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International Policy Coordination in a World with Model Uncertainty

Author(s):
International Monetary Fund. Research Dept.
Published Date:
January 1988
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There is now a substantial literature that establishes the conditions under which international coordination of economic policies can be expected to lead to better outcomes, and a few empirical studies that attempt to quantify those gains.1 The consensus of these empirical studies is that there exist clear gains to policy coordination, but that their magnitude is not enormous (Oudiz and Sachs (1984), McKibbin and Sachs (1988)). Most empirical studies to date have not, however, attempted to take into account the fact that the true econometric model describing the economy is not known.2 An exception is Frankel (1987), who shows that in this case coordination may reduce welfare.

Indeed, policymakers have long doubted the value of international macroeconomic coordination in an environment characterized by uncertainty with respect to the econometric models being used by the various national policymaking groups. Typifying this view, Martin Feldstein (1983, p. 44), then Chairman of the Council of Economic Advisers, wrote:

Economists armed with econometric models of the major countries of the world can, under certain circumstances, identify co-ordinated policies that, quite apart from the balance-of-payments constraints, are better than the outcome of unco-ordinated country choices. But, in practice, the overwhelming uncertainty about the quantitative behaviour of individual economies and their interaction, the great difficulty of articulating policy rules in a changing environment … all make any such international fine tuning unworkable.

Consistent with this view, Frankel (1987) shows that if policymakers choose their policies on the basis of one set of empirical macromodels, each one having an equal chance of being correct, and policymakers simply ignore the reality of model uncertainty, then coordinated policies may be inferior to the uncoordinated, or Nash, equilibrium.3

There are at least two grounds for criticism of such an approach. First, it attributes irrationality to the policymakers, since they must certainly recognize the reality of model uncertainty. Second, although such an argument appears to be an indictment of international policy coordination, it is rather an argument against the pursuit of any activist policy—coordinated or uncoordinated—when the effects of such policy are not known. A lesson of the literature on policy choice in the presence of parameter uncertainty4 is that, in general, policy instruments should be used more conservatively (Brainard (1967)). Evaluation of gains to policy coordination should be done in a context in which policymakers take this lesson into account. In such a context, the ex ante gains to policy coordination must be positive; indeed, model uncertainty may provide an additional incentive to coordinate macroeconomic policies.5

In this paper the focus is on the case in which governments are assumed to take explicit account of any model uncertainty when forming optimal policies. A theoretical section shows that whether model uncertainty increases or decreases the attractiveness of coordination depends on the source of the uncertainty in the reduced form of the model. Uncertainty about the effects of the foreign instruments on the home country (that is, the transmission effects) invariably raises the gains from coordination relative to the Nash equilibrium. In contrast, uncertainty regarding the effects of the country’s own instruments on itself has ambiguous implications for the benefits from coordination.

The effects of model uncertainty on the ex ante expected gains from policy coordination are then assessed in a two region econometric model of the world economy. Uncertainty is introduced at the level of the structural model rather than the reduced form. Uncertainty is considered with regard to parameters such as the interest elasticity of money demand, price elasticities in import and export equations, the slope of the Phillips curve, the degree of substitution between government and private consumption, and the strength of accelerator effects on investment. This approach allows tracing the effect of uncertainty about econometric estimates of key structural parameters on the reduced-form multipliers and, ultimately, on the welfare implications of coordinated versus uncoordinated macroeconomic policies.

Assessment of gains to coordination involves a comparison of dynamic programming solutions that maximize expected utility (either independently or jointly) under uncertainty about the effects of the instruments. The welfare criterion used is the expected (ex ante) value of an objective function in which the expectation is calculated over the prior probability weights assigned to each model. The analysis, therefore, may be viewed as an extension of the seminal work of Brainard (1967) to a two-player, strategic setting. It should also be stressed that since we focus on the expected gains from coordination, we do not have to specify, arbitrarily, the “true” model of the world economy.

The plan of the paper is as follows. Section I reviews some of the theoretical arguments about the effects of model uncertainty on the gains from policy coordination. Section II describes the econometric model, MINIMOD, used for the simulation work and specifies confidence regions for the key structural parameters. Section III explains the procedure used to calculate the gains from coordination under model uncertainty. Section IV presents estimates of those gains, and Section V offers some brief concluding remarks. An Appendix gives the derivation of both Nash and cooperative strategies under model uncertainty.

I. Theory of Model Uncertainty and Policy Coordination

To begin the discussion, it is useful to consider the effects of model uncertainty on the gains from policy coordination in the simplest possible theoretical structure—one that has been widely used in the literature on policy coordination.6 The discussion here follows Ghosh (1987). Let policymakers have the quadratic objective function defined over the price level:

subject to the linear reduced-form equation

where p is the price level, ∈ is an inflation shock, M is the home country’s monetary policy, and M* is the foreign country’s monetary policy. It is assumed that θ1 >0 and θ1 + θ2>0. The coefficient θ1 represents the effects of the home country’s monetary policy on its own price level, whereas θ2 gives the effects of the foreign instrument on the home country’s price level; these coefficients are termed domestic and transmission multipliers, respectively. A similar objective function and model apply to the symmetric foreign country:

Note that θ1 and θ2 again represent domestic and transmission multipliers, respectively. Under a cooperative regime a single planner maximizes a weighted average of the two countries’ welfare functions:

In a world with symmetric countries, it is natural to choose the relative welfare weight of ω = 0.5 so that any gains from coordination are shared equally. In the absence of coordination, each country maximizes its respective objective function, V or V*, choosing its policies under the assumption that the other country’s policy settings are given. Because countries are mirror images, we may concentrate on the home country with the understanding that the effects on the foreign country are identical.

In the absence of model uncertainty, the unique symmetric Nash equilibrium is characterized by the following policy settings:

Similarly, under cooperation,

The Nash and cooperative equilibria coincide so that there are no gains from coordination in this case. Here each country has one instrument and one target so that the two countries can achieve their first-best outcomes without coordination (see Oudiz and Sachs (1984)).

Now suppose that the multipliers θ1 and θ2 are uncertain, with means μθ1, μθ2 and variances σθ12,σθ22, respectively. For expositional purposes it is convenient to separate the effects of uncertainty about domestic multipliers from the effects of uncertainty about transmission multipliers.7 The objective function is now

where the expectation is taken over the uncertain parameters. The symmetric Nash strategies are now given by

whereas the cooperative equilibrium is given by

Therefore, unless

the cooperative and Nash equilibria differ, and there would be gains from coordination.8 When there is no uncertainty (that is, σθ1 = σθ2 = 0), then equation (1c) necessarily holds, but in general this will not be the case.

The reason that model uncertainty introduces gains from coordination has been discussed in some detail by Ghosh (1987). As Brainard (1967) has shown, under instrument uncertainty the “efficiency” of policy instruments decreases, and the optimal program calls for the modified use of policy.9 The use of policy instruments brings the mean values of the target variables closer to their “bliss” points; however, it also increases their variances around their means. The planner therefore faces a mean-variance trade-off in the use of his instruments.10 In the Nash equilibrium each government treats the variance from the foreign instrument as exogenous additive uncertainty; that is, as independent of its own instrument. In equilibrium, however, the foreign instrument is a function of the home instrument (at the symmetric equilibrium dM*/dM = 1), so that the Nash player incorrectly estimates the efficiency of his instrument and chooses an inappropriate degree of intervention.11

From an initial equilibrium in which there are no gains from coordination, therefore, the introduction of uncertainty about either the domestic or the transmission multipliers unambiguously raises the incentive to coordinate. More generally, the effects of introducing model uncertainty into a model in which there already exist gains from coordination depend on whether there is uncertainty about the domestic or the transmission multipliers.

Consider a model in which each country has two targets, the price level and the level of output, but only one instrument, the money supply. It can be shown that increasing the degree of domestic multiplier uncertainty has ambiguous effects on the gains from policy coordination (Ghosh (1987)). For example, in the limiting case of an infinite increase in the home multiplier uncertainty (that is, where σθ12), the optimal policy, both under Nash and under cooperative behavior, is to refrain from any intervention whatsoever:

The Nash and cooperative strategies therefore converge at this point, eliminating all gains from coordination. For intermediate values, the expected gains from coordination may either be enhanced or diminished when domestic multiplier uncertainty is increased. An increase in parameter uncertainty will lead to a less active use of the policy instrument; if the Nash equilibrium is too interventionist, then an increase in domestic multiplier uncertainty by reducing the use of policy will bring the Nash equilibrium closer to the cooperative solution, thereby reducing gains from coordination. Conversely, if the Nash equilibrium is biased toward too much intervention, then an increase in domestic multiplier uncertainty may tend to increase gains from coordination.

In contrast, the effect of transmission multiplier uncertainty is unambiguous. The divergence between the Nash and cooperative cases is always an increasing function of the degree of transmission uncertainty—and so, correspondingly, is the gain from coordination. The difference in impact of domestic and transmission multiplier uncertainty will figure prominently in the discussion of quantitative estimates of gains to coordination presented below.

II. Structure of the Econometric Model

The model we have used to estimate the gains of policy coordination is a small, two-region macroeconomic model called MINIMOD (see Haas and Masson (1986)). This model contains blocks for the United States and an aggregate of the rest-of-the-world (ROW) countries, with about 30 equations per region. The regions are linked through uncovered interest parity and bilateral trade flows.

Since MINIMOD is nonlinear, it was first linearized around a point on a steady-state growth path. A linear model is necessary to make the policy optimization tractable; the essential conclusions about the effects of model uncertainty on the gains from coordination are not, however, specific to such a model.12 The linearized model exhibits saddle-point stability and has as many unstable eigenvalues as jumping variables (five of them, corresponding to the exchange rate, two long-term bond rates, and two inflation rates).13 For these nonpredetermined variables, terminal rather than initial conditions are imposed such that they do not grow without bound. As a result, a unique, stable rational expectations path exists. The linearized version of the model is a function of the structural parameters of the original, nonlinear model, so we can give an intuitive, economic interpretation to the effects of uncertainty about parameters.

An argument often given for policy coordination is that, because under cooperation governments could choose the same policies they were pursuing at the Nash equilibrium, by revealed preference, coordination must improve welfare. In MINIMOD, however, expectations are formed rationally, and governments arc assumed to be unable to commit their actions in advance, so that private sector reaction functions faced by the cooperative planner will, in general, differ from those faced by the governments acting independently. As Rogoff (1985) has demonstrated, the standard argument that coordination must be beneficial thus breaks down in the presence of a forward-looking private sector. In particular, monetary expansion may be more tempting when pursued jointly, since exchange rate depreciation would not result. Therefore, if governments are unable to commit themselves in advance not to pursue expansionary policies, the private sector will include a (rational) forecast of looser monetary policy in its wage demands, and the inflation-output trade-off will be less favorable in the coordinated regime than in the Nash. Whether this effect is sufficiently strong to render the gains from coordination negative depends on the specific parameters of the model. In the simulations reported below, the gains from coordination always do turn out to be positive, so that the possibility raised by Rogoff does not apply to MINIMOD, at least for the policies and shocks considered.

The key structural parameters that govern both the domestic effects of policy changes and their transmission to the foreign economy are listed in Table 1, together with their values in MINIMOD and in two alternative models, intended to capture the upper (model 1) and lower (model 2) bounds of 95 percent confidence regions. The parameters of M1NIMOD were not in most cases directly estimated; rather, they were derived from other empirical work, primarily from the U.S. Federal Reserve’s Multicountry Model (MCM; see Stevens and others (1984)). Uncertainty in the parameters cannot therefore be directly associated with estimated standard errors. In any case, parameter uncertainty in empirical models is likely to be considerably greater than estimated standard errors because specification searches have usually been used to achieve a good fit and to accord with prior assumptions about the size of coefficients. A better measure of model uncertainty is the range of estimates resulting from different estimation strategies and different maintained hypotheses. Parameter changes will also result from different sample periods, and if parameters do change over time then this variation, which may be important, also may not be correctly captured by estimated standard errors in a model with constant coefficients.

Table 1.Parameter Values of U.S. Model for Different Simulations
Parameters Changed,

by Simulation
Model 1Model 2Memorandum

Item:

Parameter

Values

in MINIMOD
1.None: Baseline
2.Semi-elasticity of money demand with respect to current and lagged interest rate
Current-0.4842-0.1948-0.3390
Lag 1-0.1859-0.0748-0.1303
3.Degree of direct crowding out of private consumption by government expenditure0.3-0.30.0
4.Slope of the Phillips curve: effects of current and lagged capacity utilization rate on gross national product (GNP) deflator
Current0.04050.00450.0225
Lag l0.08660.00960.0481
Lag 2-0.0580-0.0064-0.0322
5.Elasticity of import demand with respect to activity2.92731.62732.2773
6.Elasticity of import demand with respect to real exchange rate-0.8226-0.3226-0.5726
7.Elasticity of export demand with respect to foreign activity1.31480.21480.7648
8.Elasticity of export demand with respect to real exchange rate1.33390.87391.1039
9.Effect of gross domestic product (GDP) on investment activity0.04270.01340.0294

Another issue is whether parameters vary endogenously as a result of a change in the policy regime, a possibility raised by Lucas (1976). The Lucas critique is obviously relevant here, since policy coordination would constitute a clear change in regime relative to the historical period. Parameter ranges should be wide enough to allow for this possibility.

The first key parameter that is considered in Table 1 is the interest elasticity of money demand in the United States. The demand for narrow money—the Ml aggregate—has been affected in recent years by two major structural changes: financial deregulation and technological innovations that have changed the costs of carrying out financial transactions. Simpson (1984, pp. 261–62), described some of these changes and concluded that

… rapid financial change continues to affect the behavior of M1 and thus the setting of Ml growth objectives. Considerable uncertainty surrounds the contours of the relationship that now exists among M1, income, interest rates, and other economic developments and that will exist once the transition phase has drawn to a close… The preceding discussion strongly suggests that the interest elasticity of Ml demand will be much lower once the deregulation process has ended and that the demand to hold Ml balances will continue to be “noisy,” especially over short periods…

The equation favored by Simpson includes an own rate of interest (in particular, the rate on NOW accounts, which are included in Ml) as well as the market rate. He predicted that the elasticity with respect to the overall level of the interest rates will fall over time—relative to the “standard” elasticity of 0.10—as the proportion of M1 that is subject toan unregulated own rate grows. He cited in opposition, however, a model developed at the Federal Reserve Bank of San Francisco, which explains strong M1 growth in the 1982–83 period not by deregulation but rather by a high interest elasticity—equal to 0.15—that is asserted not to have changed in recent years. Thus we quantify uncertainty about money demand elasticities by postulating a range of 0.05 to 0.15 for the interest elasticity: this range includes both a substantial decline from the standard elasticity and the opposing view of the San Francisco Federal Reserve Bank. This range, plus or minus half the point estimate, was applied to the MINIMOD coefficients, which are semi-elasticities and hence cannot be directly compared with Simpson’s estimate. It should be noted that, since Simpson’s 1984 article, the Federal Reserve Board abandoned Ml targets, citing instability in the money demand, income, and interest rate relationship.

Another parameter that is of direct relevance to the effects of policy is the degree to which government spending is a direct substitute for private consumption expenditure. If government programs can just as easily be provided directly by the private sector, then there need be no real effects from increases in government spending (provided also that the financing of that spending has no real effects—that is, Ricardian equivalence holds). In contrast, if government spending is of quite a different nature, then it should not affect the utility of private consumption. There is a parameter in MINIMOD that measures this degree of substitutability; it is provisionally set to zero so that there is no direct offset through private consumption of government spending.

Empirical estimates of the degree of offset are quite mixed. It should be emphasized the government and private spending can be either substitutes or complements. If complements, then increases in government spending tend to increase private consumption; for instance, improved roads or parks may stimulate expenditures on automobiles and induce people to travel. Several authors have discovered complementarity (Buiter and Tobin (1979), Feldstein (1982), and Kormendi (1983)); in contrast, Ahmed (1986) found substitutability (that is, a direct offset on private spending). Although the data with which the models were estimated and also the precise specification differ considerably, the studies cited above suggest a range of −0.3 to 0.3, with a midpoint at the MINIMOD estimate of zero.

Clearly a crucial feature of any model is the degree of wage-price stickiness or, alternatively, the slope of the Phillips curve. In the limiting case of perfect flexibility, monetary policy has no real effects. In MINIMOD, the degree of flexibility is captured by coefficients on the rate of capacity utilization in the equation for the rate of change of the gross domestic product (GDP) deflator. Uncertainty about these parameters is directly related to uncertainty concerning the “sacrifice ratio” between the cumulative output loss relative to the potential needed to achieve a given reduction in the rate of inflation. The higher are the coefficients on capacity utilization, the lower is the sacrifice ratio. There is a considerable range of estimates of the sacrifice ratio; according to Sachs (1985), traditional estimates ranged from 6 to 18, with Arthur Okun’s best guess (cited in Sachs) being 10. Sachs himself calculated the sacrifice ratio to be 3 in the 1982–84 disinflationary period. A reasonable range, therefore, might be 2 to 18, or plus or minus 0.8 times the point estimate. Such a range was applied to the MINIMOD coefficients in Table 1.

Trade equations have been the object of numerous empirical studies, and most have used the standard specification, or a variant of it, embodied in MINIMOD—-imports by each region in volume terms depend on real domestic expenditure and on relative prices; that is, on the real exhange rate. Despite the consensus about this specification, there is still controversy about the relevant elasticities, especially about their implications for future U.S. current account deficits.14 A comparison of seven detailed models of the U.S. current account revealed that, for both imports and exports, activity and price elasticities varied widely. For non-oil imports, income elasticities of demand ranged from 1.2 to 2.5, whereas price elasticities ranged from 0.7 to 1.2 (see Bryant and others (1987, pp. 133–34)). Similarly, for nonagricultural goods exports, foreign income elasticities range from 1.0 to 2.1, and relative price elasticities from 0.5 to 1.0. These elasticities apply to data other than those in MINIMOD: trade flows include only a subset of merchandise trade, whereas MINIMOD’s trade flows include all merchandise trade as well as nonfactor services. Nevertheless, the elasticities give a good measure of the uncertainty relating to the parameters in trade equations. On the basis of the above results, the range of absorption elasticities was taken to be MINIMOD’s estimate plus or minus 65 percent of the mean for imports, 55 percent for exports. Relative price elasticities were taken to include values 25 percent on either side of MINIMOD’s coefficient for imports, 23 percent for exports.

A final important linkage for both monetary and fiscal policies is the accelerator effect of income variations on investment. MINIMOD’s equation for the change in the capital stock is based on a conventional lagged adjustment to the optimal level, with that level depending on GDP and the user cost of capital. The accelerator effect thus depends on the speed of adjustment to the optimal capital stock. Several alternative specifications give a similar positive effect of changes in GDP on investment. Clark (1979) estimated equations with different theoretical underpinnings; implied estimates for the contemporaneous effects of output ranged from 0.03 to 0.08 when his coefficients for producers’ durable equipment and for nonresidential structures were weighted together using 1986 relative shares. Applying a similar range to the point estimate in MINIMOD gave the range presented in Table 1 for that parameter, The ranges of parameter estimates are very large—larger in general than those implied by estimated standard errors from any single model. Although the ranges do not allow for a change in sign of the structural parameter (with the exception of the direct effect of government expenditure on consumption), reduced-form multipliers do in fact change signs. The range of resultant policy effects is indeed wider than that implied by the different models considered by Frankel (1987). Therefore, the parameter uncertainty treated here is quite fundamental to the macroeconomic outcomes from the model.

III. Optimal Strategies Under Model Uncertainty

Calculation of the gains from policy coordination involves comparison of the expected utility achievable under the dynamic time-consistent Nash equilibrium (Basar and Olsder (1982)) with that enjoyed by each country when a single planner maximizes a weighted sum of the two countries’ welfares. In the simulations reported below, equal weights were assigned to each country,15 and the gains from coordination were assessed by reference to this global welfare function.

In the baseline simulations—where there is no model uncertainty—we use the standard dynamic programming solution for the optimal control of an economy with forward-looking variables (see Oudiz and Sachs (1985)). Once model uncertainty is introduced, however, the optimal control solution must be modified to take into account the risks associated with the use of policy. We therefore extended the algorithms of Chow (1975) and Kendrick (1981) to deal with forward-looking variables and strategic behavior.

The logic of the model is as follows. In period t, policymakers observe the inherited state vector xt, which is predetermined; the timing convention of the linearized MINIMOD is such that all nonjumping endogenous variables are determined at the beginning of the period and are thus included in xt, Policymakers then set their instruments with a view to influencing xt+1 and any jumping variables in period t, et At the time that policies must be decided, the true model describing the world economy is unknown, and policymakers must use their prior beliefs about the possible models. In period t + 1, xt+1 is observed, so that the model that applied to period t may be inferred. Because the realization of the true model is assumed to be independent in each period, however, policymakers will use the same priors in the next period; updating of priors is not required. The home government’s welfare function is assumed to be given by

where yt is an n vector of endogenous variables, measured as deviation from the bliss point; Ω is an (n × n) matrix of utility weights; and E(.) is the expectation operator. All of the model’s endogenous variables are contained within yi and the nonzero terms of Ω assign weights to those variables targeted by the home-country planner. The foreign government maximizes a similar objective function.

The world economy’s dynamics are given by

where i indexes the k possible models describing the world economy; (Ai, Bi, Ci, Di, Fi, Gi) are constant coefficient matrices for model i;16xt is a p vector of predetermined variables; et is an m vector of forward-looking variables; et+1 is the expectation, as of period t, of the jumping variables in period t + 1; and Ut is an r vector of home and foreign policy instruments.

In each time period there are four vectors of potentially different probability priors assigned to each of the k possible models. The home and foreign governments use the subjective priors Πt, and Π*t to derive their optimal policies. The private sector assigns the priors Π^t, to the models. Finally, there is an objective probability, Π¯t, that any given model is the correct description of the world economy. The priors are exogenously given and are not updated either through active or passive learning.

The specification of the objective probability vector, Π¯t depends on the treatment of model uncertainty. If it is assumed that there is a unique correct model—say, model j—that describes the world economy over the relevant horizon, but that policymakers have yet to discover it, then Π¯t is degenerate; that is, Π¯t is a vector of zeros except for the j th element, which is unity. An alternative treatment of model uncertainty, and one that is implemented in what follows, assumes that the world economy is too complex to be captured by a single model. Rather, the correct model changes from period to period and may be viewed as a truly stochastic realization from the set of k possible models, where each outcome has the associated objective probability i˜t

The objective priors are used to calculate the expected welfare for each country:

where the notation yit reads “the value of yt implied by model i,” so that V represents the true mathematical expectation of the country’s welfare. Rational expectations by the private and public sectors in each country obtain when their subjective priors coincide with the objective probabilities:

In the simulation analysis below, it is assumed that these conditions of rational expectations obtain. In contrast, Frankel (1987) explicitly assumed that policymakers do not exhibit such rational expectations.

Although equation (4) postulates an infinite horizon, it is convenient to start with a finite-horizon problem and then let the number of periods go to infinity.17 Let the horizon be T periods, and consider the final period. It is assumed that by period T the world economy has attained a steady state and that all forward-looking variables have stabilized. The world economy has inherited a state vector xT, and the objective function for the home country is

A corresponding expression gives the foreign country’s objective function, vT* Because the forward-looking variables have stabilized by period T,

where, again, ei reads “the value of the forward-looking variables implied by model i.” Substitution into the respective models yields

where JiT and KiT are matrices specific to model i. In period T, xT does not have an i superscript because it represents the (observed) inherited state vector. The vector of endogenous variables, YiT is given by

where, again, MiT and NiT are matrices specific to model i. Substituting equation (8) into equation (5) yields the objective functions.

When the countries are in a cooperative agreement, a single global planner chooses the entire vector of instruments, U, to maximize a weighted sum of the utility functions:

Note that the global planner uses each country’s probability priors to evaluate the respective expected utilities. If Π ≢ Π*, then at least one of the governments does not have rational expectations. As long as the global planner uses each country’s own priors in evaluating ν and ν*, however, disagreements over the true model do not preclude the possibility of coordination. Indeed, such disagreements over the prior vector can, by themselves, give rise to gains from coordination in models in which there are no other conflicts between the countries.18 Given the linear-quadratic structure of the model, the solution to the planner’s optimization problem may be written as

where the C superscript denotes the cooperative solution.

When countries are acting noncooperatively, each chooses the subset of U under its control—u and u*—to maximize νT(·) and ν*T(·), respectively, taking the actions of the other government as given. The resultant equilibrium may be written as

Hence, from equation (7),

where Γ denotes ΓC or ΓN as appropriate. Rational expectations of the forward-looking variables are

Finally, the period-T value functions for the home country are

Similarly, substituting for ΓN yields the value under noncoopcrative behavior:

Correspondingly, for the foreign country,

The functions ΓTC,ΓTN,θ˜T,STC,STC*,STN,STN* give the optimal policies, the evolution of the forward-looking variables, and the value functions, given that the state vector is xT.

Now consider period T −1, to which the state vector xT-1 has been bequeathed. At T − 1 the home government faces the following optimization problem:

where ν(.) represents the value function under cooperative or non-cooperative behavior, as appropriate. Note that xTi now has an i superscript: in period T—1 it is not known whichz χT will be bequeathed, since the model pertinent to period T—1 is not known. Hence, in period T − 1, ET-1T()} is found by taking the expectation over all possible xTi Expanding equation (12) makes this clearer:

The inner summation—over πT—yields the expected utility in period T, given that the inherited state vector is xt. The outer summation—over πT-1—yields the expectation of vT(XiT) over the possible xiT bequeathed from period T −1.

The optimization now yields St-1(), and so forth. As described in the Appendix, this procedure is repeated until stationary functions [θ(χ),S(χ),S*(χ)] are obtained. When these functions become stationary, the truncation error from solving a (large) but finite period problem rather than the true infinite horizon becomes negligible. From an initial state vector x0, the present value of the gains from coordination accruing to the world are then given by

IV. Simulation Results

Model uncertainty about the structural parameters for the U.S. block in MINIMOD was introduced to examine its effects on the gains from coordination. Table 1 indicates which parameters were assumed uncertain in each simulation. The simulations also required specification of the utility weights and probability priors. The choice of the utility function parameters is likely not to be critical here because the primary focus is on the effects of model uncertainty relative to the baseline (no model uncertainty) results. It was assumed that policymakers in each country target GDP, the rate of change of prices, and the current account balance. The square of each of these variables, expressed as a percentage of its value at the point of linearization, enters the objective function; there are no cross-product terms.19 The relative weights on the targets were taken to be 1, 10, 1.5, respectively. In addition, we included penalties for the use of monetary and fiscal policies, measured as a percentage deviation from the levels of the policy variables in the baseline, and assigned a relative welfare weight of 0.5 to each quadratic term. The discount factor, β, for each government was chosen to be β = 0.95, and the weight assigned to each country in the global planner’s objective function was ω = 0.5. Throughout the simulation exercises, we assumed that all policymakers, together with the private sector, exhibit rational expectations, and that the priors are not updated:

In each of the simulations, there were two competing models (models 1 and 2 in Table 1) with probability weights ¯πi = 0.5, i = 1, 2.

As discussed in Section I, the implications of model uncertainty for the gains from coordination depend, among other things, on whether it is the domestic multipliers that are uncertain or the transmission effects from abroad that are unknown. It is useful, therefore, to gain some understanding of how the structural parameter uncertainty translates into multiplier uncertainty in the reduced form of the model. Because the forward-looking variables are functions of the entire future sequence of policy settings, however, it is not possible—in a rational expectations setting—to define a measure of a multiplier uncertainty that is independent of whether governments are acting cooperatively or noncooperatively. As a partial solution to this problem, we solved for a rational expectations path along which future monetary and fiscal policies are at their baseline values, and concentrated on uncertainty about the impact effects of policy. Using a recursive form of the Blanchard-Kahn (1980) solution method, we obtained the dynamic systems:

where Λi and Φi are semireduced-form coefficients, derived under the assumption that Ut+j = 0, for all j >0. The mean value of the effect of an instrument uMt on a target variable Zjt+1 is:

where ΦijM is the (j, M) element of Φi, and the average has been taken over the k possible models (here k = 2). Similarly, the variance of each model’s multipliers is given by

For each target Zjt, we then defined a measure of the parameter uncertainty20 regarding the use of instrument uM as

so that ζjM ∈ (0,1) is a decreasing function of the severity of model uncertainty. Note that ζ is a dimensionless statistic, satisfying ζ = 1 when there is no uncertainty and ζ→0 as the effectiveness of the instrument worsens.

Table 2 gives the ζ values for each combination of the instruments and targets in each set of simulations. In the baseline simulation, of course, ζ is always unity because, by construction, there is no model uncertainty. More generally, the prevalence of model uncertainty implies that ζ will be less than unity, at least for the effects of some of the instruments on some of the targets. A number of points emerge from this table. First, there is usually greatest disagreement among the models with respect to the effects of policies—domestic or foreign—on the current account. The greater is the welfare weight placed on the current account, the more important will be the implications of this form of model uncertainty for the gains from coordination. Second, any uncertainty about parameters of the home country, such as the degree of fiscal crowding out, or the slope of the Phillips curve, is often magnified in the transmission process. For example, in simulation 3, where the degree of fiscal crowding out in the United States is the only uncertain parameter, uncertainty about the effects of U.S. fiscal policy on ROW GDP and on ROW inflation is greater than uncertainty surrounding the effects on U.S. GDP or inflation.

Table 2.Degree of Reduced-Form Model Uncertainty, ζ, of Policy Instruments on Selected Variables
EndogenousPolicy Variables
SimulationVariablesaUGaUMaRGaRMa
1.BaselineRGDP1.000*1.000*1.0001.000
UGDP1.0001.0001.000*1.000*
UCURRBAL1.0001.0001.000l.000
RP11.000*1.000*l.0001.000
UPI1.0001.0001.000*1.000*
2.Interest semi-elasticityRGDP0.999*0.859*0.9990.999
of U.S. moneyUGDP0.9990.8550.997*0.995*
demand uncertainUCURRBAL0.9830.8540.9980.999
RPI0.975*0.850*0.9860.999
UPI0.9840.8500.997*0.999*
3.Direct crowding outRGDP0.913*1.000*1.0001.000
effect of U.S. gov-UGDP0.9201.0001.000*1.000*
ernment spendingUCURRBAL0.938l.0001.0001.000
uncertainRPI0.869*1.000*1.000l.000
UPI0.879l.0001.000*l.000*
4.Slope of U.S. PhillipsRGDP0.179*0.431*0.9980.162
curve uncertainUGDP0.9660.8980.912*0.555*
UCURRBAL0.6130.2650.7400.366
RPI0.471*0.419*0.4670.414
UPI0.4930.4100.379*0.412*
5.Activity elasticity ofRGDP0.915*0.999*0.9990.996
U.S. importsUGDP0.9960.9980.999*0.999*
uncertainUCURRBAL0.9640.9680.9940.978
RPI0.827*0.949*0.9170.967
UPI0.7300.9670.983*0.970*
6.Real exchange rateRGDP0.997*0.995*0.9990.981
elasticity of U.S.UGDP0.9990.9990.999*0.992*
imports uncertainUCURRBAL0.9640.9940.9960.942
RPI0.940*0.994*0.7730.929
UPI0.9080.9900.927*0.939*
7.Activity elasticity ofRGDP0.999*0.994*0.9990.999
rest-of-world (ROW)UGDP0.9990.9990.682*0.951*
imports uncertainUCURRBAL0.9980.9950.4900.991
RPI0.995*0.984*0.4170.994
UPI0.9970.9870.999*0.995*
8.Real exchange rateRGDP0.998*0.996*0.9990.996
elasticity of ROWUGDP0.9990.9990.999*0.981*
imports uncertainUCURRBAL0.9910.9980.9920.983
RPI0.958*0.993*0.8800.971
UPI0.9720.9940.968*0.973*
9.Effect of GDP onRGDP0.996*0.999*0.9990.996
U.S. investmentUGDP0.9150.9980.999*0.999*
uncertainUCURRBAL0.9640.9680.9940.978
RPI0.827*0.967*0.9170.967
UPI0.7300.9490.983*0.970*
Note: A value of unity (ζ = 1) signifies absence of model uncertainty for that policy instrument.

Variables are defined as follows: UG, U.S. fiscal policy; UM, U.S. monetary policy; RG, ROW fiscal policy; RM. ROW monetary policy; UGDP, U.S. GDP; RGDP, ROW GDP; UCURRBAL, U.S. current account balance; UPI, U.S. consumer inflation; RPI, ROW consumer inflation. A single asterisk

(*) denotes transmission effect.
Note: A value of unity (ζ = 1) signifies absence of model uncertainty for that policy instrument.

Variables are defined as follows: UG, U.S. fiscal policy; UM, U.S. monetary policy; RG, ROW fiscal policy; RM. ROW monetary policy; UGDP, U.S. GDP; RGDP, ROW GDP; UCURRBAL, U.S. current account balance; UPI, U.S. consumer inflation; RPI, ROW consumer inflation. A single asterisk

(*) denotes transmission effect.

In the symmetrical two-country model without expectations that was discussed in Section I, uncertainty about transmission effects—the effects of home policy on the foreign country and vice versa—always increased the gains from coordination, whereas uncertainty about the domestic effects had ambiguous implications for the incentive to coordinate. (Although in the limiting cases where ζ = 0 for domestic instruments, the gains from coordination are necessarily eliminated.) In Table 2, uncertainty about the transmission multipliers has been indicated by an asterisk(*). In simulations 3 and 4, the degree of transmission uncertainty is almost always greater than the corresponding uncertainty for the domestic targets. For these simulations, therefore, the analysis of Section 1 suggests that the presence of model uncertainty should serve to raise the gains from coordination. For the remaining simulations, however, because domestic multiplier uncertainty is large, there is no clear presumption about the effects of model uncertainty. The gains from coordination were calculated for a standard shock of a 1 percent decline in GDP. relative to baseline, together with a 1 percent rise in the consumer price level, again relative to baseline, in both the U.S. and ROW sectors.21 The shock is purely transitory, occurring in the initial period and returning to zero immediately. Because of the dynamics of the model, however, effects on endogenous variables may persist for a considerable number of periods. To focus on the effects of model uncertainty per se, the gains from coordination under each set of parameter values must be normalized by the corresponding “certainty equivalent” model. Otherwise, the effects of model uncertainty cannot be distinguished from the fact that different models, when known to be realized with unit probability, imply different gains from coordination. For example, suppose that the two possible models obtain with probability π1 and π2, respectively,22 and let the gains from coordination be denoted by Ψ(π1, π2). The certainty equivalent gains from coordination, ¯Ψ(π1, π2), are given by

It should be emphasized that ¯Ψ will not in general equal the gains under the baseline parameters in Table 1. The effects of model uncertainty are beneficial to the case for coordination if and only if

Table 3 reports the gains from coordination under model uncertainty, Ψ(π1, π2), and for the certainty equivalent model, ¯Ψ(π1, π2), under each of the pairs of parameter values listed in Table 1. The gains reported are the total benefit accruing to the world economy over the infinite horizon and are expressed in terms of the one-quarter U.S. GDP equivalent. Thus the estimated gains in the baseline simulation, with the original MIN1MOD parameters and no model uncertainty, are equivalent to the utility associated with a 19.5 percent increase in U.S. GDP sustained over one quarter. This estimate of the gains from coordination is roughly commensurate with previous empirical estimates (for example, Oudiz and Sachs (1984), who found gains of about 1 percent, but sustained over three years), although of course they depend on the specific utility function and shocks under consideration.

Table 3.Gains from Coordination Under Model Uncertainty
Gain UnderGain Under

Certainty
Uncertainty,Equivalent,Ratio,
SimulationΨ(π1, π2)¯Ψ(π1, π2)Ψ(π1, π2)/¯Ψ(π1, π2)
1.Baseline19.519.51.00
2.Interest elasticity of
U.S. money demand
uncertain17.09.91.42
3.Direct crowding out
effect of U.S. gov-
ernment spending
uncertain29.212.22.38
4.Slope of U.S. Phillips
curve uncertain25.110.92.29
5.Activity elasticity of
U.S. imports
uncertain23.919.21.24
6.Real exchange rate
elasticity of U.S.
imports uncertain3.412.50.27
7.Activity elasticity of ROW imports
uncertain23.916.81.41
8.Real exchange rate
elasticity of ROW
imports uncertain19.417.11.13
9.Effect of GDP on U.S.
investment uncertain9.617.80.53

From Table 3 it is apparent that the presence of model uncertainty in general raises the gains from policy coordination—in this model, and for this particular shock and objective function. Only for simulations 6 and 9, in which there is uncertainty about the real exchange rate elasticity of import demand by the United States and about the effect of U.S. GDP on U.S. investment demand, respectively, does model uncertainty serve to reduce the gains from coordination. Inspection of Table 2 reveals that in these simulations domestic multiplier uncertainty in general exceeds the corresponding transmission uncertainty. Specifically, the effects of ROW government expenditure on ROW inflation exhibit considerable parameter uncertainty in simulation 6, whereas in simulation 9 all the transmission effects are less uncertain than are the corresponding domestic multipliers. In the other simulations, however, the effect of model uncertainty is to raise the gains from coordination. Indeed, in simulations 3 and 4, where there is in general much greater transmission uncertainty relative to domestic uncertainty, the gains from coordination are more than doubled; whereas for the other simulations the effects of model uncertainty are far from negligible, at least in terms of proportionate increases in the certainty equivalent gains. It is still true that, in absolute terms, gains from policy coordination are estimated to be relatively modest.

V. Conclusions

This paper has examined the desirability of international policy coordination in an environment characterized by model uncertainty. Rather than use the reduced forms of different models, we chose a single two-region macroeconometric model and introduced parameter uncertainty at the structural level of the model. The primary advantage of such an approach is that it allows tracing the effect of uncertainty about structural parameters on reduced-form multipliers and, in turn, on the gains from coordination.

The major finding in this paper has been that model uncertainty, far from precluding policy coordination, may in fact provide a strong incentive for countries to coordinate their macroeconomic policies. By contrast, in a series of articles Frankel reached exactly the opposite conclusion (Frankel (1987) and Frankel and Rockett (1986)).

The essential difference between Frankel’s approach and ours is that Frankel explicitly assumed that policymakers do not exhibit rational expectations. Policymakers were assumed to ignore the presence of model uncertainty in choosing their optimal plans, even though they disagree on which is the correct model of the world economy. Within our framework, therefore, the probability priors used by Frankel’s policymakers, Π and Π*, do not coincide with the objective probabilities ̃Π. Because the mathematical expectation of each country’s welfare is evaluated at the true probability weights ̃Π, policymakers are effectively maximizing the wrong objective function. If in the Nash equilibrium policymakers are undertaking incorrect policies, and if under cooperation they become more efficient at making mistakes, it is scarcely surprising that coordination may reduce welfare.

But the general argument that ignorance of the correct model is likely to negate gains from coordination is problematic. First, it attributes an extraordinary degree of irrationality to policymakers to suggest that they simply ignore model uncertainty. Second, this argument has very little to do with policy coordination: it is simply an indictment of activist policy when the precise effects of policy are unknown. If there is some doubt about the effects of policies, it is clearly desirable for the authorities to be cautious in setting those policies.

Although the gains from coordination in the baseline simulation without model uncertainty were rather modest, in simulations where policymakers explicitly recognize model uncertainty when framing their policies the gains from coordination relative to the Nash equilibrium increased considerably. Policy setting in the latter does not properly take into account the feedback effect on overall uncertainty that results from induced changes in foreign policies. In terms of GDP-utility equivalents, model uncertainty can increase the gains from coordination by as much as a factor of 2. Furthermore, in line with theoretical arguments, the implications of uncertainty about MINIMOD parameters for the incentive to coordinate depend crucially on the source of that uncertainty. When transmission effects are unknown, the greater is the uncertainty, the greater are the gains from coordination. When the effects of policies on the domestic economy are uncertain, however, model uncertainty can either increase or decrease the gains from coordination, depending on several other factors.

This paper has been in part methodological—exploring the design of coordination regimes in dynamic rational expectations models in the presence of model uncertainty—and there remain several possible extensions of our work. First, we intend to undertake more extensive simulation analyses to see how structural parameter uncertainty is embodied in the reduced-form multipliers of such models. This extension is of considerable interest because the disagreements between multicountry models may be reducible to econometric estimates of certain key structural parameters. Second, although we have employed a dynamic model, we have not included any learning mechanism, whether active or passive. It seems reasonable, however, that policymakers may start with some vector of priors that are slowly updated until they converge to the rational (objective) probabilities over the models. If policymakers learn about the domestic effects of their policies faster than they do about transmission effects, an increase in the gains from coordination may occur over time. Conversely, if policymakers attempt active learning—that is, perturbing the economy in order to learn about it—then the Nash equilibrium may become even more inefficient, since each government will introduce excessive noise into the world economy.23

APPENDIX
Nash and Cooperative Strategies Under Model Uncertainty

In this Appendix the Nash and cooperative strategies under model uncertainty are derived, by using a dynamic programming solution technique. Assume, in period t, that

where ν(·) is the home country’s value function, and ν*(·) is the foreign country’s value function; χt is the p - dimensional state vector in period t; et is the q-dimensional vector of jumping variables; and S and S* are matrices. The k models are given by

where Ut is an r - dimensional vector of controls. Substituting equation (17) into equations (18) and (19) yields

and

The home country’s value function is given by

where Ω is a matrix of objective function weights, and St+1 summarizes the future discounted utility stream. The foreign country’s expected value function is given by a similar expression. Under the Nash equilibrium, each country chooses a subset of Ut, either u*t or u*t. Let lowercase letters denote conformable partitions of M, N, H, and L. Solving for the Nash equilibrium yields

Under coordination, the corresponding first-order condition is

where ΩG and SG are matrices of the global planner’s value function:

The home country’s value function therefore becomes

where Γ denotes ΓC or ΓN as appropriate. Similarly, the foreign country’s value function may be written as

The expected value of the forward-looking variables is

Replacing St+1 by St, S*t+1 by S*t and θt+1 by θt in equation (23), we obtain recursive rules to compute the equilibria. The recursion is started by choosing St=ST=0 and by requiring that eT+1 = eT.

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Mr. Ghosh is a graduate student at Harvard University; this paper was written while he was visiting the Research Department.

Mr. Masson is an Advisor in the Research Department. He holds a doctorate from the London School of Economics and Political Science.

The authors thank Guillermo Calvo and Swati Ghosh, as well as colleagues in the Fund, for helpful comments. Any opinions expressed are those of the authors and not those of the International Monetary Fund.

For surveys of the theoretical and empirical literature, see Cooper (1986), Horne and Masson (1987), and Fischer (1987).

Canzoneri and Minford (1988) consider uncertainty about other countries’ actions but do not treat model uncertainty.

In this paper, parameter and model uncertainty are treated as being the same; this is valid provided the models can be nested in a more general form, with, for instance, some parameters being zero or unity depending on which model is correct. As an example, suppose that there are two reduced-form models y = aX and Y = bZ; they can be nested in Y = αaX + (1 − α)bZ, where α equals unity if the first model is correct, and zero if the second one is correct.

See Ghosh (1986a, 1986b) and Ghosh (1987).

If parameter uncertainty is introduced at the level of the structural (as opposed to the reduced-form) model, however, the presence of domestic multiplier uncertainty will in general be associated with uncertainty about the transmission effects, so that this separation is not possible.

If θ1 and θ2 are correlated, the expressions for the optimal strategies become more complicated, but the condition for efficiency of the Nash equilibrium remains the same. Note that the Nash strategy can be over-or under-interventionist relative to the cooperative policy.

Brainard (1967) defined the efficiency of a policy instrument as the ratio of its mean effect to its variance.

Ghosh (1987) used a modified mean-variance portfolio analysis diagram to illustrate the inefficiency of the Nash strategy.

There are forms of noncooperative behavior—for instance, the consistent conjectural variation equilibrium—that do account for foreign government reactions. It can be shown, however, that all noncooperative strategies will be inefficient in the face of uncertainty.

See Ghosh (1987) for a discussion of the effects of model uncertainty on the gains from coordination in a model with general functional forms.

See Masson (1987) for details.

A recent conference at The Brookings Institution compared the recent tracking and future projections of several models of the U.S. current account; the implications of these results are discussed in Bryant and Holtham (1987) (see also Krugman and Baldwin (1987)). For a general survey of empirical work on trade equations, see Goldstein and Khan (1985).

For convenience, in MINIMOD the ROW region is referred to as a country.

These matrices have dimension [(p×p),(p×q), (p×r),(q×p),(q×q), (q×r)] respectively.

This algorithm is an extension to that used by Oudiz and Sachs (1985) in their deterministic model; wherever possible, a similar notation has been adopted. The Appendix provides a fuller derivation.

See Frankel (1987) and Ghosh (1987) for discussion.

See Oudiz and Sachs (1984) for a similar objective function.

This measure is a simple transformation of the coefficient of variation, which Brainard (1967) used as a measure of policy effectiveness.

In general, the joint distribution describing the error terms in the model’s equations will be related to the distribution describing the parameters. With an estimated model, one could choose a drawing from these distributions that was consistent with the historical data. This was not done here, however.

In particular, it is assumed that π1 = π2 = 0.5.

Ghosh (1986b) explores this possibility in a theoretical model.

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