Veronica’s preference for consumption and leisure is given by U(C, L) = (C200)(L

Question

Veronica’s preference for consumption and leisure is given by U(C, L) = (C200)(L80) so her marginal utility of leisure MUL = (C 200) and her marginal utility of consumption is given by MUC = (L80). There are 168 hours available in the week to split between work and leisure. She earns 5 dollars per hour after tax.

a) what is the budget constraint

b) Optimal numbe rof leisure and consumption

c) Now she receives 320 dollars worth of welfare benefits each week regardless of how much she works as part of the government program. Graph her new budget constraint.

d) How many hours would she work under the new welfare program?

Explanation / Answer

a) The worker’s consumption C is given by

            C = wh + N

Where C is the worker’s consumption

            h is the number of hours worked

            N is the non-labor income

            w is the hourly wage rate

Since the number of hours worked is total endowment of time minus leisure hours, the worker’s consumption C is given by

            C = w (168 – L) + N

            C = 168(5) – 5L + N

            C + 5L = 840 + N                               …. (1)

which is the budget constraint. Here, N is zero.

b) Calculate MRS.

            MRSCL = MUC/MUL

                                = (L – 80)/(C – 200)

At the optimal level,

            MRSCL = - slope of the budget constrain

            (L – 80)/(C – 200) = - dC/dL

            (L – 80)/(C – 200) = - (-5)

            (L – 80) = 5C - 1000

            L = 5C – 920                                       ….. (2)

Substitute this result in eq. (1).

            C + 5(5C – 920) = 840 + N

            C + 25C – 4600 = 840 + N

            26C = 5440 + N

Since N is zero, the above equation solves to C = $209.23.

Substitute 209.23 in (1).

            209.23 + 5L = 840 + N

Since N = 0,

            5L = 630.77   

            L = 126.15

Therefore, the optimal level of leisure is 126.15 hours and the optimal level of consumption is $209.23.

c) The non-labor income is not zero now; it is $320. So the budget constraint is

C + 5L = 840 + 320

C + 5L = 1160                                                …. (3)

The following figure shows the budget constraint.

When the non-labor income is zero, then the maximum consumption can be found by substituting 0 for leisure and 0 for N in eq. (1).

            C + 5(0) = 840 + 0

            C = 840

The maximum leisure is of course 168 hours. Therefore AB is the budget line when the non-labor income is zero.

When the non-labor income rises from zero to 320, then the budget line shifts upwards by $320. Therefore, the new budget line is DE.

d)

To find the optimal bundle, Substitute eq. (2) in eq. (3).

            C + 5(5C – 920) = 1160

            C + 25C – 4600 = 1160

            26C = 5760

The above equation solves to C = $221.54. This is not possible, as the consumption cannot be less than $320 now. This means there is a corner solution.

The two corners of the budget constraint are (C = 320, L = 168) and (C = 1160, L = 0).

At (C = 320, L = 168), the utility is

            U         = (320 – 200) (168 – 80)

                        = 10560

At (C = 1160, L = 0), the utility is

            U         = (1160 – 200) (0 – 80)

                        = -76,800

Utility is greater at (C = 320, L = 168). Therefore under the new welfare program, the consumer shall not work, as he wants a leisure equivalent to its total time endowment.

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