STOCHASTIC

B-662 ROBERT WIt£ON
pi, pa, etc. are the expected marginal utilities at the solution, and where yiT(x) =
Sye(x)/dxn , y"(x) = dyi(x)/dx„ , etc. Observe that (27) corresponds closely to the
exclusion criterion (23) developed in the case of discrete projects: the principle difference
is that in the present case the intensity levels are varied continuously until the
expected discounted valuation is zero, as in (27), since a positive value would imply
that a further increase in the intensity level would be desirable.
In practice, discrete projects are the rule rather than the exception (except in
financing), which limits the usefulness of the formulation (26). For this reason, specialization
of the formulation to more structured situations (e.g., specialization of the
event tree to the case of a Markov chain) appears to be of little practical value and is
not dealt with here.
9. Summary
In summary, the formulation developed above deals directly with the pervasive
effects of uncertainty in investment analysis. The basic concept underlying the formulation
is the notion of a project's cash flow description superimposed on an event tree
or state description. The formulation explicitly embodies measures of risk aversion
and intertemporal income preferences. Projects of nearly any sort can be handled,
and, in particular, heterogeneous sources of financing are considered.
Seeking the optimal design of a complementary financing program for a project
yields inclusion criteria for its acceptance, and analysis of the borderline case between
risk aversion and risk preference yields an exclusion criterion.
The practicality of the methods proposed is limited mainly by the difficulties of
data collection, and for the next few years, by the size and speed of computing equipment.
Appendix. Construction of Preference Measures for a Firm
Two methods for constructing the preferences of a firm from the preferences of its
owners have been developed recently. A direct extension of the economic theory of
risk markets using a state description of uncertainty provides one approach [1], and
the theory of cooperative games with sharing provides the other [8, 9, 10, 11]. Each
approach offers certain advantages, so we will describe both very briefly.
The Economic Theory of Risk Markets.*
Assume a state description of uncertainty as in the text except that all of the states
that might occur (for all time periods) are indexed by the single index j = 1, • • • .
Then the investment and financial policies of a firm describe a state distribution of
returns to the various securities (common stock, bonds, etc.) issued by the firm; say,
Tjt is the return in state j to securities of type k. Hence, the market value v of the firm
is a function v ({r,t\) of the state distribution of returns (r,-»), and
(Al) v = £*p*
where p* is the market value of the securities of type k. Investors are assumed to optimize
their consumption programs and investment portfolios, with the result that each
investor assesses a marginal rate of substitution between present income and future
income in state j from a security of type k, denoted by mtp, (the dependence upon k
* I am indebted to Jacques Dreze and Jack Hirshleifer for discussions on this topic, although I
bear sole responsibility for my conclusions.
MODELS THAT HAVE A SINGLE DECISION POINT