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1 American Economic Association Dynamic Strategic Monetary Policies and Coordination in Interdependent Economies: Comment Author(s): Alain de Crombrugghe, Nouriel Roubini, Jeffrey D. Sachs Reviewed work(s): Source: The American Economic Review, Vol. 81, No. 5 (Dec., 1991), pp Published by: American Economic Association Stable URL: Accessed: 13/04/ :07 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. American Economic Association is collaborating with JSTOR to digitize, preserve and extend access to The American Economic Review.

2 Dynamic Strategic Monetary Policies and Coordination in Interdependent Economies: Comment By ALAIN DE CROMBRUGGHE, NOURIEL ROUBINI, AND JEFFREY D. SACHS* In the June 1988 edition of this Review, and under the above title, Stephen Turnovsky, Tamer Basar, and Vasco D'Orey presented numerical simulations of dynamic monetary policy games under alternative assumptions of Nash, Stackelberg, and cooperative play between two countries. The context that they use is a Rudiger Dornbusch (1976)-style perfect-foresight model of floating exchange rates. Turnovsky et al., however, rely upon an erroneous solution concept for this type of model. They actually exclude instantaneous responses of the exchange rate to changes in current and expected policy variables. This critically affects the results that they present and entirely accounts for the dynamic path of policy variables that the authors describe. I. The Model Turnovsky et al.'s (1988) model is described by the following equations: (1) Yt *dyt* - d2[it (Pt d3(pt* + Et-Pt) Pt)] ( 1') yt= lt - 2 [ It* -(Pt*+ 1 - Pt)] - d3(t + Et -Pt) * Department of Economics, Harvard University, Cambridge, MA 02138, Department of Economics, Yale University, New Haven, CT 06520, and Department of Economics, Harvard University, Cambridge, MA 02138, respectively. We are grateful to Peter Boone for comments and to two referees for their suggestion to expand the scope of this comment. Alain de Crombrugghe acknowledges the financial support of the Belgian National Fund for Scientific Research. (for aggregate demand), (2) Mt - Pt = elyt - e2it (2') Mt* - Pt* = e,yt* - e2 It* (for money demand), (3) Et+ - Et = It -It* (for uncovered interest parity), (4) Ct SPt + (- 6)(Pt* +Et) (4') Ct* = SPt* + (1-6) ( Pt -Et) (for the consumer price index), and (5) P~Pt+,- Pt=Yt (5') Pt+ -1Pt = Yt (for sticky prices and the Phillips curve). In this model, the only true state variables are the prices (Pt and Pt*), while the exchange rate Et is a jumping variable. The money supplies (Mt and Mt*) are control (forcing) variables. All the other variables are endogenous. Thus, as is standard, the model can be reduced to a three-dimensional system of difference equations in the two prices and the exchange rate, with two roots inside the unit circle (corresponding to the state variables) and one root outside the unit circle (corresponding to the jumping variable). This three-dimensional system is shown by Turnovsky et al. in equation (10) of their paper (p. 345). With suitable transformations, the model can be rewritten as a one-dimensional system in the real exchange rate st = Et + Pt* - Pt, with the forcing variables given by the 1439

3 1440 THE AMERICAN ECONOMIC REI7EW DECEMBER 1991 real money stocks m, = M, - P, and m*= Mt* - Pt*. Turnovsky et al. show this reduced-form system in their equation (9) (p. 345): (6) St+1= cst + bmt - bm*. This system has one root outside the unit circle, corresponding to the fact that the real exchange rate is a jumping variable. II. Solution Concept: Rational Expectations The economics of the model imply that the reduced-form equation (6) determines St given expectations of st+1, and not the reverse. Referring to Bennett T. McCallum's (1983) methodology, the "minimal state variables" (i.e., the only state (predetermined) variables that cannot be eliminated) are the prices (Pt and Pt*). Policy rules for the nominal money supplies (Mt and Mt*) and the equilibrium path consistent with rational expectations for the nominal (Et) and, hence, for the real exchange rate (St) can be found on the basis of those two state variables only.' Other solutions, of the form st = f(st- ), can be accommodated by the model but are bubbly (explosive or damped). There is no reason to focus on those here since the purpose of Turnovsky et al. is not to look at the behavior of bubbly paths, but rather to look at optimal strategic policies with rational agents and flexible exchange rates. The erroneous approach of Turnovsky et al. is their definition of the real exchange rate, St, as an additional state variable. They explicitly claim that the model will show no endogenous jumps of the forward-looking nominal exchange rate, on the grounds that monetary policy is able to prevent such 1Given the state variables Pt and P*, it readily appears from (6) that a possible policy rule is to keep the real money supplies (mi and m*) constant by choosing fully accomodative nominal money supplies M, and Mt*. The corresponding rational-expectations path for the exchange rate is a constant real exchange rate s,, with endogenous jumps in the nominal exchange rate E,. jumps (pp. 345, 348). However, the jumps of the exchange rate are actually instantaneous responses to current and expected changes in the policy variables. Thus, Turnovsky et al.'s claim amounts to the negation of the perfect-foresight nature of the model. As is well known from the rational-expectations literature, the uniqueness of rational-expectations model solutions is guaranteed by the restriction that the system return to its longrun equilibrium. This is justified by the ability of economic agents to foresee this equilibrium (e.g., McCallum, 1983; Olivier Blanchard and Stanley Fischer, 1989). The monetary policies discussed by Turnovsky et al. are choices of nominal M, and Mj* that minimize an intertemporal quadratic loss function in output and inflation. The home-country loss function is (7) VO= Ep[ ay2 + ( a) t2 t =0 where Trt = t+1- Ct. The foreign country has an identical loss function defined over the foreign values of output and inflation. From these functions, rational agents know that fixity of the exchange rate is not an objective of the authorities. Even if it were, it would still be economically and technically inappropriate to treat the exchange rate as a predetermined (state) variable. The set of relevant "minimum state variables" that would determine the monetary policy rule would still be limited to the prices Pt and Pt*. Simply, the policy rule would take a different form. The rationalexpectations nature of the model would be unaffected by a change in the objective function. III. Solution Technique The solution algorithm cited by the authors (Tamer Basar and Geert J. Olsder, 1982) does not allow for the presence of forward-looking jumping variables in the feedback solutions and so cannot handle the case in which st is properly specified as a function of the expectations of st+1. Appropriate algorithms for dynamical feed-

4 VOL. 81 NO. 5 DE CROMBRUGGHE ETAL: DYNAMIC MONETARY POLICIES, COMMENT 1441 TABLE 1-POLICY RESPONSE TO A PRICE SHOCK (FORWARD-LoOKING EXCHANGE RATE) Period Variable Home economy: Output (Yd) Inflation (Ct+1- Cd) Price level (Pt) Nominal interest rate (It) Nominal exchange rate (Et) Real exchange rate (st) Money control (Md) Foreign economy: Output (Yt*) Inflation (C*1- C*) Price level (Pt*) Nominal interest rate (It*) Money control (Mt*) Sources: Turnovsky et al. (1988) for the model and parameter values; Oudi and Sachs (1985) for the simulation technique. back solutions with forward-looking variables have been described by Gilles Oudiz and Sachs (1985) for the case of Nash and cooperative equilibrium and by ourselves (de Crombruggh et al., 1988) for the case of Stackelberg leadership. A feedback computation of a time-invariant policy rule takes care of its time consistency. A stationary rule governing the forward-looking variables ensures that they are on the stable path determined by rational expectations. IV. Dynamics In the specific game analyzed by Turnovsky et al., the two-country economy starts with a "unit positive shock in the relative prices s" (pp ). The authors define the shock as a shock to st, which is obviously problematical, since st should be a forward-looking endogenous variable. The initial shock should properly be defined as a shock to the true state variables of the system, Pt and Pt*. Presumably, the shock can be properly defined as a joint movement in prices such that Pt - Pt* drops by one percent. For concreteness, let us consider a shock in which Pt drops by 0.5 percent while Pt* rises by 0.5 percent. It should be readily apparent that both countries can remain at their bliss points (zero intertemporaloss) after such a shock if the home country simply reduces M, in proportion to the drop in P, and the foreign country raises M,* in proportion to the rise in P,*, so that m, and m* remain unchanged. Future values of M and Mt* should also be changed by the same proportion, so that the money-supply changes are permanent ones. The nominal exchange rate would then immediately appreciate by one percent. The real exchange rate st = Et + Pt* - Pt would be unaffected by the shock, since Et would rise by the same amount as Pt - Pt*. Yt would be unaffected (remaining at zero), and future inflation, defined as Ct+1- Ct, would also remain at zero, since both current and future prices would rise in proportion to the shock to p.2 Table 1 2If 7t were defined as C, - Ct l1, instead of Ct+ 1 - Ct, then the money supplies would not be totally accomodative in the first period (Ct-1 = 0, Ct > 0), but in the second and all the following periods the optimal policy would be the one just described. Thus, there would be dynamics in the real variables for the first period only.

5 1442 THE AMERICAN ECONOMIC REVIEW DECEMBER 1991 shows these results for the first three periods. Thus, there are no true dynamics introduced by a relative price shock in the world defined by (1)-(7), since the money stocks and the exchange rate can immediately adjust to neutralize the real effects of changes in P, and P,*. Turnovsky et al. derive real dynamics in their simulations by the implicit assumption that Et is fixed at time t, so that changes in Pt and Pt* automatically translate into changes in the real exchange rate St. In practice, the authors use the onedimensional reduced-form equation for the real exchange rate, where st is incorrectly treated as a state variable (see Turnovsky et al., 1988 p. 345) and is thereby forced to adjust by 1 percent, the size of the initial shock. This accounts entirely for the dynamics of the equilibrium feedback policy rules for mt and m* that Turnovsky et al. calculate under the alternative assumptions of Nash, Stackelberg, and cooperative play by the monetary authorities of the two countries. V. Conclusion Following Dornbusch (1976), Turnovsky et al. (1988) assume sticky domestic prices but a clearing exchange-rate market governed by rational expectations. However, when they claim that optimizing monetary policy allows them to treat the exchange rate as a predetermined variable, they exclude any instantaneous exchange-rate response to current and expected policies. This exclusion of forward-looking jumps of the exchange rate contradicts the rationalexpectations nature of the model. This claim, moreover, is the only source of dynamics in Turnovsky et al.'s exercise. Judicious adaptations of their model or objec- tive function could produce some interesting dynamics, provided the shock and the solution concept can be appropriately formulated. REFERENCES Basar, Tamer and Olsder, Geert J., Dynamic Non-cooperative Game Theory, New York: Academic Press, Blanchard, Olivier J. and Fischer, Stanley, Lectures on Macroeconomics, Cambridge, MA: MIT Press, de Crombrugghe, Alain, Roubini, Nouriel and Sachs, Jeffrey D., "The Relevance of Stackelberg Leadership in Monetary Policy Games," unpublished manuscript, Harvard University, August 1988; Chap- ter 2 in Alain de Crombrugghe, "Leadership in International Monetary Policy," Ph.D. dissertation, Harvard University, July Dornbusch, Rudiger, "Expectations and Exchange Rate Dynamics," Journal of Political Economy, December 1976, 84, McCallum, Bennett T., "Non-uniqueness in Rational Expectations Models: An Attempt at Perspective," Journal of Monetary Economics, March 1983, 11, Oudiz, Gilles and Sachs, Jeffrey D., "International Policy Coordination in Dynamic Macroeconomic Models," in Willem Buiter and Richard Marston, eds., International Economic Policy Coordination, Cambridge: Cambridge University Press, 1985, pp Turnovsky, Stephen J., Basar, Tamer and d'orey, Vasco, "Dynamic Strategic Monetary Policies and Coordination in Interdependent Economies," American Economic Review, June 1988, 78,