Question

Consider an individual that has a utility function: U = 3 G^5 L^5 where G is units of goods consumed in a day and L is hours of leisure enjoyed in a day. The individual has a time endowment of 16 hours for the day. The price of goods is $5/unit. a) What is the reservation wage of this individual when non-wage income is $10? b) What is the reservation wage of this individual when non-wage income is $100? c) How much labor will this person offer to the market and how many goods will be consumed when non-wage income is $10 and the wage is $10/hour? d) How much labor will this person offer to the market and how many goods will be consumed when non-wage income is $100 and the wage is $20/hour? e) Now consider that the individual (with non-wage income of $100 and a wage of $20/hour) receives an overtime premium of 50% after 6 hours of work per day. How much labor will this person offer to the market and how many goods will be consumed?

Answer 1

The reservation wage is $130 when non-wage income is $10 and $0 when non-wage income is $100. When non-wage income is $10, the person will offer 14 hours of labor and consume 48 units of goods. When non-wage income is $100, the person will offer 6 hours of labor and consume 44 units of goods. If the wage is $20/hour with an overtime premium of 50% and the non-wage income is $100, the person will offer 16 hours of labor and consume 98 units of goods.

Step-by-step explanation:

To find the reservation wage of this individual, we need to equate their utility from work to their utility from leisure.

a) When non-wage income is $10:

U = 3G^5L^5

Non-wage income = wL

10 = 5L

L = 2

If the individual works for 16 - L = 14 hours, their total income will be 14w = 140. Subtracting non-wage income, their reservation wage will be 140 - 10 = $130.

b) When non-wage income is $100:

Non-wage income = 100

100 = 5L

L = 20

Working hours = 16 - L = 16 - 20 = 0 (The person does not need to work at all because non-wage income is enough to meet their needs.)

Reservation wage: $0

c) When non-wage income is $10 and the wage is $10/hour:

Non-wage income = 10

10 = 10L

L = 1

G = (non-wage income + wage * hours worked) / price of goods = (10 + 10 * 14) / 5 = 48

The person will offer 14 hours of labor to the market and consume 48 units of goods.

d) When non-wage income is $100 and the wage is $20/hour:

Non-wage income = 100

100 = 10L

L = 10

G = (non-wage income + wage * hours worked) / price of goods = (100 + 20 * 6) / 5 = 44

The person will offer 6 hours of labor to the market and consume 44 units of goods.

e) When non-wage income is $100 and the wage is $20/hour with an overtime premium of 50%:

Overtime hours = Total hours worked - 6 = 16 - 6 = 10

Regular wage = 20 * 6 = 120

Overtime wage = 20 * 1.5 * 10 = 300

Total wage = Regular wage + Overtime wage = 120 + 300 = 420

Non-wage income + Total wage = 100 + 420 = 520

G = (non-wage income + wage * regular hours worked + overtime hours * overtime wage) / price of goods = (100 + 20 * 6 + 10 * 30) / 5 = 98

The person will offer 16 hours of labor to the market and consume 98 units of goods.