# What is the mathematical definition of the autarky interest rate?

Consider an economy that lives for two periods and is inhabited by a continuum of identical individu Show more 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 households 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. 2.1. State the economys lifetime budget constraint in terms of Ct Yt Tt and r. 2.2. Restate the economys lifetime budget constraint now in terms of Ct Yt Gt and r. 2.3. Use your answer to (2.2) to solve for C1 as a function of C0 Gt Yt and r. 2.4. The households 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 youve 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)) 2.5. Given your answer to (2.3) what is the economys 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? 2.6. Use your answers to (2.4) and (2.5) to draw the following in the (C0 ; C1 ) plane: the economys 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 economys optimal open-economy consumption point (make note of the slope of an indifference curve). 2.7. 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? 2.8. 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. 2.9. When will consumption smoothing be optimally desired by the household? Explain mathematically. 2.10. Explain the implications for international financial transactions of (r^A > r r^A < r and r^A = r.) Show less

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