Strategic complementarity slows macroeconomic adjustment to temporary shocks.

Oh, Seonghwan ; Waldman, Michael

I. INTRODUCTION

Starting with the work Of Diamond [1982], Hart [1982], and Bryant [1983], a number of authors have employed models which use potential coordination failure to show that many features of the Keynesian framework can be captured in models consistent with the microfoundations approach.(1) For example, in the context of a search model Diamond demonstrates the existence of equilibria with "too low" a level of aggregative activity, Hart captures this feature and the simultaneous existence of multipliers in a model of monopolistic competition, while technological interactions are the crucial element of Bryant's analysis. One reason to refer to these seemingly diverse set of papers as a single literature is that all the models are in fact driven by the presence of the same factor: strategic complementarity. A macroeconomic model which exhibits strategic complementarity is simply one where, the larger is aggregate production, the larger is the incentive for any particular agent to produce. Cooper and John [1988] first demonstrated that the presence of strategic complementarity is a characteristic of all of the models mentioned above, and persuasively argue that this is the critical feature which lies behind the finding of Keynesian-type results.

In this paper we argue that strategic complementarity is also important in understanding the dynamic response of economies to temporary shocks. In particular, we find that strategic complementarity is potentially an important factor in understanding why an economy may return only slowly to steady-state behavior after a temporary shock. That is, given the presence of any of a variety of factors which would cause the economy not to instantaneously return to full employment after a temporary shock (e.g., adjustment costs associated with changing the capital stock), the speed with which the economy returns to steady-state behavior is negatively related to the degree of strategic complementarity in the environment.

Related analyses are those of Haltiwanger and Waldman [1989], Bomfim and Diebold [1992], and Baxter and King [1990]. The first two papers consider models wherein some agents are characterized by adaptive or regressive expectations while others am characterized by rational expectations. Among other findings, these analyses demonstrate that the speed with which the economy returns to steady-state behavior after a temporary shock is negatively related to the degree of strategic complementarity in the economy.(2) We argue that the relationship between strategic complementarity and the dynamic response to temporary shocks is more general than is suggested by these papers. In their analyses temporary shocks have long-term effects because some of the agents make systematic mistakes when forming expectations. Here it is argued that, given any of a variety of factors which would cause temporary shocks to have long-term effects, there is a negative relationship between adjustment speed and the degree of strategic complementarity.

Baxter and King consider a world where, because of productive externalities, the economy exhibits increasing returns at the economy-wide level. Using simulation techniques they show, first, that the presence of increasing returns serves to increase the output volatility exhibited by the economy in response to shocks and, second, that after a one-time shock the return to steady-state behavior is slower in a world characterized by increasing returns. Baxter and King do not discuss their results in terms of strategic complementarity. However, it is easy to show that one way of introducing strategic complementarity is to assume that the economy exhibits increasing returns at the economy-wide level. Our paper suggests that what is central in Baxter and King's study is not the introduction of increasing returns per se, but rather that the introduction of increasing returns is one way of introducing strategic complementarity. That is, Baxter and King's findings can be thought of as a manifestation of the central point of the current paper.

The outline for the paper is as follows. Section II presents a simple stylized model that illustrates the relationship between strategic complementarity and the speed of return to steady-state behavior at the most basic and abstract level. Section III analyzes a richer model characterized by strategic complementarity and capital-stock adjustment costs. Section IV discusses models in which temporary shocks have long-term effects for reasons other than capital-stock adjustment costs. Section V presents some concluding remarks.

II. A STYLIZED MODEL

In this section we develop a simple stylized model which illustrates in a clear fashion the relationship between strategic complementarity and speed of return to steady-state behavior.3 Let firm i' s frictionless reaction function be given by [y.sub.i,t] = [ay.sub.t] + [e.sub.t], 0 < a < 1. The variable [y*.subi,t] is firm i' s output in period t in the absence of adjustment costs, [y.sub.t] is the average output per firm in period t, a is the parameter that measures strategic complementarity, and et is the value of an exogenous parameter in period t. Equation (1) is derived by imposing the partial-adjustment equation [y.sub.i,t] = [by.sub.i,t] + (1-b)[y.sub.i,t], 0 < b < 1, onto this frictionless reaction function, where b is the parameter that measures the slowness of the individual firm's adjustment path.

[MATHEMATICAL EXPRESSION OMITTED]

We can solve for the symmetric equilibrium by setting [y.sub.i,t] = [y.sub.t]. This yields equation (2) as the condition which must be satisfied in short-run equilibrium.

[MATHEMATICAL EXPRESSION OMITTED]

Further, substituting et = e for all t and [y.sub.t] = [y.sub.t-1] yields equation (3) as the condition which characterizes long-run equilibrium or steady-state behavior.

(3) [y.sub.t]: [1/(1 - a)]e.

A comparison of equations (1) and (2) illustrates the central argument of the paper. Equation (1) states that at the individual- firm level the slowness of the adjustment path--captured by the coefficient b--is independent of the degree of strategic complementarity. In contrast, equation (2) tells us that at the macro level the slowness of the adjustment path--captured by the term b/[1-(1-b)a]--depends not only on b but on the value of the parameter which measures the degree of strategic complementarity. In particular, if there is a one-time temporary shock to the value of the exogenous parameter e, then the greater is a the larger is b/[1- (1- b)a] and thus the slower is the return to steady-state behavior.

The intuition behind this result is as follows. Suppose there is a one-time temporary shock which causes average output per firm to fall below the steady-state average output level. During the adjustment period all firms will have low output, and the representative firm will want to keep its output low for two reasons. First, because of its own adjustment costs it does not want to go immediately to the steady-state output level. Second, given the presence of strategic complementarity, the representative firm will want to keep its output below the steady-state level because all other firms' outputs are below the steady-state level. It is this second factor which at the macro level causes the speed of return to steady-state behavior to be negatively related to the degree of strategic complementarity.

To further illustrate the argument consider the following numerical example. Let b = 1/2 and suppose that through period T-1 the economy is in the steady-state equilibrium where et = 0 for all t. Further, in period T assume e rises such that [y.sub.T] = 1, and then (e.sub.t) = 0 for all t > T. (4) Figure 1 depicts the adjustment path for [y.sub.t] following the one-time shock to e for various values of a. There are two points to note about the diagram. First, the return to steady-state behavior is slower when strategic complementarity is present, i.e., when a > 0. Second, as discussed above, the higher is the degree of strategic complementarity the slower is the return to steady-state behavior.

III. A MODEL WITH CAPITAL-STOCK

ADJUSTMENT COSTS

In this section we demonstrate that the results discussed above hold in a more fully specified macroeconomic model.

The Model

We construct a macroeconomic model similar to one analyzed in Haltiwanger and Waldman [1989]. The main differences are, first, that all agents here are assumed to have rational expectations and, second, agents (i.e., firms) now face adjustment costs for changing their capital stock.

The model is also related to the analyses of Bryant [1987], Kiyotaki [1988], and Well [1989]. Their papers consider environments characterized by investment opportunities and strategic complementarity and demonstrate the possibility for multiple equilibria which vary in terms of the rates of investment and growth in the economy. Our model considers the same type of economy, but rather than examining the possibility of multiple equilibria, we focus on the adjustment path which the economy follows in response to a temporary shock.

Consider a continuum of risk-neutral agents who must decide each period on an output level. Let [Y.sub.i,t] denote agent i' s production level in period t. The cost to agent i of producing [Y.sub.i,t] is denoted [c.sub.i,t], where [c.sub.i,t] = (y.sub.i,t])[sup.2]/(1 + k.sub.i,t]). The term [k.sub.i,t] is the size of agent i' s capital stock in period t. The specification therefore states that the larger is the capital stock in period t, the smaller is the cost of producing any fixed output level in that period. The manner in which [k.sub.i,t] is determined is described below.

Let [Y.sub.t] be period t' s aggregate production. The gross return to an agent from producing an amount [y.sub.i,t] is given by [Gamma.([Y.sub.t]) [y.sub.i,t], where [Gamma'] > 0. The assumption [Gamma'] > 0 means that the economy exhibits strategic complementarity, i.e., an increase in aggregate production raises the incentive for each individual firm to produce.(5)

Given the assumption [Gamma'] > 0, we can interpret this model in terms of a number of existing macroeconomic models of coordination failure. For example, consider Diamond [1982]. In that model the key restriction on behavior is that each individual is better off trading rather than consuming what he produces. Under this interpretation, [Gamma'] > 0 indicates the presence of positive trading externalities. That is, the larger is aggregate production, the higher is the probability that any particular trader will successfully complete a trade.

One can also interpret [Gamma'] > 0 as arising from demand linkages between imperfectly competitive producers in a multisector economy as in Hart [1982]. Under this interpretation, [Gamma(Y.sub.t]) denotes the marginal revenue from production, and [Gamma'] > 0 indicates that demand linkages cause the marginal revenue function facing a producer in a particular sector to shift out as the outputs of other sectors increase.

We now discuss the determination of [k.sub.i,t]. Agent i must split his net return in each period between consumption and investment. For simplicity we assume that it takes one period to construct capital. To be specific, [k.sub.i, t+1] is given by where (1-6) is a depreciation term and thus falls in the interval (0,1), and [w.sub.i,t] is agent i' s investment expenditure in period t.(6) It is assumed that a firm faces adjustment costs for changing its capital stock, i.e., m(0) = 0, m'(0)--[infinite], and m'(w) > 0, m"(w) (0 for all w > 0. The term [u.sub.t +1] is a parameter shared by all the agents, which captures the productivity of investing in capital. It is assumed that each agent chooses his expenditures on capital to maximize the discounted expected value of his consumption stream, where each agent discounts the future by a factor [Beta].

The model we have described above may display multiple steady-state equilibria because strategic complementarity is present. Since we want to abstract away from this possibility, we impose the following conditions:

[MATHEMATICAL EXPRESSION OMITTED]

where Y(k) is defined by the equation Y(k)/r[Y(k)] = (1 + k)/2 for all k [greater than or equal to] 0 and n = [m.sup.-1). Equation (5) guarantees that for a fixed value of k, the resulting value for Y is unique. Equation (6) ensures that the return to investing in capital is sufficiently concave to guarantee a unique and stable steady-state equilibrium.

Analysis

As in section II, the focus of our analysis is on how the economy responds to temporary shocks, in particular on how the path of adjustment following a shock is affected by the degree of strategic complementarity in the economy. We will consider shocks to the productivity of capital investment. Below (Y.sup.S (micro) denotes the steady-state value for Y when [u.sub.t], the productivity of capital, equals [micro], its steadystate value, in every period.

We begin by investigating how our model economy responds to an anticipated temporary shock. In particular, proposition 1 states the response of the economy to a shock of the following form. Up to period T-2 the economy is in a steady-state where p = [micro]. In period T - 1 all the agents learn that Pt = P for all periods except T, while [micro.sub.T]= [micro]. (All proofs are contained in the appendix.)

PROPOSITION 1. Given an anticipated temporary shock, then (i) and (ii) hold if [micro] > (<) [micro]

[MATHEMATICAL EXPRESSION OMITTED]

That is, following an anticipated temporary shock the economy gradually returns to the original steady-state production level. What happens is that the shock causes capital-stock holdings to change both because agents alter their investment plans, and because of a direct change in the productivity of investing in capital. In turn, the changed values for capital-stock holdings result in the economy only slowly returning to the original steadystate production level.

Proposition 1 is not very surprising. Given the specification of the function m(.), i.e., that each agent faces adjustment costs for changing his capital stock, it would be surprising if the model did not exhibit long-term effects of temporary shocks.(7) We now turn our attention to a more interesting question, specifically what is the relationship between the speed with which the economy returns to steady-state behavior and the degree of strategic complementarity. In order to investigate this issue we characterize a transformation of r(.) that increases the degree of strategic complementarity. Suppose [^Gamma(Y*)= r(Y*). Then [^r(-)] represents an increase in the degree of strategic complementarity if [^r(Y)] > r'(Y) for all Y. In other words, an increase in the degree of strategic complementarity involves an increase in the slope of r(.) around some fixed point.

PROPOSITION 2. Suppose the economy experiences the type of anticipated temporary shock considered in proposition 1. A transformation of r(.) which increases the degree of strategic complementarity but leaves [Y.sup.S(micro)] unchanged will cause [Y.sub.t]-[Y.sup.S(micro)] to increase for every t > T.

Proposition 2 considers the following hypothetical case. Suppose there is an increase in the degree of strategic complementarity which leaves steady-state behavior unchanged. The speed with which the economy returns to steady-state behavior is negatively related to the degree of strategic complementarity. The logic behind this result is similar to that given in the previous section. Suppose there is a shock which causes capital stock holdings in period T to fall below their steady state values. Should an agent invest so as to quickly return capital stocks to the steady-state level? Since during the adjustment period aggregate production is below its steady-state value, the incentive to invest is negatively related to the degree of strategic complementarity. In turn, because the incentive for investing is reduced by an increase in strategic complementarity, we find that an increase in strategic complementarity serves to decrease the speed of adjustment following the shock.

One question of interest concerns the robustness of our findings to different types of shocks. An increase in strategic complementarity increases the initial impact of an anticipated shock, while decreasing the speed with which the economy returns to steady-state behavior. We would like to know whether this relationship between speed of adjustment and strategic complementarity holds for shocks whose initial impact is not increased by an increase in strategic complementarity. To investigate this issue we now consider an unanticipated shock. We assume that the economy is in a steady state where [micro] = [(-)/micro] up to period T - 2.

In period T - 1 the agents choose expenditures on capital assuming that the economy will remain in the steady state. In fact, however, [micro.sub.T] = [micro], and the agents only find this out in period T. Finally, for every period subsequent to T it is the case that Pt = g, and agents make their investment decisions in those periods knowing this is the case.

It is easy to demonstrate that in response to this type of unanticipated shock the economy works in a fashion similar to the anticipated case considered in proposition 1. That is, an unanticipated shock causes a change in capital-stock holdings and because of adjustment costs there is only a slow return to the steady state. The reason the unanticipated shock is of interest is that in this case an increase in the degree of strategic complementarity has no effect on how current activity is affected by a temporary shock (see proposition 3 below). Thus, we can investigate the significance of strategic complementarity in an environment where the initial impact of the shock does not depend on the degree of strategic complementarity.

PROPOSITION 3. Suppose the economy experiences the type of unanticipated temporary shock described above. A transformation of r(.) which increases the degree of strategic complementarity but leaves [Y.sup.s](micro) unchanged will also leave [Y.sub.T]- [y.sup.s] (micro) unchanged, but will cause [Y.sub.T] - [Y.sup.S.(micro)] I to increase for every t>T.

Proposition 3 indicates that what is driving our results is not simply that an increase in strategic complementarity increases the initial impact of a temporary shock. Rather, even in a world where the initial impact is unaffected by the degree of strategic complementarity, it is still the case that an increase in strategic complementarity decreases the speed with which the economy returns to steady state behavior. The intuition is as given above. An increase in strategic complementarity decreases (increases) the incentive to invest after the economy is hit by a negative (positive) shock. Hence, an increase in strategic complementarity serves to inhibit the economy's return to steady-state behavior.

As a final point, we would like to mention that it would be easy to extend our results to the type of persistent shocks which are currently popular in the real business cycle literature. For example, suppose that [micro.sub.T] = [micro] and [micro.sub.T+j]= [Lambda.sup.j]([micro] - [- micro) + - micro] for all j > 0, where 0 < [Lambda] < 1. It would be easy to show that for this type of shock one could derive propositions similar to propositions 2 and 3.

IV. OTHER FACTORS WHICH CAUSE TEMPORARY SHOCKS TO HAVE LONG-TERM EFFECTS

In the previous section we considered a model where each agent faces adjustment costs for changing his capital stock, and demonstrated that the presence of strategic complementarity may be an important factor in explaining the speed of recovery after a temporary shock. In this section we argue that this conclusion is quite general. That is, given the presence of any one of a variety of factors which would cause the economy not to instantaneously return to full employment after a temporary shock, the presence of strategic complementarity will be an important determinant of the speed of the economy's return to steadystate behavior. In what follows, so as to avoid redundancy with propositions derived in the previous section, the argument will proceed on an informal rather than on a formal basis.

Adaptive Expectations

Another reason why temporary shocks may have long-term effects is because some or all of the agents are characterized by adaptive expectations. This factor was analyzed in Haltiwanger and Waldman [1989], and a related analysis appears in Bomfim and Diebold [1992]. In particular, Haltiwanger and Waldman considered a variant of the model presented above wherein a subset of agents form their expectations for [Gamma(Y.sub.t)] in an adaptive fashion, and capital is not part of the production process. They also found that an increase in the degree of strategic complementarity serves to decrease the speed of return to steady-state behavior after a temporary shock.

One can understand their result by considering an economy populated solely by agents with adaptive expectations. Suppose that in such an economy there is a temporary shock which causes [Gamma(Y.sub.t)] to fall below its steady-state value by some fixed amount. This will lower the expectation agents have for [Gamma(Y.sub.T+1)], which in turn will reduce aggregate output in T+1. However, for any fixed decrease in output in T+1, the realized value for [Gamma(Y.sub.T+1)] will be smaller the higher is the degree of strategic complementarity. What this implies is that, since the realized value for affects the expectation agents have for [Gamma(Y.sub.T+2)] aggregate output in period T+2 will be smaller the higher is the degree of strategic complementarity. In turn, continuously repeating this argument one finds that for every t > T + 2, the deviation from steady-state output will be higher the higher is the degree of strategic complementarity.

Sticky Prices

Another factor frequently used to generate long-term effects from temporary shocks is the existence of sticky prices. This notion has been formalized in the literature in a number of different ways. For example, the idea of prices being set in a staggered fashion goes back to the work of Fischer [1977] and Taylor [1980], menu costs have been explored in the work of Akerlof and Yellen [1985] and Mankiw [1985], while Ball and Cecchetti [1988] and Ball and Romer [1989] focus on whether sticky prices arise in models where the timing of price changes is assumed to be endogenous. We argue here that, if temporary shocks have long-term effects because of sticky prices of one type or another, the speed with which the economy returns to steady-state behavior will be negatively related to the degree of strategic complementarity.

We put forth our argument in the context of a monopolistic competition model where price setting is staggered, i.e., half the firms set their prices every even period while the other half set their prices every odd period. We also assume that the economy exhibits strategic complementarity in prices--that is, a firm's optimal price is an increasing function of the aggregate price level? Starting from a steady state, consider how this economy responds to a one-time unanticipated increase in the money supply which occurs in, say, period T. Group A will refer to the set of firms which set their prices in periods T, T+2, T+4, etc., while group B will refer to the set of firms which set their prices in periods T+I, T+3, etc. In period T+1 group B firms will want to increase their prices, but not all the way to the new steady-state levels. The reason is that in period T+1 group A firms will remain at the old steadystate prices, and given strategic complementarity, this provides an incentive for the group B firms to only partially adjust. Further, the higher is the degree of strategic complementarity the higher will be the incentive for group B firms to only partially adjust, and thus the lower will be the prices set by group B firms in period T+1. In turn, repeating this argument for periods T+2, T+3, etc., one has that for every t > T + 1, the difference between the actual price level and the eventual steady- state price level will be an increasing function of the degree of strategic complementarity in the environment. In other words, just as was true under the capital-stock adjustment-cost assumption and the adaptive-expectations assumption, the speed with which the economy adjusts to the new steady state is negatively related to the degree of strategic complementarity.

Finally, although the argument above is put forth in the context of a specific model of sticky prices, our conjecture is that the result is quite general. That is, given almost any reasonable specification under which sticky prices cause temporary shocks to have long-term effects, we conjecture that the speed with which the economy adjusts to the new steady state will be negatively related to the degree of strategic complementarity.

Other Types of Adjustment Costs

As a final point, we would like to make clear that although the formal model of section III focuses on adjustment costs associated with a firm changing its capital stock, the argument applies much more generally. In particular, our feeling is that almost any adjustment-cost model consistent with temporary shocks having longterm effects will be such that the speed with which the economy returns to steady-state behavior after a temporary shock will be negatively related to the degree of strategic complementarity in the environment.

Consider for example a model of monopolistic competition and inventories, where each firm faces adjustment costs for changing its level of inventory holdings. Suppose further that in period T there is an unanticipated temporary shock to the economy which causes aggregate inventory holdings to rise. Blinder and Fischer [1981] consider just such a model and demonstrate that the response of the economy to this temporary shock will be an immediate fall in aggregate output, and then a gradual return to the original steady- state level. We can now consider the role that strategic complementarity would play in such an environment. Consider period T+1. Since during the adjustment phase aggregate production is below its steady-state value, in period T+1 the incentive to produce will be smaller the larger is the degree of strategic complementarity. In addition, since the incentive to run down excess inventories is positively related to the incentive to produce, there will also be a negative relationship between the incentive to run down inventories and the degree of strategic complementarity. The result is that aggregate output in period T+1 will be negatively related to the degree of strategic complementarity, while aggregate inventory holdings at the end of period T+I will be positively related to the degree of strategic complementarity. In turn, repeating this argument for periods T+2, T+3, etc., yields that, just as was true for the capital-stock adjustment-cost model of section III, for every t > T + 1 the deviation from steadystate behavior will be positively related to the degree of strategic complementarity in the environment.

V. CONCLUSION

The presence of strategic complementarity in macroeconomic models has been used to explain a host of important phenomena, e.g., multipliers, multiple equilibria, and the possibility of underemployment equilibria. In this paper we argue that the presence of strategic complementarity in the macro setting is also important for understanding why an economy may exhibit a slow return to steady-state behavior after a temporary shock. In particular, we argue that given the presence of any one of a variety of factors which would cause the economy not to instantaneously return to full employment after a temporary shock, the speed of adjustment is negatively related to the degree of strategic complementarity.

One way in which the analysis of this paper might fruitfully be extended is to consider the concept of the automatic stabilizer. In an earlier literature on economic fluctuations the concept of the automatic stabilizer was a central element for explaining why the post-World War II economy seems to be less prone to large fluctuations than the pre-World War II economy. The logic was that there were changes in the system of governing, e.g., changes in the tax system, which reduced economic instability in response to shocks. Our claim is that the concept of the automatic stabilizer can be at least partially understood in terms of strategic complementarity. Specifically, our analysis suggests that any change which serves to reduce the degree of strategic complementarity should be a type of automatic stabilizer. This follows from our finding that adjustment speed after a temporary shock is negatively related to the degree of strategic complementarity. In other words, exactly consistent with the definition of an automatic stabilizer, any change which reduces the degree of strategic complementarity should reduce the instability the economy exhibits in response to shocks. In future work we hope to more fully investigate this idea. In particular, our plan is to formally investigate the links between strategic complementarity and those aspects of the post-World War II economy that earlier authors have identified as automatic stabilizers.

[TABULAR DATA OMITTED]

APPENDIX

PROOF OF PROPOSITION 1. To prove proposition 1 we must first demonstrate that the economy is characterized by a unique and stable steady-state equilibrium. Since we assume a continuum of agents in the unit interval, total output as a function of the period t capital stock, [Y(k.sub.t)], is defined by the expression [Y(k.sub.t)]/[Gamma[Y(k.sub.t)]] = (1 + [k.sub.t])/2. This follows from each agent producing to the point where the marginal cost of production equals the marginal revenue of production. Given a fixed value for [k.sub.t], we now have that equation (5) is a sufficient condition for a unique [Y.sub.t].

The representative agent solves the following maximization problem.

[MATHEMATICAL EXPRESSION OMITTED]

Let n = [m.sup.-1]. Since m'> 0 and m"< 0, we have that n' > 0 and n" > 0. (Al) can be rewritten as

[MATHEMATICAL EXPRESSION OMITTED]

We can easily demonstrate that (A2) is a well-defined dynamic optimization problem, and thus sufficient conditions for an interior solution are the Euler equation and the transversality condition.

[MATHEMATICAL EXPRESSION OMITTED]

Note that for deriving the equations above the representative agent takes the sequence of [Y.sub.'t s] as given, rather than as a function of his own actions. Given equation (6), (A3) and (A4) yield that there exists a unique steady state which is defined by (A5), where k* denotes the steadystate value of the capital stock.

[MATHEMATICAL EXPRESSION OMITTED]

Given n'(0) = 0 and [gamma](0) > 0 (see footnote 5), (AS) yields that k* > 0.

To consider stability we first take a linear approximation of the Euler equation around k*

[MATHEMATICAL EXPRESSION OMITTED]

Rewriting (A6) we obtain

[MATHEMATICAL EXPRESSION OMITTED]

The characteristic polynomial for A is

[MATHEMATICAL EXPRESSION & TEXT OMITTED]

It is clear that equation (6) is sufficient for one of the roots to lie between zero and one. The other root exceeds one. Thus by Theorem 6.9 in Stokey and Lucas [1989] the above steady state is stable. Also, the fact that 0 < Lambda.sub.1! < 1 implies that adjustment paths are monotonic. Finally, [Lambda.sub.1] is given by equation (A9).

PROOF OF PROPOSITION 2. (A3) and (A9) yield

[MATHEMATICAL EXPRESSION OMITTED]

PROOF OF PROPOSITION 3. This proof follows along the same lines as that of proposition 2 except that agents now choose [w.sub.T-1] given a belief that [micro.sub.T] = [micro]

REFERENCES

Akerlof, George, and Janet Yellen. "The Macroeconomic Consequences of Near-Rational Rule-of-Thumb Behavior." Quarterly Journal of Economics, Supplement 1985, 823-38.

Ball, Laurence, and Stephen Cecchetti. "Imperfect Information and Staggered Price Setting." American Economic Review, December 1988, 999-1018.

Ball Laurence, and David Romer. "The Equilibrium and Optimal Timing of Price Changes." Review of Economic Studies, April 1989, 179-98.

Baxter, Marianne, and Robert King. "Productive Externalities and Cyclical Volatility." University of Rochester Working Paper No. 245, September 1990.

Blinder, Alan, and Stanley Fischer. "Inventories, Rational Expectations, and the Business Cycle." Journal of Monetary Economics, November 1981, 277-304.

Bomfim, Antulio. "Business Cycle Dynamics, Forecast Heterogeneity, and Strategic Complementarities: When Do Inefficient Prediction Rules Start to Matter?" Working paper, Board of Governors of the Federal Reserve, March 1992.

Bomfim, Antulio, and Frank Diebold. "Near-Rationality and Strategic Complementarity in a Macroeconomic Model: Policy Effects, Persistence and Multipliers." Working paper, Board of Governors of the Federal Reserve, July 1992.

Bryant, John. "A Simple Rational Expectations Keynes-Type Model." Quarterly Journal of Economics, August 1983, 525-29.

"The Paradox of Thrift, Liquidity Preference and Animal Spirits." Econometrics, September 1987, 1231-35.

Cooper, Russell, and John Haltiwanger. "Inventories and the Propagation of Sectoral Shocks." American Economic Review, March 1990, 170-90.

Cooper, Russell, and Andrew John. "Coordinating Coordination Failures in Keynesian Models." Quarterly Journal of Economics, August 1988, 441--63.

Diamond, Peter. "Aggregate Demand Management in Search Equilibrium." Journal of Political Economy, October 1982, 881-94.

Durlauf, Steven, "Multiple Equilibria and Persistence in Aggregate Fluctuations." American Economic Review, May 1991, 70-74.

Fischer, Stanley. "Long Term Contracts, Rational Expectations, and the Optimal Money Supply Rule." Journal of Political Economy, February 1977, 163-90.

Haltiwanger, John, and Michael Waldman. "Rational Expectations and the Limits of Rationality: An Analysis of Heterogeneity." American Economic Review, June 1985, 326--40.

"Limited Rationality and Strategic Complements: The Implications for Macroeconomics." Quarterly Journal of Economics, August 1989, 46383.

Hart, Olivet "A Model of Imperfect Competition with Keynesian Features." Quarterly Journal of Economics, February 1982, 109--38.

Kiyotaki, Nobuhiro. "Multiple Expectational Equilibria Under Monopolistic Competition." Quarterly Journal of Economics, November 1988, 695-713.

Leijonhufvud, Axel. Information and Coordination: Essays in Macroeconomic Theory. New York: Oxford University Press, 1981.

Lucas, Robert. "Understanding Business Cycles," in Stabilization of the Domestic and International Economy, edited by K. Brunner and A. Meltzer. Amsterdam: North-Holland, 1977, 7-29.

Mankiw, N. Gregory. "Small Menu Costs and Large Business Cycles." Quarterly Journal of Economics, May 1985, 529-37

Oh, Seonghwan, and Michael Waldman. "The Macroeconomic Effects of False Announcements."

Quarterly Journal of Economics, November 1990, 1017-34.

Stokey, Nancy, and Robert Lucas. Recursive Methods in Economics. Cambridge: Harvard University Press, 1989.

Taylor, John. "Aggregate Dynamics and Staggered Contracts." Journal of Political Economy, February 1980, 1-23.

Weil, Philippe. "Increasing Returns and Animal Spirits." American Economic Review, September 1989, 889-94.

* Professor, Department of Economics, Seoul National University, and Professor, Johnson Graduate School of Management, Cornell University. We would like to thank Antulio Bomfim, the participants at workshops at Brown University, Georgetown University, and the Federal Reserve Bank of St. Louis, as well as two anonymous referees for helpful comments, and the UCLA Academic Senate for financial support.

1. Later papers include Bryant [1987], Cooper and John [1988], Kiyotaki [1988], Haltiwanger and Waldman [1989], Weil [1989], Cooper and Haltiwanger [1990], Oh and Waldman [1990], Durlauf [1991], Bomfim [1992] and Bomfim and Diebold [1992]. See also Leijonhufvud [1981] for a non-technical analysis which captures many of the same ideas.

2. In terms of this result, the difference between the two papers is that Haltiwanger and Waldman consider a microfoundations-type setting, while Bomfim and Diebold look at a more aggregative economic model. See also Haltiwanger and Waldman [1985] and Bomfim [1992].

3. We would like to thank one of the referees for suggesting the analysis of this section.

4. The value of [e.sub.T] which results in yr = 1 depends on the value of a. Note, if we employed the same value of et for all values of a considered, then the size of the initial impact of the shock would be positively related to a. Hence, in this case we would have a second reason for why the speed of return to steady- state behavior is negatively related to the degree of strategic complementarity. See proposition 2 of the following section for a related result.

5. To rule out the possibility of a degenerate equilibrium where the capital stock equals zero we impose the condition [Gamma(0)] > 0. One can interpret r(0) as being the gross return from a unit of output when an individual is unable to trade. Hence, r(0) ) 0 simply states there is a positive return derived from consuming one's own production.

6. Introducing a time-to-build a assumption would serve to complicate the analysis without changing the qualitative nature of the results.

7. See Lucas [1977] for an earlier discussion concerning capital- stock adjustment costs and the longterm effects of temporary shocks.

8. Ball and Romer [1989] show that if the timing of pricing decisions were made endogenous in this type of model, then price setting would not be staggered. However, Ball and Cecchetti [1988] demonstrate that staggered price setting would arise if imperfect information were added to the model.