Janet is planning for retirement and has $1000 to purchase stocks and bonds. Lets denote the money she spends on stocks as 's' and the money she spends on bonds as 'b'. Each bond has an interest rate of 0%.

That is if spends $10 on bonds, it will be worth $10 at retirement. The stocks are more risky. Stocks will be worth three times their purchase price with 50% probability and zero with 50% probability. Janet is risk averse with a utility function equal U(M) = ln (M)

a. Compare the expected value of $1 of stocks with $1 of bonds.

b. Write down Janet's budget constraint

c. Write down Janet's expected utility as a function of the stocks and bonds that she has purchased.

d. Use the constrained optimization technique that we learned in the previous section or convert it into an unconstrained optimization problem to help Janet decide how many stocks and bonds to buy to maximize her expected utility. Write the Lagrangian function

e. Solve for her optimal portfolio of stocks and bonds.