Industry dynamics with stochastic demand.

Here, for example, [[bar.[mu].sub.[tau]], is the distribution on technology along the [bar.[theta]] sequence. Weak demand has the immediate effect of inducing more inefficient firms to exit, which leads to an improved distribution of firm efficiencies, which persists at all future dates (i). The higher productivity caused by a past downturn implies greater future output once demand recovers sufficiently (ii), illustrating the cleansing effect of recessions for future output. Indeed, if there is sufficiently little persistence in demand, then a lower current-demand state must raise expected future discounted total surplus. In turn, this increased future competition implies lower future prices and hence lower profits for a firm of a given technology quality. The greater the stress of a demand downturn (i.e., the lower are the demand realizations and the longer the lower-demand realizations persist), the more fit are the survivors, and hence the more productive is the entire industry. These results are consistent with the findings in the empirical literature that correlations of exit rates with future GDP growth are positive, large, persistent, and statistically significant (Campbell, 1998).

These theorems state the results in terms of the impact of a more prolonged boom in demand. The results can also be stated in terms of their converse--more prolonged recessions lead to greater output when demand finally improves, and more sustained high demand leads to steeper declines to lower levels of output when demand finally falls.

Remark 3. The focus of the model is on a specific industry: we take factor prices as given. However, it is worthwhile to consider the extent to which the results remain valid were factor prices to vary with market conditions. Firm [alpha]'s period production problem becomes [max.sub.l,k] p([mu], [theta])f(t, k, [alpha]) - w([mu], [theta])l. - r([mu], [theta])k. This determines firm [alpha]'s profit, [pi]([theta], [mu], [alpha]). So, as when factor prices are constant, [alpha]'s profit depends on the same three variables. The feature that the analysis exploits is [partial derivative][pi]/[partial derivative][theta] > 0: better times raise period profits for all firm types. This is unequivocal if [theta] affects only demand. More generally, an application of the envelope theorem reveals that it is true if [partial derivative][pi]/ [partial derivative][theta] = [partial derivative]p([mu], [theta])/ [partial derivative][theta] x f(l, k, [alpha]) - [partial derivative]w([mu], [theta])/ [partial derivative][theta]l - [partial derivative]r([mu], [theta])/[partial derivative][theta]k > 0, [for all][alpha]. A sufficient condition for this to be so is that technologies be Cobb Douglas or quadratic, so that the profit function is a multiplicatively separable function of the firm's productivity parameter: if an increase in [theta] helps one firm type, it helps all firm types, and with a Cobb-Douglas technology, f(l, k, [alpha]) = [alpha] [k.sup.b] [l.sup.c], it does so if p/[w.sup.c][r.sup.b] rises with [theta]. The theorems all extend to general equilibrium if, in addition, (a) factor prices (weakly) increase with industry input demand (so reduced past exit further raises period profits), and (b) reinterpreting the demand states as industry period profit states (that take into account the equilibrium impacts on factor prices), there is persistence in the [theta] profit states. Theorem 2 extends because worse distributions of firms, ceteris paribus, imply lower factor prices. In turn, persistence in profit states, plus counter-cyclical exit, ensures that the other theorems extend, including a generalized version of Theorems 4 and 5 (given counter-cyclical exit holds).

Remark 4. In our model, all entry takes place through exit and renewal: there is no greenfield entry. Because we assume that the type of a new entrant does not depend on the exiting firm's type, the entry-exit process is driven by an active firm's decision to exit. With a fixed measure of possible firms, there is obviously a tight link between past exit and future entry, making our model best suited for characterizing exit. Even within our model, one can gain insights into entry by interpreting entrants as those firms in their first period of production--so that a "new" firm that does not produce in the current period (because its technology draw was poor) is not an entrant, severing the perfect link between exit and entry. With this interpretation, we predict that entry is greater when the economy leaves a recession (as is found in the data), both due to the greater exit in the past downturn and because the improved demand means that a greater fraction of potential entrants find it optimal to produce.

To address properly and distinctly both entry and exit, one needs to allow for greenfield entry--entry "by birth," so that a potential entrant can enter by creating a new firm or by taking over an existing one. Both forms of entry are important. In Canadian manufacturing, Baldwin and Gorecki (1987) found that entrants after 1970 accounted for 26.2% of total sales by 1979. Those that entered by creating a firm accounted for 14% of total sales, whereas those that entered by acquiring another firm accounted for over 12% of total sales. Modifying our model to accommodate entry through the creation of new firms would appear to reinforce the counter-cyclical movement in the distribution of firm quality. With demand persistence, a high demand state increases the expected payoff of an entrant, and hence raises entry. Because firm productivities stochastically rise with age, the immediate impact of such greenfield entry would be to worsen the distribution of productivities of operating firms, reinforcing our results. We do not consider greenfield entry solely because it complicates characterizations of the distributions of firm productivities at arbitrary future dates and states.

Even without greenfield entry, there are two useful notions of joint exit and entry--net entry (gross entry minus gross exit) and gross turnover (gross exit plus gross entry). Bilbiie, Ghironi, and Melitz (2007) establish that net entry is pro-cyclical (i.e., there are more firms in good times than in bad). Our results on exit immediately imply that net entry should rise at the onset of a boom and fall at the beginning of a downturn. Further, the predicted qualitative magnitude is increased when we interpret entrants only as new firms that produce, or allow for the possibility that exiting firms must search for potential buyers, and may not uncover them (or, equivalently, there is a matching between buyers and sellers). (14) This pattern in net entry would also seemingly be enhanced further were we to integrate the greater greenfield entry observed in booms. Davis and Haltiwanger (1990, 1992) establish that gross turnover is counter-cyclical despite the fact that net entry is pro-cyclical. In our model, greater past exit drives greater entry activity, and this "double counting" can deliver the counter-cyclical gross turnover despite the greater greenfield entry observed in booms.

[] Dynamics and expectations. We next consider the impact of an anticipated future increase in demand on current exit decisions and hence future distributions of firm productivities. We show that, ceteris paribus, better-anticipated future market conditions lead to more exit at all earlier dates. Take a transition kernel [THETA](* | [theta]) that exhibits persistence, so that [theta]' > [theta] implies that [THETA](* | [theta]') first-order stochastically dominates [THETA](* | [theta]). Now, fix a future point in time, T, and consider the impact of replacing the transition kernel at that date with one of the alternative transition kernels, [bar.[THETA]]([[theta].sub.T] | [[theta].sub.T-1]) or [??]([[theta].sub.T] | [[theta].sub.T-1]), where [bar.[THETA]]([[theta].sub.T] | [[theta].sub.T-1]) > [??]([[theta].sub.T] | [[theta].sub.T-1]) for each [[theta].sub.T-1]--Then at date T - 1, [bar.[THETA]])([[theta].sub.T] | [[theta].sub.T-1]) represents the expectation of better demand than [??]([[theta].sub.T] | [[theta].sub.T-1]--an anticipated increase in future demand relative to [??].

Theorem 7. Suppose that exit is counter-cyclical and consider two economies at date T - 1 that differ solely in anticipated future demand conditions at date T,

[bar.[THETA]]([[theta].sub.T] | [[theta].sub.T-1]) > [??]([[theta].sub.T] | [[theta].sub.T-1]), [for all] [[theta].sub.T-1],

where for t > T, the two economies have a common transition kernel [THETA]([[theta].sub.t] | [[theta].sub.t-1]) that exhibits persistence. Then, for any given demand state [[theta].sub.T-1] and distribution [[mu].sub.T-1], there is more exit in the bar economy with better anticipated future demand, so that so date T - 1 prices are higher in the bar economy and the distribution of productivities at date T is better, [bar.[mu].sub.T] > [[??].sub.T].

Proof. Because exit is counter-cyclical, for a fixed [mu], a higher demand draw [theta] implies higher prices at current and future dates and states--higher current demand makes higher future demand states more likely, and counter-cyclical exit implies that the distribution of firms is worse. Hence, integrating at date T - 1 over date T continuation payoffs, we have

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