Question

Suppose that a worker’s utility (i.e., preferences) with respect to total income (Y) and hours of leisure time per week (LT) can be represented by the following Cobb-Douglas

utility function:

U = Y0.4 ∙ LT0.6 (note: A=1; α=0.4; β=0.6)

Assume that the market wage is $25 per hour of work (H), and his/her non-labor income is $300 per week. The worker has 70 hours per week to allocate between labor market activity and leisure time (i.e., T = 70).

1. Given the above information, specify the worker’s budget constraint (i.e., the budget line equation). Illustrate the budget line graphically.

2. Given the above information, determine the worker’s equilibrium (utility- maximizing) combination of work, leisure hours, and income (total and labor market). Illustrate the worker’s equilibrium outcome graphically (be sure to show a general sketch of the I-Curve).

3. Determine and illustrate graphically what happens to hours of work when the worker’s non-labor income decreases to $200 per week. Does this information suggest that leisure time is a “normal” good? Explain.

4. Holding non-labor income at the original $300, determine and illustrate what happens to hours of work and leisure time if the market wage increases to $30 per hour. You need only show the ‘Total Effect’ of the wage change.

5. Given the information in d., illustrate the worker’s labor supply function over the $25 - $30 wage change. Explain which effect (SE or IE) is dominant.

Answer 1

a) Budget Line : Total income- income from labor+ income from non-labor: Y=wL+M, where L is labor hours and M is income from non labor

Labor hours= total time-leisure time= 70-LT

Y= 25(70-LT)+300 = Y=2050-25LT

Graph is shown below:

b) Equilibrium quantity is determined by setting up a lagrange function, where we maximize consumer utility with respect to the budget constraint:

Putting the last equation in the budget line, we get: Total LT= 49.2; L= 70-49.2= 20.8; Y= 820

Graphically shown below:

c) If M falls to 200, budget line shifts inwards. New BL is Y= 1950-25LT. Since the slope doesn't change, there is only income effect. With the new BL, we find that new LT is 46.8 and new Y is 780. Since LT falls when total income falls, we infer that LT is a normal good.

Graphically shown below (new BL, new utility curves and new numbers are shown in black):

d) Here, the slope of BL changes. Incorporating new changes, utility maximizing quantities are determined below. The mechanism is the same as previous one:

New maximizing LT is 48, new L is 70-48= 22. So, working hours have increased with increased wage rate.

Graphically shown below:

e)Labor supply function is graphically shown below. It is a positive sloping curve. As wage rate increases, leisure time becomes more expensive, so the individual is shifting to supply more labor. Hence, substitution effect is greater.

LS function: Y-30= (30-25/22-20.8)(X-22): wages= 4.167Labor-61.674