Industry dynamics with stochastic demand.

Other papers assume that all prices are constant, which simplifies the analysis greatly, as the distribution of firms in the economy does not affect an individual firm's profits, and hence decision making. In such settings, Monge (2001) and Cooley and Quadrini (2001) explore the impact of interest rate shocks on the entry and exit of firms and aggregate output. In their models, firms borrow to finance capital, and firm productivities evolve stochastically. Cooley and Quadrini show that their model is consistent with the empirical regularities that smaller/newer firms have higher and more variable growth rates because they are credit rationed, yet they are more likely to exit.

Bergin and Bernhardt (2002, 2005) derive how dynamics are affected by the time- to-build feature of capital investment under the assumption that demand is sufficiently price elastic. By incorporating this feature of capital, they can distinguish between large firms and productive firms, and explain distinctly the size, productivity, and age patterns of profit, output, and exit. Capital in place "slows" exit responses by firms: firms first tend to downsize and then exit in response to low demand or productivity realizations. As a result, the distribution of firm productivities evolves more "sluggishly" in response to demand changes. Relatedly, Lambson (1991) highlights how sunk costs dampen the responsiveness of entry and exit to market conditions.

Finally, Ericson and Pakes (1995), and Pakes and Erickson (1998) develop a parsimonious model of the industry dynamics of a small, imperfectly competitive industry in which firms' investments have stochastic outcomes. In their reduced-form model, a firm's profit depends on the relative success of its investment decisions. Their goal is to develop a flexible framework for empirical work, one that incorporates firm heterogeneity.

2. The model

* We consider the dynamics of a single industry with a continuum of risk-neutral firms that discount future period profits using a common discount factor, [beta] [member of] (0, 1). Inverse demand in a period is given by p(Y, [theta]), where Y is industry output and [theta] is a random demand state. This market price is a continuous function, declining in Y and increasing in [theta], with p(0, [theta]) > 0 and p(*, [theta]) [greater than or equal to] 0. Demand evolves according to a Markov process: given [theta], next period's demand state is drawn from [THETA](* | [theta]). We assume that [THETA](* | [theta]) is continuous in [theta]. (1)

A firm requires a plant to produce output according to the production function f(l, k, [alpha]), where k [greater than or equal to] 0 is capital, l [greater than or equal to] 0 is labor, and a captures the quality or productivity of its technology. Plants are in fixed supply, and without loss of generality, we normalize the measure of plants in the economy to one. (2) The production function, f(l, k, [alpha]), is strictly monotone increasing in its arguments, strictly concave in k and l with complementary inputs, and with f(l, k, [alpha]) = 0 if either l or k is 0. Without loss of generality, we take [alpha] to be in [0, 1]. Let [mu] denote the measure of all technologies in the economy, consisting of firms that will produce and firms that will exit the market: in any period, the market will be supplied by those firms that choose to remain--exiting firms do not produce. Take [mu] to be normalized with [mu]([0, 1]) = 1.

At the beginning of a period, given a demand state [theta] and [mu] a distribution (3) over firms in the economy, each firm type [alpha] decides whether to remain in the market and produce that period, or to exit and search for a buyer. If firm type a stays in the market, it chooses capital and labor to maximize profit,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where p is the (endogenously determined) market-clearing equilibrium price, w > 0 is the wage rate, and r > 0 is the unit price of capital. (4) The solution to this profit maximization problem gives input demands, l(p, [alpha]) and k(p, [alpha]), and supply function y(p, [alpha]) = f(l(p, [alpha]), k(p, [alpha]), [alpha]). Firm [alpha] then continues on to the next period, receiving a new technology draw, [??], from a distribution P(* | [alpha]), described below.

Because plants are in fixed supply, a potentially more productive entrant must purchase a plant from an exiting firm. (5) We assume that an exiting firm's plant must be idled for the one period that it takes to find a buyer that can better employ the plant and for the purchaser to retool the plant for its own use. Hence, a firm type [alpha] must choose whether to continue production or to exit and lose production for one period. Thus, entry in period t + 1 is completely determined by exit in period t--only plants of exiting firms are available for new entrants (6)--and as the number of firms exiting varies, so does the mass of firms supplying the market in period t. It is easy to modify the model by increasing the time it takes a new user to retool, and by introducing a positive probability that an exiting firm does not find a buyer in the period that it exits. The qualitative effects would be to (i) raise the opportunity cost of exit, (ii) reduce the sensitivity of the value of exit to current market conditions, and (iii) sever the tight link between the mass of exiting firms in one period and the mass of entrants in the next period (see Remark 4).

To capture the limited bidding for an exiting firm's plant, we assume that in negotiations with the buyer, an exiting firm receives only a share 1 - [gamma] [member of] (0, 1] of the discounted profits that a new entrant expects given current market conditions when the entrant makes production and exit decisions optimally. The value of y captures the degree to which the plant is highly specialized so that an exiting firm receives less than the plant's full value to the firm that buys it. (7)

Our modelling captures the essence of the empirical findings of Ramey and Shapiro (2001) that (i) a firm which sells its capital generally receives far less than the capital's value, especially for more specialized capital, and (ii) "there (is) a time cost to restructuring. The process of winding down operations before selling capital results in significant periods of under-utilization. (8) It is only at times when firms cease operations that they sell significant portions of capital."

Evolution of a firm's technology. If a firm's current technology is [alpha], its technology in the next period is drawn from P(* | [alpha]), a conditional distribution over technologies given [alpha]. To capture the fact that a firm with a better technology in one period is likely to have a better technology the next period, we assume that

P(* | [alpha]) = w([alpha])F(*) + [1 - w([alpha])]G(*),

where F [??] G(F conditionally first-order stochastically dominates G), F [not equal to] G, and w([alpha]) [member] [0, 1] is continuous and strictly increasing in [alpha]. (9) That is, the new technology of a firm with current productivity parameter or is drawn from a weighted distribution of a good distribution F(*) and a bad distribution G(*), where the weight w([alpha]) on the good distribution is an increasing function of the firm's current productivity. For simplicity, we assume that the technology quality of a new firm is drawn from the distribution P(* | [bar.[alpha]]), where [bar.[alpha]] [member of] (0, 1).

The only other structure that we impose is that there is some learning by doing in the evolution of a firm's technology:

[integral] P(*, [alpha])P(d[alpha], [bar.[alpha]) [??] P(*. [bar.[alpha]]).

That is, firms tend to improve over time: the technology of a firm in its second period of operation is drawn from a stochastically better distribution than the initial distribution governing the technology of entering firms. (10) The analysis only relies on a far weaker condition, [integral] w([??])d F([??], which is implied by [integral] P(*, [alpha])P(d[alpha], [bar.[alpha]]) [??] P(*, [bar.[alpha]]). Empirically, it is important to allow for learning by doing: whereas Baldwin and Gu (2004) establish that, on average, entering firms are 27% more productive than exiting firms--consistent with out central premise that firms exit to better reallocate resources--Caves (1998) highlights exit rates of new firms are 2 to 3 times that of firms 10 or more years old, and 1.5 to 2 times that of intermediate-aged firms, indicating that newer firms are less productive on average than the average firm.