Consider a special case of SP under uncertainty:

Take SP2 under uncertainty and assume the following: utility is CRRA with 1 as the coefficient of relative risk aversion. Production is linear (i.e. = 1), and the production function reads f(kt) = Atkt. Depreciation is (0,1). Assume further that productivity {At} is independent and identically distributed (iid). Hence, (At+1 | At) = (At+1). Furthermore,E[(1+At )1]<1

1. Write down the full problem in sequence notation.
2. Calculate the FOC(s).
3. Guess ct = c(kt,At) = c (1 + At )kt, where c is a constant. What does that imply for the policy function of kt+1?
4. Find a recursive solution to the FOC(s), i.e. determine c .
5. Argue that solving the FOC(s) is sufficient to find the solution to the original
6. problem.
7. Does this problem exhibit precautionary savings/investment? What would be the case for log-utility?