Problem Set 1-answers PDF

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Economics 340 Economics of the Family

Matthias Doepke Spring 2016

Problem Set 1 Answers Comparative Advantage in Home Production Question 1: Consider a married couple who currently both work and do some housework. The husband is working 60 hours a week and his salary is $1200 per week. He also does 10 hours of housework a week (cooking, laundry, etc . . . ) which is worth $100 (replacement cost that he would need to pay to outsource these activities). The wife is working 20 hours a week and her salary is $300 per week. She does most of the housework, spending 50 hours on this per week week (childcare, cooking, etc . . . ) which is worth $500. 1. Find the hourly wage of husband and wife. Who has the absolute advantage in market work? Answer: Hourly wage of husband is $20, hourly wage of wife is $15. Since husband can earn more in 1 hour, he has absolute advantage in market work. 2. Find the hourly replacement cost of housework activities for husband and wife. Answer: Hourly replacement cost of housework activities is the same for husband and wife (note that this is not the opportunity cost) and it is $10. 3. Draw the budget constraint for husband and wife (market consumption measured in dollars on one axis, value of housework measured in dollars on the other axis). Hint: The total available time is 70 hours per week.

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4. Who has comparative advantage in market work and in housework? Is this initial allocation efficient, and why? Answer: Husband has comparative advantage in market work, wife has comparative advantage in housework. Note that husband can either earn $20 in 1 hour or do housework worth $10 in 1 hour. Hence opportunity cost of $20 is $10 worth housework, so opportunity cost of $1 in the market is $1/2 worth housework. For wife, oppotunity cost of $15 is $10 worth housework, so opportunity cost of $1 in the market is $2/3 worth housework. Hence, opportunity cost of market work is lower for husband (1/2 1152 = HA + HZ . Note that even though they devote the same amount of time for cooking and they have the same income to buy ingredients, by only cooking together, they can double the total amount of food because there are increasing returns to scale in food production. Now, they think that they can benefit even more if they specialize. Suppose their objective function is total amount of homemade food (and they are sharing equally) and they are choosing how many hours each will be cooking and working. 6. Write down the maximization problem. Do not solve it! Answer: maxtA ,tZ [wA (16 − tA ) + wZ (16 − tZ )](tA + tZ ) 7. Can you find an allocation which is better than the one in 5? Explain the conditions required for a better allocation. Answer: Since Ali has a higher wage, he has comparative advantage in market production. Any allocation where Ali works more will be better for them as Zeynep can do more cooking. Note that this was not possible when they are living separately because maximization problem shows that whatever the wage is they should allocate half of their time for working, half for cooking. Now, we see returns to specialization by exploiting comparative advantage. Suppose Ali works 16 hours and does not cook at all and Zeynep only cooks (tA = 0, tZ = 16). Ht = (10 ∗ 16) ∗ 16 = 2560 > 2304. Thanks to returns to specialization (due to comparative advantage), they can have more food when they specialize. Note that the increase in total amount of food in part (5) was due to increasing reurns to scale, the increase in part (7) is due to comparative advantage. 4

The Marriage Market Question 3: Consider the marriage market model that we analyzed in class. There are 50 women and 70 men in the market, F = 50 and M = 70. The utility of being single for woman i is given by zfi = 2i, where i ∈ {1, 2, . . . , 50}. The utility of being single for man i is given by zmi = i, where i ∈ {1, 2, . . . , 70}. The joint surplus of a married couple is zf m = 120. 1. Draw the demand and supply of marriage as a function of the wife’s share of marital surplus sf . Hint: Woman i wants to marry if sf zf m ≥ zfi = 2i. To compute the supply of marriage, you may want to first consider at what sf even the most productive woman would want to marry, i.e., sf zf m = zfF = 2 × 50, and then use the fact that the supply of marriage N is linear in sf for a N < F . Answer: The most productive woman wants to marry if the return on marriage is bigger than the utility of being single, hence if 120sf ≥ 2 ∗ 50 or sf ≥ 10/12. We know that is women’s surplus share is 0, even the least productive woman does not want to marry hence F = 0, if sf ≥ 10/12 all women want to marry hence F = 50. Hence we can draw supply function by using these informaF if F < 50, sf ∈ (10/12, ∝)if F = 50. Now , think about the most tions: sf = 60 productive man: he wants to marry if return on marriage is bigger than the utility of being single. He marries if 120(1 − sf ) ≥ 70. Hence, if sf ≤ 5/12 all men want to get married, i.e. M = 70. We also know that if sf = 1, even the least productive man would not want to marry hence M = 0. Hence we can draw demand funcM if M < 70, sf ∈ (0, 5/12)if M = 70. tion by these information: sf = 1 − 120

2. Compute the equilibrium surplus shares sf and sm and the equilibrium 5

number of marriages N . Hint: If N < F and N < M, woman i = N and man i = N should be just indifferent between being married or single, so that we have the equilibrium conditions: z fN = 2N = sf 120, N zm = N = sm 120 = (1 − sf )120,

which should be solved for sf and N . Answer: By using supply and demand equations above, we can find equilibrium number of marriages where N ∗ = M = N N , hence N ∗ = 40, when we replace it either F . In equlibrium, sf = 60 = 1 − 120 demand or supply equation, we find that sf ∗ = 2/3.

3. Explain why not everyone ends up married. Who gets the higher share of marital surplus, and why? Answer: Not everyone ends up married because for the equilibrium surplus share where demand equals supply, 10 of most productive women have higher utility of being single. Women het higher share of marital surplus because they have higher utility of being single. In order to make them willing to get married, men should give a higher share of surplus. 4. Now consider the case of a higher marital surplus of zf m = 240. Draw the demand and supply of marriage and compute the marriage market equilibrium. Hint: If all women are married, the division of marital surplus is such that the last man to get married is just indifferent between marriage and single life. Is the distribution of marital surplus different than in the previous 6

F if F < 50, case? Explain why. Answer: Now, supply equation will be sf = 120 M if M < 70, sf ∈ sf ∈ (5/12, ∝)if F = 50. Demand equation will be sf = 1 − 240 (0, 17/24)if M = 70. Note that these lines intersect at N ∗ = 50, when we replace N = 50 in demand equation, we get sf ∗ = 19/24. Note that with this equlibrium share,the last man to get married is indiffirent between marriage and single life. The distribution of surplus is different than before, now women get even a higher share.

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