Industry dynamics with stochastic demand.

In the proof, we show that the exit rule does not depend on the initial demand realization if [rho] = [phi] = 1, that is, if demand is perfectly persistent, so that if, for example, [[theta].sub.t] = [bar.[theta]], then [[theta].sub.[tau]] = [bar.[theta]], for all [tau] > t. In this case, the "demand" realization enters current and continuation profits in the same multiplicative way. The persistence in the Markov demand process, that is, the values of [rho] and [phi], does not affect current profits, but it does affect future expected profits. When the current demand is low, reducing [phi] raises the probability of a future high-demand state and hence raises future expected prices. This raises the value of a better technology in the future, and hence the attraction of exit. Conversely, reducing [rho] when the current demand state is high reduces future expected prices, reducing the value of a better future technology, lowering the relative value of exit. Combining these observations reveals that there is more exit when demand is low than when demand is high, when the aggregate distribution lies in the interval [[[mu].bar], [bar.[mu]]]. The second part of the theorem notes that when the aggregate distribution is in the interval [[[mu].bar], [bar.[mu]]], then regardless of the subsequent pattern of demand realizations, the aggregate distribution stays in this interval. And, given any [epsilon], if the aggregate distribution is initially outside [[[mu].bar], [bar.[mu]]], then there is a fixed length of time t such that there is probability at least (1 - [epsilon]) of being in this interval within t periods and hence there is probability 1 that the aggregate distribution will lie in this set. Remark 2. If we add the assumption that demand is multiplicatively separable, p(Y, [theta]) = h([theta]) p(Y), then one can prove that exit is counter-cyclical for all distributions [mu]. This result is developed in the Appendix following the proof of Theorem 4.

Contemplation of the two-state Markov demand process indicates how to construct a counterexample to exit being counter-cyclical. Suppose there were three demand states, [[theta].sub.H] > [[theta].sub.M] > [[theta].sub.L], where if current demand is high, it remains high forever, and if current demand is low, it remains low forever, but if [theta] = [[theta].sub.M], then in the next period, demand will rise to [[theta].sub.H] (and remain high, thereafter). Then exit is the same when demand is high and low, but when current demand is intermediate, [theta] = [[theta].sub.M], exit is greater, as firms anticipate that future demand and hence prices will be higher.

Output, price, and profit movement. The implications of Theorem 3 (better distributions are preserved along a demand path, implying lower prices) and Theorem 4 (exit is counter-cyclical) may be combined to characterize the evolution of key variables as a demand state persists for a longer period of time.

Theorem 5. Let profit functions be multiplicatively separable. Then for [theta] [member of] {[[theta].bar], [bar.[theta]]}, for [[mu].sub.t-1] [member of] [[[mu].bar], [bar.[mu]]]:

(i) Aggregate output is higher in period t if demand is high than if it is low: [Y.sub.t]([bar.[theta]], [mu]) > [Y.sub.t], [[theta].bar], [mu]), [for all][mu].

(ii) If a period of high demand begins in period t and ends [tau] + 1 periods later, then aggregate output first rises and then falls, [Y.sub.t-1] < [Y.sub.t] > [Y.sub.t+1] > ... > [Y.sub.t+1[tau]+1], so that prices rise throughout: [p.sub.t] < [p.sub.t+1] < ... < [p.sub.t+[tau]. Hence, the output and profits of a firm of type [alpha] rise: [y.sub.t-1]([alpha]) < [y.sub.t]([alpha]) < ... < [y.sub.t-[tau]]([alpha]) and [[pi].sub.t-1]([alpha]) < [[pi].sub.t]([alpha]) < ... < [[pi].sub.t+[tau]]([alpha]).

(iii) If a low demand downturn begins in period t and ends [tau] + 1 periods later, then [Y.sub.t-1] > [Y.sub.t] < [Y.sub.t+1] < ... < [Y.sub.t+1[tau]+1] so that prices fall as the downturn continues: [p.sub.t-1] > [p.sub.t] > ... > [p.sub.t+[tau]]. Hence, the output and profits of a firm type [alpha] fall as the downturn continues: [y.sub.t- 1]([alpha]) > [y.sub.t]([alpha]) > ... > [y.sub.t+[tau]]([alpha]) and [[pi].sub.t- 1]([alpha]) > [[pi].sub.t]([alpha]) > ... > [[pi].sub.[tau]]([alpha]).

Proof Part (i) follows because there is less exit when [theta] = [bar.[theta] Part (ii) follows because the distribution worsens over time as the boom progresses (see the proof to Theorem 4), implying that output falls (Theorem 3). The implication of falling output, for prices, firm output, and profits is immediate. Part (iii) is the downturn analogue to Part (ii).

A more prolonged period of high demand leads to an increasingly inefficient distribution of firms, which, in turn, implies rising prices and hence higher individual output and profits. It is worth stressing that this result holds no matter how persistent the demand process is. With two demand states, {[bar.[theta]], [[theta].bar]}, in a demand boom, if viewed as a growth in demand occurs over one time period, output rises during the growth stage (as in [ii] above) and subsequently tails off due to the worsening distribution of firm productivities as demand remains high.

Importantly, were we to modify the model to incorporate an exogenous systematic growth in technological opportunities, then the model can generate predictions consistent with the empirical regularity that recessions are sharper and more asyrmnetric than booms. In particular, if, in periods of high demand, the exogenous improvement in the distribution of technology productivities is enough to offset the increasingly inefficient (endogenous) distribution of firm productivities, then output grows gradually as long as demand remains high. Conversely, any systematic improvements in the distributions from which technologies are drawn sharply reinforces the endogenous improvement in firm productivities caused by a downturn in demand, so that recessions are short.

Also note that incorporating stickiness into capital investment, either with a capital-in-place formulation or by modelling the putty-clay feature of capital, prolongs the industry response to downturns in demand. Bergin and Bernhardt (2005) show that firms respond to a decrease in demand by first downsizing, and then subsequently exiting. Consequently, output may fall for multiple periods following the onset of a period of low demand.

Combining the insights of Theorems 3 and 5, it follows immediately that past downturns lead to greater output, once demand has "recovered". Consider two economies that differ only in that one economy had past periods with lower demand realizations. Say that demand has recovered in the economy that had lower demand realizations if both economies have the same current demand realization. Then:

Corollary 1. Consider two economies with identical current demand, but one economy had weakly lower demand in past periods. Then the economy which has recovered from a past history of lower demand has a better distribution of firms, greater output, and lower prices.

Qualitatively, given Theorem 3, the key to the corollary is that there is more exit in bad times. That is, as long as exit rates are counter-cyclical--[theta] < [theta]' then [alpha]([mu], [theta]') > [alpha]([mu], [theta]')--(as Campbell, 1998 documents empirically),.... then the result holds without further structure on the [theta] process.

We have shown that exit is counter-cyclical in a two-state separable environment. So, too (without imposing any other structure), exit is also counter-cyclical as long as demand is not too persistent. As long as exit is counter-cyclical, we can contrast industry dynamics across economies that start out with the same distribution of firms, but one experiences higher aggregate demand states than the other.

Theorem 6. Suppose that exit is counter-cyclical. Let [[??].sub.0] = [[??].sub.0]. Consider two aggregate demand histories, [[??].sub.t], [bar.[theta].sub.0], where the hat economy has higher demand realizations than the bar economy: [[??].sub.0] > [bar.[theta].sub.0], [[??].sub.[tau]] [greater than or equal to] [bar.[theta].sub.[tau]], 0 [less than or equal to] [tau] [less than or equal to] t. Then past lower demand:

(i) Leads to better distributions of firms: [[??].sub.[tau]+1] > [[??].sub.[tau]+1], 0 [less than or equal to] [tau] [less than or equal to] t.

(ii) Raises future output, thereby reducing future prices and firm profits: [bar.[Y.sub.[tau]] > [[??].sub.[tau]], [bar.p.sub.[tau]] < [[??].sub.[tau]], and

[bar.[pi].sub.[tau]]([alpha]) < [[??].sub.[tau]]([alpha]), 0 [less than or equal to] [tau] [less than or equal to] t.

Proof Result 1 follows immediately from counter-cyclical exit and Theorem 2. Consider the first period r in which [[??].sub.[tau]] > [bar.[theta].sub.[tau]]. Then, counter-cyclical exit ensures the higher- demand state implies less exit, and hence a worse distribution of firms. From Theorem 2, even if subsequent demand states are identical, the worse distribution is preserved, and subsequent higher-demand realizations reinforce the result.

Result (ii) follows because as the distribution of firms on the "bar" path is better than the distribution on the "hat" path, price will be lower on the "bar" path unless there is (substantially) more exit in the "bar" economy. So suppose that [bar.[alpha].sub.t] > [??].sub.t]. But then,

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a contradiction. The implications for output and firm profits are immediate.