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Optimal and Sustainable Exchange Rate Regimes: A Two-Country Game-Theoretic Approach

Author(s):
International Monetary Fund. Research Dept.
Published Date:
January 1993
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In the debate over the relative merits of fixed and flexible exchange rate regimes, economists have focused on such issues as the role of exchange rate flexibility in current account adjustment, the importance of monetary policy autonomy in pursuing national objectives, the need to enhance monetary policy credibility, and the public-goods nature of exchange rate stability. The theoretical literature has established that optimal exchange rate arrangements depend on a number of criteria, including the structural characteristics of an economy (such as wage-price flexibility, factor mobility, openness of goods markets, and financial integration), the nature of shocks to the economy (such as nominal versus real shocks and country-specific versus global shocks), and the relative importance of economic growth, price stability, and exchange rate stability as policy objectives.1 Recent studies have attempted to evaluate the welfare outcomes of different international monetary arrangements (with or without international policy coordination) for individual economies and the world as a whole. These studies have often involved game-theoretical modeling.2 This paper attempts to synthesize these studies and offers a theoretical framework for evaluating welfare outcomes of alternative exchange rate regimes when monetary policy is set noncooperatively.

The paper develops a type of Barro-Gordon (1983) macroeconomic model with two interdependent countries and examines the question of how optimal and sustainable exchange rate regimes are chosen. It attempts to identify optimal and sustainable exchange rate regimes under which each sovereign national authority sets an independent monetary policy. Although policymakers in the model noncooperatively pursue their objectives under a given regime, they may coordinate their choice of regime in order to secure a Pareto-superior outcome. In this paper, an exchange rate regime is defined as sustainable when it represents a perfect Nash equilibrium in the sense of Selten (1975)—that is, a Nash equilibrium in which a policymaker has no incentive to deviate from the chosen regime in the presence of “trembling.”

This paper will employ a multi-stage game approach suggested by Hamada (1985) to identify sustainable exchange rate regimes. The last stage of the game obtains each country’s maximum noncooperative payoff for each exchange rate regime- earlier stages involve determining sustainable (or perfect Nash equilibrium) exchange rate regimes. From these sustainable regimes, the two countries’ authorities may together choose the optimal one. The paper argues that the optimal choice of regime depends fundamentally on the authorities’ ability to make monetary policy commitments.3 Other important factors include the nature of real shocks to the economy (global or country-specific), the substitutability between domestic and imported goods, and the relative importance given to exchange rate stability in a policymaker’s objectives.

The following section sets up a two-country macroeconomic model consisting of two monetary authorities and three types of private agents and explains each decisionmaker’s objective. It also defines three exchange rate regimes (flexible, managed, and fixed) and three commitment scenarios (no commitment, symmetric commitment, and asymmetric commitment). Next, the paper examines each of the three commitment scenarios under alternative exchange rate regimes, explaining solution procedures and comparing equilibrium solutions and welfare outcomes. Finally, the paper turns to the strategic choice problem of exchange rate regimes in a three-stage game framework and focuses on the role of international coordination in the choice of exchange rate regime. The paper also explores what might change if the authorities attach importance to exchange rate stability in their policy objectives.

I. The Model

This section presents a two-country model, formulates the maximization problems of decisionmakers, and defines the three exchange rate regimes and the three commitment scenarios to be examined in the paper. The model is standard, and is kept straightforward enough to allow comparison of equilibrium outcomes under different commitment assumptions and under alternative exchange rate regimes.

Two-Country Model

The model is an extension of the Barro-Gordon (1983) closed macroeconomic framework to a two-country world economy framework. The model’s structure is similar to that of Rogoff (1985),Turnovsky and d’Orey (1986, 1989), Canzoneri and Henderson (1988), and Giavazzi and Giovannini (1988,1989). The model includes many popular elements (such as nominal wage setting and Lucas supply functions) but does not consider pertinent dynamics (such as those generated by capital accumulation and sticky prices) or imperfect-incomplete information. The two economies are assumed to be symmetric in size and structure. These simplifications help to focus more sharply on the essential role of monetary policy commitment in the choice of exchange rate regimes. The model is described as follows:

where

mt =nominal money supply in logarithmic form,

yt =real output, measured as a deviation from the natural rate level and expressed in logarithmic form,

pt =price of domestically produced goods in logarithmic form,

qt =general price index in logarithmic form,

qt+i\t =one-period-ahead expectation of the general price index in logarithmic form.

it =nominal interest rate expressed in natural units,

st= spot exchange rate, measured as units of home currency per unit of foreign currency in logarithmic form,

st+1\t=one-period-ahead expectation of the logarithm of the spot exchange rate,

ut vt, wt =mutually and serially uncorrelated disturbances with zero means.

Foreign variables are expressed with asterisks and home variables without. The Greek letters are positive constants.

Equations (1) and (5) describe the money market equilibrium conditions in the two countries. Money balances on the left-hand side and output on the right-hand side are deflated by the country’s general price index, qt.The income elasticity of money demand, Φ, is assumed not to be excessively greater than unity (so that A2, which is defined later, is positive). The disturbance, u1 or ut*, summarizes serially uncorrelated shocks affecting the money market.

Equations (2) and (6) represent goods demand in the two countries. Goods demand depends negatively on the real interest rate, itqt+1\t+qt, and positively on the real exchange rate, st+pt*pt, and on foreign output relative to domestic output, yt*yt. The real interest rate, which affects business capital investment and household purchases of durables (including houses), is defined as the nominal interest rate, it minus the expectation of the one-period-ahead rate of inflation in general price indices, qt+1\tqt. Business firms and households form the expectation of general prices, qt+1\t. The terms involving the real exchange rate and relative output, δ(st+pt*pt)+β(yt*yt), capture net exports. The stochastic disturbance, vt or vt*, represents serially uncorrelated demand shocks.

Equations (3) and (7) define goods supply in terms of standard Lucas supply functions. Each economy is assumed to be specialized in the production of a distinct good. Since labor is the only variable input used for production in each country, output depends positively on the inverse of the real wage facing firms, ptqt\t1. Here, qt\t1is interpreted as the period-t nominal wage set by labor and management in a wage contract negotiation. Recall that the same one-period-forward notation, qt+1\t, is used to indicate price expectations set by businesses and households. The term, wt or wt*, denotes serially uncorrelated supply shocks.

Equations (4) and (8) define the general price index in the two countries. The definition embodies the assumption that θ and 1 − θ are fixed expenditure shares spent on domestic and imported goods. It is assumed that people have preferences for domestic goods in their spending, so that 1/2θ1.

Equation (9) is the usual condition for uncovered interest rate parity. The two economies are perfectly integrated financially so that the nominal interest rate differential, itit*, equals the expectation of the oneperiod-ahead rate of change in the nominal exchange rate, st+1\tst.4 Currency and bond market traders set this expectation, which is nothing but the forward premium from the assumed absence of an exchange risk premium.

Equations (1)-(9) contain nine endogenous variables, two monetary policy instruments, and three expectations variables: yt,yt,*pt,pt,*qt,qt,*it,it,*st,mt,mt,*qt+1\t,qt+1\t,* and st+1\t. (The expectations variables qt\t1 and qt\t1* are predetermined, though they must be rational in equilibrium.) The home and foreign monetary authorities choose two out of the three variables mt,mt*, and st as their policy instruments, letting the remaining variable be determined endogenously. A chosen combination of instruments defines the exchange rate regime (see below). The model is closed by specifying how private agents set the three expectations variables, qt+1\t,qt+1\t,* and st+1\t. An important feature of the model is that it formulates both monetary policy and private behavior explicitly, which allows an analysis of the interactions between them.

Maximization Problems of Monetary Authorities and Private Agents

The objectives of the home and foreign monetary authorities are to set their respective monetary policies to maximize the expected value of the discounted sum of utilities:

Here, Δxt=xtxt1, where xt is any variable. E0[] is the mathematical expectations operator conditional on period 0 information (period 0 being the beginning of the planning horizon), ρ (orρ*) is the subjective rate of time preference, and Ut(orUt*) is the period-t instantaneous utility. The utility function indicates that each authority has the Barro-Gordon inflationary bias; it tries to raise the growth rate of real output, Δyt(orΔyt*), as much as possible, while stabilizing domestic price inflation, Δqt(orΔqt*), at zero.5 The parameter ω (orω*) is a relative weight attached to the objective of inflation stability. At this point of the analysis, the authorities are assumed not to attach any weight to exchange rate stability in their objectives. This issue will be taken up in a later section.

Private sector agents, in each period, form their expectations of the one-period-ahead general price indices and spot exchange rates. Private individuals are aggregated into three representative decisionmakers: the home price—expectations setter, the foreign price—expectations setter, and the global currency trader. The price-expectations setter represents the interests of business firms undertaking capital investment, house- holds purchasing durable goods, and labor and management negotiating nominal wages, and setsqt+1\t(orqt+1\t*).The global currency trader represents the interests of currency and bond market participants and sets st+1\t.

Hence the objectives of private agents are to set the values of their expectations variables, in each period, so as to minimize the expected prediction errors of general price indices or spot exchange rates. That is, they maximize:

with respect to qt+1\t,qt+1\t,* and st+1|t, respectively.6

Exchange Rate Regimes

This paper examines three exchange rate regimes, depending on whether the authorities use the money supply or the exchange rate as their policy instruments, and whether or not they fix the exchange rate.

The first is a freely flexible exchange rate regime where the two monetary authorities use their respective money supplies as policy instruments, allowing the exchange rate to be determined freely in the market. This is the regime most often studied in the recent literature on international macroeconomic policy interdependence. The second is a managed exchange rate regime where one authority uses the money supply as its policy instrument and the other uses the exchange rate (see Giavazzi and Giovannini (1989)). This regime introduces asymmetry to the model since the two authorities use different monetary policy instruments. The third is a fixed exchange rate regime where one authority fixes the exchange rate and the other uses the money supply as its policy instrument. Since fixing the exchange rate requires a commitment, the regime cannot be maintained unless the rate-fixing authority is fully capable of making a policy commitment.7

The paper focuses on these three polar exchange rate regimes and does not take up other cases where the authority controls a combination of the exchange rate and the money supply.8

Monetary Policy Commitment Versus Discretion

The ability of each authority to make a monetary policy commitment has important implications for determining which exchange rate regimes are optimal and sustainable. For example, intuition suggests that an authority capable of making a monetary policy commitment should not fix or manage the exchange rate against the currency of a country experiencing high and unstable inflation. Another example is that smooth functioning of a fixed exchange rate system requires that at least one authority be committed to a stable and noninflationary monetary policy.

This paper considers the polar cases in which each authority is either fully capable of making a monetary policy commitment or not at all capable of such commitment. This leads to three scenarios of monetary policy commitments:

No commitment—Noauthority can make a monetary policy commitment;

Symmetric commitment—Bothcountries’ monetary authorities can commit themselves to their respective policy rules;

Asymmetric commitment—Onlyone authority can make a commitment to a monetary policy rule, while the other authority cannot.

The concept of “commitment” in this paper is defined in terms of the authority’s ability to carry out any given (possibly time-inconsistent) policy rule as well as other players’ belief that it will indeed implement the rule. That is, an authority capable of making a commitment is interpreted as one endowed with technologies of monetary policy commitment. Since whether an authority is endowed with commitment technologies is assumed to be common knowledge, it introduces a leader-follower relationship to international policy games.

In the case of no commitment, both the home and foreign authorities pursue discretionary monetary policy by playing a Nash-Cournot game vis-à-vis each other and private agents.9 In the case of symmetric commitment, the two authorities set monetary policy rules acting as Nash-Cournot players with each other and as Stackelberg leaders vis-à-vis private agents.10 Finally, in the case of asymmetric commitment, the authority with commitment technologies behaves as the sole leader, while the other authority and private agents act as Nash-Cournot players with each other and as Stackelberg followers vis-à-vis the former authority.

Throughout the paper, it is assumed that all decisionmakers can observe the current and past disturbances and that the authorities make their decisions noncooperatively when setting their respective monetary policies. The following sections compare noncooperative outcomes under different assumptions of commitments and different combinations of monetary policy instruments (exchange rate regimes).

II. No Commitment

Let us first consider the case where neither the home nor foreign authority can commit itself to a monetary policy rule. In the absence of commitment, each authority pursues discretionary policy, acting as a Nash-Cournot player vis-à-vis the counterpart authority and private agents. In what follows, the equilibrium solutions are obtained for the money supply, real output, the general price indices, and the exchange rate, all in the form of rates of change, under alternative exchange rate regimes.11

Flexible Exchange Rate Regime

Under the flexible exchange rate regime, the home and foreign authorities set the money supplies, Δmt and Δmt*, in each period and, as a result, the exchange rate, Δst is determined freely in the market.

Since Δmt and Δmt*are policy variables, the reduced-form solutions to equations (1)—(9) can be expressed as

where

and Δ2xt=Δ(xtxt1)=xt2xt1+xt2, where xt is any variable.

To find equilibrium, first the home and foreign monetary authorities’ maximization problems are solved. Under the flexible exchange rate regime with no commitment, the two authorities set the values of the money supplies, Δmt and Δmt*, simultaneously in each period t to maximize their current objectives, Ut and Ut* defined by equation (10), taking as given each other’s money supply and private expectations, qt+1|t,qt+1|t,* and st+1|t, and subject to the reduced-form solutions (12).12 The first-order conditions of the maximization problems yield

which constitute the authorities’ reaction functions.

Next, the private agents’ maximization problems are solved. In each period t.the private agents maximize Vt,Vt*, and Wt defined by equation (11), over qt+1|t,qt+1|t,* and st+1|t, respectively, using equation (12) and taking Δmt and Δmt* and other private agents’ expectations as given. These maximization problems yield the following reaction functions of the private agents:

where Etxt+1 is the mathematical expectation of xt+1 (any variable in period t + 1) conditional on information available in period t. These are the usual rational expectations requirements used in many macroeconomic models.

Combining the reaction functions, equations (13) and (14), the equilibrium solutions under the flexible exchange rate with no commitment are obtained as 13

and,

Here, μt,μt*,ηt,ηt*, and ξt depend only on real shocks (Δvt,Δvt*,Δwt, and Δwt*) and are defined in the Appendix.

The rate of inflation in each country is stabilized at a positive rate, and real output and the exchange rate fluctuate randomly over time. Since private sector expectations are rational in equilibrium, monetary policy cannot raise the equilibrium growth rate of real output systematically above the natural rate. Real output, general prices, and the exchange rate depend only on real shocks, Δvt,Δvt*,Δwt, and Δwt*, because the money supply is set always to offset nominal shocks, Δut and Δut*. This property holds in all exchange rate regimes considered in the paper.

Each authority inflates as in a Barro-Gordon type of closed-economy model, because discretionary policymaking leads the authority to expand the money supply at a positive rate (without affecting equilibrium real output in a systematic way). However, the inflation rate, (Ψ1/Π1)/ωor (Ψ1/Π1)/ω*, is not too excessive in comparison with the rate that would be observed in a closed-economy model, which is γ/ω[>(Ψ1/Π1)/ω] or γ/ω*[>(Ψ1/Π1)/ω*].14 The reason is that in a two-country world under flexible exchange rates, each authority controls the money supply in an attempt to reduce price inflation indirectly through competitive appreciation of the respective currency. Although both authorities cannot appreciate their currencies simultaneously, the equilibrium is characterized by positive but moderate rates of money growth and price inflation. In a closed economy, there is no competitive appreciation by definition and the authority is forced to inflate at a higher rate.

The period-t utility of each authority, then, turns out to be

Taking the unconditional expected value,

Each authority’s welfare is negative reflecting the Barro-Gordon inflation bias.

The private agents’ expected utilities are

where Ξ is the variance of ξt defined as

The home and foreign price-expectations setters achieve the maximum (bliss) welfare of zero because general price inflation is stabilized completely at a known positive rate in each country. The global currency trader’s welfare is negative owing to unpredictable exchange rate fluctuations as long as Ξ>0. When Ξ=0, which is the case if the two countries’ real shocks are identical (vt=vt*andwt=wt*) or if goods produced in the two countries are perfect substitutes (δ,sothatΠ3Π40,Π1Π21/(1+λ+γΦ),andΓ20), the exchange rate is completely stabilized and the global currency trader achieves the maximum (bliss) welfare.

Managed Exchange Rate Regime

The managed exchange rate regime considered in this paper is asymmetric, in the sense that one country’s authority uses the exchange rate and the other country’s authority uses the money supply as policy instruments.15 If the home authority controls the money supply, Δmt, and the foreign authority manages the exchange rate, Δst the reduced-form solutions to equations (1)—(9) should be obtained as functions of Δmt and Δst (Hence the foreign money supply, Δmt*, is now determined by Δmt and Δst.) On the other hand, if the home authority manages the exchange rate, Δst and the foreign authority controls its money supply, Δmt*, the reduced-form solutions should be expressed as functions of Δst and Δmt*.

The equilibrium is presented here on the assumption that the home authority controls the money supply, Δmt and the foreign authority manages the exchange rate, Δst. The precise reduced-form expressions for this case are reported in the Appendix. When the policy instruments are reversed between the two authorities, the equilibrium solutions have only to be reversed, because of the assumed symmetry of the two-country model.

The home authority, the foreign authority, and private agents maximize Ut over Δmt,Ut* over Δst and Vt,Vt*, and Wt over qt+1,qt+1|t*, and st+1|t, respectively, each taking all other players’ choice variables as given, subject to the reduced-form expressions. These maximization problems yield

and

where μt,μt*,ηt,ηt*, and ξt are the same as those which appear in equation (15).

The equilibrium solutions indicate that the two authorities stabilize inflation at positive, but different, rates; home inflation, γ/ω is excessively high while foreign inflation, (Ψ1/Π1)/ω*, is moderate (since γ>Ψ1/Π1). Both countries’ equilibrium inflation rates are positive because of the Barro-Gordon inflation bias under discretion. Foreign inflation becomes moderate because the foreign authority can appreciate its own currency by managing the exchange rate and, thus, alleviate the Barro-Gordon inflation bias; it directly sets the exchange rate in such a way as to keep imported-goods prices from soaring. Home inflation becomes excessive because the home authority lacks a direct instrument to counteract home-currency depreciation and thus ends up importing inflation from abroad. Essentially, exchange rate management has a beggar-thy-neighbor effect in that the rate-managing authority is made better off, and the counterpart authority worse off.

The welfare of each authority per period is summarized as

It is clear that the rate-managing (foreign) authority enjoys a higher welfare than does the money supply—controlling (domestic) authority.

The private agents’ welfare levels turn out to be the same as those under the flexible exchange rate regime; they are summarized in (16).

Fixed Exchange Rate Regime

The fixed exchange rate regime of this paper requires an authority to make a fixed rate commitment. This means that when no authority is capable of making a monetary policy commitment, as assumed in this section, a fixed exchange rate regime cannot be maintained.

III. Symmetric Commitment

Next, it is assumed that both the home and foreign authorities can make monetary policy commitments. This case is not only of theoretical interest but also highly relevant; it allows the authority to behave as a Stackelberg leader vis-à-vis private agents and opens up room for a fixed exchange rate arrangement.

Flexible Exchange Rate Regime

Under the flexible exchange rate regime with symmetric commitment, each authority can precommit itself to a money supply rule. This is a contingency rule that relates the money supply growth rate in all future periods, Δmt or Δmt*, to the shocks to the economies. This section focuses on a linear money supply rule given the linearity of the two-country model and the quadratic objectives of the decisionmakers.16

To find equilibrium, first the private agents’ optimal responses to Δmt and Δmt* are obtained. They are the same as equation (14), except that the agents’ information set now contains the money supply rules. Then, the two monetary authorities, as Stackelberg leaders vis-à-vis private agents, take the private optimal responses into consideration and set the linear contingency rules for the money supply growth rate in all periods to maximize the unconditional expected utility, E0Ut or E0Ut*.17 The authorities set the rule in period 0 as Nash-Cournot players with each other. Given the rule, real output growth no longer depends on money supply growth and the authority’s objective now simply becomes stabilizing the rate of inflation.

Complete price stabilization is accomplished by setting the money supply rule as

where μt and μt* are the same as in equation (15a).18 Then, the equilibrium solutions for real output, prices, and the exchange rate are

The best contingency rule prescribes zero inflation for both countries in all periods, although real output and the exchange rate fluctuate randomly over time. In comparison to the case of no commitment, each authority can eliminate the Barro-Gordon inflation bias altogether through commitment to a stable. zero-inflationary money supply rule. Note that the differences in solutions between the case of symmetric commitment and no commitment are observed only in the constant terms of the nominal variables and that the random terms in all variables are identical in the two cases.

Hence, the authorities achieve the maximum welfare of zero:

The private agents’ expected utilities are the same as those in the case of no commitment (equation (16)).

Managed Exchange Rate Regime

Under the managed exchange rate regime, one country’s authority now makes a commitment to a linear exchange rate rule, while the other makes a commitment to a money supply rule.

It turns out that the equilibrium solutions are identical to those of the flexible exchange rate regime, equation (17). This is because each authority can set a stable zero-inflation policy rule regardless of the instruments chosen. Hence, with symmetric commitment, the flexible and managed exchange rate regimes are effectively the same. Unlike in the case of no commitment, exchange rate management by one authority does not exert an adverse effect on the other. Each authority can accomplish complete price stability and obtain the maximum possible level of welfare.

Fixed Exchange Rate Regime

Under the fixed exchange rate regime with symmetric commitment, one authority fixes the exchange rate and the other makes a money supply commitment. The authority fixing the rate abandons monetary policy autonomy while the other retains it.19

On the assumption that the home authority sets a contingency rule for the money supply, Δmt, and the foreign authority fixes the exchange rate, Δst = 0, the equilibrium solutions for real output and prices are obtained as

The home authority completely stabilizes inflation at zero in all periods, while the foreign authority on averageachieves the zero rate of inflation with some fluctuations around zero. Foreign averageinflation is zero because the fixed rate commitment allows the foreign authority to import zero inflation, on average, from the home country; the home country provides a nominal anchor for the world economy. However, the fixed rate commitment prevents the foreign authority from pursuing a stabilization rule, and hence causes unpredictable inflation fluctuations.

This result indicates that the home authority’s welfare is at its maximum, while the foreign authority’s welfare is generally lower:

where Ξ has been defined in equation (164 The expression for E0Ut* indicates that the foreign authority’s welfare is negative unless Ξ = 0. When Ξ = 0, which is the case if the two countries’ real shocks are identical or if goods produced in the two countries are perfect substitutes, the foreign authority does not lose anything by abandoning its monetary policy autonomy and pegging the exchange rate to the nominal-anchor home currency. Put differently, an authority experiencing country- specific shocks or producing goods distinct from goods abroad has nothing to gain from sacrificing monetary policy autonomy to undertake fixed rate commitments.20

The fixed exchange rate regime results in the following welfare to private agents:

The home price—expectations setter and the global currency trader achieve the bliss welfare of zero because of perfect stabilization of home prices and exchange rates. The foreign price—expectations setter, on the other hand, receives a lower level of welfare because of unpredictable inflation variability, unless Ξ = 0.

IV. Asymmetric Commitment

What will be the equilibria when only one authority can make a monetary policy commitment while the other cannot? In this section, it is assumed that the home authority makes a commitment, and hence sets a policy rule in period 0 as the sole Stackelberg leader. The foreign authority and private agents set their decision variables in each period tas Nash-Cournot players with each other and as Stackelberg followers vis-à-vis the home authority.21

Flexible Exchange Rate Regime

Under the flexible exchange rate regime, the home authority in period 0 sets a linear money-supply rule for all future periods, while the foreign authority sets the value of the money supply by discretion in each period.

The solutions are obtained in two procedures. First, the reaction functions of the foreign authority and the private agents are derived from their respective maximization problems. The foreign authority’s reaction function is nothing but the first-order condition of its period-t maximization problem:

The private agents’ reaction functions are the rational expectations conditions already derived in equation (14), with the home authority’s money supply rule included in the information set.

Second, the home authority sets the contingency rule for Δmt to maximize its unconditional expected utility subject to the rational expectations conditions and the foreign reaction function. This maximization problem yields the equilibrium solutions.

Thus, the home authority completely stabilizes the rate of inflation at zero while the foreign authority stabilizes it at a positive rate. Discretionary policymaking forces the foreign authority to inflate, though the inflation rate is not too excessive, as explained in the case of no commitment. The ability to make a commitment allows the home authority to set a zero-inflation money supply rule without being adversely affected by foreign inflation.

The welfare outcome is that the home authority achieves the maximum welfare of zero, while the welfare level of the foreign authority is negative at (Ψ1/Π1)2/2ω*. The private agents’ welfare levels are given by equation (16).

Managed Exchange Rate Regime

Under the managed exchange rate regime, the home authority sets a money supply or exchange rate rule in period 0 and the foreign authority sets the value of the remaining policy instrument by discretion in period t. This case deserves special attention since both the choice of monetary policy instruments and the degree of commitment are asymmetric between the two authorities.

Money Supply Commitment

First consider the case where the home authority makes a money supply commitment and the foreign authority sets the value of the exchange rate.

The equilibrium solutions turn out the same as those for the flexible rate regime above, equation (19). The home authority completely stabilizes inflation at zero, while the foreign authority stabilizes inflation at a positive moderate rate, (Ψ1/Π1)/ω*. Essentially, as long as the leader authority sets a money supply rule Δmt), the follower authority’s choice of policy instrument (whether Δmt* or Δst) does not matter. Hence, the flexible and managed exchange rate regimes are identical in outcome, if the authority capable of commitment sets a money supply rule and plays the role of a nominal anchor for the world economy.

Exchange Rate Commitment

Next, suppose the home authority makes an exchange rate commitment and the foreign authority sets the value of the money supply by discretion.

To obtain these solutions, first note that the reaction functions of the foreign authority and private agents are respectively expressed as equations (18) and (14). Then, the home authority sets the contingency rule for Δst, to maximize its unconditional expected utility subject to these two equations. This yields the equilibrium solutions:

The solutions indicate that the home authority completely stabilizes inflation at zero, while the foreign authority stabilizes inflation at an excessively high rate of γ/ω*. Foreign inflation becomes excessive because the home authority appreciates its own currency rapidly, thus exporting inflation to, and exacerbating the existing Barro-Gordon inflation bias in, the foreign country. The foreign authority cannot effectively counteract the rapid depreciation of its currency and ends up importing inflation. As in the case of no commitment, exchange rate management has an adverse effect on the counterpart authority when the latter is incapable of making a commitment and pursues a discretionary monetary policy.22 This is in sharp contrast to the case where the counterpart authority can make a commitment to a money supply rule (the case of symmetric commitment under managed exchange rates or the case of asymmetric commitment under a money supply commitment), in which exchange rate management has no beggar-thy-neighbor effect.

It turns out that the welfare outcomes of private agents are the same as those for a money supply commitment by the home authority or for the flexible exchange rate regime (equation (16)).

Fixed Exchange Rate Regime

Under the fixed exchange rate regime with asymmetric commitment, the home authority must make a fixed rate commitment and the foreign authority sets the value of the money supply in each period. This means that the home authority sacrifices its ability to pursue an anti-inflationary monetary policy and that the foreign authority exercises its autonomous (potentially inflationary) monetary policy. The anticipated outcome is not desirable.

The equilibrium solutions are

The home authority suffers from excessively high and variable inflation and the foreign authority experiences equally excessive but stable inflation. Ironically, the welfare of the home authority is lower than that of the foreign authority, though the former is capable of commitment and the latter is not. This is because the foreign authority, which stabilizes inflation by adjusting its money supply to shocks in each period, is pushed to inflating excessively at a constant exchange rate, just as in a closed- economy framework. The home authority, on the other hand, completely forgoes the opportunity to use an autonomous monetary policy rule; by fixing the exchange rate, it is forced not only to import the high inflation abroad but also to abandon monetary policy autonomy for stabilization.23

The welfare of private agents under fixed exchange rates is similar to the case of symmetric commitment. Since home inflation is now variable, the home price—expectations setter receives a negative welfare (when Ξ>0), while other private agents receive the maximum bliss welfare of zero.

V. Choice of Exchange Rate Regimes

A number of interesting results emerge from the above analytical exercises. This section summarizes the results concerning the payoffs to the authorities, explains how the authorities choose optimal and sustainable exchange rate regimes, and explores the situation in which the authority assigns some utility weight to exchange rate stability.

Summary of Payoffs

Table 1. summarizes the results. It presents the cases of no commitment, asymmetric commitment, and symmetric commitment, respectively. Three strategies are available to each authority: controlling the money supply (Δm), managing the exchange rate (Δs) and fixing the exchange rate (Δs=0). A combination of these strategies determines the exchange rate regime.24 The expressions in each cell of the table are the payoffs to the home and foreign authorities (their unconditional expected utilities per period, E0Ut and E0Ut*). Note that the payoffs involving both Δs and Δs*(Δs,Δs*),(Δs,Δs*=0),(Δs=0,Δs*), and (Δs=0,Δs*=0)—are not defined because the two authorities cannot simultaneously manage or fix the exchange rate to determine the endoge-nous variables of the economies. Also note that Table 1 does not report the payoff under the fixed exchange rate regime (involving Δs=0 or Δs*=0) because of the absence of an authority capable of fixed rate commitments. Since the authorities choose the exchange rate regime, the payoffs to the private agents are irrelevant, and hence not reported in the table.

Table 1.Summary of Payoffs to the Home and Foreign Authorities
No commitment
Foreign authority (discretion)
Δm*Δs*
Home authority(discretion)ΔmE0Ut=12ω(Ψ1Π1)2

E0Ut*=12ω*(Ψ1Π1)2
E0Ut=12ωγ2

E0Ut*=12ω*(Ψ1Π1)2
ΔsE0Ut=12ω(Ψ1Π1)2

E0Ut*=12ω*γ2
Undefined
Asymmetric commitment
Foreign authority(discretion)
Δm*Δs*
Home authority (rule)ΔmE0Ut=0

E0Ut*=12ω*(Ψ1Π1)2
E0Ut=0

E0Ut*=12ω*(Ψ1Π1)2
ΔsE0Ut=0

E0Ut*=12ω*γ2
Undefined
Δs=0E0Ut=ω2(γω*)21ω(Π1Π2Γ1)2Ξ

E0Ut*=12ω*γ2
Undefined
Symmetric commitment
Foreign authority(rule)
Δm*Δs*Δs*=0
Home authority(rule)ΔmE0Ut=0

E0Ut*=0
E0Ut=0

E0Ut*=0
E0Ut=0

E0Ut*=1ω*(Π1Π2Γ1)2Ξ
ΔsE0Ut=0

E0Ut*=0
UndefinedUndefined
Δs=0E0Ut=1ω(Π1Π2Γ1)2Ξ

E0Ut*=0
UndefinedUndefined

Notes:

When both authorities set discretionary monetary policies (no commitment); the payoffs are strictly negative under the flexible or managed exchange rate regime. (The fixed exchange rate regime cannot be maintained.) Each authority’s payoff turns out to he determined completely by the counterpart authority’s choice of instrument. An authority is made worse off if its counterpart uses the exchange rate as a policy instrument; exchange rate management has a beggar-thy-neighbor effect. The question is: How can each authority keep its counterpart from managing the exchange rate when the latter may have no private incentive to do so? This question is addressed below in a three-stage game framework.25

When the home authority can make a commitment to a policy rule and the foreign authority resorts to discretion (asymmetric commitment), the home authority always obtains the maximum payoff of zero under the flexible or managed exchange rate regime, because it can ensure stable zero inflation. The foreign authority’s payoff depends on whether the home authority makes a money supply or exchange rate commitment. Foreign inflation is moderately high and its payoff moderately low if the home authority makes a money supply commitment, but is excessively high and the payoff very low if the home authority makes an exchange rate commitment. Exchange rate management by the home authority has an adverse effect on the foreign authority, though exchange rate management by the latter has no such effect on the former. The home authority can accomplish stable zero inflation through commitment. The foreign authority would be better off if the home authority stays away from exchange rate management and sticks to money supply control.

Under the fixed exchange rate regime, the home authority’s payoff is extremely low and the foreign authority’s payoff is the same as that when the home authority manages the exchange rate. The reason the home authority’s payoff is extremely low is twofold: first, the home authority imports excessive inflation from the foreign country by fixing the exchange rate, and second it abandons monetary policy autonomy, thereby forgoing the opportunity to counteract shocks to the economy (the case of Ξ>0). Even when Ξ=0, the fixed rate regime is not attractive to the home authority because of the importation of foreign excessive inflation. The home authority can only lose if it decides to fix the exchange rate against the currency of an inflationary country. The result is intuitively plausible.

When both authorities make commitments to policy rules (symmetric commitment), they attain the highest payoff of zero under the flexible or managed exchange rate regime. This is because each authority can set a policy rule to ensure stable zero inflation under either regime. Therefore, the authorities are indifferent between the two regimes. (Exchange rate management has no beggar-thy-neighbor effect since the counterpart authority is capable of commitment and can thus control its own inflation.) Under the fixed exchange rate regime, inflation of the rate-fixing country is zero on average but is variable so that its payoff is negative, unless Ξ=0. When Ξ=0, the fixed rate regime is as attractive as other regimes to the rate-fixing authority. Regardless of the value of Ξ, the payoff to the authority controlling the money supply is always at its maximum: zero.

Strategic Choice of Regime: The Three-Stage Game

This section examines the question of the strategic choice of exchange rate regimes. The analysis offers the first formal attempt in the literature to apply a multi-stage game approach, suggested by Hamada (1985), to the choice ofexchange rate regime in a popular macroeconomic framework.26 The approach is useful in finding the game’s equilibrium regimes and focusing on aset of Pareto-dominant ones (that are equilibria) when there is a multiplicity of equilibrium regimes.

Table 1 has provided necessary information for this purpose. However, the fact that the payoffs of instrument combinations involving both Δs and Δs* are undefined means that it is inappropriate to use these tables directly to find the game’s equilibria. To avoid such instrument combinations, consider the following three-stage game.

In the first stage of the three-stage game, one country’s authority (the home authority) chooses one of the three strategies: controlling the money supply, managing the exchange rate, or fixing the exchange rate if it can make fixed rate commitments. In the second stage, the other (foreign) authority chooses a strategy from a feasible set. If the home authority chooses to manage or fix the exchange rate in the first stage, then the foreign authority’s only possible choice in the second stage is to control the money supply: managing or fixing the exchange rate is no longer included in its feasible set of strategies.27 The last stage of the three-stage game consists of a game to set the rule or value of the instrument chosen in the first and second stages. The task, then, is to find equilibrium as well as optimal combinations of strategies by using information given by Table 1.

To do so, it is convenient to examine the normal form of the game as described in Table 2. The home authority is assumed to move first and so choose to control the money supply Δm), manage the exchange rate (Δs), or fix the exchange rate (Δs=0) if it can. Since the foreign authority chooses its strategy after observing the home authority’s choice of policy instrument, a pure strategy for the foreign authority must specify its response to any possible choice of the home authority. Thus, in the case of no commitment, the foreign authority has two pure strategies: either always control the money supply (Δm*Δm*) or manage the exchange rate if the home authority has chosen the money supply and control the money supply (Δm*) if it has chosen the exchange rate

Table 2.Normal-Form Representation of the Three-Stage Game
No commitment
Foreign authority
Δm*Δm*Δs*Δm*
Home authorityΔm-1,-1-2,-1
Δs-1,-2-1,-2
Asymmetric commitment
Foreign authority
Δm*Δm*Δm*Δm*Δm*Δm*
Home authorityΔm-0,-1-0,-1
Δs-0,-2-0,-2
Δs=0-3,-2-3,-2
Symmetric commitment
Foreign authority
Δm*Δm*Δm*Δs*Δm*Δm*0Δm*Δm*
Home authorityΔm-0,-0-0,-0-0,-1
Δs-0,-0-0,-0-0,-0
Δs=0-1,-0-1,-0-1,-0

Δs*Δm*. In the case of asymmetric commitment, the foreign authority has two pure strategies: either always control the money supply Δm* or manage the exchange rate if the home authority has chosen the money supply and control the money supply Δm* if it has managed or fixed the exchange rate (Δs*Δm*Δm*). In the case of symmetric commitment, the foreign authority has three pure strategies: Δm*Δm*Δm*,Δs*Δm*Δm*,or0Δm*Δm*, the last of which means that the foreign authority fixes the exchange rate Δs*=0 if the home authority has chosen the money supply and controls the money supply Δm* if it has managed or fixed the exchange rate.

The payoffs to the home and foreign authorities are shown numerically in each cell of Table 2 the left-hand-side number is the payoff to the home authority and the right-hand-side number is the payoff to the foreign authority. Note that the numerical values of the payoffs are examples only but the cardinal ranking of the outcomes, which matters, is preserved. Nash equilibria are shown by boldface payoffs and Selten’s (1975) perfect Nash equilibria are indicated with circles.28 We call an exchange rate regime “sustainable” if it is a perfect Nash equilibrium in the sense of Selten.

In the case of no commitment, there are three Nash equilibria, and out of the three the two strategy combinations involving Δs are perfect Nash equilibria in the sense of Selten. Note that the flexible exchange rate regime (Δm,Δm*Δm*)is Pareto superior to the managed rate regime but is not a perfect Nash equilibrium. Hence, there is no exchange rate regime that is both optimal and sustainable. The implication is that the Pareto-superior outcome is not likely to be chosen if left entirely to the monetary authorities’ selfish decisionmaking about policy instruments. This suggests that international coordination and mutual agreements on the choice of policy instruments (and hence exchange rate regimes) is necessary to achieve a Pareto-superior outcome and avoid the undesirable beggar-thy-neighbor effects of exchange rate management.29 Since the resulting flexible exchange rate regime is at least a Nash equilibrium, each authority has no incentive to deviate from it as long as it is certain that the counterpart authority is fully committed to the flexible rate regime.

In the case of asymmetric commitment, the flexible and managed rate regimes, where the home authority chooses Δm or Δs, are perfect Nash equilibria. Comparing these equilibria, the strategy combinations involving Δm are Pareto comparable to each other and Pareto superior to the combinations involving Δs. That is, the regime where the home authority sets a money supply rule is Pareto superior to the regime where it sets an exchange rate rule. To achieve a Pareto-superior outcome, therefore, the home authority should agree to avoid exchange rate management and set a money supply rule as a nominal anchor in the world economy. Some pre-play communication or international coordination concerning the choice of exchange rate regime is called for because such an outcome is not automatically guaranteed by the home authority’s unilateral choice.

Once the home authority agrees to control the money supply, the foreign authority may control the money supply or manage the exchange rate, so that either the flexible or managed exchange rate regime can emerge. A fixed exchange rate regime never emerges; it is neither a Nash equilibrium nor Pareto optimal.

When symmetric commitment exists, there are five Nash equilibrium combinations of strategies, including the flexible and managed exchange rate regimes, which are all Pareto comparable to each other. Out of these five, four are perfect Nash equilibria. Since the four equilibria are both optimal and sustainable, neither pre-play communication nor international coordination of regime choice is necessary to ensure Paretoefficient outcomes. If the real shocks to the two economies are identical (global shocks) or if the goods produced in the two countries are prefect substitutes (the law of one price, or purchasing power parity) so that Ξ=0, the fixed rate regime is also a perfect Nash equilibrium and Pareto comparable to the other regimes. This result is consistent with the literature focusing on the importance of the symmetry of the shocks across countries and the similarity of industries as prerequisites for a system of fixed exchange rates.30

It is important to reemphasize that the type of international coordination mentioned above does not require coordinated (or joint) determination of the values or rules of the particular instruments chosen; it only requires that policymakers agree on the choice of instruments. The purpose of international coordination here is to establish an international monetary framework in which the national authorities can pursue their independent policy objectives noncooperatively and, at the same time, obtain desirable welfare outcomes. Thus, it must be distinguished from the widely discussed notion of international policy coordination where the authorities set monetary policy cooperatively for joint maximization of world welfare.31

Exchange Rate Stability as a Policy Objective

In our theoretical model above, it is assumed that an authority’s objectives are to raise real output growth and stabilize price inflation as much as possible. Exchange rate stability is not considered as an objective.

However, one of the policy issues concerning the flexible rate experience since the advent of generalized floating in 1973 is the potentially adverse effects caused by serious misalignments and excessive volatility of exchange rates. Many authors have argued that real exchange rates often move for reasons unrelated to changes in underlying economic factors, resulting in bandwagon effects and speculative bubbles. Exchange rate instability is alleged to raise exchange risk, inhibiting international trade and investment. It undermines the role of money as an international unit of account and thus generates inefficiency in global markets. Thus a case could be made that a welfare gain accrues from exchange rate stability.32 Indeed, the industrialized countries have increasingly geared their monetary policies toward exchange rate stabilization in recent years. A notable example is the EMS (European Monetary System) members’ attempts to limit exchange rate fluctuations within narrow bands and their plan to form a monetary union by the end of this century.

This section examines briefly the possible implications of including exchange rate stability as one of the policy objectives.

In the above analysis, when both authorities are fully capable of making a commitment (symmetric commitment), a fixed exchange rate regime is neither a Nash equilibrium nor Pareto optimal as long as Ξ>0. Only when Ξ=0 does the fixed rate regime become optimal and sustainable. The implication is that if Ξ is sufficiently small and the utility weight attached to exchange rate stability in the policy objectives is sufficiently large, the fixed rate regime may Pareto-dominate other regimes.

When only one authority is capable of making a commitment (asymmetric commitment), scope for fixed exchange rates is much more limited. Because fixing the rate is very costly to the rate-fixing authority, it not only imports excessive inflation from abroad but also forgoes the opportunity to pursue low and stable inflation. This means that to prefer a fixed rate regime the utility weight attached to exchange rate stability must be very large to offset the cost. To the extent that the resulting exchange rate stability inhibits the objective of price stability, however, the outcome may not be desirable and may reduce the credibility of a fixed rate commitment.

In the case of no commitment, the fixed exchange rate regime cannot be maintained since no authority is capable of making a fixed rate commitment. No matter how much utility weight is attached to exchange rate stability, therefore, the fixed exchange rate arrangement is not viable.

To summarize, there is some scope for fixed exchange rates when one country has a strong anti-inflationary commitment, which provides a nominal anchor for the international monetary system. When no authority can make a commitment, there is no such scope.

VI. Concluding Remarks

This paper has focused on the question of how to identify optimal and sustainable exchange rate regimes in a world economy of two interdependent countries. For this purpose, a simple game-theoretic model of the Barro-Gordon type is used to compare noncooperative welfare outcomes under different assumptions of commitments and under alternative exchange rate regimes. A three-stage game approach to the strategic choice of exchange rate regimes turns out to be very useful; it helps identify optimal (in a Pareto sense) and sustainable (or perfect Nash equilibrium) exchange rate regimes. The basic message is that it is in the best interest of policymakers to choose optimal exchange rate regimes that are sustainable while pursuing monetary policies noncooperatively under the agreed regime. This type of international coordination can secure the best welfare outcomes for individual economies and the world as a whole without endangering national sovereignty in the conduct of monetary policy.

The paper is an extension of the previous game-theoretic literature on exchange rate regimes (such as Canzoneri and Gray (1985),Canzoneri and Henderson (1988), and Giavazzi and Giovannini (1989)). It is a multi-period model with the Barro-Gordon inflation bias; it includes country-specific supply and demand shocks; it considers both monetary rules and discretion; and it compares noncooperative welfare outcomes under different commitment scenarios and under alternative exchange rate regimes. By using the multi-stage game approach by Hamada (1985) and extending the analysis by Turnovsky and d’Orey (1989), it offers a solid theoretical framework in which one can examine how selfish policy- makers choose exchange rate regimes from strategic viewpoints.

Some of the detailed, innovative results of the paper can be summarized as follows. The first, general result is that regardless of the commitment scenarios examined, exchange rate regimes that are Nash or perfect Nash equilibria (or sustainable) are found. International coordination would allow a Pareto-superior regime to be chosen from these Nash or perfect Nash equilibria. For example, the IMF, the OECD, the G7, and the Economic Summits could help shape a desirable exchange rate regime by providing a forum in which the choice of industrial countries’ policy instruments is appropriately coordinated. They do not have to coordinate the conduct of monetary policy, but have only to agree on an exchange rate regime in which monetary policies can be set noncooperatively.

Second, when no authority is capable of making a commitment (no commitment) the flexible exchange rate regime is both a Pareto-optimal and Nash equilibrium regime, though it is not perfect Nash. Although the managed rate regime is a perfect Nash equilibrium, it is Pareto inferior because of its beggar-thy-neighbor effect. Hence, strong international coordination is required to ensure that the authorities make commitments to the flexible exchange rate regime and avoid exchange rate management. No scope exists for fixed exchange rates as an optimal and sustainable regime.

Third, when one authority is capable of commitment and the other is not (asymmetric commitment), money supply control by the former ensures an outcome that is both Pareto optimal and sustainable (or perfect Nash equilibrium). Once the authority capable of commitment agrees to set a money supply rule, the other authority may control the money supply or manage the exchange rate. Thus, either a flexible or a managed exchange rate regime, with the anchor country pursuing a stable and zero-inflation policy, emerges as an optimal, sustainable regime. The fixed exchange rate regime is neither a Nash equilibrium nor optimal.

Fourth, when both authorities are equally capable of commitment (symmetric commitment), the flexible and managed exchange rate regimes yield identical welfare outcomes for the authorities and, as a result, coordination of the choice of exchange rate regimes is unnecessary. The fixed exchange rate regime is not a Nash equilibrium as long as Ξ>0. When Ξ=0, the fixed rate regime becomes a perfect Nash equilibrium and Pareto comparable to the other regimes. In fact, a fixed exchange rate regime can only become optimal and sustainable when symmetric commitment exists and Ξ=0,—that is, when both authorities can make commitments to policy rules andreal shocks to the economies are global or goods produced in both countries are perfect substitutes. Alternatively, two anchor countries facing different (country-specific) real shocks or dissimilar product mixes should not be tied together through fixed exchange rates.

Although the theoretical predictions of the model arequite plausible, the model itself may contain biases that favor a flexible exchange rate regime over a fixed rate regime. It did not include those elements that potentially work against a flexible rate regime, such as currency misalignments, speculative bubbles, imperfect or incomplete information, the public goods aspect of exchange rate stability, and the explicit role of fixed exchange rates as a device to enhance the credibility of low-inflation commitment. Inclusion of these elements may bolster the case for a fixed rate regime.

In the discussion of symmetric and asymmetric commitment, the paper did not examine the question of how an authority endows itself with commitment technologies. It is possible to analyze the situation where each authority has an incentive to build up unilaterally a reputation of being credibly committed and thus change the structure of the payoff matrix in its own favor, if the benefit of doing so exceeds the cost. Such an analysis would reveal that the lower the cost of pursuing an anti- inflationary monetary policy and the lower the subjective rate of time preference, the more incentive an authority has to enhance the credibility of its commitment in the eyes of other players. It would be useful in future work to incorporate these rational calculations by the authority.

Although the growing strains in intra-EMS currency relations, which began in the fall of 1992, will undoubtedly slow down the process of monetary integration, it is likely that the European Community countries will resume their move toward European Monetary Union once Germany overcomes the fiscal difficulties associated with the reconstruction of the eastern part of its economy. With the emerging strength of the deutsche mark and the Japanese yen, the world appears increasingly headed toward a multiple-currency system. What will the future international monetary system look like? The theoretical framework presented in this paper offers a useful guide for examining the factors that shape exchange rate regimes and thus may assist in designing a desirable future international monetary system.

APPENDIX

This Appendix provides the reduced-form solutions of the model under the managed exchange rate regime and the equilibrium solutions under different assumptions of commitments and under alternative exchange rate regimes.

Reduced-Form Solutions Under the Managed Exchange Rate Regime

The reduced-form solutions under the flexible exchange rate regime are presented in equation (12) of the text. The reduced-form solutions to equations (1)—(9) under the managed exchange rate regime where the home authority controls the money supply, Δmt and the foreign authority manages the exchange rate, ΔSt can be expressed as

where Ψj,Γj, and Πj are defined following equation (12) in the text. On the other hand, if the home authority manages the exchange rate and the foreign authority controls its money supply, the reduced-form solutions are the same as above except that the home and foreign variables have to he reversed because of the assumed symmetry of the economic structures of the two countries.

Equilibrium Solutions in the Case of No Commitment

The equilibrium solutions under the flexible exchange rate regimein the case of no commitment are expressed as follows:

where

and

The equilibrium below is based on the assumption that the home authority controls the money supply and the foreign authority manages the exchange rate:

and

Equilibrium Solutions in the Case of Symmetric Commitment

When both the home and foreign authorities can make a money supply commitment, the flexible exchange rate regimeyields the following equilibrium solutions:

The equilibrium solutions under the managed exchange rate regimeare identical to those under the flexible exchange rate regime:

When the home authority makes a commitment to a money supply rule and the foreign authority fixes the exchange rate,the equilibrium solutions are:

Equilibrium Solutions in the Case of Asymmetric Commitment

When the home authority makes a money supply commitment and the foreign authority sets the money supply in each period, the equilibrium solutions under the flexible exchange rate regimeare

When the home authority makes a money supply commitment and the foreign authority sets the exchange rate in each period, the equilibrium solutions for the managed regimeturn out to be the same as those under the flexible rate regime above:

When the home authority makes an exchange rate commitment and the foreign authority sets the money supply in each period, the equilibrium solutions under the managed regimeare

When the home authority makes a fixed exchange ratecommitment and the foreign authority sets the money supply in each period, the equilibrium solutions are

and

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The author is Professor of Economics at the Institute of Social Science, University of Tokyo. He was a Visiting Scholar in the Research Department when this paper was written. He is grateful to an anonymous referee. Alex Cukierman, Nicholas Economides, Donald Mathieson, Mark Spiegel, and participants at seminars at the IMF and New York University for comments and suggestions; to Morris Goldstein for helpful discussions; and to Charles Iceland for editorial assistance. The views expressed are those of the author and do not necessarily reflect those of the International Monetary Fund.

See the literature on optimum currency areas: Mundell (1961),McKinnon (1963),Kawai (1987), and Tavlas (1992).

As explained below, the word “commitment” refers to a poiicymaker’s ability to abide by prescribed rules, such as a stable zero–inflationary money supply rule and a fixed exchange rate rule. It also assumes that other players believe the prescribed rule will indeed be enforced.

See Mathieson and Rojas–Suarez (1990) for a model without perfect capital mobility and its implication for the choice of exchange rate regimes.

The objective function is standard in the literature; see Barra and Gordon (1983). With the authority’s utility function given by equation (10), it is well known that the Barro–Gordon economy is characterized by an inflation bias in the absence of precommitment. A similar inflationary bias arises when the utility function is given by

a variation of which is used by Rogoff (1985) and Canzoneri and Henderson (1988). The basic qualitative results of the paper do not hinge on which form is used as the authority’s objective function.

This formulation of private agents’ objectives is a direct extension of Rogoff (1987). See also Barra and Gordon (1983) and Backus and Driffill (1985). Alternatively, all the private agents may be aggregated into a single decision– maker or treated as an infinitely large number of independent decisionmakers. In the former case, the private agents as a whole may be assumed to set the values of their expectations, qt+1|t,qt+1|t*, and st+1|t, simultaneously so as to maximize a single objective Et[v1(qt+1qt+1|t)2v2(qt+1|*qt+1|t*)2v3(st+1st+1|t)2] where 0vj1 and Σvj=1. In the latter case, private individuals may be classified into three groups, that is, home price—expectations setters h(h=1,2,...),foreign price—expectations setters f(f=1,2,...),and global currency traders k(k=1,2,...),Home price—expectations setter hmay be assumed to set qt+1|thto maximize Et(qt+1qt+1|th)2,foreign price—expectations setter fto set qt+1|t*f to maximize Et(qt+1*qt+1|t*f)2,and global currency trader k to set St+1|tk to maximize Et(st+1st+1|tk)2,each taking all other agents’ choice variables as given. These alternative formulations yield results that are identical to those in the text.

In this paper, an authority incapable of making a commitment cannot fix the exchange rate, because fixing the rate is not necessarily the hest policy at the time of implementation. The paper does not consider the situation where an authority with limited credibility of fixed rate commitments may attempt to maintain fixed exchange rates.Cukierman (1992) and Cukierman, Kiguel, and Liviatan (1992) offer a useful guide in extending the analysis in this direction.

Also the paper does not consider the possibility of using the interest rate as a policy instrument. See Turnovsky and d’Orey (1989), Henderson and Zhu (1990), and Kawai and Murase (1993) for models with the interest rate as a possible policy instrument.

In a Nash–Cournot game under no commitment, players set their decision variables (the values of policy instruments and expectations variables) simultaneously in each period so as to maximize their individual objectives by taking other players’ decision variables as given.

The two authorities act as Nash–Cournot players with each other in that they set their policy rules simultaneously to maximize their objectives by taking each other’s policy rule as given. They act as Stackelberg leaders vis–à–vis private agents in that the former move first by committing themselves to policy rules, which the latter are forced to take as given when making their decisions.

The solutions are all expressed in first differences (that is, rates of change) for ease of exposition.

Even though each authority has an infinite time horizon, the maximization of the discounted sum of expected utilities reduces to the period–by–period maximization problem. The reason is that (a) the authority cannot make a policy commitment at the beginning of the planning horizon (period 0), and hence maximizes the objective function backward in time, and (b) the authority does not have to look beyond the current period t, when solving the period–r maximization problem, owing to the lack of backward–looking state variables, and hence memory. The resulting equilibria are memoryless (or open–loop Nash) equilibria.

To arrive at the equilibrium solutions one must first find the rational expectations of general prices and spot exchange rates by assuming away “rational bubbles.” The rational expectations equilibrium solutions are then obtained.

The two–country model is reduced to a closed–economy model if δ=β=1θ=0 and equation (9) is deleted. In a closed–economy Nash–Cournot game between an authority and private agents, the authority would be pushed to inflate at the rate γ/ω since it cannot make a monetary policy commitment.

Giavazzi and Giovannini (1989) study the managed exchange rate regime in a two–country two–period model without the Barro–Gordon inflation bias. The two countries in their model are subject to global supply shocks only. Our model extends their approach in several directions: it is a multi–period model with the Barro–Gordon inflation bias: it includes country–specific supply and demand shocks: and it compares equilibrium solutions under different commitment assumptions and different exchange rate regimes.

However, the linear money supply rule may not be the unique equilibrium rule since the linearity is not necessarily a generic feature of the model.

This is equivalent to the maximization of E0Σ(1+ρ)tUtorE0Σ(1+ρ*)tUt*because no memory is required to maximize the discounted sum of expected utilities. When setting the money supply rule, each authority takes the counterpart authority’s rule as given.

To arrive at this money supply rule, one must first find the rational expectations of general prices and spot exchange rates under a linear money supply rule, which turn out to be

Next, using these expressions, money–supply contingency rule (17a) is derived.

Canzoneri and Gray (1985) interpret fixed exchange rate regimes in a leader– follower framework where an authority controlling the money supply is the leader and an authority fixing the rate is the follower. They assume such a fixed–rate leader–follower relationship, while allowing both authorities to make commitments; see Canzoneri and Henderson (1991). This paper only assumes that an authority fixing the rate must make a fixed rate commitment: hence it must be a leader vis–à–vis private agents, and is not necessarily a follower vis–à–vis the counterpart authority. Obstfeld (1991) examines a fixed exchange rate regime with “escape clauses” (devaluation or revaluation options). The model in this paper does not allow currency realignments once the exchange rate is fixed. Nor does the model consider the role of fixed rates as a device to enhance a low– inflation reputation (Giavazzi and Pagano (1988)).

One policy implication is that countries capable of making monetary policy commitments should not fix, or adopt rigid target zones for, exchange rates when they are dissimilar in shocks and product mix. However, this does not preclude the desirability of exchange rate smoothing in extreme cases “when there are shared perceptions that exchange rates are very badly misaligned” (Goldstein and others (1992)). The last situation is not considered in this paper.

The case of asymmetric commitment is relevant when the world economy consists of countries with different degrees of commitment. Examples include the Bretton Woods system, with the United States as the leader and many other peripheral countries as followers, or the European Monetary System, with Germany as the leader and some of the “soft–currency” member countries as followers.

Giavazzi and Giovannini (1989) also discuss such an adverse effect, but they do not consider the case of asymmetric commitment nor compare results under different assumptions of commitment.

This might eventually lead to a loss of commitment reputation and the breakdown of such a scenario.

Though the exchange rate (Δs) is common to both authorities, a separate notation (Δs*) is used to indicate it as an instrument chosen by the foreign authority.

Such a question arises because, as pointed out by Canzoneri and Henderson in a similar context, an authority’s choice of instrument does not affect its own payoff but it does affect its counterparts’. “This question leads us naturally to the problem of how to model instrument selection in a game theoretic context … it appears to have no resolution that is completely satisfactory” (Canzoneri and Henderson (1988, p. 121)).

Turnovsky and d’Orey (1989) use a two–stage game approach in their study of the optimal choice of instrument in a two–country framework, when each country’s policy instruments include the money supply and the interest rate but not the exchange rate. Hence, they do not address the choice of optimal exchange rate regimes. In addition, they assume perfect substitutability between domestic and foreign goods(δ, so that Π3Π40,Π1Π21/[1+λ+γΦ], and Γ20, and hence Ξ0), which is a special case of the model in this paper.

The model thus avoids the potential problem of the instability of managed exchange rates suggested by Giavazzi and Giovannini (1989).

Selten’s (1975) perfect Nash equilibrium is often called the “trembling–hand perfect Nash equilibrium” to distinguish from his earlier concept of subgame perfectness. Trembling–hand perfectness reflects the idea that for a strategy to be an equilibrium it must continue to be optimal for the player even when the other player picks some out–of–equilibrium action with an arbitrarily small probability— that is, when the other player’s hand “trembles.” All perfect Nash equilibria in Table 2 turn out to he the same as “weakly dominant” strategies. See also Kreps (1990, chap. 12) for a concise discussion of these concepts.

Giavazzi and Giovannini (1989) argue that if one authority is sufficiently large in size relative to the other, the authority of the larger country is likely to have less incentive to manage the exchange rate. If the home country is large, its authority would have an incentive to control the money supply.

See Frenkel, Goldstein, and Masson (1991) and Tavlas (1992).Mundell (1961) argues that if two countries face identical disturbances, symmetrical policy responses will suffice to adjust their economies, rendering the exchange rate instrument ineffective. In addition, in a world where the law of one price prevails for goods, there is not much merit in maintaining monetary policy autonomy as discussed by McKinnon (1963). These are part of the criteria for optimum currency areas; see also Kawai (1987) for a survey.

Nevertheless it is useful to examine if the noncooperative outcome in this model replicates the efficient (or world planner’s) outcome. In the case of symmetric commitment, the noncooperative outcome is as efficient as the cooperative outcome. In the case of no commitment, the noncooperative outcome may never be dominated by the cooperative outcome as argued by Rogoff (1985). The case of asymmetric commitment may require the world welfare to be defined in a specific way; see Kawai and Murase (1993).

Many advocates of exchange rate stabilization regard the costs of large exchange rate fluctuations as very high; see, for example, Williamson and Miller (1987), McKinnon (1988), and Krugman (1988).

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