Question

Shelly’s preferences for consumption and leisure can be expressed as U(C, L) = (C – 100)  (L – 40). This utility function implies that Shelly’s marginal utility of leisure is C – 100 and her marginal utility of consumption is L – 40. There are 110 (non-sleeping) hours in the week available to split between work and leisure. Shelly earns $10 per hour after taxes. She also receives $320 worth of welfare benefits each week regardless of how much she works.

a) Graph Shelly’s budget line.

b) What is Shelly’s marginal rate of substitution when L = 100 and she is on her budget line?

c) What is Shelly’s reservation wage?

d) Find Shelly’s optimal amount of consumption and leisure.

Answer 1

U(C, L) = (C - 100) × (L - 40)

Marginal Utility of Leisure = C-100(wrongly given as C-200 in question)

Marginal Utility of consumption = L-40

Total available non sleeping hours in a week = 110

A)

Welfare benefits received = $320

If Shelly does not works, she has 110 hours for leisure. However, she will get $320 as welfare benefits.

If she decides to work for the complete 110 hours. her wage income would be = 110*wage rate = 110*10 = $1100. Hence her total consumption, in this case, would be = wage income + welfare benefits = $1100+$320 = $1420

Moreover, the budget equation can be written as C = 320+10(110-L). This implies that consumption equals $320(welfare benefits) plus the wage income received 10*(110-L)

B)

Marginal Rate of Substitution when L = 100 and she is on her budget line

If Shelly enjoys the leisure of 100 hours, she will work for 10 hours. Her consumption(C), in this case, would be = 10*10+320 = $420

Thus MRS = MUL/MUC = (C-100)/(L-40) = (420-100)/(100-40) = 320/60 = 5.33

C)

When Shelly does not work and enjoys leisure for 110 hours, her consumption is $320. Thus the reservation wage can be defined as MRS when she is not working. Thus the wage rate must be equal or greater than this to induce her to work.

MRS = (C-100)/(L-40) = (320-100)/(110-40) = 220/70 = $3.14

D)

The optimal amount of consumption and leisure can be calculated by equating the MRS with the wage rate.

MRS = w

(C-100)/(L-40) = 10

C = 10L-300

Put the value of C in the budget equation, C = 320+10(110-L)

10L-300 = 320+10(110-L)

10L-300 = 320+1100-10L

20L = 1720

L* = 86

Put the value of L* in budget line we get,

C = 320+10(110-86)

C = 320+240

C* = 560