Industry dynamics with stochastic demand.

Thereafter, demand realizations are drawn from the same distribution, so [bar.v].sub.T](x) = [[??].sub.T](x). For any given [[mu].sub.T-1], current profits for any marginal exiter [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the same in both the hat and bar economies, but the value to exit is greater in the bar economy. Hence, for any given [[mu].sub.T-1] there is more exit in the bar economy than the hat economy, and the distribution of firm productivities at date T is better, [bar.[mu].sub.T]([[mu].sub.T-1]) > [[??].sub.T]([[mu].sub.T-1]). In turn, this increased exit in the bar economy implies that for any given [[mu].sub.T-1] and [[theta].sub.T-1], date T - 1 prices are higher in the bar economy than the hat economy.

Hence, anticipation of different future market conditions can generate the positive correlations of exit rates with future GDP growth found by Campbell (1998). The key to Theorem 7 is that an anticipated improvement in future demand does not directly affect current profits; it only raises the future value of having a better technology--as counter-cyclical exit plus demand persistence imply a stochastically better distribution of prices for a given [mu]--and hence the value of exit. In turn, the greater exit at date T - 1 implies prices are higher at that date. One would like to extend this result to dates [tau] < T, but the greater exit at T - 1 provides a countervailing impact on prices from the perspective of those dates. That is, at date T - 1, from the perspective of the marginal exiter in the hat economy, the prices in the bar economy would be higher, and so there must be more exit in the bar economy, and whereas the value function at date T - 1 of this marginal exiter must be higher in the bar economy, we cannot extend this result to earlier dates to order the value functions of all [alpha].

4. Thin resale markets

* The discussion so far has not considered how industry dynamics vary across industries distinguished by different depths of the market for the plants of exiting firms. That is, how do variations in [gamma] affect outcomes? The degree of plant specialization varies significantly across industries. We now provide conditions under which the smaller is [gamma], that is, the stronger an exiting firm's bargaining position is with a buyer, the greater is exit. Finally, we show by construction that when [gamma] > 0 so that the social and private incentives to exit do not coincide, then not only can a recession weed out "bad" firms but a downturn in demand can actually raise welfare (total surplus) by inducing more efficient exit.

The direct effect of a thinner market is to reduce exit. To see this, suppose there is a one-time increase in thinness in the market for a bankrupt firm's plant, so that [gamma] increases to [gamma]', an increase that is unanticipated at earlier dates. This increase to [gamma]' reduces the value of exit relative to the value of remaining in the industry, leading to a worse distribution of firms at all future dates.

The net effect of a thinner resale market is more subtle because of the indirect effect of reduced exit on future competition. If [gamma] < [gamma]' it does not follow that for all ([theta], [mu], [alpha]), that [v.sub.[gamma]]([theta], [mu], [alpha]) > [v.sub.[gamma]]'([theta], [mu], [alpha]). Whereas sufficiently unproductive firms are worse off facing [gamma]', because bankrupt firms sell their plants for less, the indirect consequence of reduced exit is that the future distribution of firms is worse. In turn, this reduced future competition implies higher future prices, so that high [alpha] firms that are unlikely to enter bankruptcy may be helped if firms are more reluctant to exit. Consequently, for [gamma] < [gamma]', it may not be that [v.sup.c.sub.[gamma]]([theta], [mu], [alpha]) - [v.sup.e.sub.[gamma]]'([theta], [mu], [alpha]) > [v.sup.c.sub.[gamma]]'([theta], [mu], [alpha])-- [v.sup.e.sub.[gamma]]'([theta], [mu], [alpha]) for all ([theta], [mu]). Yet, it must be true for some ([theta], [mu]), else the future distribution of firms would always be better in the [gamma]' economy, so that both the direct and indirect effects would encourage more exit in the [gamma]' economy, a contradiction of the premise. To highlight dependence of the equilibrium exit rule on [gamma], write [[alpha].sup.[gamma]] ([theta], [mu]). If there is no aggregate uncertainty, so that p(Y, [theta]) = p(Y) (and [[alpha].sup.[gamma]]([theta], [mu]) becomes [[alpha].sup.[gamma]]([mu])), then one can derive the effects of an increase in [gamma] on industry dynamics.

Theorem 8. Suppose there is no aggregate uncertainty, so that p(Y, [theta]) = p(Y). Then, for all [gamma], given [[mu].sub.0], there is a unique equilibrium. The distributions of firm productivities converge monotonically over time to the unique stationary equilibrium, [[mu].sup.[gamma].sub.[infinity]] along the sequence, either [[mu].sup.[gamma].sub.t] [greater than or equal to] [[mu].sup.[gamma].sub.t-1], [for all]t or [[mu].sup.[gamma].sub.t- 1] [greater than or equal to] [[mu].sup.[gamma].sub.t], [for all]t. In the limiting stationary economies, for [gamma] > [gamma]',

(i) Firms are less willing to exit in the economy with a thinner resale market, [[alpha].sup.[gamma]] ([[mu].sup.[gamma].sub.[infinity]]) < [[alpha].sup.[gamma]'] ([[mu].sup.[gamma].sub.[infinity]]) < [[alpha].sup.[gamma]'] ([[mu].sup.[gamma]'.sub.[infinity]])

(ii) The limiting distribution is better in the economy with the thicker resale market, [[mu].sup.[gamma]'.sub.[infinity]] > [[mu].sup.[gamma].sub.[infinity]].

(iii) Any given [alpha] operates at a larger and more profitable scale in the economy with thinner markets.

The characterization of the evolution of the distribution of firms to the invariant steady state extends Hopenhayn (1992a)-who analyzes the behavior of individual firms in the limiting economy. In particular, this theorem shows that the economy does not cycle over time as it approaches the limiting steady state.

If [gamma] is interpreted as reflecting the degree to which inputs are specialized in an industry, then inter-industry comparisons can be made. Industries with more specialized inputs will tend to have more unproductive, but larger, firms and lower rates of exit. This is consistent with the finding of Dunne et al. (1989a) and Asplund and Nocke (2006) that substantial and persistent differences in entry and exit rates across industries exist, and that industries with higher entry rates also have higher exit rates. (15)

Further insights on the effects of thin markets for a bankrupt firm's plant can be gleaned from the polar case of perfectly elastic demand, p(Y, [theta]) = p([theta]). The qualitative findings hold as long as demand is "sufficiently" elastic. If p(Y, [theta]) = p([theta]), then [v.sub.[gamma]]([theta], [mu], [alpha]) = [v.sub.[gamma]]([theta], [??], [alpha]), [for all][mu], [??]: only the direct negative effect of [gamma] on exit remains. When p(Y, [theta]) = p([theta]), the marginal exiting firm equates

[pi]([theta], [[alpha].sup.*]([theta)) = [beta] [integral] [v.sub.[gamma]]([??], [??])[THETA](d[??]|[theta])[(1 - [gamma])P(d[??]|[bar.[alpha]]) - P(d[??]|[[alpha].sup.*])].

Because the right-hand side is decreasing in y, it follows immediately that a reduction in y raises exit, and hence the distribution of firm technology qualities. Individual firms exit according to their private incentives, 1 - [gamma], so there is too little exit from a total surplus maximization perspective. The exit decision corresponds to the exit choice made by a social planner when the exiting firm is destroyed with probability [gamma]; the fact that next period's distribution of firms does not reflect this destruction is irrelevant because the distribution does not affect a firm's payoffs. The greater is [gamma], the greater is the wedge between equilibrium exit and efficient exit.

We now show constructively that when [gamma] > 0, so that firms are too reluctant from a surplus-maximizing standpoint to exit, downturns in demand can increase welfare by inducing inefficient firms to sell their plants to firms that could use them more productively. Because downturns in demand raise the attraction of exit, they induce more inefficient firms to act in accordance with maximizing social surplus, so that privately optimal actions are more closely aligned with socially optimal actions.

Example. Let technologies be given by [alpha]f(l), where [alpha] [member of] {[alpha].bar], [??]}, [alpha].bar] < [??]. Suppose that the productivity of a firm's technology does not vary with time, [alpha].sub.t] = [alpha].sub.t-1], and that entrants are equally likely to draw each technology: Pr([alpha] = [??]|entry)= .5. Demand is perfectly elastic: p(Y, [theta]) = [theta], [theta] [member of] {[[theta].bar], [bar.[theta]]}, [[theta].bar] < 0, and demand states are i.i.d, over time. Markets for a bankrupt firm's plant are thin: 1 > [gamma] > 0. The discount factor is [beta] > .5. There are as many good firms as bad.