In HTML, it is impossible (for me) to put in greek letters; hence a stands for alpha, b for beta.
2(a) First, the present value of income is Y = Y0 + Y1/(1 + i) = 100 + 90/1.05 = 185.71. Then, the "demand functions" from a Cobb Douglas utility function are C0 = [.6/(.6 + .4)]Y/p0 and C1/(1 + i) = [.4/(.6 + .4)]Y/p1. Since p0 = p1 = 1 by assumption, C0 = .6 x 185.71 = 111.43 and C1 = 1.05 x .4 x 185.71 = 77.99. The person borrows 11.43 in the fist period which is repaid in the second period plus interest of .57 for a total of 90 - 77.99 = 12. [10 points]
2(b) It is necessary that C0 = 100 = a x 185.71. Solving for a = .538 means that the utility function must be U = C0.538C1.462. [10 points]
2(c) Now, Y = 100 + 90/1.08 = 183.33, which is lower. Next, C0 = .6 x 183.33 = 110.00 and C1 = 1.08 x .4 x 183.33 = 79.20. Thus the person lowers present consumption and increases future consumption. [10 points]
2(d) Calculate the utility levels for each of the two interest rates, using natural logs. With 5% interest rate, lnU = .6 x ln111.43 + .4 x ln77.99 = 4.57. For an interest rate of 8%, lnU = .6 x ln110 + .4 x ln79.2 = 4.569, which is slightly smaller. Hence, the person is better off with lower interest rates since he is a borrower. [10 points]
5(a) Her budget constraint is (80 - 20)Ct = (65 - 20)Yt. Thus Ct = .75Yt and her marginal propensity to consume is .75. [10 points]
5(b) In every year that she works she spends $7500 and saves $2500. After 45 years of work she has 45 x 2500 = 112,500. During her remaining 15 years, she spends this at the rate of $7500 a year. Thus, she has the same consumption pattern during her lifetime. [10 points]
5(c) If she knows about the inheritance now her life-time resources are 45 x 10,000 + 1,000,000 = 1,450,000 which she spends over 60 years at the annual rate of $24,166.67; on the other hand, if the inheritance simply arrives at 40 years of age, she will have an annual consumption of $7500 before that and 25 x 10,000 + 1,000,000 + 20 x 2500 divided by 40 or $32,500 for the last 40 years. [10 points]
7(a) To derive the demand equations we equate the marginal product of K to its real cost, R/P, i.e., YK = aKa - 1Nb = R/P. Multiply and divide the left side by K to obtain aY/K = R/P or K = aY/(R/P). In natural logs (lower case letters), this can be written as k = ln a + y - (r - p). Hence the output elasticity is dk/dy = 1. Also the price elasticity is dk/d(r - p) = -1. [10 points]
7(b) If R/P falls by 1% then the demand for K increases by 1%. To determine the substitution of labour for capital along an isoquant, we differentiate the production function in natural logs, which produces dY = adk + bdn = 0; solving for dn = a/b. [10 points]
7(c) K1 - K0 = [5/(5 + 16))][K* - K0]. If K* increased by 1%, then K will increase by 5/21 or .238% in the first period. [10 points]