In a two-period model, suppose the Alfred's lifetime utility function is U(c1; c2) = u(c1) + u(c2), where u(.) is a concave function. The market interest rate is constant, r. Alfred's income are y1 and y2 in Period 1 and 2, respectively. The initial wealth endowment is w0.

(a) Derive the Euler equation in this case.

(b) Further assume that = 1 and r = 0, solve for the optimal consumption (c1, c2) in Period 1 and 2.

(c) Further assume that income in Period 2 is a random variable, ~y2, which takes two values, yh and yl, with equal probability, i.e., E[~y2] = (yh+yl)/2 =y2. What is the agent's optimal consumption in Period 1, if the utility function takes the quadratic form, i.e. u(c) = nc-1/2c^2?

Is there any precautionary saving? Why or why not?