Question

Consider a matching model where the marital output is h(x,y) = 5x + 2xy, where x is the male characteristic, and y is the female characteristic. The utility from being single is assumed to be zero. Suppose that in this marriage market there are only three men, x1 = 8, x2 = 6 and x3 = 4, and two women, y1 = 9 and y2 = 2. Consider a non-transferable utility (NTU) framework where the marital output is shared as such: men get 75%, women get 25%. a. i. Compute the appropriate matrix of marital outputs. Show your computations. ii. Explain who marries whom, assuming that women propose: does the equilibrium lead to PAM (positive assortative mating) or NAM (negative assortative mating)? (50%) b. Will there be any men or women who remain unmarried? Who and why? (15%) c. Now assume that a new woman enters the market, and that her y3 = 3. i. Find the new equilibrium assuming that women propose: who marries whom, whether anybody remains unmarried. ii. Show whether men or women are made better off or worse off by the above change, and explain why. (35%)

Answer 1

Using the given matching model, the appropriate marital outputs have been computed, leading to positive assortative mating where each woman matches with the man that provides her the best outcome. With the addition of a third woman in the market, all individuals become matched, with men relatively worse off and women better off due to the new arrangements.

Step-by-step explanation:

To compute the marital output, we use the formula h(x,y) = 5x + 2xy, and the sharing rule in the non-transferable utility framework which gives men 75% and women 25% of the output. The men have characteristics x1 = 8, x2 = 6, and x3 = 4, and the women have characteristics y1 = 9 and y2 = 2. Below is the computed matrix of marital outputs:

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• x1, y1: h(8,9) = 5(8) + 2(8)(9) = 40 + 144 = 184 (Men: 138, Women: 46)
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• x1, y2: h(8,2) = 5(8) + 2(8)(2) = 40 + 32 = 72 (Men: 54, Women: 18)
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• x2, y1: h(6,9) = 5(6) + 2(6)(9) = 30 + 108 = 138 (Men: 103.5, Women: 34.5)
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• x2, y2: h(6,2) = 5(6) + 2(6)(2) = 30 + 24 = 54 (Men: 40.5, Women: 13.5)
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• x3, y1: h(4,9) = 5(4) + 2(4)(9) = 20 + 72 = 92 (Men: 69, Women: 23)
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• x3, y2: h(4,2) = 5(4) + 2(4)(2) = 20 + 16 = 36 (Men: 27, Women: 9)

Assuming women propose, y1 will prefer x1 over x2 as her outcome with x1 is higher. Similarly, y2 will prefer to match with x2 over x3 for a higher outcome. This results in positive assortative mating (PAM) as the woman with the highest characteristic matches with the man with the highest characteristic. x3 remains unmatched as there are more men than women.

Introducing a new woman y3 with characteristic 3 into the market, y1 still marries x1, and now y3 marries x2 (as y3 has a better outcome with x2 than with x3), and y2 with x3. In this case, all individuals are matched, but whether men or women are made better or worse off depends on the specific utility gained or lost due to the changes. Introducing y3, men are relatively worse off as they end up with partners of lower characteristics than before, while women are relatively better off as they secure a match.