Nazarbayev University

• EMEM 253

• ECON 319 - Matching Theory and Applications Nazarbayev University Department of Economics Spring 2021 Quiz 3 – Solution Suppose that we have a set of woman W and a set of man M where |W| = |N| = n. Suppose that individuals differ only in their education level, which can be denoted by x1 < x2 < · · · < xn for the women, and y1 < y2 < · · · < yn for the men; that is, higher indices denote higher types. Let f(xi , yj ) denote the payoff of woman with type xi when coupled with man of type yj , and g(yj , xi) denote the payoff of man with type yj when coupled with woman of type xi . Suppose that f is increasing in y and g is increasing in x; that is everyone prefers higher types as partners. 1. [2 pts] Suppose that there is no possibility for transfers; i.e. we are under strict non-transferable utility environment. What is the stable matching? What is relation between the assumptions on f and g and the stable matching. Is this also optimal given that the environment is non-transferable. (Hint: Optimality requires it to be not dominated by any other matching; keeping non-transferable utility assumption on.) Solution: Notice that with strict non-transferable utility case, the ordinal preferences would simply yield the stable matching. Note f is increasing in y, which means, any women x prefers man with higher y. Thus the preference of woman x would be P(x) ≡ yn x yn−1 x · · · x y1 for any x ∈ {x1, · · · , xn}. With a parallel argument, the preference of man y would be P(y) ≡ xn y xn−1 y · · · y x1 for any y ∈ {y1, · · · , yn}. One can simply see that there is a unique stable matching µ of this environment where µ(xi) = yi for all i = 1, 2, · · · , n which is Positive Assortative Matching (PAM). Note that the driving result for the stable matching to exhibit PAM is that the individual output functions f and g are increasing in the partner’s type. Recall that with nontransferable utilities, stable matchings are optimal as the stable matchings are in the core in such case. 2. [3 pts] Now, define h(xi , yj ) = f(xi , yj ) + g(yj , xi) as the total surplus of the matching between xi , and yj . Suppose also that h is increasing in xi and in yj (separately). Is this enough information to determine what the optimal (or stable) matching of the transferable utility case? If not, what else is necessary to conclude the optimal (stable) matching to exhibit PAM, and what is needed for NAM. Please briefly explain. Solution: Notice the fact that f is increasing in y does not guarantee that h also does, as we are not given with the information about how g moves with y. For the same reason, we could not conclude that h is increasing in x either. Thus the assumption h(xi , yj ) increasing in xi , and also increasing in yj is not redundant. We know that the necessary and sufficient condition for the optimal matching to exhibit PAM and NAM that h is supermodular, and submodular, respectively. Now, h(xi , yj ) increasing in xi only means that h(x, yj ) >