Jenny has preferences given by the utility function U(C, L) = C^2 L. So the slop

Question

Jenny has preferences given by the utility function U(C, L) = C^2 L. So the slope of her indifference curve is C/2L .

Johnny has the preferences given by the utility function U(C, L) = C L. So the slope of his indifference curve at any point is C/L .

(a) Which of them has the relatively stronger preference for consumption over leisure? Explain.

(b) They can both earn $10 per hour, they both have a non-labor income of $300 per week and they have 110 hours per week of non-sleeping time. Who works the most hours? How much do each of them make per week?

(c) What are their reservation wages?

(d) Starting from their preferred choice of work hours at $10 per hour (from part 2), suppose they were offered overtime at $20 per hour how many hours of overtime would each of them want to put in? How much would each of them earn?

Explanation / Answer

a) The slope of an indifference curve is known as its Marginal Rate of Substitution (MRS). Jenny faces an MRS equals to –C/2L while Johnny faces an MRS equals to –C/L. This implies that Jenny’s MRS = ½ of Johnny’s MRS. Jenny is willing to give up 2 units of leisure to have one unit of Consumption while Johnny is willing to sacrifice only 1 unit of leisure. This suggests that Jenny’s preferences for consumption are stronger.

b) Total time of 110 hours can be used as leisure or as work and consume or a mix of both. If all that time is spent in working (and consuming) both earns and consume a weekly amount of $1,100 + $300 = $1400. If all that time is spent in attaining leisure, the total income earned would only be $300.

This implies that both face the same budget constraint which is M = wC + 300. Here w = wage rate which is $10 per hour and C is the number of working hours. Note that C = 110 – L so substitute this in the budget equation to get M = 10(100 – L) + 300. The final equation of the budget is M = 1400 – 10L which has a slope -10.

Now optimum bundle would be having the slope of budget line and slope of the indifference curve, MRS equal to each other:

MRS = w

Jenny -> – C/2L = – 10 or C = 20L

Johnny -> – C/L = – 10 or C = 10L

Substitute this allocation rule in the time constraint to get the optimal bundle:

Jenny -> C + L = 110 so 20L + L = 110 or L* = 110/21 and C* = 2200/21

Johnny -> C + L = 110 so 10L + L = 110 or L* = 10 and C* = 100

c) The reservation wage gives the minimum increase in income that would make a person indifferent between remaining at the endowment point and working that first hour. Reservation wages are the lowest wages that the worker is willing to accept for the given job. It is the absolute value of the slope of the indifference curve at the endowment point.

Jenny's reservation wage = C/2L = 300/220 and Johnny's reservation wage = C/L = 300/110

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