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Question 4 [MIT] Consider a society S consisting of 2M students, and M schools each of size 2. Each student has a preschooling human capital 6,. The post-schooling human capital of a student 3' with pre-schooling human capital 8,; attending school 3' is NIB: l (643) 2 where em- is the pre-schooling human capital of the student other than 2' attending school 3'. (Thus if two students each with pre-schooling capital of 2 enter a school, they each leave with post-school capital of 4. hi = (81;) a. Show that, to maximize the sum of post-schooling human capitals in society, 21' E Shi, the two students with the highest preschooling human capital should be allocated to one school, then the next two students with the highest pre-schooling human capital should be allocated to another school and so on. b. Suppose that parents maximize wh p, where w is the wage-rate and p is their cost of schooling. Show that the following pricing strategy will maximise the sum of post- schooling human capitals: each school chooses some cost of attendance (tuition) pj and accepting all students who want to attend at this price. [Suggestionz Consider a given student with pre-school capital :12. Find the increase in his post-school income when he goes to school with a peer of ability 3;, and when he goes to school with a peer of ability :1; + A. How much is he willing to pay for the extra A? How does this willingness change with 8?] c. What could go wrong with this decentralization scheme? d. Now suppose that the post-schooling capital is given by 1 1 h, = 26,- + e_, e?ei Can this allocation be decentralized in the same manner as in part b.? If not, why not