Industry dynamics with stochastic demand.

We study the dynamics of an industry subject to aggregate demand shocks where the productivity of a firm's technology evolves stochastically over time. To characterize the intertemporal evolution of the distribution of firms, we discuss in particular how exit decisions, aggregate output, profits, and distributions of firm productivities vary (a) across different demand realization paths; (b) along a demand history path, detailing the effects of continued good or bad market conditions; and (c) for different anticipated future market conditions. We show how poor demand conditions can lead to increased exit of low-productivity firms at all future dates and states and raise welfare due to the impact on exit decisions.

1. Introduction

* This article integrates aggregate demand uncertainty into a dynamic stochastic model of firm entry and exit, and derives the consequences both for the evolution of the distribution of firms and for individual firm decision making. The research builds on the dynamic stochastic equilibrium model of firm entry and exit developed by Hopenhayn (1990, 1992a, 1992b). Hopenhayn analytically characterized the individual exit and production decisions of firms according to their age, size, and productivity in the unique invariant steady-state equilibrium.

We extend that work to an environment with aggregate demand uncertainty. In particular, we determine how individual firm exit decisions, aggregate output, profits, and distributions of firm productivities vary (i) across different demand realization paths; (ii) along a demand history path, detailing the effects of continued good or bad market conditions; and (iii) for different anticipated future market conditions. We identify sufficient conditions under which the theoretical model can reconcile empirical regularities regarding counter-cyclical exit, correlations of exit rates with future GDP growth, and the relative length and extent of recessions and expansions.

Incorporating aggregate uncertainty together with individual stochastic heterogeneity-both necessary features of a rich model of industry dynamics--introduces formidable technical and modelling challenges. A significant contribution of this article is to characterize the distribution of firms, rather than simply calculate selected higher-order moments. In particular, we characterize the evolution of the distribution of firms following arbitrary histories of demand realizations, deriving conditions under which the distribution of firms can be ordered conditional on equilibrium exit decisions. We then consider such key questions as: does an economy with a better distribution of firms produce more output in all states? Will a more protracted period of high demand lead to a better distribution of firms, or worse?

A second contribution is to endogenize the value of exit by building in an opportunity cost to exit: a firm can exit and sell its resources to another firm, but this requires that the firm's resources go unutilized for a period while the resources are retooled so that they can be used by a potentially more efficient entrant. The amount an entrant is willing to pay for those resources reflects the profits that it expects to earn, and hence will vary with market conditions.

Endogenizing the value of exit complicates the characterization of the equilibrium evolution of the distribution of firms. Both the endogenous value of production and the endogenous value of exit vary pro-cyclically, both rising in higher aggregate demand states. Two basic issues must then be addressed: (i) does firm exit rise or fall in higher demand states?, and (ii) is the immediate impact of these exit decisions on the distribution of firm productivities preserved over time? In particular, does the effect of reduced exit on the distribution of firms persist, so that the distribution of firms is worse at all subsequent dates and states, or could this effect be reversed? Answering this question is fundamental to addressing the impact of recessions on the long- run productivity of the economy.

Obtaining analytical answers to these questions is difficult. The standard analytical tool for this class of models is to prove that the competitive economy corresponds to the solution of a social planner's problem. Here, even when the social planner's problem characterizes equilibrium, it is of limited help: the fact that the equilibrium solves a surplus maximization problem, does not ensure, in particular, that an improvement in the distribution of firms adversely affects all firms because of the endogenous effects on exit. Consequently, the social planner characterization does not help to shed light on issues such as whether an improvement in the distribution of firms is preserved at all future dates and states.

This leads us to identify conditions on the transition process for a firm's technology that ensure the aggregate distribution is always totally ordered in stochastic dominance terms, and we use this to prove that an improvement in the distribution of firms is preserved along every future demand path. This yields a strong characterization result: ceteris paribus, an economy with greater past exit produces more output at every future date and state.

With aggregate distributions comparable in stochastic dominance terms, it is possible to derive sufficient conditions for exit rates to be counter-cyclical. For standard production technologies (e.g., constant elasticity of substitution [CES]) and any two-state Markov demand process, the endogenous value of production varies more with market conditions than the endogenous value of exit. It follows that higher demand causes exit to fall. Combining this with the result that the effect of increased exit on future distributions of firm productivities is always preserved, yields the result that demand downturns increase future output and lead to better future distributions of firm productivities (ordered by stochastic dominance) at every future date and state.

These results permit a characterization of the effect of aggregate demand shocks on industry dynamics. We contrast outcomes across different demand realization paths, comparing exit decisions and their consequences for aggregate output, profits, and productivity distributions when one history of demand realizations is uniformly better than another. Following this, we study outcomes along a demand history path in order to describe the effect of continued good or bad market conditions on exit decisions and aggregate variables. In particular, we prove that as a demand contraction continues, the distribution of firms grows ever better, setting the stage for greater future output once demand improves. Conversely, firms in booming economies rest on their laurels, so that the distribution of firms grows ever worse as a boom continues, sowing the seeds for a greater fall in output when the demand boom ends. We then derive how anticipated future market conditions affect exit decisions and aggregate outcomes. Better anticipated future market conditions induce more exit and give rise to better distributions of firm productivities in the current period.