Industry dynamics with stochastic demand.

Theorem 3. Consider two period t distributions, [[mu].sub.t] and [[??].sub.t], over firm technologies. Suppose that [[mu].sub.t] [??] [[mu].sub.t] implies that [[mu].sub.t+1] [??] [[mu].sub.t+1]. Then along a common demand path, for [tau] [greater than or equal to] t, output is higher in the hat economy, [[??].sub.[tau]] > [Y.sub.[tau]], so that prices are lower, [[??].sub.[tau]] < [p.sub.[tau]], and the profit of each firm type [alpha] is lower, [??][([alpha]).sub.[tau]].

Proof By assumption, an improvement in the distribution at time t ([mu] [up arrow] [mu]')improves next period's distribution (taking into account the resultant change in the current exit rule), and increases current period's output. Proceeding inductively, the improvement in next period's distribution leads to an increase in that period's output and an improvement in the distribution for the following period. And so on. Thus, the initial distribution in every period (prior to the exit decision) is better, and the output in every period greater. The patterns for prices and profits follow.

Thus, improved competition raises current output with associated consequences for prices and the profitability of different firm types. (12) Improved competition may raise exit, but not by enough to offset the output consequences of the improved firm quality. Again, Bergin and Bernhardt (2005) show that more structure is required if firms choose capital prior to demand and technology shock realizations. This is because firms respond to a worse distribution over technologies by increasing capital, and capital and technology productivities have distinct impacts on output and exit decisions. However, Bergin and Bernhardt show that analogous results follow if demand is sufficiently elastic.

[] Cyclical fluctuations. The previous discussion considered the impact of different initial distributions of firms for productivity, output, price, and profitability along a given demand path, [{[[theta].sup.[tau]-1]}.sub.[tau][greater than or equal to]t]. We next study the impact of fluctuations in [theta] on exit decisions, output, and profitability of firms, characterizing the consequences of demand shocks for current and future output, and current and future aggregate productivity. The analysis is complicated by the fact that when demand improves, there are competing influences on a firm's decision to exit: operating is more profitable, but the firm is also worth more if sold. That is, improved demand raises incentives to remain in the market which would reduce the average efficiency of firms in the market next period. However, the persistence of improved demand also raises the benefit to having a better technology next period, producing a "counter-incentive" for inefficient firms to exit.

Cyclical exit. We first derive conditions under which, ceteris paribus, downturns in demand induce more (unproductive) firms to exit (worsening the immediate effect of the downturn), to be replaced by firms that are stochastically better. This result is combined with Theorem 2 to show that worse current demand conditions always imply better future distributions of firm technologies. That is, not only do recessions have the cleansing effect of weeding out more firms with unproductive technologies, but past recessions also reduce the number of unproductive technologies at each future date and state. Exit decisions thus mitigate the impact of a prolonged downturn in demand.

The equilibrium exit threshold, [[alpha].sup.*], is determined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and rearrange the expression to give

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

With [[alpha].sup.*] fixed, raising the current demand state raises current profit, but if there is persistence in demand, raising the current demand state also makes higher future demand states more likely, raising the relative future payoff from exit (and being replaced by a stochastically better technology). Whether [[alpha].sup.*] rises with [theta] (i.e., whether there is less exit), then depends on which rises more.

Inspection reveals that the value of remaining in the industry rises more rapidly with [theta] than does the value of exit, as long as there is sufficient mean reversion in the demand process. Most transparently, this is so if the demand process is sufficiently close to being independently distributed. (13) For a broad class of production technologies, it tunas out that there is always "enough" mean reversion in any two-state Markov demand process for the relative value of remaining in the industry to rise with demand.

Specifically, we now focus attention on technologies that give rise to multiplicatively separable profit functions, so that [pi]([alpha], p([mu], [theta])) = g(p)h([alpha]) for some functions g and h. This class of production functions includes such standard production technologies as the CES with [alpha] entering multiplicatively, that is, [pi](l, k, [alpha]) = m([alpha])[bar.f](l, k), and quadratic production functions of the form f(l, [alpha]) = al - b/2[alpha]l2.

We consider a general two-state stationary Markov demand process where [theta] [member of] {[bar.[theta]], [[theta].bar]} with [[theta].bar] [less than or equal to] [bar.[theta]] and let Pr([theta] = [bar.[theta]]|[bar.[theta]]) [equivalent to] [rho] and Pr([theta] = [[theta].bar]|[[theta].bar] [equivalent to] [phi]. That is, [rho] and [phi]b capture the extent of persistence of good and bad states: higher values of [rho] and [phi] mean greater persistence in the good and bad states, respectively. Although we do not impose it, presumably [rho] [greater than or equal to] 1 - [phi]: future demand is more likely to be high when current demand is high.

If [rho] or [phi] equals 1, the state [bar.[theta]] or the state [[theta].bar] is absorbing, that is, once reached, it is never left. Then, with the state constant over time, the aggregate distribution converges to [[mu].sub.[infinity]], the state-independent, long-run aggregate distribution. This follows because with price constant and the multiplicative structure of profits, the exit threshold is independent of the constant state of demand. Given ([rho], [phi]), define M to be the set of possible aggregate distributions when the initial distribution is [[mu].sub.[infinity]]: [??] [member of] M means that there is a time t, and a finite history of aggregate demand realizations, {[[theta].sub.tau]}.sup.t.sub.[tau]=1, such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where for any distribution [??], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denotes the aggregate distribution t periods later following demand realization sequence {[[theta].sub.tau]}.sup.t.sub.[tau]=1, starting from the distribution [??].

Let [bar.[mu]] and [[mu].bar] be, respectively, the distributions of firm qualities induced by an arbitrarily long sequence of [theta] = [[theta].bar] and [theta] = [bar.[theta]] realizations starting from [[mu].sub.[infinity]], the unique stationary distribution when demand is independent of [theta]. Equivalently, as the proof of Theorem 4 shows, these distributions can be identified with the upper and lower bounds of the set M: because distributions over technologies are totally ordered, we can think of [[mu].bar] and [bar.[mu]] as defining an interval or region of distributions [[[mu].bar], [bar.[mu]]], with [[mu].bar] the lowest and [bar.[mu]] the highest. The next theorem establishes that in this environment, there is more exit if demand is low than if it is high.

Theorem 4. Let profit functions be multiplicatively separable. Then for [theta] [member of] {[theta].bar], [bar.[theta]]}, for any distribution of firm technologies [mu] [member of] [[[mu].bar], [bar.[mu]]], there is more exit if demand is low than if it is high: [alpha]([mu], [theta]) > [alpha]([mu], [bar.[theta]]). Furthermore,

(i) The interval [[mu].bar], [bar.[mu]] is absorbing in the sense that given any initial distribution [mu]' [member of] [[mu].bar], [bar.[mu]], for and any sequence ([[theta].sub.1], [[theta].sub.2], ..., [[theta].sub.t], if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [mu]" [member of] [[mu].bar], [bar.[mu]] and

(ii) For any [??] [not member of] [[[mu].bar], [bar.[mu]]],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof See the Appendix.

Thus, if the technology distribution is in [[[mu].bar], [bar.[mu]]], exit is greater in the low than the high state. Over time, the distribution moves within but never leaves the set, regardless of the history of demand realizations. Furthermore, starting from any distribution outside this set, with probability 1 a sequence of demand shocks will occur that moves the aggregate distribution into this set.