Question

Consider a couple (a husband and a wife) that jointly represents their collective preferences between combinations of household production time (X) and purchased goods and services (Y) according to the formula W = X²Y, where W represents the level of welfare. Suppose the maximum time available in a day is 18 hours and currently the wife devotes 6 hours to market work (H) at a wage of $15 per hour.

a. What is the level of welfare associated with the wife’s current situation?

b. How much additional purchasing power would the wife contribute if her market work hours increased to 10?

c. How much of an increase in purchased goods and services would be necessary to compensate for the additional 4 hours of lost household production?

d. Should this couple choose to have the wife increase her market work by 4 hours?

e. If this couple is raising a child, suppose that combinations of household production and purchased goods are now ranked according to the formula U = X³Y. Would the additional 4 hours of market work be a good move for the family in this situation?

Answer 1

a) Currently, the wife is working for 6 hours out of the available 18 hours in a day. Her per hour wage is $15. So, for 6 hours she earns $90. This is Y.

The remaining after she has worked for 6 hours = 18-6 = 12 hours, she devotes to the household production time (X).

W = X²Y = 144*90 = 12,960.

b) Now, the market work hours has increased from 6 hours earlier to 10 hours. In 10 hourse, she earns $15*10 = $150.

So, additional purhcasing power that she contributed when she works for 10 hours= $150-$90 = $60.

c) How much of an increase in purchased goods and services would be necessary to compensate for the additional 4 hours of lost household production?

Now, time left with her after working for 10 hours = 18-10 = 8 hours. In order to be as well off as they were earlier, W = 12,960

12,960 = X²Y

12,960 = 8²Y

Y = 12,960/84 = 202.5

Therefore, in order have same welfare, purchasing power should increase from $90 (income when she worked for 6 hours) to $202.5 (income when she works for 10 hours) , or an increase of $112.5

d) In order to be as well off as they were when the wife worked for 6 hours, their purchasing power should increase by $112.5. But as we found out in part b of this question, in reality the purchasing power goes up by only $60. When purchasing power is $150 and X= 8, then

W= X²Y = 64*150 = 9,600

Since welfare has decreased, this means that the wife should not work for additional 4 hours.

e) Now, U = X³Y

Initially When X = 12 hours and Y = $90, then W = 1728*90 = 155,520

Now, X=8 hours.

155,520 = 512*Y

Y= 303.75

In order have same welfare, purchasing power increase from $90 (income when she worked for 6 hours) to $303.75 (income when she works for 10 hours) , or an increase of $213.75

If she works for 4 hours more, she gets $60 but this is less than what she needs (increase of $213.75) in order to have same welfare. Therefore, the additional 4 hours of market work would not be a good move for the family in this situation.

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