Consider an economy that lives for two periods and is inhabited by a continuum of identical individuals grouped into and aggregate risk-sharing household. Each period, aggregate output Yt is dropped from the sky (so this is a "coconut" endowment economy like in class) and a government sector taxes the economy Tt and consumes Gt, running a balanced budged in each period. There is no uncertainty about the level of output in any one period or government consumption decisions, which the household takes as given when solving its intertemporal utility maximization problem. Furthermore, as in class, if output is not consumed or loaned out in any one period, then it spoils and cannot be carried over to a following period. The economy exists for two periods only, 0 and 1. Furthermore, this is a small open economy, so it can borrow and lend freely at the constant-across-periods international interest rate r. The household's lifetime utility is given by

\(U = \sum_{t=0}^{1} \beta^t * C_t^{1-\sigma}/(1-\sigma)\)

where Ct is consumption, \(1/\sigma\) is the elasticity of intertemporal substitution ( \(\sigma\) is a parameter), and \(\beta\) is the (constant) subjective discount factor. For simplicity, we omit work/leisure choices.

1. State the economy's lifetime budget constraint in terms of Ct, Yt, Tt, and r

2. Restate the economy's lifetime budget constraint, now in terms of Ct, Yt, Gt, and r.

3. Use your answer to (2.2) to solve for C1 as a function of C0 , Gt, Yt, and r.

4. The household's intertemporal utility maximization problem is

\(Max_C U= \sum_{t=0}^{1} \beta^t * C_t^{1-\sigma}/(1-\sigma)\)

such that the intertemporal budget constraint holds. Expand the summation term from above, substitute in the expression for C1 that you obtained in (2.3), and take first-order conditions with respect to C0 (by solving for C1 in terms of C0 you've effectively set up the problem so that C0 is the only choice variable). Also, show mathematically that in (C1 ; C0 ) space the slope of an indifference curve is given by \(-C^{-\sigma}_0 / (\beta * C^{-\sigma}_1)\)

5. Given your answer to (2.3), what is the economy's budget line in (C1 ; C0 ) space? Show mathematically that in (C1 ; C0 ) space the slope of the budget line is -(1 + r). What is the maximum amount that the household can consume in period 0? Also, what is the maximum amount that the household can consume in period 1?

6. Use your answers to (2.4) and (2.5) to draw the following in the (C0 ; C1 ) plane: the economy's budget line (make note of what the ordinate and abscissa are, mark the closed- economy private consumption point, and make note of the slope of the budget line) and an indifference curve showing the economy's optimal open-economy consumption point (make note of the slope of an indifference curve).

7. Given the assumptions made in problem 2, what is the mathematical definition of the autarky interest rate? How does the autarky interest rate depend on each periods endowment and government consumption?

8. As in class, assume that the capital account is equal to zero, as are capital gains on external wealth. Using the notation we worked with in class and given the assumptions made in problem 2, state the "generic" account (that is, the current account that must hold in any period t). Then, assume that the economy is born with no predetermined asset claims so A0 = 0 and state the current account in period 0 in terms of that periods output, consumption, and government spending. Also, state the current account in period 1 in terms of that periods output, consumption, government spending, and the interest rate r.

9. When will consumption smoothing be optimally desired by the household? Explain mathematically.

10. Explain the implications for international financial transactions of \(r^A > r, r^A < r, and r^A = r.\)