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.
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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.
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.
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,
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
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
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
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
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).