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A policy-game approach to the high inflation equilibrium
JOURNAL OF
Development ECONOMICS
Journal of Development Economics Vol. 45 (1994) 135-140
ELSEVIER
A policy-game
approach to the high inflation equilibrium
Miguel A. Kiguel a,*, Nissan Liviatan a,b a The World Bank, 1818 H Street, N. W., Washington, DC 20433, USA ’ The Hebrew Unir,ersity of Jerusalem, Jerusalem,
Israel
Received April 1992; final version received August 1993
Abstract The high inflation equilibrium (HIE) refers to a steady state solution where inflation is higher than the one which maximizes seigniorage (i.e. inflation is on the falling segment of the Laffer curve>. It has been claimed that the HIE can be stable when the adjustment of expectation is fast. However, adjustment is fast because inflation is high; so the argument is circular. In this paper we rationalize the HIE using the policy game approach which distinguishes between discretionary and precommitment regimes. We expand, and demonstrate rigorously (using a technique developed by Obstfeld), Barro’s suggestion that a HIE may result from a discretionary regime. We show that this is not possible under precommitment. In addition, we demonstrate that a HIE may result from non-fiscal factors (such as the motivation to reduce unemployment) in a discretionary regime. We conclude with a simple diagrammatic exposition of the difference between our approach and the previous one. Keywords:
High-inflation
JEL classification:
equilibrium;
Policy game approach
E5, E6
1. Introduction Many economists rate that is justified
exceeds the share the view that inflation in some countries by the needs to finance the budget deficit through money
* Corresponding author The views expressed herein are those of the authors and do not necessarily reflect those of the World Bank or its affiliated organizations. 0304.3878/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0304-3878(94)00030-G
136
M.A. Kiguel, N. Liuiatan/Journal
of DeLzlopment
Economics 45 (1994) 135-140
creation (seigniorage). This is especially true in high inflation countries, such as Israel or Brazil, where high inflation (above 200 percent per year) has not been associated with unusually high seigniorage. One explanation for this phenomenon relies on the inflation tax model which is characterized by dual steady state equilibrium (Sargent and Wallace, 1981, 1987). The same level of seigniorage could be obtained at a low inflation rate, with an inelastic demand for money, or instead at a high inflation rate, above the revenue maximizing rate (we shall refer to the latter as High Inflation Equilibrium, HIE), at points on the elastic side of the demand for money. According to some models, the HIE can be explained by the fact that this equilibrium is stable while the low inflation equilibrium is unstable. Bruno and Fischer (1990) show that the HIE is stable when expectations are rational in the sense of myopic perfect foresight, and the money market adjusts instantaneously. In the framework of the adaptive expectations model authors have drawn a distinction between slow and fast adjustments of expectations. It has been shown that the HIE tend to be dynamically stable when expectations adjust rapidly (see Auernheimer (1973), Evans and Yarrow (1981)), but it is unstable when expectations adjust slowly. Since in high inflation economies expectations adjust quickly (because slow adjustment is costly), one may obtain a stable HIE. An entirely different approach to inflation has been formulated recently in the policy game literature (as in the papers by Barro and Gordon). This approach stresses the role of surprise inflation tactics and of non-fiscal motivations in generating high inflation rates. The inability of the policy maker (PM) to make credible commitments about his monetary policy rules (a phenomenon which is quite common high inflation economies) may then bring about a HIE as suggested by Barro (1983a, p. 12). This outcome is not possible when the PM is capable of making credible commitments about his policies. In the present paper we shall first present a rigorous and yet simple confirmation of Barro’s conjecture. We shall then show that non-fiscal motivations may lead to a HIE even when the fiscal motivation by itself will not yield this result.
2. High inflation
equilibrium
In order to see how a HIE can be generated define seigniorage (S,) as
in an inflation
tax model let us
where M, is the nominal stock of money (which is assumed to equal base money) and P, is the price level in period t. Inflation and expected inflation are defined as rrt = (P, - P,_ l)/Pt and nfe = (Pte - P,_ l)/Pte when P: is the price level P, expected in period t - 1. The demand for real balances (assuming a constant real
of Development Economics 45 (1994) 135-140
137
interest rate) is given by L, = ~5(n,~+1) (with L’ < 0) and it is assumed money market is in equilibrium. We may then express (11 as
that the
M.A. Kiguel, N. Liuiatan/Journal
s, =+-te+,
) -+-XI
(I’)
- r,).
In this equation rte is predetermined from period t - 1, while nt is treated as a policy variable in the discretionary regime. Following Barro (1983) we express the PM’s utility function for any period t as U,=~,Sr-f(7T,)--(~TT,e+,),
fl,r’>O,
(2)
where 13 represents the utility from seigniorage while (minus) f and r represent the loss from actual and expected inflation (the latter can be due to loss of liquidity through L). We shall assume, for simplicity, that 8 is constant, so that our system contains no random elements. We assume that the PM maximizes a discounted sum of utilities u, = &~U,, lY=t
O
(3)
where r is the discount factor. Let us assume that the policy function in the discretionary regime, specifying the optimal n, for any specified L,-, and 8 is given by the time-invariant function rrf = g(L,_ ,, 0). The existence of this function will be established below. This function will ensure time consistency in the sequential optimization for the infinite horizon model at hand (in using this methodology we follow Obstfeld (1989)). We assume that g( - ) is common knowledge so that the rational public, having observed L,, must form its expectations in t according to $+,
=g(‘%,
0)
(4)
Combining (4) and L(nF+ 1) we can determine 7~~~1. In fact, nT must be a constant (say rc) for all 72 t + 1 and L, is constant (say Lc) for T 2 t. In a steady state L,_ 1 is equal to L”. (It is understood that rrc and L’ depend on 0.1 Let us maximize (3) subject to (1’) and (2) treating rr, as a free variable, taking n,” (or L,_ 1> as given and imposing the constraint nTe = n7* = rrc for T 2 t + 1. The first-order condition yields 8L( 7r;> -f’(
Tr,) = 0.
We assume, along with Barro, that f” > 0 which enables 01, recalling that L,_i policy function rr,? = g(L,_,, information about the government’s objective function, quire that in the absence of shocks rrp = rr,?. This condition, Eq. (5) and the fact that n-T = rr, equilibrium for rr given 0 when L,_ 1 = Lc. Thus, using in (5) we have n=~(e>,
g=(f”-ez-‘L>o.
(5) us to derive from (5) the =L(nF). Given the full rational expectations redetermine a steady state rrl = r: = r,” = n-’ = rr
(6)
138
M.A. Kiguel, N. Liuiatan/Journal of Development Economics 45 (1994) 135-I40
Under the foregoing assumptions rr can be increased, in steady states, in an arbitrary manner by raising 8. It is, therefore, clear that the demand for money in steady states can be reduced sufficiently so as to reach the elastic region, which corresponds to the downward sloping part of the Laffer curve.
3. Equilibrium
under precommitment
In the precommitment regime the PM is bound by credible commitments with regard to monetary policy. Suppose that in every period t the PM makes a credible announcements about 7rt+ 1 so that rite+ 1 = rTTl+1 at the optimization stage. Given that he honors his commitment, he will treat 7~, as predetermined. If he chooses to increase rr,+ 1 this will reduce S, (since L(rrF+ 1> decreases) but S,, 1 will increase, as can be seen from (I’) moved one period forward. Assume, tentatively, that the PM’s policy function for any period 8 is ?r7+ i = G(0) and maximize (3) with respect to rr,+ , subject to rrt+ 1 = T,“, 1 and applying r+ 1 = G( 0) for T 2 t + 1 together with the rational expectations requirement $+ 1 = G( 0) for r 2 t + 1. Treating L(rrJrte)and rr, as given we obtain the optimality condition f3Z( 1 - I) + i+( L + 7rL’) - 7’ - rf’ = 0,
(7)
where the argument of all the functions is n = rr(+ 1, which is consistent with G(0). Note also that since L’, - 7’ and -f’ are negative, it follows that (L + rrL’) > 0, which means that L is inelastic. Thus the precommitment equilibrium is on the rising part of the Laffer curve. Comparing (7) with (5) it is readily verified that the solution value of r is smaller in the precommitment case than in the discretionary one.
4. Non-fiscal
cause for HIE
Non-fiscal considerations can be introduced as follows. Suppose that the PM considers the level of employment to be too low as a result of excessive unemployment benefits, over which he has little control (as in Barro and Gordon (1983b)) or because of labor monopolies. These factors can produce a HIE even when the fiscal side by itself, would not lead to this state. Let ur be the labor-unions’ real wage target so that the nominal wage is set at where p: is the expected price level for period t (all in terms of UL + P,", logarithms). The actual real wage is then given by vr + p,’ -p, which we approximate by vL + rrp - r,. The PM’s real wage target uo is assumed to be smaller than vL. We may then suppose that the PM wishes to minimize the deviations of the actual real wage from vo. Thus we add to the u, function (2) the term - (6/2)(7rF - ~~ - ~1)~ where u = uL - vG > 0.
M.A. Kiguel, N. Liuiatan /Journal
Using condition
of DeLxelopment Economics 45 (1994) 135-140
a similar reasoning to that of Section for rrt (analogous to (9)):
fX+b(a:-
?r,+rJ)
-f’=O.
In equilibrium rr, = n-:, Applying this condition expression only, we can express (8) as 0L( rrl”) + bu -f’(
rt)
2 we obtain
=
provisionally
139
the optimality (8) to the bracketed
0.
(9) The relation of the policy game approach to the HIE with the traditional can be explained by means of Fig. 1. The LL curve is the demand curve for money and S,S, represents the seigniorage equation (l’), under conditions of steady state, for given level of seigniorage (S,), i.e. S, = rrL(n-TT).The curve S,S, represents a higher level of S and the point A,,, represents the maximum steady-state level of seigniorage. On S,S, the HIE is at the intersection with LL at A,, and the low inflation equilibrium is at A,. In the traditional analysis SS is treated as exogenous and the appropriate equilibrium point is determined by the dynamic properties assumed by the model. The policy game approach can be brought into this framework by means of Eq. (9) which can be rewritten, in steady states, as
(9’)
nf
L Fig. 1.
140
M.A. Kiguel, N. Liviatan/Joumal
of Development Economics 45 (1994) 135-140
The value of L corresponding to the right-hand side of (9’) is represented by YY in Fig. 1. The determination of rr is then represented by the intersection of LL with W at A’n which corresponds to a HIE. The level of seigniorage (or the ‘fiscal deficit’) is determined endogenously according to the intersection point. Thus the curve S,S, is drawn to be consistent with the intersection of LL and YY. The intersection point may of course occur at a low inflation equilibrium. The factors which will shift YY upward, towards a HIE, are high values of 8, bu and a low aversion to inflation (f’). The solution which corresponds to the precommitment regime is given by the horizontal line rrf which was derived from (7). This must always lie below A,. It must also be below the discretionary equilibrium even if the latter corresponds to a low inflation (as explained earlier).
References Auernheimer, Leonardo, 1973, Essays in the theory of inflation, Ph.D. Thesis (University of Chicago, Chicago, IL). Barro, Robert J., 1983, Inflationary finance under discretion and rules, Canadian Journal of Economics 16, 1-16. Barro and David B. Gordon, 1983, A positive theory of monetary policy in a natural rate model, Journal of Political Economy 91, 589-610. Bruno, Michael and Stanley Fischer, 1990, Seigniorage, operating rules and the high inflation trap, Quarterly Journal of Economics CV, 353-374. Evans, T.L. and G.K. Yarrow, 1981, Some implications of alternative expectations hypotheses in the monetary analysis of hyperinflations, Oxford Economic Papers 33, 61-80. Obstfeld, Maurice, 1989, Dynamic seigniorage theory: An exploration, Working paper no. 2869 (NBER, Cambridge, MA). Kiguel, Miguel and Nissan Liviatan, 1988, Inflationary rigidities and orthodox stabilization policies, The World Bank Economic Review 2, no. 3, 273-298. Sargent, Thomas J. and Neil Wallace, 1987, Inflation and the government budget constraint, in: A. Razin and E. Sadka, eds., Economic policy in theory and practice (MacMillan, London).
Development ECONOMICS
Journal of Development Economics Vol. 45 (1994) 135-140
ELSEVIER
A policy-game
approach to the high inflation equilibrium
Miguel A. Kiguel a,*, Nissan Liviatan a,b a The World Bank, 1818 H Street, N. W., Washington, DC 20433, USA ’ The Hebrew Unir,ersity of Jerusalem, Jerusalem,
Israel
Received April 1992; final version received August 1993
Abstract The high inflation equilibrium (HIE) refers to a steady state solution where inflation is higher than the one which maximizes seigniorage (i.e. inflation is on the falling segment of the Laffer curve>. It has been claimed that the HIE can be stable when the adjustment of expectation is fast. However, adjustment is fast because inflation is high; so the argument is circular. In this paper we rationalize the HIE using the policy game approach which distinguishes between discretionary and precommitment regimes. We expand, and demonstrate rigorously (using a technique developed by Obstfeld), Barro’s suggestion that a HIE may result from a discretionary regime. We show that this is not possible under precommitment. In addition, we demonstrate that a HIE may result from non-fiscal factors (such as the motivation to reduce unemployment) in a discretionary regime. We conclude with a simple diagrammatic exposition of the difference between our approach and the previous one. Keywords:
High-inflation
JEL classification:
equilibrium;
Policy game approach
E5, E6
1. Introduction Many economists rate that is justified
exceeds the share the view that inflation in some countries by the needs to finance the budget deficit through money
* Corresponding author The views expressed herein are those of the authors and do not necessarily reflect those of the World Bank or its affiliated organizations. 0304.3878/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0304-3878(94)00030-G
136
M.A. Kiguel, N. Liuiatan/Journal
of DeLzlopment
Economics 45 (1994) 135-140
creation (seigniorage). This is especially true in high inflation countries, such as Israel or Brazil, where high inflation (above 200 percent per year) has not been associated with unusually high seigniorage. One explanation for this phenomenon relies on the inflation tax model which is characterized by dual steady state equilibrium (Sargent and Wallace, 1981, 1987). The same level of seigniorage could be obtained at a low inflation rate, with an inelastic demand for money, or instead at a high inflation rate, above the revenue maximizing rate (we shall refer to the latter as High Inflation Equilibrium, HIE), at points on the elastic side of the demand for money. According to some models, the HIE can be explained by the fact that this equilibrium is stable while the low inflation equilibrium is unstable. Bruno and Fischer (1990) show that the HIE is stable when expectations are rational in the sense of myopic perfect foresight, and the money market adjusts instantaneously. In the framework of the adaptive expectations model authors have drawn a distinction between slow and fast adjustments of expectations. It has been shown that the HIE tend to be dynamically stable when expectations adjust rapidly (see Auernheimer (1973), Evans and Yarrow (1981)), but it is unstable when expectations adjust slowly. Since in high inflation economies expectations adjust quickly (because slow adjustment is costly), one may obtain a stable HIE. An entirely different approach to inflation has been formulated recently in the policy game literature (as in the papers by Barro and Gordon). This approach stresses the role of surprise inflation tactics and of non-fiscal motivations in generating high inflation rates. The inability of the policy maker (PM) to make credible commitments about his monetary policy rules (a phenomenon which is quite common high inflation economies) may then bring about a HIE as suggested by Barro (1983a, p. 12). This outcome is not possible when the PM is capable of making credible commitments about his policies. In the present paper we shall first present a rigorous and yet simple confirmation of Barro’s conjecture. We shall then show that non-fiscal motivations may lead to a HIE even when the fiscal motivation by itself will not yield this result.
2. High inflation
equilibrium
In order to see how a HIE can be generated define seigniorage (S,) as
in an inflation
tax model let us
where M, is the nominal stock of money (which is assumed to equal base money) and P, is the price level in period t. Inflation and expected inflation are defined as rrt = (P, - P,_ l)/Pt and nfe = (Pte - P,_ l)/Pte when P: is the price level P, expected in period t - 1. The demand for real balances (assuming a constant real
of Development Economics 45 (1994) 135-140
137
interest rate) is given by L, = ~5(n,~+1) (with L’ < 0) and it is assumed money market is in equilibrium. We may then express (11 as
that the
M.A. Kiguel, N. Liuiatan/Journal
s, =+-te+,
) -+-XI
(I’)
- r,).
In this equation rte is predetermined from period t - 1, while nt is treated as a policy variable in the discretionary regime. Following Barro (1983) we express the PM’s utility function for any period t as U,=~,Sr-f(7T,)--(~TT,e+,),
fl,r’>O,
(2)
where 13 represents the utility from seigniorage while (minus) f and r represent the loss from actual and expected inflation (the latter can be due to loss of liquidity through L). We shall assume, for simplicity, that 8 is constant, so that our system contains no random elements. We assume that the PM maximizes a discounted sum of utilities u, = &~U,, lY=t
O
(3)
where r is the discount factor. Let us assume that the policy function in the discretionary regime, specifying the optimal n, for any specified L,-, and 8 is given by the time-invariant function rrf = g(L,_ ,, 0). The existence of this function will be established below. This function will ensure time consistency in the sequential optimization for the infinite horizon model at hand (in using this methodology we follow Obstfeld (1989)). We assume that g( - ) is common knowledge so that the rational public, having observed L,, must form its expectations in t according to $+,
=g(‘%,
0)
(4)
Combining (4) and L(nF+ 1) we can determine 7~~~1. In fact, nT must be a constant (say rc) for all 72 t + 1 and L, is constant (say Lc) for T 2 t. In a steady state L,_ 1 is equal to L”. (It is understood that rrc and L’ depend on 0.1 Let us maximize (3) subject to (1’) and (2) treating rr, as a free variable, taking n,” (or L,_ 1> as given and imposing the constraint nTe = n7* = rrc for T 2 t + 1. The first-order condition yields 8L( 7r;> -f’(
Tr,) = 0.
We assume, along with Barro, that f” > 0 which enables 01, recalling that L,_i policy function rr,? = g(L,_,, information about the government’s objective function, quire that in the absence of shocks rrp = rr,?. This condition, Eq. (5) and the fact that n-T = rr, equilibrium for rr given 0 when L,_ 1 = Lc. Thus, using in (5) we have n=~(e>,
g=(f”-ez-‘L>o.
(5) us to derive from (5) the =L(nF). Given the full rational expectations redetermine a steady state rrl = r: = r,” = n-’ = rr
(6)
138
M.A. Kiguel, N. Liuiatan/Journal of Development Economics 45 (1994) 135-I40
Under the foregoing assumptions rr can be increased, in steady states, in an arbitrary manner by raising 8. It is, therefore, clear that the demand for money in steady states can be reduced sufficiently so as to reach the elastic region, which corresponds to the downward sloping part of the Laffer curve.
3. Equilibrium
under precommitment
In the precommitment regime the PM is bound by credible commitments with regard to monetary policy. Suppose that in every period t the PM makes a credible announcements about 7rt+ 1 so that rite+ 1 = rTTl+1 at the optimization stage. Given that he honors his commitment, he will treat 7~, as predetermined. If he chooses to increase rr,+ 1 this will reduce S, (since L(rrF+ 1> decreases) but S,, 1 will increase, as can be seen from (I’) moved one period forward. Assume, tentatively, that the PM’s policy function for any period 8 is ?r7+ i = G(0) and maximize (3) with respect to rr,+ , subject to rrt+ 1 = T,“, 1 and applying r+ 1 = G( 0) for T 2 t + 1 together with the rational expectations requirement $+ 1 = G( 0) for r 2 t + 1. Treating L(rrJrte)and rr, as given we obtain the optimality condition f3Z( 1 - I) + i+( L + 7rL’) - 7’ - rf’ = 0,
(7)
where the argument of all the functions is n = rr(+ 1, which is consistent with G(0). Note also that since L’, - 7’ and -f’ are negative, it follows that (L + rrL’) > 0, which means that L is inelastic. Thus the precommitment equilibrium is on the rising part of the Laffer curve. Comparing (7) with (5) it is readily verified that the solution value of r is smaller in the precommitment case than in the discretionary one.
4. Non-fiscal
cause for HIE
Non-fiscal considerations can be introduced as follows. Suppose that the PM considers the level of employment to be too low as a result of excessive unemployment benefits, over which he has little control (as in Barro and Gordon (1983b)) or because of labor monopolies. These factors can produce a HIE even when the fiscal side by itself, would not lead to this state. Let ur be the labor-unions’ real wage target so that the nominal wage is set at where p: is the expected price level for period t (all in terms of UL + P,", logarithms). The actual real wage is then given by vr + p,’ -p, which we approximate by vL + rrp - r,. The PM’s real wage target uo is assumed to be smaller than vL. We may then suppose that the PM wishes to minimize the deviations of the actual real wage from vo. Thus we add to the u, function (2) the term - (6/2)(7rF - ~~ - ~1)~ where u = uL - vG > 0.
M.A. Kiguel, N. Liuiatan /Journal
Using condition
of DeLxelopment Economics 45 (1994) 135-140
a similar reasoning to that of Section for rrt (analogous to (9)):
fX+b(a:-
?r,+rJ)
-f’=O.
In equilibrium rr, = n-:, Applying this condition expression only, we can express (8) as 0L( rrl”) + bu -f’(
rt)
2 we obtain
=
provisionally
139
the optimality (8) to the bracketed
0.
(9) The relation of the policy game approach to the HIE with the traditional can be explained by means of Fig. 1. The LL curve is the demand curve for money and S,S, represents the seigniorage equation (l’), under conditions of steady state, for given level of seigniorage (S,), i.e. S, = rrL(n-TT).The curve S,S, represents a higher level of S and the point A,,, represents the maximum steady-state level of seigniorage. On S,S, the HIE is at the intersection with LL at A,, and the low inflation equilibrium is at A,. In the traditional analysis SS is treated as exogenous and the appropriate equilibrium point is determined by the dynamic properties assumed by the model. The policy game approach can be brought into this framework by means of Eq. (9) which can be rewritten, in steady states, as
(9’)
nf
L Fig. 1.
140
M.A. Kiguel, N. Liviatan/Joumal
of Development Economics 45 (1994) 135-140
The value of L corresponding to the right-hand side of (9’) is represented by YY in Fig. 1. The determination of rr is then represented by the intersection of LL with W at A’n which corresponds to a HIE. The level of seigniorage (or the ‘fiscal deficit’) is determined endogenously according to the intersection point. Thus the curve S,S, is drawn to be consistent with the intersection of LL and YY. The intersection point may of course occur at a low inflation equilibrium. The factors which will shift YY upward, towards a HIE, are high values of 8, bu and a low aversion to inflation (f’). The solution which corresponds to the precommitment regime is given by the horizontal line rrf which was derived from (7). This must always lie below A,. It must also be below the discretionary equilibrium even if the latter corresponds to a low inflation (as explained earlier).
References Auernheimer, Leonardo, 1973, Essays in the theory of inflation, Ph.D. Thesis (University of Chicago, Chicago, IL). Barro, Robert J., 1983, Inflationary finance under discretion and rules, Canadian Journal of Economics 16, 1-16. Barro and David B. Gordon, 1983, A positive theory of monetary policy in a natural rate model, Journal of Political Economy 91, 589-610. Bruno, Michael and Stanley Fischer, 1990, Seigniorage, operating rules and the high inflation trap, Quarterly Journal of Economics CV, 353-374. Evans, T.L. and G.K. Yarrow, 1981, Some implications of alternative expectations hypotheses in the monetary analysis of hyperinflations, Oxford Economic Papers 33, 61-80. Obstfeld, Maurice, 1989, Dynamic seigniorage theory: An exploration, Working paper no. 2869 (NBER, Cambridge, MA). Kiguel, Miguel and Nissan Liviatan, 1988, Inflationary rigidities and orthodox stabilization policies, The World Bank Economic Review 2, no. 3, 273-298. Sargent, Thomas J. and Neil Wallace, 1987, Inflation and the government budget constraint, in: A. Razin and E. Sadka, eds., Economic policy in theory and practice (MacMillan, London).