Problem
An individual has utility from consumption in period 1 c1 and consumption in period 2 c2: u(c1, c2) = log(c1) + ß log(c2), where 0 < ß = 1. Their budget constraint in the first period sets consumption in period 1 and savings s equal to the endowment in period 1, I1: c1 + s = I1. Their budget constraint in the second period sets consumption equal to the endowment plus savings (which has accrued interest at a rate r > 0): c2 = I2 + (1 + r)s.
Maximize the utility function by:
Combine the two budget constraints by canceling out s.
Solve the budget constraint for c2 as a function of c1 and the endowments.
Solve for U(c1) by plugging in your expression above in the utility function for c2. Now utility is just a function of c1.
Maximize utility by taking the partial derivative of U(c1) with respect to c1. Solve for c*1 as a function of ß, r, and the endowments I1 and I2. Use the budget constraint to solve for c*2.
What is the relationship between c1 and c2 if I1 =I2 and ß= 1 ? 1+r
What happens to c1 when r increases? What happens to c2?
The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.