Consider an example of a household consumption-savings problem with income uncertainty. Assume that the household utility function is u(c) = c^1/1 . The household problem is to choose first period savings or borrowing s to maximize its expected welfare or discounted sum of periodic utilities given by u(c1) + Eu(c2) subject to

c1 + s = y1

c2 = y2 + (1 + r)s + e

where > 0 is the discount factor, E expectations operator, y1 first period income, y2 second period income, r interest rate, and e a stochastic variable that takes value e1 with probability 0.5 and value e2 with probability 0.5. Notice that given savings or borrowing s, Eu(c2) can be determined as 0.5u(y2 + (1 + r)s + e1) + 0.5u(y2 + (1 + r)s + e2)

i) Derive a first-order condition that characterizes optimal savings or borrowing.

ii) Assume y1 = 1, y2 = 1, = 1, r = 0, and = 5, e1 = 0.5, e2 = 0.5. Solve the household problem (with e.g. Matlab, R, Python, Julia, or Excel). What is the optimal savings or borrowing s (with 2 digits precision)? What would it be without income uncertainty (with e1 = e2 = 0)? How does income uncertainty affect household welfare?

(Hint: To solve the household problem, you may simply consider a large number of different feasible savings and pick the one that generates the highest expected welfare. Alternatively, use the first-order condition.)