I. Introduction

In recent years there has been a marked increase in interest in international economic policy coordination, as evidenced by the proliferation of meetings of officials and of academic publications. Though the presumption is that policy coordination among the major industrial countries is a good thing, there exist valid doubts concerning the possibility of designing appropriate intervention and coordination rules when the effects of policies are uncertain. Speaking at the American Economic Association meetings in December, 1987, Martin Feldstein noted (Feldstein [1988, p. 10]):

Uncertainties about the actual state of the international economy and uncertainties about the effects of one country’s policies on the economies of other countries make it impossible to be confident that coordinated policy shifts would actually be beneficial.

Though coordinated policies may, ex post, turn out to be have been ill advised, the relevant question is whether they are likely to result in higher welfare on average than uncoordinated policies, despite the presence of such uncertainties.

The issue has been discussed in several recent papers, though a consensus on the implications of model uncertainty for the desirability of coordination has yet to be achieved. Frankel, in a series of articles (Frankel [1987], [1988], Frankel and Rockett [1988]) argues that model uncertainty makes coordination too risky and that, on average, countries are as well off pursuing non-cooperative policies as they are under coordination. Holtham and Hughes Hallett [1987], in contrast, show that criteria can be applied that diminish the likelihood that coordination will be bad. Ghosh and Ghosh [1986] and Ghosh and Masson [1988] have argued that the existence of model uncertainty does not necessarily preclude a beneficial role for the coordination of macroeconomic policies; indeed, it may in fact provide an additional incentive to coordinate policies internationally.

The differences in conclusions stem from two essential differences in approach. First is the question of whether to evaluate gains ex ante or ex post. Ghosh and Masson focus on the ex ante expected gains from coordination while Frankel considers the ex post actual gains after arbitrarily specifying which is the true model. The second issue concerns the nature of expectations formation. In Ghosh and Masson, policymakers are assumed to have rational expectations across the set of possible models and to take due account of the presence of model uncertainty in formulating policies whereas in Frankel’s approach policymakers are assumed to have different subjective priors with respect to the different models, but each believes (wrongly) that he/she knows the correct model. ^1/

In this paper we re-examine the issue of model uncertainty and policy coordination, highlighting the effects of the differences in approach described above. We use variants of a simple consensus open-economy model presented in Oudiz and Sachs [1985], which is based on the Dornbusch [1976] extension to the Mundell-Fleming model. ^2/ If the subjective priors of policymakers (and of private agents) equal the objective probabilities, the average of welfare values achieved ex post—presumably the relevant criterion—will equal the ex ante expected welfare value. This was the case in our earlier paper; here we relax that assumption, and allow subjective priors and objective probabilities to differ, as does Frankel. We estimate alternative versions of the Oudiz-Sachs model, and perform stochastic simulations on the assumption that one or the other version is the true one, and agents assign non-zero probabilities to each of the models; we use the ex post welfare criterion in evaluating the gains from coordination. Our work differs from most of Frankel’s work, however, in that we continue to maintain that policymakers take account of the model uncertainty, and maximize expected utility over the range of models.

This paper also makes a significant departure from all of the earlier coordination literature in that it abandons the purely static view of model uncertainty which has hitherto been adopted. Instead, it is assumed that agents update their priors over the set of possible models in a Bayesian fashion. ^3/ There is an extensive literature on “learning rational expectations” (Brock [1972], Cyert and de Groot [1974], De Canio [1979], Bray [1982], Blume, Bray and Easley [1982], Bray and Savin [1986], and Marcet and Sargent [1988]). The conclusion of this literature is that least squares learning seems to converge to rational expectations in a wide variety of circumstances, but that it may also converge to an incorrect model. In particular, Blume and Easley [1982] is closest to the setup of our paper, as the authors consider Bayesian updating of prior probabilities applied to a finite set of models. They construct examples of cases where both the true model and the wrong model are locally stable, i.e. where it is possible that agents do not converge to rational expectations. It is thus of interest to analyze in an empirical macroeconomic model whether convergence to rational expectations occurs, both in a cooperative and in a Nash equilibrium. Furthermore, since the process of learning affects both policy setting and the private sector’s expectation formation, it will change the economy’s equilibrium and hence the estimated gains from coordination.

We evaluate three regimes: a cooperative equilibrium in which policies are jointly chosen to maximize a weighted average of the two countries’ utilities; a non-cooperative, or Nash, equilibrium in which each country maximizes its own utility, taking as given the actions of the foreign government; and finally a non-interventionist “pure float” exchange rate policy, where each country keeps the money supply at its exogenous target level. The models we use are based on an empirically-estimated, two country (U.S. and an aggregate rest-of-OECD) model with a number of structural variants. Although these structural differences are seemingly minor and innocuous, the differences in the reduced-form multipliers of the models are substantial, with the degree of model uncertainty similar to that in Frankel’s study. ^4/ Policy conflicts between the countries arise from structural shocks to the world economy—including money demand shocks, aggregate demand shocks and inflation shocks. In our simulation analysis we use drawings from a joint distribution describing these shocks that is based on the empirically-estimated covariance matrix.

Our main conclusions may be briefly summarized. First, relatively minor structural differences in the models translate into substantial differences in the reduced-form multipliers. Second, if optimal policies—whether cooperative or non-cooperative—assign sufficiently little weight to the true model, and there is no learning, then the economy can become dynamically unstable with potentially huge gains or losses from coordination. In contrast, uncoordinated policies involving freely-floating exchange rates and exogenous money targets never result in dynamic instabilities in our estimated version of the Oudiz-Sachs model. Therefore, in the absence of model learning, floating exchange rates may be optimal because they are more robust to model misspecification. We have argued previously (in Ghosh and Masson [1988]) that the lesson from model uncertainty is not that coordination is bad, but that policy—whether it is coordinated or not—should not be too activist. Our results here support that conclusion. Third, once Bayesian learning is allowed, optimal policies never result in dynamic instability and, even when very little initial probability is assigned to the true model, the ex post gains from coordination (when discounted back to the present) are always positive in our simulation exercises.

The reason for the latter result is simple. If policies are set to maximize ex ante expected utility, coordination only results in welfare deterioration ex post if the models are very different; in that case, however, it becomes very easy to distinguish between the models and learn which is the true model. The subjective priors therefore quickly converge to the true model and coordination is welfare improving. We would not want to exaggerate the relevance of this result to real-world policy choice; instead, it highlights the inadequacy of the assumption that one model is the “true” one. ^5/ Nevertheless, experience of the past quarter century does provide evidence that policymakers abandon views of the world that can be seen to be wrong—such as the view that there exists a long-run tradeoff between unemployment and inflation that can be exploited by aggregate demand policy—and thereby avoid the more disastrous consequences of their actions.

The paper is organized as follows. Section II presents some of the theoretical arguments concerning the effects of learning on the gains from coordination. Section III describes the empirical model used for the simulations. Section IV describes the procedure used to evaluate the gains from coordination under model uncertainty and Bayesian learning. Section V presents the results of the simulation analysis. Section VI discusses some caveats to the results, while section VII offers some brief concluding remarks.

II. Gains From Policy Coordination: Ex ante and Ex post

In discussing the benefits of policy coordination it is useful to distinguish two concepts of the welfare gains. The first criterion is the ex ante expected gain from coordination—that is, the difference between the expected utility under the cooperative regime and the expected utility under the non-cooperative regime. The second is the ex post actual gain from coordination—the difference in actual welfare level after realizations of all random variables are known. In most of the literature, where the gains from coordination arise from policy interdependence and the inefficiency of the Nash equilibrium (see Cooper [1968], Hamada [1974], Oudiz and Sachs [1984], Canzoneri and Gray [1983]), the distinction is immaterial. As long as the models are deterministic, or at least certainty equivalence obtains, both criteria give the same ranking of strategic regimes. ^6/ The main conclusion of this literature is that the gains from coordination are usually positive though not necessarily so (Rogoff [1985]).

There is some recent literature on coordination under model uncertainty, in which it is assumed that policymakers are uncertain about the effects of their instruments on the target variables. In this context certainty equivalence does not apply, and the use of the ex ante welfare criterion can give very different results from the ex post measure. We will term “ex ante expected utility approach” the use of subjective priors to evaluate welfare gains; examples of this approach are Ghosh [1986], Ghosh and Ghosh [1987], Ghosh and Masson [1988], all of which assume that optimal policies are set on the basis of expected utility maximization (as in Brainard [1967]) using their subjective probability priors. The use of subjective priors to compute actual average level of welfare is legitimate as long as the subjective priors also correspond to the objective probabilities that the models are true. In this approach, the structure of the economy is viewed as changing over time and at any instant is a random realization from the set of possible models. Aside from the possibility that other distortions in the economy negate the gains from coordination (as in Rogoff’s example mentioned above) the ex ante level of welfare can never be lower with coordination than without coordination.

In contrast, what we will call the “ex post welfare criterion” evaluates welfare using the “true” model: the economy is assumed to be correctly characterized by a single model, which policymakers however have not yet discovered. In this context, coordination may actually reduce welfare. Frankel and Rockett [1988], for instance, assume in the main part of their paper that policymakers ignore the presence of model uncertainty when choosing policies but assume (possibly wrongly) that they know the true model. Since the subjective priors of policymakers (which are unity for the model they believe and zero for all others) do not in general coincide with the objective probabilities (which are unity for the true model and zero otherwise) the average ex post level of welfare differs from the ex ante expected utility and coordination may decrease welfare.

In order to illustrate the issues involved, it is useful to consider the following, static, reduced-form theoretical model (the discussion is based on Ghosh and Masson [1988]). Let the equations describing inflation (p) and output (y) of the home economy be:

and

where ϕ is a positive shock to inflation which is observed at time t, M and M are the domestic and foreign money supplies respectively, $\begin{array}{cc}\begin{array}{cc}\mathrm{ϵ}\begin{array}{c}\mathrm{p}\\ \mathrm{t}\end{array}& \mathrm{and}\end{array}& \mathrm{ϵ}\begin{array}{c}\mathrm{y}\\ \mathrm{t}\end{array}\end{array}$ are i.i.d. zero mean shocks which are not directly observable, and λ, θ, γ, ν are all positive constants. Policymakers are assumed to have a quadratic objective function V defined over output and inflation, with relative weight ϕ, and discounted using factor β:

A similar objective function and reduced-form model applies to the fully symmetric foreign country.

In an uncoordinated equilibrium each country maximizes its utility taking as given the policy choice of the foreign country. In a cooperative equilibrium, a global social planner maximizes a weighted average of the two countries’ objective functions:

In what follows we will drop the time subscript and only consider optimal policies in period t. Moreover, since the countries’ parameters are the same and the shocks are symmetric, the two countries’ policies are identical. We will report only the home country’s values. The Nash, or uncoordinated equilibrium of this model is given by:

M^N = -(λ + θ)ϕ/[λ(λ + θ) + ϕγ(γ + ν)]

Assuming a symmetric cooperative equilibrium (ω=0.5), the corresponding cooperative policy is:

M^C = -λϕ/[λ² + ϕ(γ + ν)²]

The assumptions on the parameter values are sufficient (though not necessary) to ensure that the Nash equilibrium is too contractionary in response to an inflation shock:

M^N = M^N* < M^C = M^C*

The intuition is straightforward: in the uncoordinated regime, each country assumes that tighter monetary policy relative to the foreign country will result in an exchange rate appreciation and therefore a fall in the inflation rate. Since both countries attempt this simultaneously, their efforts are vitiated and the world simply inherits a contractionary bias (see Sachs [1983]). The cooperative planner recognizes the futility of competitive appreciations and therefore avoids this contractionary bias.

We now introduce model uncertainty into the framework by assuming that the parameters (θ,γ/,λ,ν) are uncertain, with means (μ_θ,μ_γ,μ_λ,μ_ν) and variances $\left({\mathit{\sigma }}_{\mathrm{\theta }}^2,{\mathit{\sigma }}_{\mathit{\gamma }}^2,{\mathit{\sigma }}_{\mathrm{\lambda }}^2,{\mathit{\sigma }}_{\mathit{\nu }}^2\right)$; covariances between pairs of parameters are assumed to be zero. Optimal policies under model uncertainty in each regime are given by:

Although a “small” degree of model uncertainty about domestic multipliers raises the incentive to coordinate, as the degree of domestic multiplier uncertainty increases $\left({\sigma }_{\lambda }^2\begin{array}{cc}\begin{array}{cc}& \mathrm{or}\end{array}& {\sigma }_{\gamma }^2\Rightarrow \infty \end{array}\right)$, the deflationary bias of the Nash regime is eliminated and the Nash and cooperative strategies converge (Ghosh and Masson [1988]). By contrast, greater uncertainty about the transmission effects $\left(\begin{array}{c}{\sigma }_{\nu }^2\Rightarrow \infty \end{array}\right)$ ^7/ causes a larger divergence between the Nash and cooperative strategies. These results indicate how model uncertainty affects the ex ante expected gains from coordination; since these policies are chosen to maximize the ex ante expected utility, by construction expected welfare cannot be lower under cooperation than under non-cooperation.

However, for any given realization from the set of possible models, coordination may result in lower welfare than non-cooperative behavior; therefore, ex post welfare may be lower if governments have cooperated. For instance, in Frankel [1987], since there is one true model of the economy, but policymakers do not know which of several alternatives is the correct one, then, ex post, the cooperative regime may yield lower welfare than the non-cooperative regime. This is all the more likely if policymakers completely ignore model uncertainty in choosing policies, and hence are likely to be more activist than is optimal.

To simplify the discussion assume that the uncertainty about the parameters arises because there are a two possible models of the economy:

pⁱ = αⁱM - θⁱM^* + ψ i=1,2

where α=λ+θ, and that policymakers have a single objective, inflation (ϕ=0). Agents assign prior probabilities of Π and (1-Π) to each model. The non-cooperative strategy is:

M^N = M^N* = -ψ[Πα¹ + (1-Π)α²]/[Πα¹(α¹-θ¹) + (1-Π)α²(α²-θ²)]

and the cooperative strategy is:

M^C = M^C* = -ψ[Π(α¹-θ¹) + (1-Π(α²-θ¹)]/[Π(α¹-θ¹)² + (1-Π(α²-θ²)²]

A third possible policy is to refrain from intervention entirely, or rather to maintain the money supply at the level prevailing before the shock:

M^F = M^F* = 0

Such a policy corresponds to the combination of freely flexible exchange rates and a fixed target for the money supply.

Suppose model 1 is the true model; then the value function associated with each regime is given by:

V^N = -(1/2)ψ²{(1-Π)²[α²(α¹-θ¹)-α¹(α²-θ²)]²/[Πα¹(α¹-θ¹) + (1-Π)α²(α²-θ²)])

V^C = -(1/2)ψ²{(1-Π)²[(α²-θ²)((α¹-θ¹)-(α²-θ²)]²/[Π(α¹-θ¹)² + (1-Π) (α²-θ²)²]}

V^F = -(1/2)ψ²

The level of welfare under cooperation is necessarily an increasing function of Π, the probability associated with the true model; whereas the second-best nature of the Nash equilibrium implies that welfare may be an increasing or a decreasing function of the probability assigned to the true model. Since the pure float regime does not require any policy intervention, the actual level of welfare is independent of policymakers’ subjective priors. From the expressions for the value functions it is simple to find specific parameter values such that coordination is welfare inferior to non-cooperative policies as long as Π ≠ 1. However, it is also possible to find parameter values such that the floating exchange regime is superior to both cooperative and non-cooperative policies, indicating that the use of any activist policy in the face of uncertainty may make matters worse. As argued in Ghosh and Masson [1988], however, this shows that caution is needed in setting policy in these conditions, and is not an indictment of coordination per se.

It is to be expected from the discussion above that the effects of learning on the ex ante gains from coordination will depend upon whether initially there is a large degree of domestic multiplier or transmission multiplier uncertainty. If uncertainty is primarily related to transmission effects then as agents learn about the model the gains from coordination should decrease, and conversely for uncertainty about the domestic effects of policies. The effects on ex post gains from coordination are more difficult to characterize: since welfare under cooperation must rise as agents learn about the model, but may either increase or decrease in the non-cooperative equilibrium, the effects on the gains from coordination are ambiguous.

It is an empirical question, to which we now turn, as to whether coordination is likely to reduce welfare either relative to a Nash strategy or to a pure floating exchange rate. In the absence of model learning, coordination may certainly reduce welfare ex post, and, in a dynamic model, may lead to dynamic instability of the economy if the wrong model is used to set policies. On the other hand, if learning is sufficiently fast, and ex ante optimal policies are sufficiently robust to model errors, coordination could still be welfare superior to non-cooperative policies, both ex ante and ex post.

III. The Empirical Model

The empirical model we adopt is a general, two-country Mundell-Fleming model with forward-looking exchange rate expectations (see Dornbusch [1976], Oudiz and Sachs [1985]). The model was estimated using the data that are the basis for the IMF’s MULTIMOD model (Masson and others [1988]). Results are presented in Table 1, using the same notation as in Oudiz and Sachs [1985]. Our two countries are the United States and the rest of the world; data for the latter region resulted from aggregation of the remaining industrial countries in MULTIMOD. ^8/ The estimation period, using annual data, was 1966 to 1986. Aggregate demand equations were estimated using instrumental variables: the instruments included money stocks, government spending, time, and lagged prices and output.

Table 1.

Parameter Estimates for Oudiz-Sachs Model, 1966-86. (standard errors in parentheses)

Table 1.

Parameter Estimates for Oudiz-Sachs Model, 1966-86. (standard errors in parentheses)

Aggregate demand
	$\begin{array}{lllll}{\mathrm{q}}_{\mathrm{t}}& =& \underset{\left(.051\right)}{-.110}({\mathrm{p}}_{\mathrm{t}}-{\mathrm{e}}_{\mathrm{t}}-\mathrm{p}\begin{array}{c}*\\ \mathrm{t}\end{array})\underset{\left(.206\right)}{+.037}\mathrm{p}\begin{array}{c}*\\ \mathrm{t}\end{array}-\underset{\left(.301\right)}{.178}({\mathrm{i}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}+1}+{\mathrm{p}}_{\mathrm{t}})& & \\ & & \begin{array}{cc}\begin{array}{c}\underset{\left(.009\right)}{+.021}\mathrm{t}+\underset{\left(.167\right)}{.288}{\mathrm{g}}_{\mathrm{t}}+\underset{\left(2.302\right)}{5.391}\end{array}& \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& \begin{array}{ccc}{\overline{\mathrm{R}}}^2=.984& \mathit{\sigma }=.021\mathrm{DW}=1.78& \end{array}\end{array}\end{array}& & \\ \mathrm{q}\begin{array}{c}*\\ \mathrm{t}\end{array}& =& \underset{\left(.096\right)}{.171}({\mathrm{p}}_{\mathrm{t}}-{\mathrm{e}}_{\mathrm{t}}-\mathrm{p}\begin{array}{c}*\\ \mathrm{t}\end{array})+\underset{\left(.496\right)}{.514}{\mathrm{q}}_{\mathrm{t}}-\underset{\left(.207\right)}{.384}(\mathrm{i}\begin{array}{c}*\\ \mathrm{t}\end{array}-{\mathrm{p}}_{\mathrm{t}+1}^{*}+{\mathrm{p}}_{\mathrm{t}}^{*})& & \\ & & \underset{\left(.025\right)}{-.030\mathrm{t}}+\underset{\left(.468\right)}{1.599}{\mathrm{g}}_{\mathrm{t}}^{*}-\begin{array}{cc}\begin{array}{cc}\begin{array}{c}\underset{(6.0680}{5.726}\end{array}& \end{array}& \end{array}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{ccc}\begin{array}{cc}& \end{array}& {\overline{\mathrm{R}}}^2=.990& \mathit{\sigma }=.020\mathrm{DW}=1.15\end{array}& \end{array}& \end{array}& \end{array}& & \end{array}$
Money demand
	$\begin{array}{lllllll}{\mathrm{m}}_{\mathrm{t}}-\mathrm{pt}& =& \underset{\left(.066\right)}{.225}{\mathrm{q}}_{\mathrm{t}}-\underset{\left(.413\right)}{1.419}{\mathrm{i}}_{\mathrm{t}}+\underset{\left(.503\right)}{3.453}& & {\overline{\mathrm{R}}}^2=.382& \mathit{\sigma }=.038& \mathrm{DW}=1.02\\ {\begin{array}{c}\mathrm{m}\end{array}}_{\mathrm{t}}^{*}-{\mathrm{p}}_{\mathrm{t}}^{*}& =& \underset{\left(.078\right)}{.700}{\mathrm{p}}_{\mathrm{t}}^{*}-1.077{\mathrm{i}}_{\mathrm{t}}^{*}+\underset{\left(.616\right)}{.100}& & {\overline{\mathrm{R}}}^2=.853& \mathit{\sigma }=.051& \mathrm{DW}=0.67\end{array}$
Consumer price index
	$\begin{array}{ll}{\mathrm{p}}_{\mathrm{t}}^{\mathrm{c}}& =.899{\mathrm{p}}_{\mathrm{t}}+\left(1-.899\right)\left({\mathrm{p}}_{\mathrm{t}}^{*}+{\mathrm{e}}_{\mathrm{t}}\right)\\ {{\mathrm{p}}_{\mathrm{t}}^{\mathrm{c}}}^{*}& =.758{\mathrm{p}}_{\mathrm{t}}^{*}+\left(1-.758\right)\left({\mathrm{p}}_{\mathrm{t}}-{\mathrm{e}}_{\mathrm{t}}\right)\end{array}$
Output price change
	$\begin{array}{ccc}{\mathrm{p}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}-1}& =& \left({\mathrm{p}}_{\mathrm{t}-1}^{\mathrm{c}}-{\mathrm{p}}_{\mathrm{t}-2}^{\mathrm{c}}\right)+\underset{\left(.119\right)}{.095}\left[{\mathrm{q}}_{\mathrm{t}-1}-.027\left(\mathrm{t}-1\right)-7.373\right]\hfill \\ & & +.309\left({\mathrm{q}}_{\mathrm{t}-1}-{\mathrm{q}}_{\mathrm{t}-2}\right)-\underset{\left(.004\right)}{.009}\begin{array}{cc}\begin{array}{cc}& {\overline{\mathrm{R}}}^2=.351\end{array}& \begin{array}{cc}\begin{array}{c}\mathit{\sigma }=.011\end{array}& \mathrm{DW}=2.33\end{array}\end{array}\hfill \\ {\mathrm{p}}_{\mathrm{t}}^{*}-{\mathrm{p}}_{\mathrm{t}-1}^{*}& =& \left({\mathrm{p}}_{\mathrm{t}-1}^{\mathrm{c}*}-{\mathrm{p}}_{\mathrm{t}-2}^{\mathrm{c}*}\right)+\underset{\left(.088\right)}{.040}\left[{\mathrm{p}}_{\mathrm{t}-1}^{*}-.033\left(\mathrm{t}-1\right)-7.803\right]\hfill \\ & & +\underset{\left(.204\right)}{.880}\left({\mathrm{p}}_{\mathrm{t}-1}^{*}-{\mathrm{p}}_{\mathrm{t}-2}^{*}\right)-\underset{\left(.008\right)}{.031}\begin{array}{cc}\begin{array}{ccccc}& {\overline{\mathrm{R}}}^2=.351& \begin{array}{c}\mathit{\sigma }=.017\end{array}& \mathrm{DW}=1.74& \end{array}& \end{array}\hfill \end{array}$
Exchange rate
	$\begin{array}{cc}{\mathrm{e}}_{\mathrm{t}+1}={\mathrm{e}}_{\mathrm{t}}+{\mathrm{i}}_{\mathrm{t}}-{\mathrm{i}}_{\mathrm{t}}^{*}+\mathrm{ϵ}& \begin{array}{cc}& {\mathrm{\sigma }}_{\mathrm{ϵ}}=.026\end{array}\end{array}$
Variable definitions (U.S. variables are unstarred, non-U.S. variables are starred)
e	= nominal exchange rate (dollars per foreign currency)
g	= real government spending on goods and services
i	= nominal short-term interest rate
m	= monetary base
p	= GDP deflator
pc	= deflator of domestic absorption
q	= real GDP
t	= time trend

View Table

Table 1.

Parameter Estimates for Oudiz-Sachs Model, 1966-86. (standard errors in parentheses)

Aggregate demand
	$\begin{array}{lllll}{\mathrm{q}}_{\mathrm{t}}& =& \underset{\left(.051\right)}{-.110}({\mathrm{p}}_{\mathrm{t}}-{\mathrm{e}}_{\mathrm{t}}-\mathrm{p}\begin{array}{c}*\\ \mathrm{t}\end{array})\underset{\left(.206\right)}{+.037}\mathrm{p}\begin{array}{c}*\\ \mathrm{t}\end{array}-\underset{\left(.301\right)}{.178}({\mathrm{i}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}+1}+{\mathrm{p}}_{\mathrm{t}})& & \\ & & \begin{array}{cc}\begin{array}{c}\underset{\left(.009\right)}{+.021}\mathrm{t}+\underset{\left(.167\right)}{.288}{\mathrm{g}}_{\mathrm{t}}+\underset{\left(2.302\right)}{5.391}\end{array}& \begin{array}{cc}\begin{array}{cc}\begin{array}{cc}& \end{array}& \end{array}& \begin{array}{ccc}{\overline{\mathrm{R}}}^2=.984& \mathit{\sigma }=.021\mathrm{DW}=1.78& \end{array}\end{array}\end{array}& & \\ \mathrm{q}\begin{array}{c}*\\ \mathrm{t}\end{array}& =& \underset{\left(.096\right)}{.171}({\mathrm{p}}_{\mathrm{t}}-{\mathrm{e}}_{\mathrm{t}}-\mathrm{p}\begin{array}{c}*\\ \mathrm{t}\end{array})+\underset{\left(.496\right)}{.514}{\mathrm{q}}_{\mathrm{t}}-\underset{\left(.207\right)}{.384}(\mathrm{i}\begin{array}{c}*\\ \mathrm{t}\end{array}-{\mathrm{p}}_{\mathrm{t}+1}^{*}+{\mathrm{p}}_{\mathrm{t}}^{*})& & \\ & & \underset{\left(.025\right)}{-.030\mathrm{t}}+\underset{\left(.468\right)}{1.599}{\mathrm{g}}_{\mathrm{t}}^{*}-\begin{array}{cc}\begin{array}{cc}\begin{array}{c}\underset{(6.0680}{5.726}\end{array}& \end{array}& \end{array}\begin{array}{cc}\begin{array}{cc}\begin{array}{cc}\begin{array}{ccc}\begin{array}{cc}& \end{array}& {\overline{\mathrm{R}}}^2=.990& \mathit{\sigma }=.020\mathrm{DW}=1.15\end{array}& \end{array}& \end{array}& \end{array}& & \end{array}$
Money demand
	$\begin{array}{lllllll}{\mathrm{m}}_{\mathrm{t}}-\mathrm{pt}& =& \underset{\left(.066\right)}{.225}{\mathrm{q}}_{\mathrm{t}}-\underset{\left(.413\right)}{1.419}{\mathrm{i}}_{\mathrm{t}}+\underset{\left(.503\right)}{3.453}& & {\overline{\mathrm{R}}}^2=.382& \mathit{\sigma }=.038& \mathrm{DW}=1.02\\ {\begin{array}{c}\mathrm{m}\end{array}}_{\mathrm{t}}^{*}-{\mathrm{p}}_{\mathrm{t}}^{*}& =& \underset{\left(.078\right)}{.700}{\mathrm{p}}_{\mathrm{t}}^{*}-1.077{\mathrm{i}}_{\mathrm{t}}^{*}+\underset{\left(.616\right)}{.100}& & {\overline{\mathrm{R}}}^2=.853& \mathit{\sigma }=.051& \mathrm{DW}=0.67\end{array}$
Consumer price index
	$\begin{array}{ll}{\mathrm{p}}_{\mathrm{t}}^{\mathrm{c}}& =.899{\mathrm{p}}_{\mathrm{t}}+\left(1-.899\right)\left({\mathrm{p}}_{\mathrm{t}}^{*}+{\mathrm{e}}_{\mathrm{t}}\right)\\ {{\mathrm{p}}_{\mathrm{t}}^{\mathrm{c}}}^{*}& =.758{\mathrm{p}}_{\mathrm{t}}^{*}+\left(1-.758\right)\left({\mathrm{p}}_{\mathrm{t}}-{\mathrm{e}}_{\mathrm{t}}\right)\end{array}$
Output price change
	$\begin{array}{ccc}{\mathrm{p}}_{\mathrm{t}}-{\mathrm{p}}_{\mathrm{t}-1}& =& \left({\mathrm{p}}_{\mathrm{t}-1}^{\mathrm{c}}-{\mathrm{p}}_{\mathrm{t}-2}^{\mathrm{c}}\right)+\underset{\left(.119\right)}{.095}\left[{\mathrm{q}}_{\mathrm{t}-1}-.027\left(\mathrm{t}-1\right)-7.373\right]\hfill \\ & & +.309\left({\mathrm{q}}_{\mathrm{t}-1}-{\mathrm{q}}_{\mathrm{t}-2}\right)-\underset{\left(.004\right)}{.009}\begin{array}{cc}\begin{array}{cc}& {\overline{\mathrm{R}}}^2=.351\end{array}& \begin{array}{cc}\begin{array}{c}\mathit{\sigma }=.011\end{array}& \mathrm{DW}=2.33\end{array}\end{array}\hfill \\ {\mathrm{p}}_{\mathrm{t}}^{*}-{\mathrm{p}}_{\mathrm{t}-1}^{*}& =& \left({\mathrm{p}}_{\mathrm{t}-1}^{\mathrm{c}*}-{\mathrm{p}}_{\mathrm{t}-2}^{\mathrm{c}*}\right)+\underset{\left(.088\right)}{.040}\left[{\mathrm{p}}_{\mathrm{t}-1}^{*}-.033\left(\mathrm{t}-1\right)-7.803\right]\hfill \\ & & +\underset{\left(.204\right)}{.880}\left({\mathrm{p}}_{\mathrm{t}-1}^{*}-{\mathrm{p}}_{\mathrm{t}-2}^{*}\right)-\underset{\left(.008\right)}{.031}\begin{array}{cc}\begin{array}{ccccc}& {\overline{\mathrm{R}}}^2=.351& \begin{array}{c}\mathit{\sigma }=.017\end{array}& \mathrm{DW}=1.74& \end{array}& \end{array}\hfill \end{array}$
Exchange rate
	$\begin{array}{cc}{\mathrm{e}}_{\mathrm{t}+1}={\mathrm{e}}_{\mathrm{t}}+{\mathrm{i}}_{\mathrm{t}}-{\mathrm{i}}_{\mathrm{t}}^{*}+\mathrm{ϵ}& \begin{array}{cc}& {\mathrm{\sigma }}_{\mathrm{ϵ}}=.026\end{array}\end{array}$
Variable definitions (U.S. variables are unstarred, non-U.S. variables are starred)
e	= nominal exchange rate (dollars per foreign currency)
g	= real government spending on goods and services
i	= nominal short-term interest rate
m	= monetary base
p	= GDP deflator
pc	= deflator of domestic absorption
q	= real GDP
t	= time trend

There are several things to note in Table 1, which can be compared to Table 7.2 in Oudiz and Sachs [1985, p. 297]. First, the aggregate demand equation contains additional exogenous terms: the log of government spending (labeled g and g^*) and a time trend. The former is intended to capture other influences on demand, and, in particular, the effect of fiscal policy (which is not, however, considered in the policy coordination experiments discussed below). The time trend is present because aggregate demand variations should be relative to potential output, which is here exogenous and assumed to grow smoothly over time.

Second, since in the Oudiz-Sachs model domestic output prices depend solely on wages, the latter are substituted out of the model; the same is done for the rate of inflation. The resulting price change equation also has a time trend, so that the level effect of output is relative to potential output, which was generated from a separate equation where the log of output was regressed on a constant and a time trend. Finally, the equation for the exchange rate, based on uncovered interest parity, has an error term. Even though the equation is not estimated, an allowance is made in the stochastic simulations below for the fact that the equation does not hold exactly. The interpretation given to errors in this equation is that of shifts in portfolio preferences. Given a proxy for the expected exchange rate—in particular, assuming that in the historical data expectations of the exchange rate were based on a random walk model—the standard deviation of that error is seen to be sizable.

The coefficient estimates are all of the right signs, and generally fairly well determined—in particular the effects of the real exchange rate on U.S. aggregate demand and of U.S. GDP on rest-of-world demand, the money demand parameters, and the change in GDP effects on the two regions’ output price changes. Despite its simplicity, the model seems to fit the data fairly well. Nevertheless, residual serial correlation is evident in the money demand equations, and the specification embodies arbitrary constraints that are open to discussion. We proceed to relax some of these restrictions below, and estimate the resulting alternative models.

The computational burden of calculating optimal policies with model learning severely restricted the number of alternative models we could introduce; in this paper we consider three possible models. The two alternative models, models II and III, differ from the baseline model (model I) in two respects: in model II money balances are deflated by the consumer price index rather than the GDP deflator, while in model III there is a lagged dependent variable in the money demand equation and a non-vertical Phillips curve.

A first arbitrary feature of model I relates to the proper deflator for real money balances. As Branson and Buiter [1983] point out in their discussion of the Mundell-Fleming model, the effects of monetary and fiscal policies can be importantly different depending on whether the domestic output price or a broader index that includes foreign goods is used. The Oudiz-Sachs specification, excluding as it does the effects of terms of trade, conforms in this respect to the original Mundell-Fleming model. Deflating money balances not by ρ but by p^c yields the following equations:

The fit is approximately the same as the equations in Table 1 (the dependent variable is of course not the same so that they cannot be compared directly), and the coefficients are almost identical and equally well determined. Given our short sample period and high collinearity between the two price series, tests of one model against the other (using either nested or non-nested tests) are unlikely to distinguish between them. Instead, we take the equations with money balances deflated by p^c as an alternative model: this specification is included in model II.

A second major area of arbitrariness in the Oudiz-Sachs model presented in Table 1 is the dynamic specification. The money demand equation, whether specified with the output or consumption deflator, shows evidence of residual serial correlation. Most studies have allowed for the possibility that money balances adjust with a lag, and have included a lagged dependent variable (see for instance Goldfeld [1973]). If we do so, then the following equations result, which are included in model III:

Model III also relaxes the unitary coefficient on the lagged rate of inflation in the domestic price change equation. Relaxing the restriction in the inflation equation lowers the standard errors of estimate in both the U.S. and rest-of-world equations, and the standard errors for the inflation coefficients imply rejection of the unitary coefficient for the United States (but not for the rest of world). Furthermore, the change in output becomes insignificant, so we dropped this variable and reestimated, yielding:

Though the fit of the equations is only marginally superior, the two specifications have quite different long-run properties. The Oudiz-Sachs model exhibits no long-run tradeoff between output and inflation, as in steady state both output prices and consumer prices grow at the same, steady rate. In contrast, with a non-unitary coefficient, different rates of inflation are associated with different rates of output growth; this possibility would be disputed by many economists, however (see Friedman [1968], for instance). On the basis of the estimation results, it may be reasonable to attribute some non-zero probability to the existence of such a tradeoff.

Models II and III therefore differ from model I in several respects: model II deflates money balances by the consumption deflator, while model III allows for lags in money demand and for a non-vertical Phillips curve. Although these structural differences seem minor, the reduced-form multipliers differ considerably across the models, as do the dynamic properties (see Table 2 for the eigenvalues of the three models). There are 5 eigenvalues in each model, with one unstable eigenvalue in each case, corresponding to the one non-predetermined variable, the exchange rate.

Table 2.

Eigenvalues of Models I-III

Table 2.

Eigenvalues of Models I-III

Model I	Model II	Model III
1.122	1.415	1.428
.331±.850i	.412±.875i	–.183±.554i
.404	.022	–.104±.188i
–.773	–.285

View Table

Table 2.

Eigenvalues of Models I-III

Model I	Model II	Model III
1.122	1.415	1.428
.331±.850i	.412±.875i	–.183±.554i
.404	.022	–.104±.188i
–.773	–.285

A simple summary statistic for the degree of model uncertainty is given by the ratio:

ζ = μ² /(μ² +σ²)

where μ is the mean multiplier of an instrument on a particular target and σ² its variance (across the three models). This gives a dimensionless statistic which equals unity if there is no model uncertainty and approaches zero as the effectiveness of the policy instrument deteriorates, in the sense of Brainard [1967]. In a model with forward-looking variables the multipliers depend upon the anticipation of future policies as well as current policies and therefore are not independent of the regime under consideration. The ζ values reported in Table 3 are for an exogenous, permanent increase in the money supply of 4 percent. For the sake of comparison, we also report the ζ values for the models in Frankel’s study, also calculated for such a money supply shock. As can be seen, his study of 12 different models incorporates about the same degree of uncertainty about the reduced-form multipliers as our three alternative models. ^9/

Table 3.

Measures of Reduced-Form Model Uncertainty (Zeta Values in percent)

^1/Average of dynamic multipliers over the first 10 periods.

^2/Second-year effects.

Table 3.

Measures of Reduced-Form Model Uncertainty (Zeta Values in percent)

Effects of U.S. Money Supply on:				Effects of ROW Money Supply on:
US Y	ROW Y	US P	ROW P	US Y	ROW Y	US P	ROW P
Models I - III 1/
28	46	57	44	23	48	51	45
Frankel [1988]2/
68	–-	45	52	19	72	24	25

^1/Average of dynamic multipliers over the first 10 periods.

^2/Second-year effects.

View Table

Table 3.

Measures of Reduced-Form Model Uncertainty (Zeta Values in percent)

Effects of U.S. Money Supply on:				Effects of ROW Money Supply on:
US Y	ROW Y	US P	ROW P	US Y	ROW Y	US P	ROW P
Models I - III 1/
28	46	57	44	23	48	51	45
Frankel [1988]2/
68	–-	45	52	19	72	24	25

^1/Average of dynamic multipliers over the first 10 periods.

^2/Second-year effects.

IV. Optimal Policies under Model Uncertainty and Model Learning

In order to calculate the average ex post gains from coordination with model learning we adapt the algorithm developed by Ghosh and Masson [1988]. The logic of the model is as follows. In period t, the state vector x_t and a ector of subjective priors Π_t (with elements ${\mathrm{\pi }}_{\mathrm{t}}^{\mathrm{i}}$) are inherited. Policymakers choose a vector of controls (i.e. policy instruments) u_t in order to influence their target vector τ_t. They do not attempt active learning, i.e. performing policy experiments in order to discover which of the models is correct. At the end of period t a vector of endogenous variables ω_t is observed which allows agents to update their priors, yielding Π_t+1.

The dynamics of the world model are assumed to be given by:

where x is a vector of state variables, e is a vector of jumping variables, u is the vector of controls, and ε is an unobserved vector white noise shock, distributed N(0,Σ), where Σ is the variance-covariance matrix; $\left\{{\mathrm{A}}^{\mathrm{i}},{\mathrm{B}}^{\mathrm{i}},{\mathrm{C}}^{\mathrm{i}},{\mathrm{D}}^{\mathrm{i}},{\mathrm{F}}^{\mathrm{i}},{\mathrm{G}}^{\mathrm{i}},{\mathrm{\theta }}_{\mathrm{x}}^{\mathrm{i}},{\mathrm{\theta }}_{\mathrm{e}}^{\mathrm{i}}\right\}$ are constant matrices associated with model i.

In addition, structural equations of the models map the state variables and the forward-looking variables into a vector of targets τ:

Policymakers in each country are assumed to have preferences over the target vector which are represented by:

and

respectively, where the expectation is taken with respect to both the uncertainty about parameter values and the current realization of the shock ε_t. We assume that although agents do not know the true model of the world economy, they do know the variance-covariance matrix of the additive shocks. ^10/ Moreover, we do not allow heterogeneity of agents; both the private sectors and the governments start with the same priors across models and update them in the same fashion.

Following Oudiz and Sachs [1985], we derive the optimal decision rules by first calculating the dynamic programming solution for a (finite) T period horizon, and then letting T ⇒ ∞. Assume that by period T the exchange rate has stabilized and the agents in each country have the vector of priors Π_T ${\mathrm{\Pi }}_{\mathrm{T}}^{*}$ over the k possible models. Since the exchange rate has stabilized, the matrix equation given above for the global model implies:

Therefore:

The home country’s objective function is therefore:

and the foreign country’s is given analogously. As shown in Ghosh and Masson [1988], the solutions under Nash and cooperative behavior may be written: