Suppose we have a household with the following (non-differentiable) utility function:

U = min(Ct, Ct+1)

With this utility function, utility equals the minimum of period t and t + 1 consumption. For example, if Ct =3 and Ct+1 =4, then U =3. If Ct =3 and Ct+1 = 6, then U = 3. If Ct = 5 and Ct+1 = 4, then U = 4.

(a) Since this utility function is non-differentiable, you cannot use calculus to characterize optimal behavior. Instead, think about it a little bit without doing any math. What must be true about Ct and Ct+1 if a household with this utility function is behaving optimally?

(b) The period t and t + 1 budget constraints are 9.4 and 9.5 respectively. Use the condition from (a) and the intertemporal budget constraint to derive the consumption function.

(c) Is the MPC between 0 and 1? Is consumption decreasing in the real interest rate?