Consider a society S consisting of 2M students, and M schools each of size 2. Each student has a pre-schooling human capital ej. The post-schooling human capital of a student i with pre-schooling human capital e; attending school j is hi = (e.)}(e-is)} where eij is the pre-schooling human capital of the student other than i attending school j. (Thus if two students each with pre-schooling capital of 2 enter a school, they each leave with post-school capital of 4. a. Show that, to maximize the sum of post-schooling human capitals in society, Eie Shi, the two students with the highest pre-schooling 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 uwh - 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) p; and accepting all students who want to attend at this price. [Suggestion: Consider a given student with pre-school capital z. Find the increase in his post-school income when he goes to school with a peer of ability y, and when he goes to school with a peer of ability y+A. How much is he willing to pay for the extra A? How does this willingness change with e?] c. What could go wrong with this decentralization scheme? d. Now suppose that the post-schooling capital is given by 1 h = 2e; + e-i- Can this allocation be decentralized in the same manner as in part b.? If not, why not?