This appendix contains proofs of theorems 1, 5, 7, and 8. Theorem 2 is essentially proved in the body of the paper. Theorem 3 follows directly from equation (5). Theorem 4 is proved in the body of the paper. The proof of theorem 6 is similar to the proof of theorem 5, and relies on equation (5). Additional commentary pertains to theorem 2 in Shefrin-Statman (1994), and the uniqueness of state prices.

**Proof of theorem 1**

The plan of the proof is to derive expressions for P_{R}, θ_{R}, and δ_{R}, by equating two different expressions for g(x_{t}). The first expression for g(x_{t}) stems from the equilibrium condition

where ch is given by (4). The second expression for g(xt) stems from (5) which expresses equilibrium prices v in terms of g(x_{t}) and the representative trader’s parameters P_{R}, θ_{R}, and δ_{R}.

I begin the proof by defining

and

which is equivalent to the expression for (x_{t}) that appears just before the statement of Theorem 1. Next turn to the two expressions for g(x_{t}). The first uses (4) to compute the equilibrium value of g(x_{t}). By definition,

and substituting for c_{h} from (4), obtain

which, using the definition of Υ_{h}, yields

by the definition of Υ The second expression for g(x_{t}) is obtained by inverting equation (5). Doing so yields:

Now equate the two expressions for g(x_{t}) to obtain:

In the remainder of the proof, I use the last set of equations to establish the expressions for P_{R}, δ_{R}, and θ_{R} that appear in the statement of theorem 1. Notice that the preceding equation, equating the two terms for g(x_{t}), implies that:

Hence, the preceding equation defines δ^{t}_{R,t} P_{r}(x_{t}) in terms of θ_{R}. Define δ^{t}_{R,t} as the following sum, for fixed t:

Then define P_{r}(x_{t}) as the ratio

In view of the normalization implicit in the last equation,

for each t. Nonnegativity of P_{r} follows from the fact that all variables involved in the construction are nonnegative. With δ^{t}_{R,t} and P_{r}(x_{t}) defined in terms of θ_{R}, it remains to specify θ_{R}. In this respect, let θ_{R} be (8). Notice that (8) is a function of Π, and is derived from Benninga- Mayshar (1993). In particular, (8) is not dependent on P_{r} or θ_{R}. Hence there is no issue of simultaneity in determining δ^{t}_{R,t} , P_{r}(x_{t}), and θ_{R}.

The preceding argument establishes the first part of the theorem. In order to establish the nonuniqueness claim, observe that we are free to specify the function θ_{R}. Instead of choosing (8), the Benniga-Mayshar expression, we could also have set θ_{R}(x_{t}) equal to an arbitrary constant for all xt, and solved for P_{r} and θ_{R}, again, as functions of θ_{R}. Equation (11) follows from (5). For sake of completeness, I sketch the proof provided by Benninga and Mayshar (1993) characterizing θ_{R}. Note that (4) and

imply that

for all x_{t}. Based on (4), the equilibrium condition

0, and the kernel variable V = v/Π, Benninga-Mayshar define the implicit function

They note that by the principle of expected utility maximization, the representative trader’s marginal utility at C will be proportional to V . In turn, this implies that θ_{R}, the Arrow- Pratt coefficient of relative risk aversion can be defined locally by −CV 0(C)/V (C). By

computing *ðF/ðC* and *ðF/ðV* , they observe that

which, taken together with the local Arrow-Pratt measure, completes their proof. Case of log-utility: beliefs of representative trader and objective prices Consider the nature of theorem 1 in the special case when θ_{h}= 1 for all h, this being the case of log-utility. Shefrin and Statman (1994) use (3) to establish that the representative trader’s beliefs are given by the weighted sum:

and the representative trader’s normalized discount factor is:

These expressions are obtained by using (3) to equate ∑_{h}C_{h}(x_{t}) and C_{r}(x_{t}) to obtain the expression for P_{r}, and equating ∑_{h,xt} C_{h}(x_{t}) and ∑_{xt} C_{r}(x_{t}) for fixed t to obtain δ^{1}_{r}(t). By definition of ε and the preceding expression for P_{r}, obtain

so that

It follows that in order for the efficiency condition P_{r}(x_{t}) = Π(x_{t}) to hold, the sum

∑_{h} W _{h}ε_{h}(x_{t}) must be zero. Define the covariance cov{∑_{h} W _{h}ε_{h}(x_{t}) } by:

where

Multiplying out the terms in the covariance, yields the equivalent covariance expression:

for each x_{t}. Recall that prices are efficient if and only if ∑_{h} W _{h}ε_{h} = 0 for all x_{t}. This is equivalent to

Multiplication of this last expression by PhWh yields the equivalent condition

which implies that prices are efficient if and only if the sum of the error-wealth covariance and product of the average error and mean wealth is equal to zero.^{[53]}

**Uniqueness of state prices **

Arrow and Hahn (1971) survey conditions under which the equilibrium state price vector v is unique. Their theorem 5, p. 220 establishes uniqueness for the case of homogeneous traders. More generally, their theorem 7, p. 222, establishes that equilibrium state prices are unique if the gross substitute condition holds. The gross substitute condition states that

Consider (3), the demand function for an individual trader. In this regard, note that Wh is a dot product of v and ω_{h}. Differentiation of (3) with respect to v(y_{τ}) yields an expression that is the sum of two terms. The first term is

and the second term is

Note that the first sum is nonnegative, and when θ_{h} = 1, the second term is zero. Summation over h leads to

. Computation reveals that the derivative is strictly positive for θ_{h} ≤ 1, and is therefore positive in an open set of 1.

**Proof of theorem 5**

From the perspective of x_{t}−1, η(x_{t}) is the future value of a contingent x_{t} real dollar payoff. Given x_{t}−1, the future value of a contract which delivers a certain dollar at date t must be one dollar. This is why

In other words, the future value of y_{t}−claims are nonnegative and sum to unity. Hence they constitute a probability distribution. Since they deal with the transition from x_{t}−1, {η(y_{t})} are one-step branch probabilities of a stochastic process.

Under the stochastic process, the probability attached to the occurrence of xt is obtained by multiplying the one-step branch probabilities leading to x_{t}. To interpret this product, consider the denominator of (20). This term can be matched with the numerator of the x_{t}−1 one-step branch probability to form

The latter term is simply one plus the single period risk free interest rate i1(x_{t}−1) that applies on the x_{t}−1-market. Therefore the probability of the branch leading to x_{t} is the product of the single period stochastic interest rates and the present value of an x_{t}-claim: i1(x_{0})i1(x_{1}) ... i1(x_{t}−1)v(x_{t}). The product of the single period interest rates defines the cumulative return it c(x_{t}) to holding the short-term risk-free security, with reinvestment, from date 0 to date t. A call option pays q_{z}(x_{t})−K at date t,

the set of date-event pairs where the option expires in-the-money. The present value of the claims that make up the option payoff is computed using state prices v. But the present value of an x_{t}−contingent dollar is its future value discounted back by the product of the one-period risk-free rates. The discounted contingent future dollar is simply the ratio of a risk-neutral probability η(x_{t}) to a compounded interest rate i_{c}(x_{t}). Finally, the risk-neutral probability η(x_{t}) is unconditional. To convert to a distribution conditional on exercise, divide η(x_{t}) by P_{η}{A_{E}|x_{0}}. Using the conditional expectation in place of the unconditional expectation leads to the appearance of P_{η}{A_{E}|x_{0}} in (21).

**Proof of theorem 7**

To prove theorem 7, compute the first-order-condition associated with maximizing expected quadratic utility.

(31)

where ξ is the Lagrange multiplier for the optimization and has the form:

(32)

Next observe that

Substitution into (32) completes the proof.

**Proof of theorem 8 **

The proof of this theorem is computational. The one-period return to the market portfolio is the sum of the date 1 dividend and date 1 price, divided by the date 0 price, i.e.

Use (7) to compute the present values two future aggregate consumption stream: the value of the unconditional process under v, and the value of the process, conditional on x_{1}. The present value of these two streams appear respectively, in the denominator and numerator of (28), with the numerator value divided by v(x_{1}). This completes the proof.

Prof. Hersh Shefrin

**Summary: **Index