Dynamic Economics: Optimization by the Lagrange Method

Consider a model for the investment decision of a firm that treats the decision as exercising an option to invest. Once the option is taken or exercised, it cannot be reversed. Such theories of investment are surveyed and discussed in Pindyck ( 1991), Dixit ( 1992), and Dixit and Pindyck ( 1994). In the simplest case, assume that to exercise the option, it would cost the firm I dollars. The present value v(t) of the investment project at time t is assumed to vary through time according to the stochastic differential equation

dv = αvdt + σvdz (8.1)

in which z is a Wiener process with var(dz) = dt. The problem is to determine the optimum time T to invest or to exercise the option. When the option is exercised, the firm gains v(T) - I, but loses the opportunity to invest in a future time T + s when v(T + s) may be larger than v(T).

Formulate this optimization problem starting from time 0 by using the Lagrangean expression

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The state variable is v(t). The control function u(v) could be viewed as a step function, with u(v) = 1 meaning to undertake the investment when v reaches v(T) = v* and u(v) = 0 meaning not to undertake the investment when v 〈 v*. To maximize (8.2), consider the two cases u = 0 and u = 1. Keeping u fixed, first find a first-order condition by differentiating the Lagrangean expression with respect to the state variable v. Then, find a second condition to determine v*.

To find the function λ(v), set the derivative of e^{β(1 + dt)}ℒ with respect to the state variable v = v(t) equal to zero. To obtain ∂ℒ/∂v, evaluate dλ by Ito's lemma:

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