Academic journal article Economic Inquiry

The Expectational Stability of Underemployment Equilibria

Academic journal article Economic Inquiry

The Expectational Stability of Underemployment Equilibria

Article excerpt


One of the most important recent developments in macroeconomic theory has been the demonstration of how external economies of scale may generate multiple perfect-foresight equilibria. A positive externality, such as that studied by Diamond [1982], may generate two saddle-path equilibria, one converging to a high-employment steady state and the other converging to an inferior low-employment steady state. For given initial conditions there are, therefore, two equilibrium choices of employment. In Howitt and McAfee's [1988] model, which has both a positive and a negative externality, any one of a continuum of employment choices will satisfy optimality conditions and converge to the low-employment steady state. Thus, the literature on externalities demonstrates that there may exist a stable (or saddle-path stable), Pareto-inferior, low-employment equilibrium in a perfect-foresight model with market clearing and flexible prices.(1) An intriguing feature of these models is the possibility that economic fluctuations might be driven by waves of optimism or pessimism that move the economy from one equilibrium to another.

Along with the possibility of multiple equilibria there is, however, a problem of indeterminacy. How do we say which equilibrium will be chosen? The idea that waves of optimism or pessimism move the economy is provocative and intuitively appealing, but a more explicit account of how individuals come to focus on a particular equilibrium, a selection criterion, is needed.

One criterion that has been used successfully to limit the multiplicity of equilibria in rational expectations and perfect-foresight models is the stability of an adaptive learning procedure.(2) Under adaptive learning the agents in the model learn about the economy in much the same way as an econometrician. They have a (perfect-foresight equilibrium) model of the economy, and they use the available macroeconomic data to infer the value of that model's parameters. Their parameter estimates determine their forecasts, and these forecasts determine their choice of labor inputs. If a given steady-state equilibrium is unstable under a broad class of plausible learning rules, it is unlikely that the economy will converge to that equilibrium, even if it is stable under perfect foresight. If a steady state is stable under learning, then we are more confident that it may be observed in an actual economy.

In this paper I apply the concept of expectational stability to Howitt and McAfee's [1988] model of aggregate employment in the presence of externalities.(3) Expectational stability is a simple, and therefore highly tractable, iterative learning scheme that has been used by Evans [1985] and by Bray [1982, Proposition 4] to study the convergence of learning to rational expectations. In addition to being tractable, the procedure is general. Indeed, the local stability conditions of the procedure used in this paper are the same as the conditions for local stability of least squares learning as derived by Marcet and Sargent [1989a; 1989b]. These same conditions are necessary for the stability of a broad class of learning algorithms, including least squares estimation, Kalman filtering, and iterative Gauss-Newton procedures, as is shown in Ljung and Soderstrum [1983].

I study learning in Howitt and McAfee's [1988] model because it is also highly general. It incorporates both thick market (positive) and congestion (negative) externalities. It allows for multiple perfect-foresight steady states that are, alternately, stable and saddle-path stable.(4) Further, I show that the widely cited model of Diamond [1981; 1982] is a special case of the Howitt and McAfee model.

In this paper I show that the learning rule eliminates the problem of indeterminacy. It will, however, also rule out those steady states that are unambiguously Pareto-inferior. Stable low-employment steady states may exist, but they cannot be ranked. …

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