More on the time consistency of monetary policy

Journal of Monetary Economics 41 (1998) 333—350

More on the time consistency of monetary policy

Juan Pablo Nicolini*Universidad Torcuato Di Tella, Minones 2159/77, 1428 Buenos Aires, Argentina

Received 8 May 1995; received in revised form 25 August 1997; accepted 9 September 1997

Abstract

We introduce costs of unexpected inflation in a general equilibrium monetary modelby changing the timing of the constraints faced by consumers. We show that in thisenvironment monetary policy is still time inconsistent, but the nature of the inconsistencyis very different from the standard result found in the literature. In particular, we find thatthe government may find optimal to deviate by choosing inflation rates lower thanexpected. By making a brief review of the monetary literature, we argue that the model ofthis paper is more attractive than the ones proposed before to study the time consistencyof optimal monetary policy. ( 1998 Elsevier Science B.V. All rights reserved.

JEL classification: E58; E61

Keywords: Monetary policy; Time inconsistency

1. Introduction

The purpose of this paper is to reconsider the time consistency of the optimalmonetary policy problem. We argue that the current view is based on modelsthat share a particular assumption regarding the way agents are constrained atthe time monetary policy is executed. We show that if we replace that assump-tion with more plausible ones, the nature of the time inconsistency problemsubstantially changes.

The literature on the time inconsistency of optimal monetary policy buildsupon the seminal contribution of Calvo (1978).1 The basic insight of his time

*Tel.: 54 1 784 0081; fax: 54 1 784 0089; e-mail: [email protected].

1See also Lucas and Stokey (L&S)(1983) and Persson et al. (1987).

0304-3932/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved.PII S 0 3 0 4 - 3 9 3 2 ( 9 7 ) 0 0 0 7 9 - 2

inconsistency result is that in the absence of lump-sum taxation, a welfareoptimizing government may have incentives to modify the planned monetarypolicy and increase the money supply at a higher rate than previously an-nounced. Unexpected inflation acts as a lump-sum tax on real money balancesand allows the government to collect revenue without distorting the economy.In fact, it is optimal for the government to always increase the current price level,in order to make the value of private sector’s money holdings approach zero. Weshow that to obtain this extreme result it is crucial to assume that all agents areable to freely exchange bonds for currency at the time the government increasesthe money supply. If this is not the case, the results are very different, eventhough the time inconsistency problem does not vanish in general. In the modelwe consider, surprise inflation is not lump sum, because at the time the moneysupply is increased, there is a subset of agents that cannot freely exchange bondsfor currency.

The common feature of the models we consider is that newly injected moneycannot be used by all consumers for transactions during the current period.Thus, an unexpected increase in the money supply may squeeze the purchasingpower of some consumers during the current period, and increase the shadowvalue of liquidity. In the models where the standard result is obtained, like inL&S, the representative consumer can use the newly injected currency in thecurrent period, so there is no change in the value of liquidity.

It turns out that the most natural way to address this issue is in a model withheterogenous agents so we focus the discussion in a version of the Grossman andWeiss (1983) or Rotemberg (1984) model. However, as we will show, the point caneasily be made in a standard representative consumer monetary model.

There exists some independent work related to this paper. Persson et al. (1989)suggested a way to introduce costs of unexpected inflation in Sidrausky-typemodels which is similar in spirit to the one we propose. However, they did not— it was not their goal — explore the theoretical implications of this assumptionand failed to see the most striking result of this paper, namely, that thegovernment may want to surprise the private sector by choosing an inflationrate lower than expected.

In Section 2, we present the baseline model and show the nature of the timeinconsistency problem. In Section 3 we show that the same result arises inversions of representative consumer models and discuss the issue in othermonetary models. Section 4 considers the case of non-indexed debt and Section5 concludes.

2. The GWR model

The model of this section is a simple version of the models developed inGrossman and Weiss (1983) and Rotemberg (1984). There are two types of

334 J. Nicolini / Journal of Monetary Economics 41 (1998) 333–350

agents, with identical preferences over the single perishable consumption goodand leisure given by

=+t/1

bt[º(cit)!ani

t] (1)

for i"a,b, where º is increasing, concave and satisfies standard Inada condi-tions, a is a positive constant, c and n are consumption and labor effort and b isthe discount factor. We assume that there are N consumers of each type, whereN is large so that they behave competitively. The assumption of linear utilitywith respect to leisure is made to sharpen the discussion, no substantive resulthinges on this assumption. There is a linear technology that relates totalavailable consumption to labor effort

c!t#c"

t#g"n!

t#n"

t, (2)

where g is exogenous government consumption, which, for simplicity, has beenassumed constant over time.

Cash must be used to purchase consumption goods, but agents can only go tothe bank to exchange assets that yield a nominal return R per period (bonds) formoney once every two periods. The only source of heterogeneity arises from therestrictions agents face regarding trips to the bank. Agent a visits the bank inodd periods, and agent b in even periods. The first-period agent a is at the bank,and must withdraw enough money to finance consumption for the first andsecond period, because he cannot return to the bank till the third period. On theother hand, agent b must finance first period consumption out of initial moneybalances. In the second period, he can visit the bank and withdraw money tofinance second and third-period consumption. Both agents are credited theirwages at the bank every period.

To the extent that there is non-indexed government debt, surprise inflationmay have effects on the real value of tax revenues required to finance govern-ment obligations and affect the optimal inflation rate at period 1. As we show inSection 4, this is indeed the case, but to sharpen the discussion, it is better toassume that all government bonds are good denominated claims.

The restrictions that reflect these assumptions are

=+t/1

Qtptn!t5

=+t/1

Q2t~1

M!2t~1

!b!1p1, (3)

c!2t~1

p2t~1

#c!2tp2t4M!

2t~1, (4)

t"1,2,3,2 for agent a, where

Qt"

1

<tj/2

(1#Rj), t52,

Q1"1,

J. Nicolini / Journal of Monetary Economics 41 (1998) 333–350 335

is the price of money at t relative to money at t"1, M!2t~1

is the amount ofmoney consumer a withdraws from the bank at period 2t!1, p

tis the currency

price of goods at time t, and b!1is the stock of government bonds initially held by

consumer a. Similarly, for agent b, it must be true that

=+t/1

Qtptn"t#Q

2(M"

0!c"

1p1)5

=+t/1

Q2tM"

2t!b"

1p1, (5)

c"2tp2t#c"

2t`1p2t`1

4M"2t, (6)

c"1p14M"

0, (7)

where variables are defined in a similar fashion. The main difference for thisconsumer is that first period consumption has to be financed by initial moneybalances. Any cash left, can be deposited at the bank in the second period. Notethat M"

0in Eq. (7) is given, so that an injection of currency in the first period that

increases p1

taxes more heavily first-period consumption for agent b andtherefore, it is not lump sum.

In solving the consumer’s optimal problem, we want to make sure that the

solution for labor effort is interior. If we let n be the total time endowment of theagents, we assume that

g/2#º{~1(a)(nN .

This condition basically ensures that labor effort is interior in the symmetricPareto optimum. As equilibrium quantities will be lower in a decentralizedequilibrium, labor effort will be interior in any equilibrium.

The competitive equilibrium conditions of this model are:

º@(c"1)"a#

d1. p

1b

, (8)

º@(c!2t~1

)"a"º@(c"2t), (9)

º@(c!2t)"a . (1#R

2t), (10)

a . (1#R2t`1

)"º@(c"2t`1

), (11)

Qtptb"Q

t`1pt`1

(12)

for all t, where d1

is the Lagrange multiplier of constraint (7). First, note thatEq. (9) implies that c!

2t~1"c"

2tfor all t, and that they are fully determined by

preference parameters. However, c"2t`1

and c!2t

do depend on the inflation rate.This is so because cash must be held before-hand in order to consume thesegoods, and therefore they bear the inflation tax. Finally, note that, if thecash-in-advance constraint is binding, Eq. (8) determines the value of the multi-plier of restriction (7). Thus, c"

1is determined by (7).


In general, then, equilibrium quantities depend on the evolution of monetaryvariables. Thus, through monetary policy, the government may be able toinduce a particular equilibrium outcome. A Ramsey problem is, precisely, tomaximize a linear combination of both agents utilities subject to the constraintthat the government collects enough taxes to finance government expendituresand to the constraint that the resulting allocation is a competitive equilibrium.

Formally, the problem is to maximize

=+t/1

bt[º(c!t)!an!

t]#h

=+t/1

bt[º(c"t)!an"

t], (13)

subject to Eqs. (2)—(12) where h is the relative preference of the government foragent b.2 This is a complicated problem that involves the choice of prices andquantities simultaneously. However, following L&S, it is possible to use thecompetitive equilibrium conditions to eliminate prices from the restrictions toreduce the number of choice variables. Thus, by replacing Eqs. (4), (9), (10) and(12) in Eq. (3), we obtain the following implementability constraint for a:

=+t/1

bt[an!t!º@(c!

t)c!

t]#bab!

1"0, (14)

while replacing Eqs. (6), (7), (9), (11) and (12) in Eq. (5) we obtain

ban"1#bab"

1#

=+t/2

bt[an"t!º@(c"

t)c"

t]"0. (15)

Note that c"1

does not show up in agent b’s implementability constraint. This isso, because with positive nominal interest rates,3 there are no incentives to saveany of the initial money balances for future periods and thus initial moneybalances exactly cancel out with first-period consumption. This feature reflectsthe fact that in the first period, agent b’s consumption is only constrained by theinitial holdings of currency and not by total wealth, so surprise inflation wouldstrongly squeeze first-period consumption for b relative to other periods con-sumption, so it cannot be lump sum.

The optimal policy problem is choosing quantities that maximize Eq. (13)subject to Eqs. (14), (15), (9) and (2).4 The solution, as we will show below,

2The resulting allocation must be a competitive equilibrium, and the only policy instrument is theinfllation tax, thus it is not clear that all the — restricted — utility frontier can be implemented.However, with positive government expenditures, there exists a non-empty interval around 1 suchthat the problem is well defined for all relative preference parameters in that interval.

3Nominal interest rates must be positive if the relative preference parameter is one, becausegovernment expenditures are positive, so revenues must be positive. This can be formally provenfrom the first-order conditions of the policy problem below. If we introduce taxes other thaninflation, we should be more careful with this argument.

4The supporting prices can be obtained from the solution to the consumer problem.


depends crucially on the behavior of the intertemporal elasticity of substitution.In order to obtain a sharp characterization of the solution, it is convenient toassume that utility derived from consumption is of the constant elasticity ofsubstitution type

º(c)"c1~p1!p

.

This assumption is by no means necessary for the results, but it ensures thatthere is a unique solution satisfying the first-order conditions.5

Let ji be the Lagrange multiplier on the implementability constraint of agenti. Then, if a solution exists and it is interior, the following conditions must besatisfied:

[h#j"]"[1#j!], (16)

º@(c!2t)(1#j![1!p])"[1#j!]a, (17)

h.º@(c"2t`1

)(1#j"[1!p])"[h#j"]a, (18)

h.º@(c"1)"[h#j"]a (19)

for t"1,2,3,2. Consider the symmetric case in which h"1. The moregeneral case allows for redistributions that are irrelevant for the present dis-cussion. Eq. (16) then implies that j!"j", so we get rid of agent-typesuperscripts.

Eqs. (17) and (18) imply that c"2t`1

"c!2t"cN for all t51. Eq. (19) then, shows

that, unless p"1, the optimum policy is to set a value for first period consump-tion different from all other ones. Note that the multiplier j is positive, as it is theshadow price of the constraint.6 Therefore, consumption in the first period willbe greater or smaller than future consumption depending on p being larger orsmaller than one. As expenditures are constant, this means that either at periodone there is a surplus and a deficit for all other periods or a deficit in the firstperiod and a surplus in the following periods, except when p"1 in which casethe budget will be balanced every period.

In order to consider the time consistency of the solution, we should reformu-late the Ramsey problem at period two and compare the solution one obtains,for instance, for c!

2with the solution in Eq. (17). The problem is very similar,

except that all period-one values will be given, so the first choice variable is c!2.

Recall that the first choice variable does not show up in the implementability

5Recall that the Ramsey problem involves non-linear constraints, so sufficient conditions foroptimality are not satisfied for general utility functions.

6This can be formally proven following Lucas and Stokey (1983).


constraints, so the first-order conditions at t"2 are:

º@(c!2(2))"[1#j(2)]a, (20)

º@(c!2t`2

(2))(1#j(2)[1!p])"[1#j(2)]a, (21)

h.º@(c"2t`1

(2))(1#j(2)[1!p])"[1#j(2)]a, (22)

where the 2 in parenthesis indicates the solution of the problem from the vintagepoint of period two which may be different from the solution from the vintagepoint of t"1.

The first-order conditions above make clear that the solution of the optimalpolicy problem from the vintage point of time t, involves two numbers

c(t), cfut(t),

satisfying

º(cfut(t))(1!j(1!p))"º(c(t)), (23)

where c(t) represents the optimal value for the period t consumption and cfut(t)the optimal value of consumption from period t#1 on (future periods), bothfrom the vintage point of period t, where it is understood that c(t) refers toconsumption at t for the agent that is not at the bank, while the agent that is atthe bank always consumes such that the marginal utility is equal to a. Clearly,unless p"1, c(t)Ocfut(t), so optimality from the vintage point of period onerequires a constant sequence of values for consumption from period two on.However, from the vintage point of period two, optimality requires that secondperiod consumption ought to be different from consumption the followingperiods. Therefore, unless p"1, the optimal policy is time inconsistent. We nowshow that the sign of the bias depends precisely on the value of p.

Consider first the problem from the vintage point of period one. Adding upimplementability constraints (14) and (15), using the fact that consumption fromperiod two on is constant and using the market clearing conditions, we obtain

aAb1#c(1)#g

1!bB"b

1!b(º@(cfut(1))!a)cfut(1), (24)

which implies that the present value of government obligations, i.e., the initialdebt plus the initial real money balances and the present value of governmentexpenditures (the left-hand side), has to be the same as the present value of taxes (the right-hand side). This must hold every period, off course, so it has to betrue that

aAb2#cfut(1)#g

1!bB"b

1!b(º@(cfut(1))!a)cfut(1), (25)

where the stock of debt at period two shows up on the left-hand side.


By the same token, when the government reoptimizes at period two, it has tobe the case that

aAb2#c(2)#g

1!bB"b

1!b(º@(cfut(2))!a)cfut(2). (26)

But Eqs. (25) and (26) imply that

b1!b

[º@(cfut(2))!a]cfut(2)!ac(2)

"

b1!b

[º@(cfut(1))!a]cfut(1)!acfut(1). (27)

The interpretation of this last equation is simple. It implies that the value, atperiod two, of the taxes from period two on, must be the same from the vintagepoint of period one as it is from the vintage point of period two. This reflects thefact that at period two, the government has the option of changing the timing oftaxes, but it cannot modify the present value of its obligations, because thegovernment does not control the expenditure side of fiscal policy.7 Now, let

H(x)"b

1!b(º@(x)!a)x,

so Eq. (27) can be written as

H(cfut(2) )ac(2)"H(cfut(1))!acfut(1), (28)

where

H@"º@(c)(1!p)!a.

But from the first-order conditions of the Ramsey problem, we have that

j(u@(c)(1!p)!a)"a!º@(c)

and the right-hand side is negative or zero, because nominal interest ratescannot be negative. Therefore, the left-hand side will be negative, in general (itwill only be zero if the nominal interest rate is zero, but with positive g, thegovernment will always put some tax to every good that has a finite elasticity).As j is positive, H@ is negative.

Now, assume that p(1. Therefore, as j is positive, (1#j(1!p))'1, soEq. (23) implies that cfut(t)'c(t), which means that the optimal policy is having

7Note that government’s debt is indexed, so changes in inflation rates do not affect its value. Onthe other hand, we are assuming away the possibility to repudiate the debt to concentrate on thetime consistency of monetary policy. For an analysis of the time consistency of government debt, seeChari and Kehoe (1993).


the first value of consumption lower than the following ones. So, the optimalinitial inflation is larger than steady-state inflation. As inflation is the only taxand optimality requires to always be on the good side of the Laffer curve, thegovernment runs an initial surplus and a deficit from the second period on. Butas H is decreasing, Eq. (28) above implies that

cfut(2)'cfut(1)'c(2),

but recall that cfut(1)"c!2(1) from the vintage point of period one, and

c(2)"c!2(2) from the vintage point of period two. Therefore, if the government is

given the chance to reoptimize at the beginning of period two, it will choosea value for consumption for agent a lower than the previously announced, whichimplies an inflation rate higher than the one previously planned. A similarargument shows that when p'1, the optimal policy is time inconsistent, butthe government will find optimal to choose an inflation rate lower than theone announced the first period. Finally, if p"1, the optimal policy is timeconsistent.

The reason why the value of p relative to one is important, is at the heart ofthe Ramsey problem. Note that from Eq. (10) we can write the demand functionof c!

2as

º@(c!2)"a . (1#R

2). (29)

Thus, p as defined above is the inverse of the price elasticity of c!2. However, the

c!2(2) demand function is determined by the cash-in-advance constraint, and it is

thus a unit price elasticity good. The Ramsey solution is to tax more heavily thegoods which are less elastic. Thus, if p is larger than one, it means that thedemand function of c!

2is less elastic than the demand function of c!

2(2). Thus, you

must tax more heavily c!2

and that is why the government has incentives todeviate from the time-one optimal plan and chose an inflation rate lower thanpreviously announced.

The reason why time inconsistency arises in this context is because over-timeconsumers make irreversible decisions, such that the elasticities of the demandfunctions change. As the optimal tax depends on the elasticity, the optimal taxalso changes. There is a sense in which there is a ‘short-run’ and a ‘long-run’demand for money balances, and the time inconsistency problem depends onthe relative sizes of the elasticities of these two demand curves. The intuitionbehind this solution is simple. Given the structure of the model, at any timeperiod, there is always one consumer who cannot exchange illiquid assetsfor currency till next period. A surprise injection of currency squeezes thepurchasing power of money balances held by that consumer and creates distor-tions measured at the margin by the multiplier of the cash-in-advance constraintof that consumer.

This model introduces costs of unexpected inflation in a very simple way. Onecould well argue that unexpected inflation may affect the economy in many


other ways that this simple model does not capture.8 The point of the paper isthat once you introduce costs of unexpected inflation, it is by no means obviousthat the inflation bias is positive. The literature started by Barro and Gordon(1983) assumed a positive inflation bias, by imposing that the marginal cost ofunexpected inflation, evaluated at zero inflation, is zero, while the marginalbenefit is positive. This model shows that in a general equilibrium model, for oneof the two possible theoretical explanations used in the literature, that may notbe true.

3. The Cash-in-advance model

The model of the previous section makes the point very naturally. However,heterogeneity is by no means necessary to obtain the result of last section. Infact, one can show that the same result holds in representative agent cash-in-advance economies, if the timing of the constraints is changed relative to themodels traditionally used to study the problem of the time consistency ofmonetary policy. Consider a representative consumer economy given by thefollowing preferences, technology and cash-in-advance constraint:

=+t/1

bt[º(ct)!an

t] (30)

ct#g"n

t, (31)

ptct4M

t(32)

for t"1,2,3,2, where variables have the same interpretation as before.To motivate this cash-in-advance constraint, consider a representative couple

— a shopper and a worker. Let us also imagine that there are two spatiallyseparated markets, the asset market where agents can trade currency and bondsand the goods market where agents trade currency and goods. In order toobtain the standard result, one must assume that the both the shopper and theworker go to the asset market to exchange money and bonds and then they split,the worker goes to produce and sell the good, while the shopper carries thecurrency to buy the consumption good. This sequence of events corresponds tothe formulation in Lucas (1980), and used by Lucas and Stokey (1983) amongothers.

In order to reproduce the result of the previous section, we will assume thatfirst the couple splits and the shopper carries the starting money balances to thegoods market while the worker goes first to the asset market to exchange money

8The literature particularly stressed the role of nominal wage rigidities.


for bonds and then to work and sell the goods. The key difference is that in thefirst interpretation the shopper also goes to the asset market so that he canexchange the initial bond holdings for money and use it for current purchases.However, in the second interpretation, the initial bond holdings cannot be usedfor current transactions because the shopper does not go to the asset market.This sequence of events is the one proposed by Svensson (1983) and used byHodrick et al. (1991) among others.9

In Lucas’s version, when monetary policy is executed, the cash-in-advanceconstraint is not binding, and the shopper can re-adjust money balances im-mediately at zero cost.10 This is the reason why monetary surprises are lumpsum. Instead, with Svensson’s timing, at the time monetary policy is executed,the cash-in-advance constraint is binding, and the shopper cannot exchangebonds for newly injected currency and use it for purchases till next period.

We do assume that the government can operate in both markets simulta-neously, so it can use the newly injected currency to finance spending on thecurrent period and to make open market operations to buy bonds to financefuture spending.

The way in which this assumption enters into the formal structure of themodel is through the budget constraints. We assume, as in Svensson, thatconsumers start with given quantities of nominal wealth and spend part or all oftheir nominal balances on goods. They receive credit for the time worked andcan exchange bonds for currency to purchase goods next period. Thus, M

1is

given, and nominal wealth, ¼, evolves according to

¼t`1

"Mt`1

#bt`1

pt`1

"pt.n

t#M

t!p

t.ct#b

tpt(1#R

t) (33)

for t"1,2,32.On the contrary, with Lucas’s version, consumers start the period with given

wealth ¼0

that they can split between currency and bonds

M1#b

1p14¼

0. (34)

Then, they attend the goods market and wealth evolves according to Eq. (33) fort"1,2,3,2 .

We will first define the Ramsey problem using Lucas version and show that itis optimal to make the initial price level arbitrarily high. Then, we will define

9 It should be noted that the interpretation given in the text is neither the one stressed by Lucasnor Svensson. Lucas assumes that asset markets open before goods market while Svensson assumesthe opposite. However, they lead to the same mathematical formulation of the paper. The cash-flowmechanism in these deterministic inflation tax models are easier to understands with the interpreta-tion given in the text. The comments of the referee helped me to clarify this issue.

10The corresponding assumption in money in the utility function models is that end of periodrather than begining of period money balances enter the utility function. See Persson et al. (1989).


and solve the Ramsey problem with the Svensson version to show that the resultis very similar to the one obtained for the GRW model.

Let jt

and dt

be the Lagrange multipliers of the budget constraint, andcash-in-advance constraint, respectively. Then, the first-order conditions ofconsumer’s problem are

bt )º@(ct)"(j

t#d

t) . p

t, (35)

bt ) a"jt.p

t, (36)

jt"j

t`1#d

t`1, (37)

jt"j

t`1. (1#R

t`1) (38)

for t"1,2,3,2 and

j1R

2"d

1. (39)

This last equation is the main difference between the Lucas and Svenssonversions of the model, and it reflects the additional choice of nominal wealthbetween money and bonds that the consumers have the first period before theysplit in the Lucas version.

The Ramsey problem is to choose the sequence of inflation rates that maxi-mize the utility level of the representative consumer subject to the constraintthat the resulting allocation must be a competitive equilibrium. As before, wecan combine all competitive equilibrium conditions in the following implemen-tability constraint:

=+t/1

bt~1[º@(ct)c

t!a . n

t]!a.(M

0/p

1)"0,

where M0

is the amount of money the consumer holds at the beginning of thesecurities market. If the Ramsey problem is not trivial, the multiplier associatedto that constraint is positive, which means that the derivative of the Lagrangianwith respect to p

1is always positive, reflecting the incentives the government has

to generate an initial hyperinflation. This initial price level is the ‘additionaldegree of freedom’ Calvo mentions in his paper.

The analysis of the time inconsistency problem is thus the one in L&S orCalvo if we allow the representative consumer to choose between M

1and

b1according to constraint Eq. (34). We show now that the nature of the result is

very different if we do not. The first-order conditions for the consumer inSvensson version are the same as with the Lucas version, except for Eq. (39). Asbefore, we can use Eq. (37) to substitute d

tfor t"2,3,4,2. However, as in the

GRW model, d1

appears only in Eq. (35) for t"1. This reflects the fact thatif the cash-in-advance constraint is binding,11 c

1is determined by it, and

11And it will be binding, because g is positive and the only available tax is the inflation tax.


Eq. (35) for t"1 is only used to know the value of d1, the shadow price of the

constraint.If we combine now the equilibrium conditions, we obtain the following

implementability constraint:

!ba . n1#

=+t/2

bt[º@(ct)c

t!a . n

t]"0. (40)

As in the GWR model, c1

does not show up in this implementability constraint,because it exactly cancels out with initial money holdings. With our specifica-tion, there is no additional degree of freedom because consumers cannot re-adjust their portfolios immediately at zero cost.

The optimal problem is then to maximize Eq. (30) subject to Eqs. (31) and(40). If we let j be the Lagrange multipliers of the implementability constraintand we assume that the solution exists and it is interior, the following conditionsmust hold:

º@(ct`1

)(1#j[1!p])"a(1#j)

for t"1,2,3,2, and

º@(c1)"a(1#j).

Note that the structure of these first-order conditions is the same as in the GWRmodel if we assume that h"1. Exactly the same conclusions can be reachedwith these equations.

The discussion above makes clear that to the extent one hopes the model togeneralize to environments with heterogenous agents, the Svensson interpreta-tion seems a better alternative when studying the time consistency of monetarypolicy. In Lucas model, the possibility of surprise inflation at a point in timesuch that all agents are at the bank and can re-adjust their portfolios for freecomes from a very particular assumption of the representative agent models. InLucas (1980), the cash-in-advance constraint is motivated as ‘. . . Individualbehavior resembles that captured in inventory theoretic models of moneydemand, as studied by Baumol (1952) and Tobin (1956). . . ’. However, in thosemodels, the individual time profile of money demand has a very particularcyclical behavior (resembling very much actual individual patterns). To generateaggregate behavior consistent with observations one must assume that differentconsumers attend the securities market at different times. But then, as the GWRmodel makes clear, there is no way the government can increase the moneysupply at a point in time in which every single consumer can re-adjust hisportfolio for free. The cash-in-advance constraint is a short hand for the frictionsthat exist in markets that make the exchange of liquid assets (money) andnon-liquid assets (bonds) a costly process. The same frictions that justify theexistence of a dominated asset cast doubt on the assumption that it is possible to


increase the money supply at a fully centralized securities market where everyagent can exchange money for bonds at zero cost.

Before concluding the paper, let us briefly discuss the time inconsistencyproblem in other monetary models. Consider first the overlapping generationsmodel of money (see Wallace, 1980). Given that in equilibrium money holdingsare unevenly distributed across agents, the effects discussed in the analysis of theGWR model apply. Consider also the Townsend (1980) model with spatiallyseparated agents. For the time inconsistency problem to be as in Calvo or L&S,it is required that the government injects the currency across markets witha symmetry that hardly resembles the way monetary policy is executed. Inaddition, the re-distribution issue also arises. Finally, consider the Kiyotaki-Wright (1989) model. Even though it is not obvious how one would introducemonetary policy in the model, it seems clear that to generate the standard result,a very strong symmetry on the way money injections reach the pairs of agents isrequired.

The issue we discuss in the paper is closely related to the neutrality of money.Note that in the Lucas’s version of the simple cash-in-advance model, where thestandard result obtains, money is neutral. However, in Svensson’s version of thecash-in-advance model, as well as in GRW model, where the sign of the biasdepends on elasticities, money is not neutral. In fact, the GRW model wasdeveloped for money not to be neutral, to capture the liquidity effect observed inthe data. Another interesting contribution to the liquidity effect literature is thepaper by Fuerst (1992). He develops a cash-in-advance model in which a repre-sentative household splits among several different types of agents that gettogether again before next period to pool resources. It is possible to show thata simplified, deterministic — and therefore uninteresting as a model of liquidityeffects — version of Fuerst model reduces to exactly the same set of equations asthe Svensson version of the cash-in-advance model.12

Departures from neutrality have been identified by a series of empiricalpapers, see, for instance, Christiano and Eichenbaum (1992). An innovation inthe quantity of money drives down nominal interest rates — the liquidity effect— and drives up consumption and output. GRW showed that their models cangenerate predictions that are consistent with the empirical findings regardingliquidity effects while the effect on total output and consumption depends onhow the monetary shocks affects both agents. However, the Svensson’s versionused in this section has an empirical implication that is clearly at odds with thefacts, namely, an increase in money reduces consumption and output, because itreduces the value of money holdings of consumers. In this paper we used verysimple models in order to sharpen the discussion of the time inconsistency issue.These simple models are clearly unable to replicate the complicated departures

12For details, see Nicolini (1993).


from neutrality observed in the data. However, they suggest that models de-veloped to reproduce the non-neutralities will qualitatively behave, regardingthe time inconsistency issue, like the ones discussed in this paper rather than theones previously used to study the time consistency of monetary policy.

4. The case of nominal government debt

The results of the paper are different if nominal debt exists. In the simpleSvensson version of the cash-in-advance model, the implementability constraintwith non-zero initial holdings of government debt would be

!ban1!ba )

B1

p1

#

=+t/2

bt[º@(ct)c

t!n

t. a]40,

where B1

is the total value of government debt, denominated in units ofcurrency, held by the private sector at the beginning of period one. Recall thatp1"M

1/c

1, so the first-order conditions of this problem will be the same as the

one before, except for the value of first-period consumption, which will be

º@(c1)#ja

B1

M1

"(1#j)a.

In this case, present inflation will not only depend on elasticities but also onthe marginal gain derived from changing the value of the existing debt. Notethat if B

1/M

1is positive, the initial inflation should be higher than if it were zero.

This reflects the additional gain obtained by lowering the real value of thenominal debt through inflation. Note also that the debt could be chosen to makethe value of the expression in brackets on the right-hand side equal to zero,eliminating the time inconsistency of optimal monetary policy. This is the effectdiscussed in Persson et al. (1987, 1989).

Several interesting issues arise which deserve further research. First, there isthe possibility to use nominal debt to reduce the temptation to change theinflation rate as suggested by Persson et al. (1987). Second, the incentives tocreate high inflation depend on the size of government debt. This interactionbetween the size of government debt and the incentives to inflate the debt awayopen a new avenue to interpret the high inflation rates experienced in the lastthirty years. Third, the interiority of the optimal deviation can have importantimplications regarding the relevance of the term structure of government debt,an issue that has often been addressed lately. Finally, the model of this paper canbe used to study reputational issues, in general equilibrium monetary models inwhich the size of the debt is a state variable.

The unbounded solution for the initial price level was a key problem that theempirical literature addressing the issue of time inconsistency of monetary


policy had to deal with. A possible way out was to claim for the existence of costsof unexpected inflation and use ad-hoc welfare functions to evaluate optimalpolicy. An example is Poterba and Rotemberg (1990), who empirically study therelationship between ordinary taxes, inflation tax and the degree of monetiz-ation in the context of a model of optimal policy. They do specify a welfarefunction that imposes a positive optimal deviation. In their empirical section,they discuss ways to separate the effect on currency from the effect on nominalbonds and they carry the two effects along the whole analysis. Another exampleis Bohn (1996), who also imposes ad-hoc costs of inflation. He shows that theincentives to inflate away nominal liabilities have substantially increased for theUS, given that the large external deficits of the last years have been largelyfinanced by nominal obligations. Bohn (1991) also argues that by far, govern-ment debt is more important than money holdings in computing the incentivesto inflate in most developed economies. In the context of our model, if therelative risk aversion coefficient is close to one, only the stock of nominal bondsshould be considered in empirical applications, and the analysis would besubstantially simpler.

5. Conclusions

We considered the robustness of the traditional time inconsistency result ofthe literature (Calvo (1978); Lucas and Stokey (1983); Persson et al. 1987). Weshowed that result to crucially depend on a particular assumption regarding theconstraints faced by the private sector at the time money injections are done.

The key intuition is that the traditional result holds only when all newlyinjected money can be used within the period by the consumers for transactions.It is only in this case that surprise money increases are lump sum and do nothave real effects. We showed that the environment required to generate thetraditional result is a knife-edge situation impossible to replicate in existingmodels that deal with heterogeneity in an interesting way.

In the models analyzed in this paper the optimal policy is not, in general, timeconsistent. However, the nature of the time inconsistency problem is quitedifferent from the traditional result. First, the optimal deviation is finite ingeneral. Second, and this is the most striking result, the optimal deviation can benegative. That is, the government can be tempted to chose an inflation ratelower than the one chosen in the planning period.

The models solved in this paper show one easy way to introduce costs ofunexpected inflation in simple general equilibrium monetary models. Contraryto the analysis of reputation in capital taxation and debt repudiation, theliterature on monetary policy has relied on ad-hoc models, and many ofthe results of the literature depend on the special features of the ad-hocpayoff function of the government. In particular, as the payoff function typically


used rules out the possibility of optimal negative deviations, the existence ofa positive inflation bias is assumed in the models. It remains an open questionwhich of the established results of the literature hold, in general equilibriummonetary models with benevolent governments.

Acknowledgements

I would like to thank the referee, Francisco Buera, Albert Marcet, RamonMarimon, Ivan Werning and Michael Woodford for comments. I particularlywant to thank Pedro Teles for many very helpful conversions. All remainingerrors are mine.

References

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Calvo, G., 1978. On the time consistency of optimal monetary policy. Econometrica 46,1411—1428.

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Nicolini, J., 1993. More on the time inconsistency of optimal monetary policy, Working paper d 56,Universitat Pompeu Fabra.

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Rotemberg, J., 1984. A monetary equilibrium model with transactions costs. Journal of PoliticalEconomy 92, 40—58.

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Wallace, N., 1980. The overlapping generations model of fiat money. In: Kareken, J., Wallace,N. (Eds.), Models of Monetary Economies. Federal Reserve Bank of Minneapolis, Minnea-polis, MN.


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