Consider a consumer with utility function U(c,cf) = logc + 3 log c\' where c is

Question

Consider a consumer with utility function U(c,cf) = logc + 3 log c' where c is consumption in present and d is consumption in future. Consumer receives real income y and y' and can save and borrow freely at real interest rate r. There is no government and no taxes. Write the consumer's budget constraints in real terms, (BC) for current period and (BC') for future period. Discount the future constraint (BC') and derive the present value lifetime budget constraint. Write the consumer's utility maximization problem. It can be shown that the optimal choice for the problem in (b) is C^* = 1 / 1 + beta (y + y' / 1 + r) C^'* = beta(1+ r) / 1 + beta Suppose that | r = 0.5, y = 45.000. y' = 33.000. Draw a diagram, showing the budget constraint, indifference curve and the optimal choice. Remember to label axes, intercepts, and the optimal point, Suppose now that the government is taxing interest income at rate r. so that the lender who saves s in current period receives (1 + (1 - r)r)s in the future period. This essentially lowers the effective real interest rate for the lender: instead of rw = r it is now r2 = (1 - t)r with r T%. The tax has no effect on a person who is a borrower, since it is only levied on savings. Suppose that T = 0.8. Calculate the new optimal choice. Show the effect of tax in a diagram with two BCs, two ICs and two optimal points (before and after the tax was introduced). Be careful about the shape of BC when interest income is taxed. Hint: that BC has a kink point. Calculate the optimal savings s before and after the tax was introduced. Discuss how c, c. s changed because of the tax and explain the economic intuition behind this change (including the substitution and the income effects that occurred)

Explanation / Answer

a) BC = y = y + y -c

BC' = y' + (1+r)(y-c) = c'

Discounting BC'/(1+r) = y'/(1+r) +y - c = c'/(1+r)

b) Max U(c,c') = logc + blogc'

logc + blog(y' +(1+r)(y-c))

d) Max U(c,c') = logc + blog(y' +(1-t)r(y-c) )

c* = 1/(1+b)[y+y'/(1-t)r]

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