Question 4 [MIT] Consider a society S consisting of 2M students, and M schools each of size 2. Each student has a pre-schooling human capital 6,. The post-schooling human capital of a student 2' with pre-schooling human capital e,- attending school j is man _1_ (tim) 2 where cm- is the preschooling human capital of the student other than 2' attending school j. hi 2 (6,) (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, 2,65 hi, 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 preschool capital 3:. Find the increase in his post-school income when he goes to school with a peer of ability 2;, and when he goes to school with a peer of ability 3; + A. How much is he willing to pay for the extra A? How does this willingness change with 6?] c. What could go wrong with this decentralization scheme? d. Now suppose that the post-schooling capital is given by 1 hi_2i i__ 6+6 6 ii 2 2 alei Can this allocation be decentralized in the same manner as in part b.? If not, why not