Solve Question 2

The analysis in the textbook is based on a two-period model. Here we generalize it to a three-period model. There is a consumer who lives for three periods, first-period second-period and third-period. There is a government that lives also for three periods. Let the consumptions in the three periods be Cj, C2 and C3. The government is committed to buying GI, G2 and Gy amounts of goods. The consumer and the government both can borrow and lend in the financial market at the same interest rate. While generally the interest rate varies from the first period to the second, for simplicity we assume it is constant over time, equal to r . Let the amount the consumer lends (borrow if negative) in first-period, second-period and third- period be S1, S2 and $3, respectively. Analogously, the amount the government borrows (lend if negative) in the three periods are B1, By and B3. The consumer earns a labor income equal to Y1, Y2 and Y3. The government collects and the consumer pays 71, 72 and 73 amount of lump-sum taxes in the three periods. Write down the constraints facing the consumer. Write down the constraints facing the government. c) Since there is no future after three periods, it does not make sense for either the government or consumer to lend or borrow in the third period. Derive the following constraints that the consumer and the government must obey: Ci + C2 - + C3 -= Yi + - + Y3 T2 - + ( 1 +r) (1+r)- (1 +r) (1+r)2 (1+r) (1+r)2 G + G2 -+ G3 =Ti+ T2 13 ( 1 +r) (1 +7)2 (1+r) (1+r)2 d) Write down the goods market equilibrium condition for the first period. e) Suppose the consumer works and pays tax in the first period, but neither pays tax nor work in the following two periods. Assume the consumer considers the consumptions in the three periods perfect complements which must be combined in equal proportions. In terms of utility function, U = Min(Ci, C2, C3). The consumer considers his own budget constraint and ignores the government's for this part of the question. Derive the optimal consumption in the first period. Show that an increase in interest rate raises optimal consumption in the first period. Explain the result using the substitution effect and income effect f) Consider the same setting as part (e), what is the effect on optimal consumption in the first period of a dollar increase in the first-period income? For r =0.02, evaluate numerically this effect g) Explain and illustrate the Ricardian Equivalence Theorem using a tax cut of AT