Industry dynamics with stochastic demand.

To prove (i), note that [[alpha].sup.*] < [bar.[alpha]], and because w is monotone increasing, the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is constant and equal to w([bar.[alpha]]) on [0, [[alpha].sup.*]), has a "downward jump" at [[alpha].sup.*] (from w([bar.[alpha]]) to w([[alpha].sup.*])), and is monotone increasing on [[[alpha].sup.*], 1]. If [bar.[alpha]] [greater than or equal to] [[??].sup.*] [greater than or equal to] [[alpha].sup.*], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus (i) holds. Now we prove (ii).

Let [mu] = [beta]F(*) + (1 - [beta])G(*). We need to show that as [beta] increases, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] increases. Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Using stochastic dominance ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is increasing on [[[alpha].sup.*], 1]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Provided [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is increasing in [beta]. A sufficient condition for this to be so is that [E.sub.F]{w} [greater than or equal to] w([bar.[alpha]]). But [integral] P(*, [alpha])P(d[alpha], [bar.[alpha]]) [??] P(*, [bar.[alpha]]) implies that [E.sub.F]{w} [greater than or equal to] w([bar.[alpha]]). To see this expand both sides of [integral] P(*, [alpha]) P(d[alpha], [bar.[alpha]]) [??] P(*, [bar.[alpha]]),

[[integral] w([alpha])P(d[alpha], [bar.[alpha]])]F(*) + [1 - [integral] w([alpha])P(d[alpha], [bar.[alpha]])]G(*) [??] P(* | [bar.[alpha]]) = w([bar.[alpha]])F(*) + [1 - w([bar.[alpha]])]G(*).

Because [ingegral] w([alpha])F(d[alpha]) > [integral] w([alpha])P(d[alpha], [bar.[alpha]]),

[[integral] w([alpha])F(d[alpha])]F(*) + [1 - [integral] w([alpha])F(d[alpha])]G(*) [??] [[integral] w([alpha])P(d[alpha], [bar.[alpha]])]F(*) + [1 - [integral] w([alpha])P(d[alpha], [bar.[alpha]])]G(*),

so the term on the left dominates w([bar.[alpha]])F(*) + [1 - w([bar.[alpha]])]G(*), implying that [integral] w([alpha])F(d[alpha]) = [E.sub.F]{w} [greater than or equal to] w([bar.[alpha]]).

Proof of Theorem 2. We prove the result inductively. Let [v.sup.c.sub.n] ([theta], [mu], [alpha]) be the payoff (present value) to agent [alpha] in an n-period problem, when the current aggregate state is [theta], the current aggregate distribution is [mu], and the agent chooses to stay in the market with n periods remaining. Similarly, [v.sup.e.sub.n] ([theta], [mu], [alpha]) is the payoff to agent [alpha] when the current aggregate state is [theta], the current aggregate distribution is [mu] and the agent chooses to exit the market with n periods remaining. The maximum of these functions is [v.sub.n]([theta], [mu], [alpha]) = max {[v.sup.c.sub.n] ([theta], [mu], [alpha]), [v.sup.c.sub.n] ([theta], [mu], [alpha])}.

At n = 1, they are defined: [v.sup.c.sub.1]([theta], [mu], [alpha]) = [pi]([theta], [mu], [alpha]) and [v.sup.e.sub.1]([theta], [mu], [alpha]) = 0. From the properties of the profit function, these are continuously decreasing in [mu]. Now, assume that the result holds for n - 1: both [v.sup.c.sub.n-1]([theta], [mu], [alpha]) and [v.sup.e.sub.n-1]([theta], [mu], [alpha]) are continuous and decreasing in [mu]. This implies that [v.sup.*.sub.n-1] ([theta, [mu], [alpha]) is decreasing in [mu].

Consider the variation [mu] [up arrow] [??]. The impact effect of this (with no change in the exit rule) is to lower both current profit and future expected profit:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

There are two cases to consider: (1) [pi]' < [[integral]'.sub.[??][??]] and (2) [pi]' > [[integral].sub.[??][??]].

Case 1. [pi]' < [[integral]'.sub.[??][??]].

In Case 1, to restore equilibrium, we require that [pi]' [up arrow] and [[integral]'.sub.[??][??]]. To raise [pi]', raise [[alpha].sub.n]([theta]) (say [[??].sub.n]([theta]) is the exit value that restores equilibrium: continuity of period profits in the exit rule, [[alpha].sub.n]([theta]), follows because output and hence price are, so that there exists such an [[??].sub.n]([theta])). This increase in exit increases the efficiency of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as the distribution moves to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As a result [[integral]'.sub.[??][??]] falls, and this is reinforced by the "fall" in [(1 - [gamma])P(d[??] | [bar.[alpha]]) - P(d[??] | [[alpha].sub.n]([alpha]))] as [[alpha].sub.n]([theta]) [up arrow] [[??].sub.n]([theta]).

These changes result in a continuous fall in [[integral]'.sub.[??][??]] to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], say. Now, write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and from the calculations, [[??].sub.[??][??]] < [[integral]'.sub.[??][??]] [less than or equal to] [[integral].sub.[??][??]]. Write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for the new equilibrium current profit. Now,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

current profit has fallen. Because [[??].sub.n]([theta]) > [[alpha].sub.n]([theta]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Therefore, price is lower and output higher. Also, because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [v.sub.n-1] is monotonic in [mu],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Consequently, [v.sup.c.sub.n]([theta], [??], [alpha]) < [v.sup.c.s.bu.n]([theta], [mu], [alpha]). Similarly, [v.sup.e.sub.n]([theta], [??], [alpha]) < [v.sup.e.sub.n]([theta], [mu], [alpha]). Because both [pi]' and [[integral]'.sub.[??][??]] are continuous in [mu] and [[alpha].sub.n]([theta]), so are [v.sup.c.sub.n]([theta], [mu], [alpha]) and [v.sup.e.sub.n]([theta], [mu], [alpha]).

Case 2. [pi]' > [[integral]'.sub.[??][??]].

In this case, we want to reduce [pi]' and raise [[integral]'.sub.[??][??]]. As [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is too high, reduce [[alpha].sub.n]([theta]) to [[??].sub.n]([theta]) (say, the new equilibrium exit rule), where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This reduces current profit further, with greater output and lower price. As [[alpha].sub.n]([theta]) declines, this reduces the efficiency of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] increases. In addition, [(1 - [gamma])P(d[??] | [bar.[alpha]]) - P(d[??] | [[alpha].sub.n]([theta]))] increases. At the new solution,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

this implies

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

And, because

[(1 - [gamma])P(d[??] | [bar.[alpha]]) - P(d[??] | [[??].sub.n]([theta]))] [greater than or equal to] [(1 - [gamma])P(d[??] | [bar.[alpha]]) - P(d[??] | [[alpha].sub.n]([theta]))],

we get for a set of positive measure of [alpha]'s that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Were it the case that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], this could hold for no [alpha]. Thus, this cannot be the case, and because the set of measures is totally ordered (from the weighted average assumption), the reverse is true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This implies that for all [alpha], [v.sup.c.sub.n]([theta], [??], [alpha]) < [v.sup.c.sub.n]([theta], [mu], [alpha]). Similarly, for all [alpha], [v.sup.e.sub.n]([alpha], [??], [alpha]) < [v.sup.e.sub.n]([theta], [mu], [alpha]). Continuity again follows immediately. Note that in both cases, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof of Theorem 4. Let Pr([theta] = [bar.[theta]]|[bar.[theta]]) = [rho]; Pr([theta] = [[theta].bar]|[[theta].bar] = [phi]. Suppose that [rho] = [phi] = 1, so that for some [theta] [member of] {[[theta].bar], [bar.[theta]]}, [[theta].sub.t] = [[theta].sub.0] = [theta], and the marginal exiting firm, [[alpha].sup.*]([theta], [mu]), is determined by the solution to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

With [rho] = [phi] = 1 the state is constant over time and equal to [theta]. It is straightforward (see Bergin and Bernhardt, 2005) to show that the aggregate distribution converges (monotonically) to some distribution [[mu].sub.[infinity]([theta]): [lim.sub.t[right arrow]] [infinity] [[mu].sub.t] [right arrow] [[mu].sub.[infinity]]([theta]), with associated price sequence, p([[mu].sub.t], [theta]) [right arrow] p([[mu].sub.[infinity]], [theta]). From the multiplicative decomposition of profits, asymptotically, the value functions are multiplicative functions of price, so that the exit rule, [[alpha.sup.*], is asymptotically independent of [theta] (and only relative prices matter). Hence, [[mu].sub.[infinity]]([[theta].bar]) = [[mu].sub.[infinity]]([bar.[theta]]) = [[mu].sub.[infinity]] with associated exit rule [[alpha].sub.[infinity]].