Industry dynamics with stochastic demand.

Clearly, [alpha]([theta], [mu]) < [bar.[theta]], because of the opportunity cost of exiting and transferring the plant to the entrant. In summary, for each ([theta], [mu]), an equilibrium exit rule is characterized by [[alpha].sup.*] = [alpha]([theta], [mu]) where [v.sup.c]([theta], [mu], [[alpha].sup.*]) = [v.sup.e]([theta], [mu], [[alpha].sup.*]): firms with [alpha] > [alpha]([theta], [mu]) remain in the market, whereas firms with [alpha] < [alpha]([theta], [mu]) exit. It is the less-productive/higher-cost firms that find it optimal to exit so that their plant can be reallocated to better uses. This result is consistent with Dunne et al.'s (1989b) finding that higher-cost plants exit first, and Baldwin's (1995) finding that entrants, whereas relatively unproductive compared to the average firm, are more productive than the exiting firms that they replace.

Bergin and Bernhardt (1995) provide sufficient conditions for the existence of an equilibrium in economies with a continuum of agents, aggregate shocks, and idiosyncratic shocks to agent types, in which an agent's payoff depends on his own type and action, the aggregate state, and the distribution over agent types and actions in the economy. The mild continuity assumptions on the transition functions and payoffs that are required for an equilibrium to exist are satisfied here.

3. Dynamics

* Our goal is to characterize how demand fluctuations affect future distributions of firm productivities and profits, as well as aggregate variables such as prices and industry output. We first investigate the possibility of characterizing dynamics in the competitive economy using a social planner's characterization, and then use a more direct approach to characterize equilibrium dynamics.

* Social planner's characterizations. The standard approach to characterizing industry dynamics is to show first that the competitive equilibrium corresponds to the solution of a social planner's problem, and then solve that social planner's problem (e.g., see Hopenhayn, 1992a). We now consider when the competitive equilibrium can be characterized as the solution to a social planner's problem, and then explain why this social planner's characterization is of limited help in facilitating an analysis of industry dynamics.

If [gamma] = 0 (the exiting firm has all bargaining power in the resale markets for plants), then exit decisions in the competitive equilibrium correspond to the exit rule of a social planner who seeks to maximize discounted social surplus. Period social surplus can be represented as the area between the demand and supply curves. Let [p.sub.s](Y,[mu]') denote the aggregate supply function when the distribution of firms in operation is [mu]'. Then if total output is [Y.sup.*], social surplus is S([Y.sup.*], [theta], [mu]') = [[integral].sub.[0, [??]][p(Y, [theta]) - [p.sub.s](Y, [mu]')]dY. The social planner program is the optimization of the present value of the social surplus stream by choice of continuation (and hence exit) distribution. The functional equation for the social planner's problem is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In determining the aggregate supply function, [p.sub.s](Y,[[mu].sub.[alpha]]), labor choices of operating firms correspond to those made by the social planner. If at price p, firm [alpha] supplies y(p, or), then total output is Y(p, [[mu].sub.[alpha]]) = [integral] y(p, [??])[[mu].sub.[alpha](d[??}). Inverting for p gives [p.sub.s](Y, [[mu].sub.[alpha]]). The solution to the program yields an exit rule, [alpha]([theta], [mu]), at each ([theta], [mu]), which determines the evolution of the aggregate distribution.

Theorem 1. If [gamma] = 0, then the exit rule in the competitive economy is unique and corresponds to the solution to the social planner's problem.

Proof See the Appendix.

The social planner characterization only holds when [gamma] = 0. When [gamma] > 0, the exiting firm's weaker bargaining position means that the purchaser of an exiting firm's plant extracts some of its value, reducing the incentive of a firm to exit. In such circumstances, exit is inefficiently less than the socially optimal level of exit, and equilibrium differs from the social planner solution.

Theorem 1 asserts that if [gamma] = 0, then the exit rule in the competitive economy is unique and corresponds to the solution to the social planner's problem. However, the social planner's approach is of limited help. It does not a priori follow that an improvement in the distribution adversely affects all firms because of the endogenous effects on exit. Some firm types may benefit from an increase in [mu] if the resulting impact on exit of other firms in some future states goes the "wrong" way (e.g., if exit is greater in some future period where a firm type expected to be better). Consequently, we must use a different approach to investigate the impact of aggregate demand shocks or distributional shifts on individual firm decision making when [gamma] > 0.

* Competition and payoff monotonicity. Fluctuations in demand directly affect exit decisions, and hence future distributions of firm productivities. In turn, differences in the distribution of firm productivities influence future exit decisions. To analyze the impact of demand fluctuations, we consider the role played by differences in the distribution of firm productivities. We first prove that (i) individual valuation functions are always monotone decreasing in [mu] (a better distribution of competitors always reduces the expected profts of all firm types), and (ii) an important dominance condition is satisfied: in equilibrium, better technology distributions are preserved over time, so that if [[mu].sub.t] > [[??].sub.t], then along any future common demand path, at any future date [tau] > t, [[mu].sub.[tau]] > [[??].sub.[tau]], where [[mu].sub.[tau]] and [[mu].sub.[tau]] are the distributions following [[mu].sub.t] and [[??].sub.t], respectively.

In Theorem 2, we consider a finite horizon version of the economy. Let [v.sup.c.sub.n]([theta], [mu], [alpha]) and [v.sup.e.sub.n]([theta], [mu], [alpha]) be the (equilibrium) values to firm [alpha] from continuing and exiting, respectively, when the current distribution is [mu] and the current state is [theta], and there are n periods remaining. When the horizon is finite, the equilibrium exit rule can be computed by backward induction. Indeed, Theorem 2 proves that, independently of whether the social planner formulation is applicable, the equilibrium exit rule is unique. That is, at each ([mu], [theta]), there is a unique equilibrium threshold. Further, [v.sup.c.sub.n] and [v.sup.e.sub.n] are monotone decreasing in [mu]. If they converge as n increases, monotonicity is preserved.

Theorem 2. Payoffs are monotonically decreasing in the technology distribution and, in equilibrium, better distributions on technologies are maintained over time. Formally, for a finite horizon economy with n periods remaining,

(i) The functions [v.sup.c.sub.n]([theta], [mu], [alpha]) are continuously decreasing in [mu] for all n.

(ii) If [[??].sub.t] [??] [[mu].sub.t] then [[??].sub.t+1] [??] [[mu].sub.t+1].

Finally, if [v.sup.c.sub.n] and [v.sup.e.sub.n] converge pointwise as n [right arrow] [infinite], then the limiting functions are continuously decreasing in [mu]. When [gamma] = 0, [v.sup.c.sub.n] and [v.sup.e.sub.n] converge because they are derived from a social planner program which converges as n [right arrow] [infinite].

Proof See the Appendix.

The result that [v.sup.c.sub.n]([theta], [mu], [alpha]) and [v.sup.e.sub.n]([theta], [mu], [alpha]) are continuously decreasing in [mu] is both subtle and important. The key is to prove that comparing two aggregate distributions [[??].sub.t], and [[mu].sub.t], at period t, if [[??].sub.t], [??] [[mu].sub.t], then [[??].sub.t+1] [??] [[mu].sub.t+1]. To do this, we show that whereas exit may be less in the economy with the better distribution, it cannot be so much less that it reverses the ordering in distributions. Thus, dominance is inherited in subsequent periods. Were the dominance property not preserved over time, then some firm type may prefer to face a better distribution of competitors in t if that better distribution implied a worse distribution in t + 1, when the firm type expected to have a higher [alpha].

This result is far from immediate. Indeed, if capital choices are made before the demand and productivity shocks are realized, more structure is required to ensure this monotonicity result. This is because capital and technology productivities have distinct effects on exit decisions (see Bergin and Bernhardt, 2005).

The fact that a better distribution of technology productivities is preserved along every future demand path is crucial for the analysis that follows. Theorem 3 exploits this directly, showing that along a common demand path, in an otherwise identical economy with a better initial distribution of firms, future output in every future demand state is greater. Consequently, prices are always lower in the economy with the better initial distribution of firms, so that any given technology, [alpha], generates lower profit both currently and at any subsequent period in the economy with the better initial distribution.