Consider the following 2-period model of saving for retirement. In Period 1, a consumer

worker has one unit of time to allocate to labor (L) or leisure (1-L). The real wage rate is w, and

labor income (wL) is taxed at the rate t (0

Period 1 and saving (S) for retirement:

(1)

S = (1-t)wL - C1

Savings grow at the real interest rate r, and the worker gets a transfer (pension) T from the

government in Period 2. Consumption in the second period (C2) is

(2)

C2 = (1+r)S +T

The consumer-worker maximizes

u(C1 ) + u(C2 ) + v(1-L)

,

0 < <1 ,

subject to (1) and (2). The functions u(.) and v(.) are strictly increasing, continuously

differentiable, and strictly concave. Assume throughout that leisure and consumption in the two

periods are normal goods. Also assume the transfer T is sufficiently small to make saving S

positive.

a.

Derive the optimality condition for the consumer's choice over C1 and L (i.e., the labor-leisure margin), and give a brief intuitive explanation of what this condition implies.

b.

Derive the optimality condition for the consumer's choice over C1 and C2 (i.e. the Euler

equation), and give a brief intuitive explanation of what this condition implies. Briefly

indicate how your results reflect (or don't reflect) a consumption smoothing motive.

c.

Briefly indicate what are the income and substitution effects of an increase in the tax rate

t on C1 , C2 , S, and L.

d.

Briefly indicate what are the income and substitution effects of an increase in the transfer

(pension benefit) T on C1 , C2 , S, and L.