey cessation in the growth of money, there are no further expected or actual changes in the money supply. In that case, output in subsequent periods will gravitate back to its natural rate in accordance with the following expression:

[y.sub.t+n] = -(1 + [lambda])[[lambda].sup.n-1]g/4. (18)

The total loss in output from cold turkey disinflation is

(1 + [lambda])(l + [lambda] + [[lambda].sup.2] + .. )g/4 = (1 + [lambda])g/4(1 - [lambda]).

The sacrifice ratio (SR) is the ratio of the total output loss to the reduction of inflation g. In this case,

SR = (1 + [lambda])/4(1 - [lambda]). (19)

With this expression, it is easy to see that greater substitutability between goods produced by different firms, which generates more persistence in prices as reflected in a higher [lambda], results in a larger sacrifice ratio. If [lambda] = 0.8, then the sacrifice ratio would be 2.25 in this simple model. (6) With y representing the log of deviations of output from its natural rate, this can be interpreted as representing a 2.25% loss of output for each 1% reduction in the inflation rate. In other words, going from a 2% inflation rate to no inflation would result in a total one-time cost of 4.5% of output.

5. A Two-Period Decline in Money Growth

What happens if a disinflation policy is introduced in two steps? Suppose that the rate of growth of money drops to g/2 in period t + 1 and will fall to zero in period t + 2. The effect on output depends crucially on whether price setters at time t + 1 believe that the money growth rate will fall in the next period.

Suppose first that this two-step disinflation is semicredible. By this I mean that price setters expect the new lower growth in money to continue into the future, but they do not believe the still lower growth rate in the future until they see it. Once again, utilize Equation 6 with money expected to increase each period by g/2. In that case,

[x.sub.t+1] = [lambda](gt + g/2) + [(1 - [lambda]).sup.2]/2{gt + g/2 + (1 + [lambda])[(gt + 2g/2) + [lambda](gt + 3g/2) + [[lambda].sup.2](gt + 4g/2) + ...]}

which reduces to

[x.sub.t+1] = gt + g/2 + (1 + [lambda])g/4.

The average price in period t will be

[p.sub.t+1] = gt + g/2 + (1 + [lambda])g/8,

and the output effect is

[y.sub.t+1] = -(1 + [lambda])g/8. (20)

Not surprisingly, the initial impact effect is half that of the cold-turkey policy.

After observing no change in the money supply from time t + 1 to t + 2, price setters beginning at time t + 2 now expect no growth in money to continue. Then, one can show

[x.sub.t+2] = gt + g/2 + [lambda](1 + [lambda])g/4, [p.sub.t+2] = gt + g/2 + [lambda][(1 + [lambda])].sup.2]g/8 and

[y.sub.t+2] = -[(1 + [lambda]).sup.2]g/8. (21)

The introduction of a semicredible disinflation over two periods depresses output less in the initial period but depresses it somewhat more in the second and subsequent periods than with the cold-turkey disinflation. (7) However, the total output loss and the sacrifice ratio can be shown to be precisely the same.

Next consider what happens if future declines in the money growth rate are fully credible. In that case,

[x.sub.t+1] = [lambda](gt + g/2) + [(1 - [lambda]).sup.2]/2 [1 + (1 + [lambda])(1 + [lambda] + [[lambda].sup.2] + ...)](gt + g/2) or [x.sub.t+1] = gt + g/2.

The average price rises from gt to gt + g/2. Since m and p have the same value in period t + 1, there is no initial real impact on y as a result of this fully credible two-step disinflation policy. In fact, from period t + 1 on, there are no real impacts because m and p stay at the same level. In the discrete-time formulation, this two-step drop in money growth is an intermediate case between the traditional view that disinflation induces a recession when there is overlapping price setting and Ball's claim that a disinflation may cause a boom despite overlapping price setting. The depressing effect on output from the initial drop in the growth rate in money is exactly offset by lower growth in prices because of the assumed expectation of an even lower money growth rate in the future.

6. A Decline in Money Growth over Three or More Periods

Suppose that the money growth rate is reduced to 2g/3 in period t + 1, to g/3 in period t + 2, and held constant thereafter. As in the case with a semicredible two-step disinflation, the sacrifice ratio with a semicredible three-step disinflation is again precisely the same as it is with the cold-turkey disinflation. In this regard, it is interesting that when Chadha, Masson, and Meredith (1992) use the MULTIMOD model to simulate estimates of output lost to bring inflation down by 4%, they get similar estimates for a (semicredible) phased-in disinflation as for a cold-turkey disinflation. In both cases, the sacrifice ratio is slightly less than two.

In the case of fully credible future reductions in money growth rates, Equation 6 can be used this time with [m.sub.t+1] = gt + 2g/3 and [E.sub.t+1][m.sub.t+n] = gt + g for n = 2, 3, ..., to show that

[x.sub.t+1] = gt + (5 - [lambda] - [[lambda].sup.2])g/6 and (22)

[p.sub.t+1] = gt + (8 - [lambda] - [[lambda].sup.2])g/12 (23)

In this case, [p.sub.t+1] is below [m.sub.t+1], and therefore real output becomes positive:

[y.sub.t+1] = ([lambda] + [[lambda].sup.2])g/12. (24)

This illustrates the thrust of Ball's claim that a credible disinflation policy may result in a boom. In the discrete-time framework, any fully anticipated steady decline in the future growth rate of money introduced over more than two periods results in a boom. Output expands whenever the effect of the expected future decline in the growth rate of money outweighs the backward-looking effect of overlapping prices as money growth decreases. If we think of the period as six months, then fully credible disinflations spread out over a year or more will not have negative output effects. (8)

7. Deflation

Ball (1994a, p. 286) writes, "The result that disinflation causes a boom may appear counterintuitive, given the usual view that disinflations cause recessions. The source of confusion is that many papers on staggering, such as Blanchard (1983), study decreases in the level of money ... such policies do reduce output ... decreases in levels and decreases in growth rates have very different effects."

More important than the level-growth dichotomy is the nature of beliefs about future policies. The way Ball analyzes deflation is analogous to the cold-turkey disinflation analyzed here. However, to illustrate further the critical nature of credibility, consider the effect of a future deflation in a noninflation environment. Suppose the money supply has been constant at m and had been expected to stay at that level. The resulting steady-state solution for the price level is m, and output y = m - p = 0. Then suppose that, before prices are set at time t, the monetary authorities credibly announce a deflationary policy of a one-unit permanent reduction in the level of the money supply beginning at time t + 1.

Formally, set [x.sub.t-1] = m, [m.sub.t] = m, and [E.sub.t][m.sub.t+j] = m - 1 for j = 1, 2 .... If these values are substituted into Equation 6, the solution for [x.sub.t] is

[x.sub.t] = m - (1 - [[lambda].sup.2])/2.

This implies that [p.sub.t] = m - (1 - [[lambda].sup.2])/4 and [y.sub.t] = [m.sub.t] - [p.sub.t] (1 - [[lambda].sup.2])/4 > 0. Thus, a future reduction in the level of the money supply, if believed in current price-setting decisions, results in an immediate stimulus to output. (9)

8. Concluding Comments

Many countries have tried to bring down high rates of chronic inflation, and countries with low inflation have been urged to aim for price stability. (10) There usually are acknowledged costs to such disinflations. The stickiness of wages or of prices because of overlapping wage contracts or staggered price adjustments has been identified as a major reason for these costs.

Ball's (1994a) analysis, which has been reformulated here in a discrete-time framework to facilitate explicit solutions for the paths of prices and output, suggests that overlapping prices are not sufficient to cause recessions if a fully credible disinflation does not proceed too rapidly. The closest analytical antecedent to the analysis here can be found in Phelps (1978). Contemporaneously with Taylor's contributions but far less frequently cited, Phelps has essentially the same model. He shows that future reductions in the money supply, if believed, can bring down inflation without lowering employment. With two overlapping contracts, one of his results corresponds to the case here of a fully credible two-period decline in the rate of growth in money. Chadha, Masson, and Meredith (1992) also calculate fully credible costless disinflation paths with their models.

There is, however, a potential tension between gradual disinflation and credibility. For example, Sargent (1983, p. 91) comments, "Gradualism invites speculation about future reversals, or U-turns, in policy." Also, Taylor (1983, p. 992) writes, "The difficulty with [a gradual reduction in money growth] in practice is that wage negotiators have to be convinced that the deceleration will come later even though it is not occurring today. ... At the heart of this credibility is a time-inconsistency problem."

Considerable evidence exists that expectations are adaptive, that people tend to extrapolate the most recently perceived rate of inflation. Therefore, credibility of a disinflation policy will need to be gained by demonstration. As Sargent notes, some people may even expect a reversal of a recent disinflation that is proceeding gradually. I have used the term "semicredible disinflation" to capture the idea that price setters do believe any lower growth rate in money will continue but do not act on future declines in money growth until they occur. In that case, in the model presented here, the cold-turkey analysis applies to a gradual disinflation. The decline in output is more spread out with a gradual disinflation, but the total cost in lost output and the sacrifice ratio are the same as with a cold-turkey disinflation.

Thus, when the monetary authorities must first demonstrate a lower growth rate in money to be convincing about continued money growth at that rate in the future, the Phelps-Taylor model retains its reputation as a rationale not only for inflation persistence but also for outputdepressing effects of disinflation.

Received January 2000; accepted May 2001.

(1.) The origin of the term itself is a bit of a mystery. The earliest I have found it in print is in Gordon and King (1982, p. 229 ff), who have a section on the sacrifice ratio as if it were already a welt-known term. I have not yet found it explicitly used earlier by Okun.

(2.) Baltensperger and Kugler (2000) discuss how the sacrifice ratio can he affected by the degree of central bank independence. Boschen and Weise (2001) examine relationships between inflation duration and the costs of disinflation.

(3.) To establish the equivalence of Equation 6 with equation 6.80 in Romer (1996, p. 271), note that Romer's coefficient [lambda](1 - 2A)/(2A) can be shown to equal [(1 - [lambda]).sup.2]/2 by substituting for A = [lambda]/(1 + [[lambda].sup.2]), which is implied by his equation 6.68: A + A [[lambda].sup.2] = [lambda].

(4.) The derivations of this and other formulas presented in the following are contained in an appendix that is available on request.

(5.) For example, see Cecchetti (1994). Also, Jordan (1997) estimates what he calls "benefice ratios" as well as sacrifice ratios.

(6.) West (1988) and Phaneuf (1990), using variants of the Taylor model, suggest that near random walk behavior in aggregate output is not inconsistent with a persistence parameter such as [lambda] in the range between .80 and .99.

(7.) For example, one can readily show, with 0 < [lambda] < 1, that the second period output in Equation 21 is greater in absolute value than second period output under cold turkey from Equation 18 with n = 2.

(8.) The continuous-time nature of Ball's analysis accounts for his figure of 0.68 years for the borderline between recession and boom being less than the one-year borderline in the discrete-time version.

(9.) This case does not completely satisfy Ball's definition of a boom, in which output never falls below its natural rate. With persistence in prices, when the money supply is eventually lowered as expected, output will then fall below its natural rate.

(10.) See, for example, Feldstein (1997) for the United States and Todter and Ziebarth (1997) for Germany.


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Gordon, Robert J., and Stephen R. King. 1982. The output cost of disinflation in traditional and vector autoregressive models. Brookings Papers on Economic Activity 1982(1):205-42.

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Phaneuf, Louis. 1990. Wage contracts and the unit root hypothesis. Canadian Journal of Economics 23:580-92.

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Sargent, Thomas J. 1983. Stopping moderate inflation: The methods of Poincare and Thatcher. In Inflation, debt, and indexation, edited by R. Dornbusch and M. H. Simonsen. Cambridge, MA: MIT Press, pp. 54-96.

Taylor, John B. 1979. Staggered wage setting in a macro model. American Economic Review 69:108-13.

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Todter, Karl-Heinz, and Gerhard Ziebarth. 1997. Price stability vs. low inflation in Germany: An analysis of costs and benefits. NEER working paper 6170.

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West, Kenneth D. 1988. On the interpretation of near random walk behavior in GNP. American Economic Review 78:202-9.

John A. Carlson (*)

(*.) Department of Economics, Purdue University, West Lafayette, IN 47907, USA; E-mail

I am grateful to an anonymous referee, to Neven Valev, and to workshop participants at Victoria University of Wellington for helpful comments on earlier versions of this paper.
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