STOCHASTIC

of the model, see Bradley and Crane (1973). The reader is asked to clarify
some aspects of the model in Exercise CR-7. Exercises CR-8 and 9 illustrate
a two-period linear programming under uncertainty approach to the bond
portfolio problem. Exercises CR-3 and 4 describe two simple models of
portfolio revision for portfolios that consist of a single asset. Taxes, transaction
costs, and liquidity considerations are shown to affect the advisability of a
switch from one asset to another.
For further work on the portfolio revision problem, the reader may consult
Chen et al. (1971, 1972), Pogue (1970), Smith (1967, 1971), and Tobin (1965).
Unfortunately none of these sources provides a very satisfying model from a
theoretical standpoint; rather their thrust is largely concerned with ad hoc
extensions of static portfolio models. See Zabel (1973) for an interesting
extension of the Phelps-Hakansson model discussed in the next section, to
include proportional transaction costs.
II. Models of Optimal Capital Accumulation
and Portfolio Selection
The seminal work of Phelps (1962) extended the Ramsey (1928) model of
lifetime saving to include uncertainty. In Phelps' problem, the choice in each
period was between consumption or investment in a single risky asset, with
the objective being the maximization of expected utility of lifetime consumption.
Phelps assumed the utility function was additive in each period's utility for
consumption, and he obtained explicit solutions when each utility function
was the same and a member of the isoelastic marginal utility family. The papers
reprinted in Chapter 2 generalize and extend the work of Phelps in several
directions: Portfolio choice is included in the Samuelson and Hakansson
papers, and more general utility functions are treated in the Neave paper.
These papers also constitute generalizations of the multiperiod consumptioninvestment
papers of Part IV.
Neave treats the multiperiod consumption-investment problem for a
preference structure given by a sum of utility functions of consumption in
different periods, including a utility for terminal bequests. Future utilities are
discounted by an "impatience" factor, compounded in time. In this paper,
the portfolio problem is neglected, with the only decisions at any time being
the amount of wealth to be consumed, the nonnegative remainder being
invested in a single risky asset. In each period, terminal wealth consists of a
certain, fixed income plus gross return on investment, and this terminal wealth
becomes initial wealth for the following period, from which consumption and
investment decisions are again to be made, and so forth. Neave is concerned
with conditions under which risk-aversion properties are preserved in the
induced utility functions for wealth. He demonstrates the important result
432 PART V DYNAMIC MODELS