Industry dynamics with stochastic demand.

Remark 1. Some remarks on these assumptions are appropriate. If P(* | [alpha]) = w([alpha]) F(*) + [1 w([alpha]]G(*) and F first-order stochastically dominates G(F [[greater than or equal to].sub.FOSD] G), then [alpha]' [greater than or equal to] [alpha] implies that P(* | [alpha]') [[??].sub.FOSD] (P* | [alpha]). Similarly, if F conditionally first-order stochastically dominates G(F [??] G), then [alpha]' [??] [alpha] implies that P(* | [alpha]') [greater than or equal to] P(* | [alpha]') [greater than or equal to] P(* | [alpha]). Thus, the weighted average specification makes all possible distributions comparable in stochastic dominance terms, and this in turn allows comparisons of such variables as profitability at different distributions. However, a central assumption of the model is that there is technological benefit to exit when a firm becomes inefficient and is subsequently replaced by a more efficient firm. In particular, increasing the exit threshold should lead, on replacement, to a better distribution of firms. Similarly, a better current distribution should lead to a better distribution next period, under the same exit threshold. With the learning-by-doing property, these natural economic assumptions are implied by F [??] G in the weighted specification (but do not follow from the assumption that F [[??].sub.FOSD] G).

To see the issue, suppose that all firms below a threshold technology, [[alpha].sup.*] < [bar.[alpha]], exit. Firms that exit are replaced by new entrants drawn according to P(* | [bar.[alpha]]). Suppose that technology distribution [mu] stochastically dominates [mu]' Then, when the same exit rule is applied to both, the resulting distribution determined by [mu]' may dominate the distribution determined by [mu], reversing the ordering of distributions. For example, this would occur if [mu] puts all mass between [[alpha].sup.*] and [bar.[alpha]] ([[alpha].sup.*] < [[alpha].sup.*]) so that no firms exit, whereas [mu]' puts all mass below [[alpha].sup.*], so that every firm exits and receives a technology drawn from P(* | [bar.alpha]), which is better than a draw from any distribution P(* | [alpha]), for [alpha] < [bar.[alpha].

Evolution of the distribution of technologies. We assume that the economy is such that the worst firm type always does best to exit. Exit depends on many factors--the distribution of demand states, the distributions from which firms draw new technologies, the share 1 - [gamma] of an entrant's value that an exiting firm can extract, and so on--so uncovering the precise conditions under which exit occurs in general contexts is difficult. However, identifying sufficient conditions for exit to occur is easy--a grossly sufficient condition is for (i) the worst firm earns no profits from producing (e.g., the technology take the form [alpha]f(k, l)), and (ii) [gamma] is sufficiently small, so that the firm gains enough of the proceeds associated with drawing a technology from P(* | [bar.[alpha]]), rather than P(* | 0). As a result, in equilibrium, firms with technology below some ([theta], [mu])-dependent threshold, [alpha]([theta], [mu]), exit, and those above [alpha]([theta], [mu]) remain in the market. Where it does not cause confusion, we omit the dependence of the exit threshold on the state, and write [[alpha].sup.*].

At any ([theta], [mu]), an exit threshold [[alpha].sup.*] = [alpha]([theta], [mu]) generates two distributions of interest: the distribution over technologies of firms that are more productive than [[alpha].sup.*] and hence produce in the current period and the distribution over technologies of firms in the next period. We use a subscript to index the distribution of operating firms in the current period, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and a superscript to index the distribution over technologies next period, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Specifically, the measure of firms supplying the market in the current period, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is the measure obtained from [mu] after all firms with technologies that are less productive than [[alpha].sup.*] exit, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; and the exit rule [[alpha].sup.*] combines with [mu] to determine the measure over technology productivities at the beginning of next period, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (before exit decisions are made in that period), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

As a matter of terminology, comparing two exit thresholds, [alpha]', [alpha]", if [alpha]' > [alpha]", say there is more exit at threshold [alpha]'. If the corresponding distributions are equal, [mu]' = [mu]", then more exit corresponds to a larger mass of firms exiting, although in general, a higher exit threshold will not imply a larger mass of exiters.

Market clearing. With measure [mu] on technologies, an exit rule [[alpha].sup.*] generates an aggregate supply of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The equilibrium price p is determined by [theta] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] via the market-clearing condition Y(p, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) = D(p, [theta]), where D(p, [theta]) is the inverse of p(Y, [theta]) for each [theta]. With price p([theta], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) determined by ([theta], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]), a firm [alpha] earns profit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[] Equilibrium exit. An exit rule, [alpha]([theta], [mu]), gives an exit threshold at each ([theta], [mu]), and from this the evolution of the aggregate distribution over time is determined. Thus, starting at ([theta]', [mu]') (and prior to the exit-remain decision), for an optimizing firm with technology or, given the exit rule, one may attach a present value to the profit flow, v([theta]', [mu]', [alpha]). And, given a present value to future profit flows, a firm can optimally chose either to exit in the present period, or remain and produce. If these individual exit decisions are consistent with the exit rule, then the market is in equilibrium. These issues are discussed next.

Exit over time. In equilibrium, a firm's exit decision maximizes expected profits given ([[alpha].sub.t], [[mu].sub.t], [[theta].sub.t]), and the distribution of firms over time is consistent with the optimization by almost all firms, for almost all [[theta].sub.t]. At any date t, equilibrium is characterized by an exit threshold that depends on [mu] and [theta], [alpha]([theta], [mu]). This fully determines the evolution of the aggregate distribution [mu] over time along any path of demand realizations, ([[theta].sub.1], [theta].sub.2], ...). Consequently, the exit rule determines the market-clearing price sequence facing firms, and hence the present value of any firm [alpha]. The exit rule, [alpha]([theta], [mu]), is an equilibrium exit rule if and only if it determines a valuation function, v, that supports the exit rule: it must be that at ([theta], [mu]) firms wish to exit if and only if their technology is below or([theta], [mu]). Letting [[alpha].sup.*] = [alpha]([theta], [mu]) be the threshold at ([theta], [mu]), the expected value to a firm [alpha] from operating in the current period (and acting optimally thereafter) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the expected value to exit is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here [v.sup.c] denotes the value of continuing to operate, and [v.sup.e] denotes the value of exiting. The value of an operating firm a facing market conditions ([theta], [mu]) is equal to the sum of its maximized operating profits plus the discounted expected value of continuing to operate given that it chooses inputs optimally and makes future operation-exit decisions optimally. The expression for [v.sup.e] reflects the weak bargaining position of an exiting firm: an exiting firm only receives fraction 1 - [gamma] of the full discounted expected value of the plant to a new firm whose technology quality is drawn according to [bar.[alpha]]. Because firms choose whether to exit or to continue in operation optimally, the value of a firm a is given by

v([theta], [mu], [alpha]) = max{[v.sup.c]([theta], [mu], [alpha]), [v.sup.e]([theta], [mu], [alpha])}.

Because [v.sup.e] is independent of [alpha], whereas [v.sup.c] is increasing in [alpha], there is a unique value, [??], at which [v.sup.c] and [v.sup.e] are equal. Firms with technologies above [??] wish to continue and firms with technologies below [??] wish to exit. A necessary condition for equilibrium is that [[alpha].sup.*] = [??]: the exit rule [alpha]([theta], [mu]) must satisfy

[v.sup.c]([theta], [mu], [alpha]([theta], [mu])) = [v.sup.e]([theta], [mu], [alpha]([theta], [mu])), [for all]([theta], [mu]).