Question

Problem 1: Labor-Consumption Choice

Consider a household with the following utility function:
U (C,N) = lnC - (b/2)N^2

where b > 0: Namely, his utility is increasing in consumption C and decreasing in labor N: His budget

constraint is the following:

C = (1 Tw )wN where N are total hours worked. w is the wage per hour, and w is the tax on labor income. Answer to the

followings.
a) Find the optimality condition that describes equates the marginal costs and benefits of working. HINT:

you have to di§erentiate U w.r.t. to N; taking into account how N affects C:

Answer 1

Given utility function be U(C,N) = Ln C - (b/2)N^2

where C represents consumption and N represents Labour

Budget Constraint is given as C = (1 Tw)wN

where w represents wages and t is tax on labor income

Optimal condition is derived when marginal cost is equal to marginal benefit.

To derive optimal position we maximise utility subject to budget constraint. i.e.

max U (C,N) = lnC - (b/2)N^2

subject to C = (1 Tw )wN

=> max U(N) = ln [(1 Tw )wN] - (b/2)N^2

differentiating Utility w.r.t.N

dU/dN = [1/((1 Tw )wN)].((1 Tw )w) - (b/2).2N

= (1/N) - (Nb)

Appling first order differentiation condition for optimality and thus equating dU/dN = 0

0 = (1/N) - (Nb)

0 = (1 - bN^2) / (N)

1 - bN^2 = 0

1 = bN^2

N = (1/b)^(1/2)

Hence optimal amount of labor is (1/b)^(1/2)

deriving optimal consumption by putting N value in C

we get

C =  (1 Tw )w(1/b)^(1/2)

Thus we get the optimal condition of having N and C dependent on b, tax paid and on wage.

optimal N = (1/b)^(1/2)

optimal C = [(1 Tw )w] / b^(1/2)