Problem
Consider the following specialization of the discussion: The consumer has $W to invest, which he must allocate between two possible investments. The first is a sure thing- put in $1, and get $r > $1 back. The second is risky - for every $1 invested, it returns a random amount $θ, where θ is a simple lottery on (0, ∞) with distribution τ. We assume that the expected value of θ is strictly greater than r but that there is positive probability that θ takes on some value that is strictly less than r. We allow this investor to sell short the riskless asset but not the risky asset. And we do not worry about the net payoff being negative; the consumer's utility function will be defined for negative arguments.
This investor evaluates his initial choice of portfolio according to the expected utility the portfolio produces for his net payoff. Moreover, this consumer has a constant coefficient of absolute risk aversion λ > 0. That is, he chooses the amount of his wealth to invest in the risky asset (with the residual invested in the safe asset) to maximize the expectation of -e-λY, where Y is his (random) net payoff from the portfolio he picks. Let us write a(W, λ) for the optimal amount of money to invest in the risky asset, as a function of the consumer's initial wealth W and coefficient of risk aversion A.
(a) Prove that a(W, λ) is finite for all W and a. That is, the consumer's investment problem has a well-defined solution. Even if you can't do part (a), assume its result and go on to parts (b) and (c)
(b) Prove that a(W, λ) is independent of W; no matter what his initial wealth, the consumer invests the same amount in the risky asset.
(c) Prove that a(W, λ) is non increasing in λ; the more risk averse the individual, the less he invests in the risky asset.
The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.