A person chooses between leisure and consumption. The utility derived from any c

Question

A person chooses between leisure and consumption. The utility derived from any combination of leisure and consumption is given by the formula:

U=LC-88C

where U is utility, L is number of leisure hours per week, and C is the number of dollars spent on consumption per week. This person can work as many hours as desired each week (T=168) at a wage of $4 per hour. There is no other source of income.

A) How many hours does this person choose to work? How much does this person spend on consumption?

B) Suppose the wage increases to $8 per hour. Would this person choose to supply more labor at $8 than at $4? How much does this person spend on consumption?

C)Now, instead of an increase in the straight-time wage, suppose overtime is offered at $8 per hour after working 40 hours at $4 per hour. Will this person accept the overtime,and, if so, for how many hours?

Explanation / Answer

The utility function is U=LC-88C and so the marginal rate of substitution is MUL/MUC = C/L - 88. The budget line has an equation C = Y or C = (168 - L)*4 which is simplified to 4L + C = 672. Now this has a slope of price coefficient of L divided by price coefficient of C which gives slope = 4/1.

a) Now this implies at the optimal choice we have C/L - 88 = 4 or C = 4L - 352. Use this in the budget equation so that we have

4L + 4L - 352 = 672 or L = 128 hours and working hours = 168 - 128 = 40 hours

For about 40 hours does this person choose to work and he spend 40*4 = $160 on consumption

b) When wage is rate is $8, we have a budget line has an equation C = (168 - L)*8 which is simplified to 8L + C = 1334. Now this implies at the optimal choice we have C/L - 88 = 8 or C = 8L - 704. Use this in the budget equation so that we have

8L + 8L - 704 = 1334 or L = 128 hours and working hours = 168 - 128 = 40 hours

For about 40 hours does this person choose to work and he spend 40*8 = $320 on consumption. He supplies the same labor hours at an increased wage rate.

c) For a number of working hours below 40, the budget equation is 4L + C = 672 for 40 < or = L < or = 128. But for working hours greater than 40, he has C = 40*4 + (128 - L)*8 or C + 8L = 1184. Again use C = 8L - 704 and find that 8L - 704 + 8L = 1184 or L = 118 hours and working hours = 168 - 118 = 50 hours. For about 50 hours does this person choose to work and he spend 40*4 + 10*8 = $200 on consumption. He accepts overtime.

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