Consider a financial market with H individual traders. Time is discrete, with a set of dates indexed t = 0, 1, 2, ... , T. At the beginning of each date, new information s is revealed. Call s a state, and assume that it belongs to a finite set S = {s_{i}}. Let s^{t} ε S denote the state revealed at date t. The public information at the beginning of t is denoted by the trajectory x_{t} = (s^{0}, s^{1}, ... , s^{t}). That is, uncertainty unfolds according to a tree whose nodes are the date event pairs {x_{T}}. Let II(x_{T}) denote the objective probability attached to the occurrence of x_{T} . I assume that the probability attached to a trajectory x_{t} is derived from the terminal node density {II(x_{T})} as follows:

where a terminal node x_{T} is in the summation if and only if x_{t} is an ancestor node of x_{T} . In this section, I assume that all information is held in common, and trading is costless. At the outset of date 0, trader h holds an initial portfolio ω_{h}. If h holds ω_{h} through date t, and date-event pair xt materializes, then h receives dividend ω_{h}(x_{t}) during

date t. The symbol ω = ∑ω_{h} denotes the unlevered market portfolio.12 In equilibrium, the consumption growth rate

.

A financial security is represented as a vector Z = [Z(x_{t})] where Z(x_{t}) is the amount which one unit of the security pays its owner at x_{t}. Assume that the following securities are available for trade at every date t:

(1) Zero coupon, risk-free bonds: these bonds underlie the term structure of interest rates. Assume that a zero coupon bond maturing at any date t is available for trade at any date before t.

(2) The market portfolio: this security is denoted by Z_{ω}, and is a scalar multiple of ω.

(3) European put and call options on the market portfolio. I assume that we can guarantee markets to be complete by allowing enough variation in the option exercise prices. A call option issued at x_{t} has an exercise price of K, expires at date t + j, and pays max{q_{ω}(x_{t+j}) − K, 0}, where q_{ω}(x_{t+j}) denotes the price of the market portfolio on the x_{t+j}-market. A put option is analogous to a call option, but returns max{0,K − q_{ω}(x_{t+j})}.

Since markets are assumed to be complete, there are state prices that underlie security prices. Let v(x_{t}) denote the price of an xt-state contingent claim, and v = [v(x_{t})]. I take x0 as numeraire: that is, v(x0) = 1. On the date 0 market, the price qz(x0) of security Z = [Z(x_{t})] is v • Z. On the x_{t} market, the price qz(xt) of Z is the v-value of the Z-payoffs from date t on, divided by v(x_{t}).[13]

A trader’s wealth at the beginning of t consists of the market value of his x_{t}−1

portfolio, including dividends paid in xt. The trader then divides his x_{t}-wealth into a portion to be consumed at t, and a portion to be saved. The saved portion is invested in the securities which comprise his xt-portfolio. Denote trader h’s net trade of the x_{t} contingent commodity by zh(x_{t}). Then the consumption vector ch = [ch(x_{t})] is given by

C_{h} = ω _{h} + Z _{h}.

The beliefs of trader h are represented by a subjective stochastic process P _{h} on the trajectories. Hence, P _{h}(x_{t}) is nonnegative, and sums to unity for each t. Moreover, for each t, P _{h}(x_{t}) is determined by conditioning on the probabilities P _{h}(x_{T} ) that attach to the

terminal date T. I assume that each trader has a utility function featuring constant relative risk aversion. That is:

where

is h’s risk tolerance parameter. Furthermore, h’s preferences are additively separable over time and trajectories. Hence preferences are representable as the sum of weighted utilities, with weights D_{h}(x_{t}), where D_{h}(x_{t}) takes the form of a discounted probability

. Here is a discount factor satisfying 0 < < 1.

Every trader is assumed to choose his consumption plan ch by maximizing the sum of weighted utilities

(2)

subject to the lifetime budget constraint

.

Denote trader h’s x0-wealth by

Then h’s demand function is:

(3)

Note that in (3) the pattern of the consumption profile is keyed from wealth W _{h}, in that (3) specifies the fraction of wealth W _{h} which is to be consumed in each date-event pair x _{t}. In the discussion below, it will be useful to consider the consumption profile as being keyed to initial consumption C _{h}(x_{0}) rather than to Wh. Note that v(x_{0}) = 1 since x_{0} is taken as numeraire. Hence the denominator of (3) is equal to W _{h}/ch(x_{0}), so that by substitution, h’s consumption growth rate is given by:

(4)

Prof. Hersh Shefrin

**Next: **A Representative Trader Characterization

**Summary: **Index