Question

Dudley has a utility function U(C, R) = CR, where R is leisure and C is consumption per day.

He has 16 hours per day to divide between work and leisure. Dudley has a non-labor income of $48 per day.

(a) If Dudley is paid a wage of $6 per hour, how many hours of leisure will he choose per day?

(b) As a result of a promotion, Dudley is now paid $ 8 per hour. How will his leisure time change?

(c) Analyze the changes in Dudley’s leisure time using the substitution effect and the income effect.

Answer 1

(a)

U(C,R) = CR

Dudley can work for 16 hours in a day, in that case, he will earn $48 + $6(16) = $170, or he can leisure the whole day, in which case he consumes only $48.

Dudley's budget constraint is

C = 48 + 6(16 - R)

When utility function U(C,R) = C*R, the marginal rate of substitution is C / R.

Therefore, Luke's marginal rate of substitution is:

MRS = C / R.

Dudley's optimal mix of consumption and leisure is found by setting MRS equal to wage and solving for hours of leisure from given the budget constrain.

w = MRS

Putting the budget constrain equation in place of C

Dudley will choose 12 hours of leisure.

His consumption will be C = 48 + 6(16 - 12) = 48 + 24 = 72

(b)

When Dudley is paid $ 8 per hour.

budget constraint is

C = 48 + 8(16 - R)

Dudley's optimal mix of consumption and leisure is found by setting MRS equal to wage and solving for hours of leisure from given the budget constrain.

w = MRS

Putting the budget constrain equation in place of C

Dudley will choose 11 hours of leisure.

His consumption will be C = 48 + 6(16 - 11) = 48 + 30 = 78

(c)

The higher wage increases the price of leisure. The substitution effect of a higher wage causes the consumer to substitute labor for leisure. Higher wage induces the individual to supply a greater quantity of labor.

But the higher wage also has an income effect. An increased wage means a higher income, and since leisure is a normal good, the quantity of leisure demanded will go up. And that means a reduction in the quantity of labor supplied.

The income effect of the wage change is negative; Substitution effect of wage change is positive.

As Dudley's leisure time decreases with an increase in his wage, the strength of the substitution effect of wage change is greater than the strength of income effect.