We consider a two-periodendowmenteconomy. Suppose you have the following utility function U （c , c ′）=lnc +βlnc ′ where c and c ′ refer to consumption in the first and second periods, respectively, and β = 0.5. The household income in the first and second periods is denoted by y and y ′. Assume government collects lump-sum tax in both periods (T, T ′)to finance the public spending in both periods (G , G ′)and issue bonds B in the first period. A real net interest rate r is 1. 1. Write down the household’s problem. Does the household perfectly smooth their con- sumption over lifetime? Show it using the optimality condition. 2. Solve for optimal c , c ′ and s (saving) when y −T > y ′−T ′. 3. Given the conditions in part (2), there is an increase in future income y ′ such that y −T = y ′−T ′. Find new optimal c , c ′ and s . Explain how c , c ′ and s change compared to part (2). 4. Focusing on part (3), (a) Suppose that the government increases T by a small amount and decreases T ′ by the same amount. Explain how c , c ′ and s change. Interpret the results. (b) Now, we assume a borrowing constraint s ≥0. Again, suppose that the government increases T by a small amount and decreases T ′ by the same amount. Explain how c , c ′ and s change. Interpret the results. 5. Write down the the household’s problem and solve for new optimal c , c ′ and s when r =0 and the utility function given by U (c , c ′)=lnc +βlnc ′ where β =1 assuming y −T =y ′−T ′ as in part (3). Here, assume that households can borrow freely. 6. Given the conditions in part (5), there is an increase in future income y ′. Explain how c , c ′ and s change.
We consider a two-periodendowmenteconomy. Suppose you have the following utility function
U （c , c ′）=lnc +βlnc ′
where c and c ′ refer to consumption in the first and second periods, respectively, and β =
0.5. The household income in the first and second periods is denoted by y and y ′. Assume
government collects lump-sum tax in both periods (T, T ′)to finance the public spending in
both periods (G , G ′)and issue bonds B in the first period. A real net interest rate r is 1.
1. Write down the household’s problem. Does the household perfectly smooth their con-
sumption over lifetime? Show it using the optimality condition.
2. Solve for optimal c , c ′ and s (saving) when y −T > y ′−T ′.
3. Given the conditions in part (2), there is an increase in future income y ′ such that y −T =
y ′−T ′. Find new optimal c , c ′ and s . Explain how c , c ′ and s change compared to part
(2).
4. Focusing on part (3),
(a) Suppose that the government increases T by a small amount and decreases T ′ by
the same amount. Explain how c , c ′ and s change. Interpret the results.
(b) Now, we assume a borrowing constraint s ≥0. Again, suppose that the government
increases T by a small amount and decreases T ′ by the same amount. Explain how
c , c ′ and s change. Interpret the results.
5. Write down the the household’s problem and solve for new optimal c , c ′ and s when
r =0 and the utility function given by U (c , c ′)=lnc +βlnc ′ where β =1 assuming
y −T =y ′−T ′ as in part (3). Here, assume that households can borrow freely.
6. Given the conditions in part (5), there is an increase in future income y ′. Explain how
c , c ′ and s change.
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