Question

2. Capital Tax: In our two period consumption-savings model, suppose that positive interest income in period 2 is taxed at rate t. Assume that Ao -0, the individual has positive endowment in both periods, and nominal prices for the good remain the same despite the ax (a) Write down the budget constraints in each period and obtain an algebraic expression for his life-time budget constraint. (b) Suppose that at the optimal choice, the representative individual is choosing not to save in period 1. i. At his current optimal choice, is his marginal rate of substitution between current and future consumption equal to one plus the real interest rate? Explain why or why not. ii. Suppose that the tax rate on interest income is lowered. Would this change in the d 1? Explain vour tax rate encourage the representative agent to save more in perio answer carefully with aid of graphs

Answer 1

Consumers’ budget constraint in the first period is:

c + s = y − t,

where

s > 0 implies that the consumer is saving (buying the bond)

s < 0 implies that the consumer is borrowing (selling the bond),

y − t is the consumer’s disposable income after tax.

A bond issued with face value s yields a return of (1 + r)s in the following period. Note that the unit here is consumption goods

Consumers’ BC in the first period is c + s = y − t.

Consumer’s BC in the second period is c 0 = (1 + r)s + y 0 − t 0

If s < 0, then the consumer pays back both interest and principal in the second period.

If s > 0, then the consumer receives the promised return on her savings in the second period.

The consumer’s problem is given by

maxc,c 0 ,s V(c, c 0 ) (1)

subject to c + s = y − t (2)

c 0 = (1 + r)s + y 0 − t 0 (3)

However, we can substitute s in the second equation by the first one:

From BC(1): s = y − t − c

Replace s in BC(2) by the equation above:

c 0 = y 0 − t 0 + (1 + r)(y − t − c)

After rearranging the equation, we have

c + c 0 1 + r = y − t + y 0 − t 0 1 + r (PVBC)

This is the consumer’s present value budget constraint (PVBC).

Note that now we have just one PVBC and two variables to solve for the consumer’s problem. We can conduct the same graphical analysis as we did for the static problem.