Introduction: Optimization in two stages

In Chapter 2 we analyzed decision making under the threat of a takeover for management whose utility depends on the growth rate of the firm. To do so, we assumed a priori that a trade-off exists between the valuation ratio v and the growth rate g of the firm, which we have called the v-g frontier. We now analyze in more detail why such a frontier exists and how it is derived.

A firm has a set of policy variables: the amount of labor to employ, the amount to produce, the amount and the means of advertising, the amount of R & D expenditure, which industry to diversify, and so on, and of course the growth rate of capital or investment. The management of the firm determines the values of these variables so as to optimize according to some criteria. Suppose that this problem is solved in two stages: (1) Given the growth rate g, determine the optimal values of other policy variables and (2) determine the optimal growth rate. Then, by the first stage of the optimization, the optimal values of all the policy variables except g are determined as functions of g and relevant prices. Hence with these optimal values substituted, the market value of the firm or the valuation ratio is a function of a single variable, the growth rate, given prices.