Industry dynamics with stochastic demand.

Suppose that [[alpha].bar] is sufficiently small relative to [??] that total surplus is maximized when [[alpha].bar] firms exit. Let [gamma] be such that firm type [[alpha].bar] is just indifferent between continuing to operate and exiting. Such a [gamma] exists because the value of exiting varies continuously with [gamma] from 0 when [gamma] = 1 to the expected value of the new entrant when [gamma] = 0. Consider the consequence for total surplus of a one-time marginal reduction in [[theta].bar]. This drop in demand induces all unproductive, [[alpha].bar], firms to exit. The marginal loss in profits to a good firm from this reduced consumer demand is [??]([l.sup.*]([??], [[theta].bar])). The marginal loss in expected discounted profits to a bad firm is zero, because it now exits and previously was indifferent between exiting and not. If we let [beta]E[[pi]([theta], [alpha])]/(1- [beta]), then the value of exit is (1 - [gamma])z/(1 - [gamma][beta]). The social gain from inducing exit of inefficient [[alpha].bar] firms is therefore [gamma](1 - [beta])z/(1 - [beta][gamma]). Because there are just as many good firms as bad firms, the one-time reduction in [theta] from [[theta].bar] increases social surplus if and only if [gamma](1 - [beta])z/(1 - [beta][gamma]) > [??] f([l.sup.*]([??], [[theta].bar])). But note that [??]f([l.sup.*]([??], [[theta].bar])) does not depend on [bar.[theta]]. Let [gamma](bar.[theta]]) be the value of [gamma] that leaves [[alpha].bar] just indifferent to exit when [theta] = [[theta].bar]: d[gamma]/d[theta] > 0 As [bar.[theta]] is increased, it must be the ease that [gamma](1 - [beta])z/(1 - [beta][gamma]) eventually exceeds [??] f([l.sup.*]([??], [[theta].bar])), so the one-time recession of a lower 8 increases total surplus.

If [gamma] is sufficiently large, bad firms fail to internalize the social cost of their failure to exit and thereby have their technologies replaced by stochastically better ones. Downturns cause more bad firms to internalize these social costs and exit. If this gain from increased exit outweighs the foregone period surplus from lower consumer demand, then total surplus is raised.

5. Conclusion

* This article explores the dynamics of an industry when there is both aggregate demand uncertainty and idiosyncratic uncertainty about the productivity of an individual firm's technology. We characterize the intertemporal evolution of the distribution of firms where firms are distinguished by the productivity of their technology. We contrast industry dynamics across different demand state histories, along a particular demand state history, and detail how anticipation of future demand shocks affects exit decisions and the industry's evolution. The theoretical predictions on cyclical patterns in exit and productivity offer a coherent explanation for the cyclical patterns exhibited in the data.

Appendix

* Proof of Theorem 1. Fix the distribution [mu] on [0, i ] and let [mu](d [alpha]) = g([alpha]) d[alpha]. If firms with technology lower than [alpha] exit, then let [[mu].sub.[alpha]] denote the truncated distribution. If the price is p, the supply of firm [alpha] is y(p, [alpha]) and total supply is [Y.sub.s],(p, [[mu].sub.[alpha]]) = [[integral].sup.1.sub.[alpha]] y(p, [[mu].sub.[alpha]])[mu](d[[mu].sub.[alpha]]). We can invert this to get [p.sub.s](Y, [[mu].sub.[alpha]],). Because the exit threshold uniquely defines the distribution [[mu].sub.[alpha]], we may write [alpha] in place of [[mu].sub.[alpha]], with corresponding supply and inverse supply functions: [Y.sub.s](p, [alpha]) and [p.sub.s](Y, [alpha]).

The current-period social surplus is S([theta], Y, [alpha]) = [[integral].sup.Y.sub.0][p(Q, [theta]) - [p.sub.s](Q, [alpha])]dQ. Maximizing this over Y gives [Y.sup.*] = Y([theta], [alpha]), satisfying p([Y.sup.*], [theta]) - [p.sub.s]([Y.sup.*], [alpha]) = 0, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The variation of the surplus with respect to [alpha] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus,

[partial derivative]S([Y.sup.*], [alpha])/[partial derivative][alpha] = - [pi]([theta], [mu], [alpha])g([alpha]).

Now, let V satisfy the recursion

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[mu].sup.[alpha]](*) = [[integral].sup.1.sub.[alpha]] P(* | [alpha])[mu](d[alpha]) + [[integral].sup.[alpha].sub.0] P(* | [bar.[alpha]][mu](d[alpha]) = g([alpha]). Let v([alpha], [[mu].sup.[alpha]]) be the continuation payoff to a firm with technology [alpha], when the aggregate distribution is [[mu].sup.[alpha]].

Suppose that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When this is O, at say [[alpha].sup.*],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

So, the exit threshold that maximizes the social surplus is the technology threshold at which a firm is indifferent between exiting and remaining in the market.

Remark. The calculations below confirm that the differential condition used above is satisfied recursively.

Put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [mu](d[alpha]) = g([alpha]) d[alpha]. Then

[mu][([gamma]).sup.[alpha](*) = [[integral].sup.1.sub.[alpha]] P(* | [alpha])[mu]([gamma])(d[alpha]) + [[integral].sup.[alpha].sub.0]] P(* | [bar.[alpha]])[mu]([gamma])(d[alpha]).

Collecting terms,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Rearranging,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this case,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Or,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Turning to the first period, the distribution is given by [mu]([gamma]). The impact of a variation in [gamma] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Substituting for [mu]([gamma]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This is equal to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Collecting all terms,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Rearranging positive and negative terms, this yields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, the derivative of the valuation function preserves the difference of valuation between a new firm and a continuing firm.

The discussion in Theorem 2 below utilizes properties of the class of distributions generated by the truncation procedures used in the article. The following lemma establishing that with the weighted-average kernel transition, either increasing the measure of exiting firms, that is, increasing the exit threshold [[alpha].sup.*] for a fixed distribution [mu], or improving the current distribution over firm technologies [mu] for a fixed exit rule [alpha], leads to an improved distribution of firm technology qualities in the next period. This result is used subsequently in the proof of Theorem 2.

Lemma 1. The measure [[mu].sup.[alpha]*] (*) satisfies:

(i) Raising the exit threshold improves next period's distribution:

[[??].sup.*] [??] [[alpha].up.*] implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(ii) For a given exit threshold, an improvement in the current distribution improves next period's distribution,

[??], [mu] [member of] W(F,G) = {[mu] | [there exists] [beta] [member of] [0, 1], [mu] = [beta]F + (1 - [beta])G}, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. We first show that for [mu] = [beta]F + (1 - [beta]) G, [mu]' = [beta]' F + (1 - [beta]') G, with [beta]' > [beta]. Then F [??] G implies [mu]' [??] [mu]. To see this, it is sufficient to show that [[phi].sub.[beta]]([alpha]) is increasing in [beta] for each [alpha] [??] [[alpha].sup.*], where

[[phi].sub.[beta]]([alpha]) = [mu]([[alpha], 1] | [alpha] [greater than or equal to] [[alpha].sup.*]) = [mu]([[alpha], 1])/[mu]([[[alpha].sup.*], 1]).

Now,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where k([??]) = [F([[??], 1]) - G([[??], 1])] and d([??]) = G([[??], 1]). Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The numerator is

F([[alpha], 1])G([[[alpha].sup.*], 1]) - G([[alpha], 1])G([[[alpha].sup.*], 1]) - F([[[alpha].sup.*], 1])G([[alpha], 1]) - G([[[alpha].sup.*]])G([[alpha], 1])].

Canceling G([[alpha], 1]))G([[[alpha].sup.*], 1]) gives the numerator as

F([[alpha], 1])G([[[alpha].sup.*], 1]) - F([[[alpha].sup.*], 1])G([[alpha], 1]) = F([[[alpha].sup.*], 1]) G([[[alpha].sup.*], 1])[F([[alpha], 1])/F([[[alpha].sup.*], 1]) - G([[alpha], 1])/G([[[alpha].sup.*], 1]).

Thus, because [F([[alpha], 1])/F([[[alpha].sup.*], 1]) - G([[alpha], 1]/G([[[alpha].sup.*], 1])] [greater than or equal to] 0 from conditional first-order stochastic dominance, [mu]' [??] [mu]. With this preliminary result in hand, consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] gives the weight on F(*) in the distribution [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].