Journal of Monetary Economics 29 (1992) 371-388. North-Holland

Why does high inflation raise inflation uncertainty?*

Laurence Ball Princeton Universiry, Princeton, NJ 08544, USA

Received August 1990, final version received May 1992

This paper presents a model of monetary policy in which a rise in inflation raises uncertainty about future inflation. When actual and expected inflation are low, there is a consensus that the monetary authority will try to keep them low. When inflation is high, policymakers face a dilemma: they would like to disinflate, but fear the recession that would result. The public does not know the tastes of future policymakers, and thus does not know whether disinflation will occur.

1. Introduction

Economists frequently argue that a rise in the level of inflation raises uncer- tainty about future inflation. This idea is a central theme, for example, in Okun’s (1971) ‘The Mirage of Steady Inflation’ and in Friedman’s (1977) Nobel Lecture. Many empirical studies provide evidence of such an effect. But the accompany- ing explanations for the effect are usually loose (Okun, for example, uses an analogy to driving over a bumpy road). It appears that economists find the inflation level-uncertainty relation plausible but have trouble pinning down why. This paper attempts to improve our understanding of this relation by presenting a model that predicts it.’

The idea behind the model is simple: high inflation creates uncertainty about future monetary policy. To understand why, consider first a period of low

*This paper is a revised version of NBER Working Paper 3224. I am grateful for excellent research assistance from Aniruddha Dasgupta and Boris Simkovich, and for suggestions from Albert0 Alesina, Alan Blinder, Stephen Cecchetti, Robert Gibbons, N. Gregory Mankiw, David Romer, Lawrence Summers, and numerous seminar participants. An anonymous referee was unusually helpful.

‘The empirical evidence for an infiation-uncertainty link is summarized in Ball and Cecchetti (1990). Cukierman and Meltzer (1986) and Devereux (1989) present other recent explanations for the link; these papers are discussed below.

0304-3932/92/%05.00 % 1992-Elsevier Science Publishers B.V. All rights reserved

J.Mon B

372 L. Ball. Inflation and inflation uncertainty

inflation, such as the early 60s in the United States. In this situation, it is a good bet that the Fed is happy with the status quo and will attempt to prolong it. Inflation may rise at some point for a reason exogenous to the Fed, such as spending on Viet Nam. It is unlikely, however, that the Fed will simply decide that it is desirable to inflate.

Contrast this situation with a time of high inflation, such as the late 70s. Now it is not obvious what the Fed will do, because it faces a dilemma: it would like to disinflate, but fears the recession that would probably result. It is likely that disinflation will occur eventually, but the timing is uncertain. It depends on factors that are difficult to gauge in advance, such as political events and the values of policymakers. In the late 70s it would have been difficult to predict that sharp disinflation would arrive in 1981-82.2

These ideas are similar to some previous discussions of inflation uncertainty. Logue and Willet (1976) argue that ‘at higher average rates [of inflation] government financial policy will tend to be less stable as it tries to bring inflation under control while avoiding steep recession’. Fischer and Modigliani (1978) suggest that ‘governments typically announce unrealistic stabilization programs as the inflation rate rises, thus increasing uncertainty about what the actual path of prices will be’. And Friedman argues that ‘[a] burst of inflation produces strong pressure to counter it. Policy goes from one direction to the other, encouraging wide variation in . . . inflation’. The common theme is that high inflation creates uncertainty about how policy will respond.

This paper formalizes this idea by applying recent advances in the positive theory of monetary policy. Specifically, I make two modifications to Barro and Gordon’s (1983) model of the repeated game between the Fed and the public. First, following Canzoneri (1985), I introduce exogenous shocks that cause low- inflation equilibria to break down occasionally. The economy alternates be- tween periods of high and low inflation, and I can compare uncertainty in the two situations. Second, following Alesina (1987), I capture policy uncertainty by assuming that there are two policymakers who alternate in power stochastically. One policymaker is a conservative (C) whose sole objective is to keep inflation low. The other is a liberal (L) who cares about unemployment as well as inflation.

These assumptions lead naturally to a link between inflation and uncertainty. Since C hates inflation, when it is low he tries to keep it low, and when it is high he disinflates. L also tries to prolong low inflation - he resists the temptation to create an inflationary boom. However, if actual and expected inflation are high, L is unwilling to create a recession to disinflate. When inflation is low, the public is certain of future policy because C and L do the same thing. High inflation

2As this example suggests, the model is meant to apply primarily to moderate-inflation countries like the U.S. The experience of high-inflation countries (e.g., in Latin America) depends largely on factors that I ignore, such as the use of seigniorage to finance budget deficits.

L. Ball. Injlation and inflation uncertainty 373

creates uncertainty because the policymakers respond differently to the disinfla- tion dilemma and the public does not know who will be in charge.

There are two versions of my model. In the first, as in previous work, policymakers attach a cost to the level of inflation. While this specification is plausible, the subject of the paper suggests an alternative. The inflation-uncer- tainty link is important because it helps to explain why inflation is costly: economists often argue that inflation has small costs if it is perfectly anticipated but larger costs if it raises uncertainty. Motivated by this view, the second model includes a cost of uncertainty as well as a smaller cost that depends on the inflation level. This specification creates a paradox: conservatives view inflation as costly because it creates uncertainty, but uncertainty arises from their efforts to disinflate (along with liberals’ resistance). It appears that conservatives would do better by accepting high inflation. I show, however, that this may not be true.

The rest of the paper contains five sections. Section 2 presents the basic model. Section 3 describes behavior that produces an inflation-uncertainty link, and section 4 determines when this behavior is an equilibrium. Section 5 considers the model with a cost of uncertainty, and section 6 concludes.

2. The model

The model combines elements of Barro and Gordon (1983), Canzoneri (1985) and Alesina (1987). There are two policymakers, C and L. L’s loss function includes both unemployment and inflation:

z, = (U, - uy2 + ml: )

where Z, is L’s loss in period t, U is actual unemployment, U” is socially optimal unemployment (a constant), n is inflation, and a is a taste parameter. In contrast, policymaker C cares only about inflation: his loss is simply

z;=n:.

[Equivalently, C’s loss is (1) with a + co .I3 Unemployment is determined by a short-run Phillips curve:

u, = UN - (?T* - n:), UN = u”+ 1,

where UN is the natural rate of unemployment and rr: is expected inflation at

3This specification of the conservative’s tastes follows Backus and Driffill (1985). The working paper version of this paper [Ball (1990)] allows less extreme tastes: C minimizes (1) with a weight on z that is larger than L’s, but finite.

374 L. Ball, InJalion and inJIation uncertainty

t given information at t - 1. As in previous work, UN > U” creates the time- consistency problem that leads to inflation. (The assumptions that UN - U” = 1 and that the coefficient on rc - rce equals one are normalizations on the units of U and n.) Substituting (3) into (1) yields L’s loss in terms of actual and expected inflation:

z, = (71, - 7c; - 1)2 + an:. (4)

The policymakers gain and lose power stochastically. Switches are a Markov process: if L is in power in period t, then with probability c he is replaced by C at t + 1, while C is replaced by L with probability 1. The public observes which policymaker is currently in power after setting expected inflation. With these assumptions, the policymakers can be interpreted as political parties with known preferences that replace each other at irregular intervals.

As in Canzoneri, the Fed does not control inflation perfectly. This assumption is needed for high inflation to arise occasionally, since (in the equilibrium below) policymakers never inflate on purpose. Each period, the policymaker in power chooses a target inflation rate rr*. Actual inflation equals the target plus a shock:

77, = n: + E, . (5)

The shock E, can be interpreted as a monetary control error or shift in money demand. It is serially uncorrelated and distributed N(0, a’). The public does not observe E, so it cannot tell whether a rise in inflation is intentional.4

The two policymakers play a simple repeated game. At the start of each period, the public sets expected inflation; expectations are assumed to be rational. Then the current policymaker is determined and he chooses the target rr*. Finally, the inflation shock arrives, determining actual inflation. In choosing rr*, the policymaker in power minimizes the expected present value of his loss, with discount factor /I, putting equal weight on periods when he is in and out of power. Each policymaker takes the other’s behavior as given.

3. A proposed equilibrium

This section proposes behavior by the two policymakers that produces a positive relation between inflation and uncertainty. Section 4 determines when this behavior is an equilibrium and discusses other equilibria.

4This specification is a simplification of Canzoneri’s model. In Canzoneri, the inflation shock is derived by assuming that the Fed chooses the money stock and that money demand is stochastic. The money demand shock is observable - the public infers it from the money stock and the price level - but the Fed has a forecast of the shock that is private information. Adopting this more sophisticated approach would not change my main results.

L. Ball, hflation and inJlation uncertainr~ 375

Table 1

The proposed equilibrium.

P,_, =c P,_, = L. KY_, 5 ji P,_, = L, x,_l > n

n:

0 0

(1 - c)n+

7-c: if P, = C 71: if P, = L

0 0 0 0 ry

3.1. Policymakers’ targets

The proposed equilibrium is presented in table 1. P, is the identity of the policymaker at t (C or L). The equilibrium is a variation on the ‘trigger strategy’ equilibrium for one policymaker in Canzoneri. In general, the public’s expec- tations and policymakers’ inflation targets depend on inflation in the previous period and on who was in charge. If the previous policymaker was C then 9 is zero regardless of past inflation, and both policymakers target zero if in power. If the previous policymaker was L, then behavior depends on whether previous inflation exceeded ii, a positive bound. If inflation was below the bound, then expected inflation and the two targets are again zero. If inflation exceeded the bound, then C still targets zero, but L targets n+ > ?c (the values of rc+ and ?c are derived below). Since L remains in power with probability (1 - c), expected inflation in this case is (1 - c)n+.

The intuition for this behavior is the following. When expected inflation is zero, both policymakers target zero to maintain the status quo. L is tempted to raise his target to reduce unemployment, but he is deterred by the fear that x will exceed it, triggering a rise in expected inflation. (As described below, i2 is chosen to provide the right amount of deterrence.) No deterrent is needed for C, who does not wish to reduce unemployment; thus rre never rises after C is in power. Finally, when rce is positive, C disinflates but L does not, because he is unwilling to accept the recession implied by n < rce.

Over time, the economy alternates between periods of low (zero) and high (positive) expected inflation. When rre is low, it stays low until there is a large shock with L in power, so that 7~ > 5 even though L targets zero. 7ce rises in the next period and stays high while L remains in power: a high rce induces L to target 7c+ > %, which keeps rce high. Eventually C arrives and disinflates, and 7re returns to zero after that. (In principle, 9 can also return to zero through a negative shock that pushes n: below 7~. However, as described below, suffi- ciently large shocks are rare.)

3.2. The inflation-uncertainty relation

What is the relation between the level of inflation and uncertainty? This relation can be defined in slightly different ways. One measure is the relation

376 L. Ball, Infiaiion and ir$ation uncertaini>

between rte -the conditional mean of inflation - and the conditional variance. In table 1, rce is either zero or (1 - c)rr +. When rce = 0, there is no uncertainty about rr*, because both policymakers target zero. When ne = (1 - c)rr ‘, rr* is zero with probability c and rr+ with probability (1 - c); its variance is c(l - c)(n+)*. The variance of the error E is o* in all states, and E is uncorrelated with n*. Combining these results, the conditional variance of inflation is

E, _ 1 [(Tc, - n;)*] = r~* if 7ce=0, f (6)

= IT* + c(1 - c)(rr+)* if 7~: = (1 - c)rc+.

A rise in expected inflation raises the variance, because there is greater policy uncertainty.

We can also determine the relation between the variance of rr, and the previous level of actual inflation, n,_ i. If x,_ 1 I it, then 7~: = 0, so the variance of II, is r~*. If rr,_ 1 > 2, then the variance depends on who was previously in charge: if P,_ 1 = C, then the variance is still G* (because rcf is still zero); if P,_ 1 = L, the variance is G* + c( 1 - c) (x+)*. Thus, looking across periods, the relation between n,_i and uncertainty is positive but not perfect. A high 7t,_ 1 raises uncertainty whenever it raises expected inflation, but this occurs only when L was in charge.5

4. When is it an equilibrium?

This section derives conditions for table 1 to be a perfect Nash equilibrium. This requires that neither C nor L can gain by deviating. I also briefly discuss other equilibria of the model. Throughout, I sketch the argument and provide details in the appendix.

4.1. The behavior of C

Clearly C cannot gain by deviating from table 1. Since E has mean zero, TI* = 0 minimizes the expectation of C’s one-period loss, x2. Further, it is optimal for C to minimize his current loss, because his behavior does not affect the future. If C is in power, 9 is zero in the following period regardless of his action.

4.2. X and n’

To see whether L deviates from table 1, I first derive conditions defining % and rrf (see appendix). Following Canzoneri, the trigger ?I is chosen to assure that L targets zero inflation when ne = 0. In particular, the gain from a surprise

5Ball (1990) presents a variation on the current model in which the public does not observe which policymaker was previously in power. In this case, there can be an equilibrium in which n: and uncertainty rise every time n,_ 1 exceeds 5. Thus the relation between 71,~ 1 and uncertainty is tighter than in table 1.

L. Ball, Injlation and inJlalion uncertainty 377

increase in n* must balance the increased risk that 7~ exceeds it, raising expected inflation. The 5 solutions for various parameter values are presented below.

nc+ is defined by the condition that L targets rc+ when rce = (1 - c)rc+. The appendix shows that rr+ > E and 7c+ < l/(a + c). l/(a + c) is L’s ‘one-shot’ target: the one he would choose if he minimized only his current loss. L holds his target below the one-shot level to raise the probability that a negative shock pushes rr below i2, reducing the next period’s +. This effect on rc+ is small, however, because such accidental disinflation is rare (it requires a very large shock). Thus rc+ is very close to l/(a + c) in the numerical results.

4.3. SufJicient conditions for the equilibrium

As described above, ii and 7c+ are chosen to induce L to obey table 1. Thus table 1 is an equilibrium as long as solutions for these constants exist. This condition does not hold in general, but the appendix derives three parameter restrictions that guarantee an equilibrium. Here I present these conditions and briefly describe why they are needed.

The three sufficient conditions are

c < 1 - u/m, (7)

g < 6, (8)

a < (1 + /?cz - fi)/(2fi - 2pc - l), (9)

where 5 > 0 is derived numerically (see appendix). Conditions (7) and (8) assure that it can be chosen to deter L from creating surprise inflation. Intuitively, (7) bounds the probability that a liberal policymaker is replaced by a conservative. If c is too large, the ‘high’ level of expected inflation, (1 - c)rc+, is low. Thus 7~~ does not rise much if L inflates. With only a small punishment for inflation, it is impossible to induce L to target zero.

Condition (8) bounds the variance of inflation control errors. If c is too large, inflation movements are dominated by shocks, and n* has little effect on whether z exceeds a trigger. In this situation, a rise in n* does not significantly increase the probability of punishment. Thus it is again impossible to induce 7r* = 0.

Condition (9) guarantees a solution for z+, L’s positive inflation target. This condition is an upper bound on a, which measures L’s distaste for inflation. Intuitively, if L hates inflation, he targets low inflation even if rce is high - like C, he disinflates. In this case, table 1 cannot be an equilibrium. Given (9), however, L believes that disinflation is not worth the cost in unemployment.6

6Note that L is unwilling to accept a recession to disinflate, but forgoes the boom that he could create by raising inflation when nc is low. This behavior arises from a crucial asymmetry: the gain from a boom is smaller than the cost of a recession. The source of this asymmetry is the convexity of L’s loss function [see Ball (1990) for details].

378 L. Ball, InJation and i@ztion uncertainty

Conditions (7)-(9) appear empirically plausible. The upper bound on c is close to 4 when the discount factor is close to one. Low values of c fit the U.S. experience: while liberals who tolerate inflation are replaced eventually, the probability that this happens in a given period (quarter or year?) is not too large. High inflation persisted for several years before the arrival of Volcker. The restriction on a also appears realistic: liberals (William Miller?) do not care enough about inflation to raise unemployment. Finally, while the bound on 0 is too complicated to interpret, the numerical results below suggest that it is not very restrictive.

4.4. Numerical results

To better understand the results, I examine the behavior of the model for various parameter values. I start with the following base case:

p = 0.9, c = 0.25, a = 0.1, 0 = 0.3.

These values are plausible if a period is a year. /3 = 0.9 is a reasonable discount factor. c = 0.25 means that a liberal on average lasts four years (l/0.25) before a conservative arrives. The parameter a is difficult to determine directly, because economists are unsure of the costs of inflation. Thus I choose an a that produces plausible equilibrium inflation. This requires choosing the units in which infla- tion is measured - the units that fit the normalizations in the Phillips curve (2). The appendix argues that one unit of 7~ should be interpreted as three percentage points of annual inflation. With this assumption and the base parameter values, l/(a + c) is 2.9, or 8.7% inflation. Recall that n + is close to l/(a + c); thus the ‘high’ inflation target is near 9%, roughly the level in the late 70s.

Finally, G = 0.3 means (again multiplying by three) that the standard devi- ation of inflation control errors is 0.9 percentage points. This assumption appears generous, since control errors contribute little to annual movements in U.S. inflation. When the Fed tries to keep inflation stable - when it neither disinflates nor accomodates major macro shocks - inflation usually changes by less than one percentage point.7

Table 2 shows the behavior of the model in the base case, and examines robustness by varying each parameter. In the base case, the bound E is 0.48, and X+ is 2.9 [i.e., it is indistinguishable from l/(a + c)]. The probability that a shock pushes 7t > ii when 7c * = 0 is 1 - F(x) = 0.06; thus rises in expected inflation are fairly uncommon. The probability that n < ?t when L targets x+, F(E - rc+), is less than 10p9: accidental disinflation is extremely rare. Finally, the three

‘Inflation was stable. for example, in the early 1960s and much of the 1980s. The errors in the model should not be interpreted as macro shocks like OPEC or Viet Nam, because these events are observable. The shocks in the model are unobservable. (As in Canzoneri, this assumption is needed for accidental increases in inflation to trigger higher expected inflation.)

B c a

0.90 0.25 0.10 0.75 0.25 0.10 0.95 0.25 0.10 0.90 0.10 0.10 0.90 0.50 0.10 0.90 0.25 0.05 0.90 0.25 0.25 0.90 0.25 0.10 0.90 0.25 0.10

L. Ball, InJation and inJation uncertainty

Table 2

Numerical results.

cl 75 1 -F(C) Il+ F(n‘-7rn+) 85

0.3 0.48 0.06 2.86 0.00 0.3 0.37 0.11 2.86 0.00 0.3 0.51 0.04 2.86 0.00 0.3 0.68 0.01 5.00 0.00 0.3 [no solution] 0.3 0.51 0.04 3.33 0.00 0.3 0.39 0.10 2.00 0.00 0.1 0.22 0.01 2.86 0.00 0.5 0.56 0.13 2.86 0.00

0.64 0.47 0.71 1.62

0.74 0.45 0.64 0.64

-

-

379

(7)~(9)?

yes yes yes yes no

yes yes yes yes

conditions for an equilibrium are satisfied in the base case. In particular, given the other parameters the bound on CJ is 0.64, well above the assumed value of 0.3.

Most variations on the base case in table 2 produce qualitatively similar results. The only exception is a rise in c to 0.5; with c too large, (7) is violated and it is impossible to deter L from inflating. Overall, the behavior described in table 1 appears robust.

4.5. Other equilibria

As in other infinite-horizon models of monetary policy, there are many perfect Nash equilibria for given parameter values. There is always a ‘discretionary’ equilibrium in which C and L target their one-shot rates every period. In addition, table 1 coexists with equilibria involving more sophisticated behavior, such as more complicated punishments for a rise in inflation [see Rogoff (1989)].

This paper will not attempt a full analysis of the multiplicity problem. A reason for focusing on table 1 is that the behavior of expectations is realistic. rce depends on inflation and political events in the recent past, as in actual economies. In particular, policymakers can keep rce low through low actual inflation. The one-shot equilibrium is implausible because L is expected to inflate regardless of his past performance.

What is the likely outcome of the model if conditions (7)-(9) fail, so table 1 is not an equilibrium? There is no definite answer, since there are still many equilibria. It is worth noting that, even without (7)-(9) there can be equilibria that are qualitatively similar to table 1. For example, even if (7) or (8) fails, so a zero target for L cannot be sustained, there can be an equilibrium in which L chooses a positive but low target when ?te is low and a higher target when rre is high. Intuitively, the temptation to cheat is smaller when the low target is positive, because the marginal cost of inflation is higher. Since C sets rc* = 0, there is some uncertainty about rc* even when nCe is low, but less than when rce is

380 L. Ball, Inflarion and inflation uncertainty

high. I conjecture that this kind of equilibrium exists under very broad condi- tions.

5. Costs of uncertainty

5.1. Motivation

The previous sections assume that policymakers attach a cost to the level of inflation. While this assumption is plausible, the subject of the paper suggests an alternative. Economists are interested in the inflationuncertainty link largely because it helps to explain why inflation is costly. The costs of anticipated inflation, such as deadweight loss from the inflation tax, appear small. But if inflation creates uncertainty, there may be significant costs, such as greater risk in long-term nominal arrangements [Jaffee and Kleiman (1977), Fischer and Modigliani (1978)]. Motivated by this view, this section assumes that pol- icymakers’ losses depend on uncertainty about inflation as well as its current level. The effect of the current level can be very small.

This version of the model raises a new issue. In the previous section, uncer- tainty arises from C’s efforts to disinflate, along with L’s resistance. C’s motiva- tion is his view that inflation is costly. But if the main cost of inflation is uncertainty, it appears that C creates the cost by trying to eliminate it! If C simply accepted high inflation, like L, then uncertainty would greatly dimin- ish, eliminating the main cost of inflation. Why does C not take this course?

This section makes two points relevant to this issue. First, table 1, with its positive inflation-uncertainty relation, can be an equilibrium even when the model includes a cost of uncertainty. Second, while there are other equilibria with more stable inflation, C may not prefer these regimes. As suggested by Fischer and Summers (1989), reducing the costs of inflation - in this case by reducing uncertainty ~ can raise the level of inflation so much that policymakers are worse off.

5.2. An inflation-uncertainty link

Here I add a cost of uncertainty to policymakers’ loss functions and show that table 1 can remain an equilibrium. Modify the losses (1) and (2) to be

Z, = (U, - U‘J2 + an: + bE,[(n,+ 1 - TC:+~)~], (10)

Z; = a’7cf + h’E,[(rc,+ 1 - ?I:+ 1)2]. (11)

This specification attaches costs to both the current level of inflation and the variance of next period’s inflation. The first cost can be interpreted as dead- weight loss from the inflation tax, and the second as increased risk in nominal

L. Ball, Inflation and infation uncerrainry 381

contracts. One should think of a and a’ as small, so that arc2 and u’rc2 are small for moderate inflation rates.

The appendix determines when table 1 is an equilibrium in this model. As in the basic model, sufficient conditions are upper bounds on c, 0, and L’s distaste for inflation. Here, L’s tastes are measured by a linear combination of a and b.

The conditions place no restrictions on the relation of a to b and a’ to b’; thus the equilibrium survives even if the level of inflation is much less costly than uncertainty. As in the basic model, the target 7~~ is slightly below l/(u + c). Intuitively, the cost of uncertainty has little effect on L’s choice of rc+ because uncertainty is the same for all rc > 2.

It is perhaps surprising that table 1 is an equilibrium without further condi- tions. In particular, if u’<<b’, C’s behavior appears paradoxical. He disinflates because he dislikes inflation, but the major cost of inflation - uncertainty - arises from his efforts to disinflate. Nonetheless, this behavior is a perfect Nash equilibrium. The public always expects C to target zero inflation, and with Nash behavior C takes these expectations as given. If C deviated from the equilibrium by producing high inflation, the public would still expect him to disinflate in the future, and L not to disinflate. Thus the deviation would not reduce uncertainty; it would simply raise the current level of inflation, increasing C’s loss for a’ > 0 (even if a’ is very small). The unconditional expectation that C targets zero means that he cannot gain from any other behavior.8

5.3. Does C prefer high inflation?

While table 1 is a Nash equilibrium, there are again other equilibria, and table 1 may not appear the most ‘natural’ outcome. In particular, there can be an equilibrium in which C targets high inflation when L does: the self-fulfilling expectation that C disinflates need not arise. It appears that C would prefer such a regime because of the reduced policy uncertainty. And it is plausible that C could guide the economy to the equilibrium he prefers. For example, he might escape table 1 by announcing that he will never disinflate and carrying out this promise until he is believed.9

Here I show that this argument need not lead us to reject table 1. Perhaps surprisingly, even if C can move the economy to more stable inflation, he may not want to. One can show this by constructing an equilibrium in which C targets high inflation and comparing C’s losses to those under table 1.

‘Note that ne falls to zero, eliminating policy uncertainty, in the period after C gains power. ne and uncertainty rise again the next time ?T exceeds % with L in power. C cannot affect any of this behavior through his choice of targets.

‘This idea is informal, because there is no formal theory of how announcements move the economy from one Nash equilibrium to another. Previous authors often assume that policymakers can move the economy to a desired equilibrium [e.g., Taylor (1983)].

382 L. Ball, Inflation and injation uncertainty

However, this exercise proves somewhat complicated (one must again construct punishments for deviations). Thus I make my basic point - that C may not prefer high but stable inflation - with the following, more heuristic, approach. One way for C to reduce uncertainty is simply to drop out of the policymaking game - to let L set rc* every period. The model without C is equivalent to the basic model with the arrival rate c set to zero. Thus table 1 (with the behavior of C irrelevant) is still an equilibrium. Since nobody disinflates, n* is almost always high (once rc* rises, it falls only through rare accidental disinflations). On the other hand, there is little uncertainty: with one policymaker, rr* is always known in advance, and the variance of inflation is just G*. Are C’s losses smaller in this regime than in table 1 with c > O?

The relation between C’s losses when he drops out and in the basic model is ambiguous. Rather than present general conditions for C to prefer staying in power, I illustrate the possibility with two special cases. The first, which is not surprising, is b,b’ --+ 0: as in section 4, the cost of inflation depends only on its level. In this case (or for b and b’ close to zero), one can show that C prefers to stay in power because, by disinflating, he keeps inflation low in many periods.

The second case, which is more surprising, is a,~’ + 0. That is, C prefers staying in power not only when the cost of inflation depends only on the level, but also when the cost of the level is small. lo To see this result, note that L’s one-shot inflation rate, l/(a + c), reduces to l/a when C is never in power, and that n+ is close to this level. Thus, one can show that n’ + co as a + 0. When L targets rc+, C’s loss includes the term a’(~+)~, which approaches infinity as a,a’ + 0 (assuming that a and a’ approach zero at the same rate). In the basic model with c > 0, C’s losses are finite in all states, and so C prefers this regime.

This case is important because, as discussed above, small values of a and a’ are realistic. The result follows from the fact that, with one policymaker, rc+ explodes as a + 0. In the basic model, by contrast, rc+ remains moderate: as a -+ 0, l/(u + c) approaches l/c < cc. Intuitively, the possibility that C will arrive and disinflate holds down Y, and hence L’s target. If C gives up power, his loss at a given level of inflation falls, because there is less uncertainty. However, for small a the level rises so much that C is worse off. This result illustrates Fischer and Summers’s (1989) point that reducing the costs of inflation - in this case by reducing the resulting uncertainty - can be counterproductive.

The role of possible disinflation in holding down rce seems realistic. Suppose that Paul Volcker, hoping to reduce uncertainty, announced in 1979 that he would accept high inflation permanently rather than disinflate. This might have led to very high inflation: as Okun (197 1) argues, inflation may rise considerably if the public believes the Fed has given up the fight against inflation.

“C may prefer not to stay in power if a, a’, b, and b’ are all large (that is, if both costs of inflation are significant and L weights them heavily relative to unemployment).

L. Ball. Inzation and injalion uncertainty 383

6. Conclusion

This paper presents a model in which a rise in inflation raises uncertainty about future monetary policy, and thus about future inflation. When actual and expected inflation are low, there is a consensus that the monetary authority will try to keep them low. When inflation is high, policymakers face a dilemma: they would like to disinflate, but fear the recession that would result. Since the public does not know the tastes of future policymakers, it does not know whether disinflation will occur.

Is policy uncertainty an important source of the inflation level-uncertainty relation in actual economies? In principle, the relation could arise instead from the reaction of the private economy to high inflation. Hasbrouck (1979) shows, for example, that high trend inflation can raise variability by making money demand more responsive to shocks. Ball, Mankiw, and Romer (1988) argue that high inflation reduces nominal rigidity and thus steepens the short-run Phillips curve; a steeper Phillips curve implies that inflation varies more as demand fluctuates. Finally, it is possible that high inflation destabilizes the relation between the money stock and the Fed’s policy instruments, magnifying monet- ary control errors.’ 1

It seems unlikely, however, that these explanations for the level-uncertainty link are the whole story. The following may be a useful thought experiment. Suppose that a Constitutional Amendment imposes severe punishment on any Fed chairman who lets inflation deviate too much from x, and that everyone therefore knows the Fed will try to produce x. Compare inflation uncertainty when x is zero to uncertainty when x is ten percent. It is possible that money demand shifts or control errors are larger when the target is ten percent, and thus that actual inflation varies more. But with a firm commitment to the target, the variance is probably small in both cases; as discussed in section 4.4, the Fed seems able to stabilize inflation when it wants to. The important difference between zero and ten percent inflation in the U.S. is not the Fed’s ability to hit these targets, but rather the degree of uncertainty about whether the target will change.

I conclude by pointing out a limitation of my model. In the model, high inflation creates uncertainty only about disinflation - about whether inflation

“Cukierman and Meltzer (1986) and Devereux (1989) present other theories of the infla- tion-uncertainty link. These papers, like the current one, use Barro-Gordon models of time- consistent policy. In both papers, an exogenous increase in the variance of a shock, which raises the variance of inflation. also raises average inflation in the discretionary equilibrium. In Cukierman and Meltzer, a larger variance of monetary control errors makes it harder for the public to detect an intentional increase in inflation, raising a policymaker’s gain from inflating. In Devereux, a higher variance of real shocks reduces equilibrium wage indexation, which increases the temptation to inflate by increasing the real effects of inflation surprises. In both models, as in Hasbrouck and Ball, Mankiw, and Romer, inflation varies more around a policymaker’s target when the target is high. In my model, high inflation raises uncertainty about whether the target itself will change (see the discussion below).

384 L. Bail. Inflation and inflation uncertainty

will return to a low level. In actual economies, it appears that high inflation also creates uncertainty about whether inflation will rise further. Okun argues that if the Fed accepts high inflation to accomodate a shock, the public fears that inflation will rise again if there is another shock. In contrast, a nonaccomodative policy shows that the Fed is committed to keeping inflation under control. Future research should try to formalize these ideas.

This appendix presents the calculations omitted from sections 4 and 5.

A.I. L’s losses

In analyzing the basic model, a preliminary step is to determine L’s expected losses when he behaves as in table 1. Let Z(rP, rc) be L’s one-period loss [eq. (4)] and Ze(rce, x*) his expected loss given the target IL*. Eq. (4) and the distribution of the control error imply

Ze(7f-, n*) = Z(Y, 7-c*) + (a + 1)02 . 64.1)

The first term on the right is the loss if actual inflation equals n*. The second is the expected loss from control errors, which is positive because Z(s) is convex in 7~.

The next step is to find the expected present value of L’s current and future losses. I consider the present value at the start of a period, assuming that L was previously in power and the current policymaker is not yet determined. There are two possible states, rrLe = 0 and rre = (1 - c)x+; let V” and I/+ denote the present value of the loss in these states, and let ic = (1 - c)x+. It proves sufficient for what follows to derive the diference between the present values, I/+ - V”. This difference is defined implicitly by

v+ - v” = [cZ’(fi, 0) + (1 - c) Z’(fi, 71f)] - Ze(0, 0)

+ (1 - c) P[F(Tc) - F(77 - n+)] [V’ - P]. 64.2)

On the right, the first line is the current loss if 7~’ = 5 minus the current loss if rce = 0. (If 9 = ir, 7c* is zero if C arrives and rc’ otherwise; if 9 = 0, X* is zero for sure.) The second line is the difference across states in expected future losses. (1 - c) [F(n) - F(ii - rc’)] is the probability that rc > ii when 7~~ = i? but not when rre = 0; fi(V” - V”) is the future loss from n > ?I, which raises ?.l*

“In the second line, (1 - c) is the probability that L is in power, so that K* is TI[+ if z? = ir and zero if xc = 0. F(e) - F(C - n’) is the probability that IT > ir if n* = K+ but not if n* = 0.

385 L. Ball, Inflation and inflation uncertainly

Finally, (A.2) leads to an explicit solution for V+ - V”:

V+ _ V0 = CZe(& X+) - ZYO, (31 + cC.T(% 0) - ZYfi, x+)1 . [l - (1 - c) /?[F(%) - F(lr - 7r’)]]

(A.3)

A.2. % and rc+

Here I derive conditions defining the Every period, L chooses n* to minimize

trigger i2 and the positive target nf.

Ze(7re, n*) + /3[1 - F(f - 7t*)] cv+ - VO] ) (A.4)

where L takes rce and ?t as given. The first term in (A.4) is L’s expected current loss; the second is the probability that rc > il times the resulting increase in future losses. The first-order condition for minimizing (A.4) is

2(a + l)n* - 27re - 2 + bJ(X - 7c*) [V’ - V”] = 0, (A3

where I use the definition of Ze(*) [see (A.l) and (4)]. The trigger ?t is defined by the condition that L chooses rc* = 0 when rce = 0.

Substituting rt* = rce = 0 into the first-order condition yields

- 2 + bf(7Y) [V’ - P] = 0. (A.6)

The condition for rc+ is that n* = rc+ when rre = (1 - c)rr+. Substituting this condition into (A.5) yields

2(a + c)7c+ - 2 + Pf(ir - Irf) [V’ - VO] = 0. (A.7)

Eqs. (A.6) and (A.7) define % and rc+ (note that V+ - V” is a function of ti and n+). More precisely, these equations define il and rc+, and table 1 is an equilibrium, if solutions exist and the targets of zero and rr+ are global optima as well as satisfying first-order conditions. These conditions are checked below.

Eq. (A.7) implies II+ < l/(a + c), as claimed in the text. (A.6) and (A.7) imply f(71) >f(e - x+), which in turn implies rc+ > 2ii [sincef( .) is normal]. Finally, rc+ > 2% implies that increases in rre when L targets zero are more likely than decreases when L targets 7c+.

A.3. Suficient conditions

Here I derive sufficient conditions for table 1 to be an equilibrium. There are two steps. First, I consider a simple limiting case of the model in which C, the standard deviation of control errors, approaches zero. In this case, conditions (7) and (9) imply that table 1 is an equilibrium. Second, I show that this result

386 L. Ball, InJIation and inq’lation uncertainty

generalizes to ~7 > 0 as long as c lies below a bound, which is derived numer- ically [condition (S)].

CJ -+ 0: In this case, ?t + 0 and F(71) + 1. These results follow from the fact that f(7C) remains positive and finite, as required by (A.6). [Note that F(it) 4 1 means ?I approaches zero more slowly than a.] (A.6) and (A.7) imply rr+ + l/(a + c) and F(5 - rrf) + 0. Finally, the expression for P’+ - I”, (A.3), reduces to

l-c

V+ - ““=(u+c)(* -p+pC) for 6-0, (A.8)

where I use the fact that F(5) - F(ti - nf) + 1. When G + 0, the loss (A.4) approaches

zy7re, 7r*) for 7c* I 0,

(A.9)

Ze(7re, 7r*) + fl[V+ - V”] for 7r* > 0,

where I use the fact that ?i -+ 0. Table 1 is an equilibrium if L minimizes this loss by choosing 7c* = 0 when 7~~ = 0 and n* = rtf when 7ce = i?. The construction of ?r and 7~’ assures that the proposed targets are local minima of (A.9). However, in each case there is another local minimum. One must compare local minima to see whether zero and x+ are global minima in the two states.

When rre = 0 (A.9) has a local minimum at 7c* = l/(a + 1) as well as at X* = 0. [l/(a + 1) minimizes Z”, and therefore minimizes (A.9) over rr* > 0.1 Which local minimum is the global minimum? If L sets rc* = 0, (A.9) is Ze(O, 0). If he sets x* = l/(a + l), (A.9) is Ze(O,l/(a + 1)) + fl[V’ - I”]. Eqs. (A.l) and (4) imply Ze(O, 0) = 1 and Ze(O, l/(a + 1)) = a/(a + 1). Using these results and (A.8) one can show that (A.9) is lower at rc * = 0 under condition (7) the bound on c. Thus (7) assures that rt* = 0 is the global optimum when rce = 0.

When rre = ii, (A.9) has a minimum at 7c* = 0 as well as at rr* = rc+. L’s loss if he targets zero is Ze(r?, 0) = (a + l)‘/(a + c)‘. His loss if he targets rc+ is Ze(72, rr’) + /?[I” - V”], where Ze(72, rr+) = a(u + l)/(a + c)*. Using these facts, one can show that n* = rr+ is the global minimum under condition (9), the bound on a. This result and the result for rre = 0 imply that table 1 is an equilibrium under (7), (9), and the assumption that d -+ 0.

CJ > 0: Given (7) and (9), L strictly prefers the targets in table 1 for o + 0. The behavior of the model is continuous in cr, so L prefers the targets, and table 1 is an equilibrium, for some positive range of cr. However, for CJ too large, there is no solution for 5. [As CJ + a, there is no solution to (A.6) becausef(ii) + 0 for any candidate 71.1 Thus, as claimed in the text, table 1 is an equilibrium under (7), (9), and the assumption that CJ lies below a bound, 5.

L. Ball, Inflation and infarion uncertainty 387

I compute 15 numerically for various combinations of a, 8, and c satisfying (7) and (9) (see table 2). For a given 6, I check whether (A.6) and (A.7) have solutions, and whether the targets in table 1 are global minima of (A.4). Given a, /?, and c, I search over c to find 0, the largest value for which these conditions are satisfied.

A.4. Choosing units for 7~

To choose parameter values in section 4.4, I must specify the units in which inflation is measured. Recall that the Phillips curve (2) contains two normaliz- ations: the impact of 7~ - r? on U is one-for-one, and the gap between the natural and optimal unemployment rates equals one. Without these normaliz- ations, (2) becomes

u = UN - y(71 - 7ce), UN=UO+K. (A.lO)

As base parameters, I choose U - N - 6% U” = 3’74 and hence K = 3%. For recent years, 6% is a reasonable estimate of the unemployment rate consistent with stable inflation. The value of U” is a guess, since economists have not yet determined the optimal unemployment rate. K = 3% means that, in the nor- malized Phillips curve, one unit of U is three percentage points of unemploy- ment.

Estimates of the U.S. Phillips curve suggest that a one-point rise in unemploy- ment reduces annual inflation by roughly one point [Ball and Mankiw (1992)]. Thus the assumption of y = 1 in the text is realistic if U and 7~ are measured in comparable units. Since one unit of U is three points of unemployment, one unit of 7~ should be interpreted as three points of inflation. This justifies the claim in the text.

AS. Costs of uncertainty

Here I sketch the derivation of conditions for table 1 to be an equilibrium in section 5, which introduces a cost of uncertainty. Again, it is trivial that C targets zero inflation: this target minimizes his current loss, and his behavior has no effect on the future. The analysis of L’s behavior parallels the basic model. The difference V+ - V” is now defined by

v+ - VO=[cZ~(~,0)+(1-C)Ze(72,7C+)] -Z=“(O,O)

+ bc(1 -c) (rc+)2]) (A.ll)

where Ze(.) is defined as before. This expression differs from (A.2) by a term reflecting the loss from uncertainty. Similarly, a new term,

388 L. Ball, Injation and injation uncertainty

[l - F(it - lr*)] bc(1 - c)(rc+)‘, is added to the loss (A.4). These results lead to first-order conditions defining rc+ and ii. As in the basic model, Al’ is slightly less than l/(a + c).

Conditions for L to obey table 1 are derived using the approach of section A.3 of this appendix. There are again three sufficient conditions. The first two are (7) and (8), the bounds on c and c in the basic model. The last is

a(2P - 2fic - 1) + bc(1 - c) < 1 - fi + fic2. (A.12)

This condition is a generalization of (9).

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