Labor Economics Mike’s utility for money income (M) and leisure (L) is U(M, L) =

Question

Labor Economics

Mike’s utility for money income (M) and leisure (L) is U(M, L) = M^1/3*L. There are 16 hours in a day that can be dedicated to work or leisure. Mike earns $12 per hour. Assume Mike has no unearned income.

Given Mike’s utility function, it can be shown that his marginal utility of leisure is MUL = M^1/3 and his marginal utility of money income is MUM = L/(3M^2/3) .

1. What is Mike’s marginal rate of substitution between leisure and money income (MRSL,C)?

2. Assuming Mike is at an interior optimal point, what is his optimal amount of money income and leisure per day? How many hours does he choose to work per day?

3. We can be sure that Mike does indeed choose to be at an interior point (that is, working a positive amount and taking a positive amount of leisure) by comparing his utility at the interior optimal point with his utility at each corner point. What is Mike’s utility at the interior optimal point in Part #2? What is his utility if he works all 16 hours a day (so that L = 0)? What is his utility if he does not work at all (so that L = 16)? As a result, are we sure that Mike will choose an interior point?

4. Now suppose that Mike’s wage increases to $18 per hour. Now what is his optimal amount of money income and leisure per day? How many hours does he choose to work per day? When Mike’s wage changes from $12 to $18, does the income or substitution effect dominate?

5. Continue to suppose that Mike’s wage is $18 per hour. Suppose that the government starts a welfare policy that pays $B per day to non-workers (and pays $0 per day to workers). At what value of B will Mike opt out of the labor force in order to go on welfare? (Hint: Mike ranks any two possible combinations of M and L by comparing the utility levels associated with each alternative).

6. (tough) Suppose the government sets B = 50. Will Mike work at $18 per hour? Now what is the lowest wage that will induce Mike to go back into the labor market? (Hint: this is the wage at which, if Mike works, he gets the same utility as when he is on welfare.)

Explanation / Answer

ans 1. Mike’s marginal rate of substitution MRSLM= MUL/MUM = M1/3/ (L/3M2/3) = 3M/L

ANS 2.For optimal amount of money income and leisure per day

MRSLM = w

3M/L = 12

M= 4L   

H+L=16

M/w + L=16 M/w= WORKING HOURS, H AND L= LEISURE HOURS

4L/ 12 +L =16

16L/12 =16

L=12

H=4

M= w*H =12* 4 = 48

L=12 AND M=48 IS OPTIMAL POINT

ANS 3 Mike’s utility at the interior optimal point in Part #2 , U =M1/3*L = (48)1/3*12= 43.61

his utility if he works all 16 hours a day (so that L = 0), U= 0

his utility if he does not work at all (so that L = 16) ,U= 0 AS M=0

SO INTERIOR OPTIMAL POINT GIVES MOR UTILITY TO HIM SO ALWAYS PREFERRED

ANS 4

For optimal amount of money income and leisure per day

MRSLM = w

3M/L = 18

M=6L

H+ L= 16

M/w + L= 16

6L/18 +L =16

24L/18 =16

L=12

M= 6(L) = 6(12)= 72

AT WAGE =18, HOURS FOR WORK = 16-L= 16-12 =4

The substitution effect theorizes that an increase in wages will increase the number of hours an individual is willing to work. According to the theory, when wages increase cost of leisure increases, so workers substitute work for leisure.

The income effect theorizes that an increase in wages will decrease the number of hours an individual is willing to work.

SO HERE INCOME EFFECT DOMINATES THAT'S WHY NO. OF WORKING HOURS DECREASES AS WAGE RATE INCREASES.

Ans 5

if mike goes for welfare policyand doesn't work, than his income M= B

utility of mike due to this will be U1= B1/3 *16

if he works at w=18 for H=4, L=12 AND M= 72

UTILITY U2= (72)1/3* 12 =49.92

FOR THAT VALUE OF B FOR WHICH U1 =U2 , MIKE WILL OPT OUT OF LABOR FORCE

B1/3 *16 = 49.92

B=30.37 IS VALUE OF AT WHICH MIKE IS INDIFFERENT

B> 30.37 IS VALUE OF B FOR WHICH   , MIKE WILL OPT OUT OF LABOR FORCE

ANS 6.

AT B=50

UTILITY, U3 = (50)1/3* 16

= 58.95

WHEREAS UTILITY AT w=18 IS 49.92

SO AT B=50 , MIKE WILL NOT WORK AT w= 18

FOR LOWEST WAGE  that will induce Mike to go back into the labor market CAN BE CALCULATED BY

KEEPING UTILITY ON THAT WAGE RATE = UTILITY WHEN HE IS ON WELFARE

M1/3*L = 58.95

(wH)1/3*L = 58.95

w1/3 = 58.95/ (L* H1/3)

w =(58.95/ (L* H1/3))3 where L+ H= 16

AFTER FINDING OUT w FROM ABOVE EXPRESSION, THIS IS WAGE RATE ATLEAST REQUIRED TO INDUCE MIKE TO WORK

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