Question

Julie employs Katrina as her assistant. Katrina is a utility maximizing worker with u(z, f) = z^1/2f^1/2 and MRS = f/z. Katrina has non-labour income 40 and allocates 24 hours per day between labour h and leisure z. Julie pays Katrina a wage of 5 dollar per hour. Provide an Indifference Curve and Budget Line Diagram to illustrate and quantify, hours of leisure, hours of work and consumption. Katrina is over working. What hourly wage would make Katrina voluntarily cut back to exactly 2 hours per day? Show this in your diagram. The wage that makes her work 2 hours per day is equal to:

Answer 1

u(z, f) = z^1/2f^1/2

leisure is denoted by f, so consumption is denoted by z

Katrina has non-labour income $ 40

If Katrina works for an entire 24 hours, then she can earn = 24* $5 = $120

Therefore, total income of Katrina = labour income + non-labour income = $120 +$40 =$160

Price of leisure (Pf) = opportunity cost of leisure = wage rate = $5 (because she has to give up one hour of working to enjoy 1 hour of leisure)

Price of consumption is taken to be $1 (Pz)

Katrina's MRS is already given MRS = MUz/MUf = f/z (z or consumption is taken on x-axis in the question that is why MRS = f/z and not z/f)

At equilibrium, MRS = price ratio,

f/z = Pz/Pf

f/z = 1/5

5f = z

Put the value of z in the budget equation of Katrina:

price of consumption * units of consumption + price of leisure * units of leisure = income

1 * 5f + 5 * f = 160

10f = 160

f=16

Therefore, Katrina works for (24-16) = 8 hours.

If Katrina works for 8 hours, then she earns = 8*$5 = $40. Adding $40 non-labour income, she can consume worth $80. This is shown in figure 1.  Indifference Curve is IC1 and Budget Line is the blue line in figure 1. When lesiure = 24 hours, still Katrina can consume non-labour income amounting $40 (hence the gap). Katrina's consumption is $80 when she works for 8 hours.

Let the wage that makes her work 2 hours per day be w.

Katrina's income now = 24*w +40

Then,

f/z = 1/w

wf = z

Put the value of z in the budget equation of Katrina:

price of consumption * units of consumption + price of leisure * units of leisure = income

1 * wf + 5 * f = 24*w +40

Since she works only two hours, then she enjoys 22 hours of leisure, or f = 22 hours

22w + 5*22= 24*w +40

2w=70

w = $35

Therefore, the wages that makes Katrina work 2 hours per day is equal to $35

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