Anne is endowed with $500 in youth and nothing when old, with utility ln c1 + ln c2. She also has an uncertain capital technology: by saving k units in youth, she has a 1/3 chance of producing Xk units when old; otherwise, she produces 1 unit when old.

(a) Write Annes first period budget constraint. Then write her expected utility, substituting for c2 using the old age constraint.

(b) Solve for Annes utility maximizing c1 and k, and then compute the expected utility from those choices as a function of X.

(c) Suppose Anne had a risk-free capital technology instead, allowing her to obtain W k in old age for sure. Solve for her decision and equilibrium utility.

(d) Comparing the equilibrium utility from the two proceeding problems, what W would make Anne indifferent between the two capital technologies? Compute the risk premium as the average output from the uncertain technology minus W k, and comment on how that risk premium changes with X.