In this chapter, we study the decision problem of investment rates and exit times for a private business in a stochastic environment. We consider a competitive firm that produces a capital good with the capital stock K(t).

The market price P(t) of the capital good is governed by a stochastic differential equation, and the market value of the capital given by X(t) ≔ P(t)K(t). The net cash flow of the firm coincides with the profit rX(t) for the profit rate r net the total cost ϕ(q(t))X(t) of investment q(t)K(t). The firm sells its whole business at any time τ, given a resale value function g(·), when the integration of capital goods is costly. The firm's fundamental value is the total of the net cash flow and its abandonment payoff g(X(τ)). The objective of the firm is to find the investment rate q*(t) and the exit time τ* simultaneously, which maximize the expected fundamental value.

We solve the nonlinear variational inequality associated with the firm's value maximizing problem. The optimal policies are shown to exist from the optimality conditions.

The Model

Consider the decision problem of investment rates and exit times. Define the following quantities:

1. P(t) = market price of the capital good at time t.

2. ν = expected decline rate of the price.

3. B(t) = the standard Brownian motion.

4. […]