Hints for solving problem set #4 (calculus hint)

In a previous “helpful” hint pertaining to the first problem in Problem Set 4, I noted (among other things) that expected utility for Investor A is E(U(W)) = .6{(105 + 25\alpha )^{.5}} + {\rm{ }}.4{(105 - 35\alpha )^{.5}}.  Since thee square root utility function is itself a function of \alpha; i.e., U(W) = W{(\alpha )^{.5}}, this means that we must apply the chain rule in order to compute the first order condition and solve for \alpha:
\displaystyle\frac{{dU}}{{d\alpha }} = \displaystyle\frac{{dU}}{{dW}}\displaystyle\frac{{dW}}{{d\alpha  }} = .5W{(\alpha )^{ - .5}}\displaystyle\frac{{dW}}{{d\alpha }}.  Applying the chain rule, the first order condition for Investor A is:

\displaystyle\frac{{\partial E(U(W))}}{{\partial \alpha }}{\rm{ }} = {\rm{  }}.3{(105 + 25\alpha )^{ - .5}}(25) - .2{(105 - 35\alpha )^{ - .5}}(35)  = 0.

In order to determine Investor A’s optimal risk exposure, all that remains to be done is to solve this equation for \alpha.

Rinse and repeat to determine Investor B’s optimal exposure to risk.  Since Investor B’s utility function is U(W) = \ln W, B’s expected utility is E(U(W)) = .6\ln (105 + 25\alpha ) + .4\ln (105 - 35\alpha).  Applying the chain rule to U(W) = \ln W(\alpha ), it follows that \displaystyle\frac{{dU}}{{d\alpha }} = \displaystyle\frac{{dU}}{{dW}}\displaystyle\frac{{dW}}{{d\alpha }} = \displaystyle\frac{1}{{W(\alpha )}}\displaystyle\frac{{dW}}{{d\alpha }}.  Given these clues, you should be able to determine optimal \alpha for Investor B. However, before you begin, keep in mind that since Investor A’s Arrow-Pratt coefficient is {R_A}(W) = .5/W and Investor B’s Arrow-Pratt coefficient is {R_A}(W) = 1/W (see pages 8, 10, and 12 of http://fin4335.garven.com/fall2019/lecture6.pdf to see how these measures were determined), we know that Investor B is more risk averse than Investor A.  Therefore, Investor B will select a lower level of exposure to risk then Investor B.

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