Suppose that a city operates two neighborhood schools, one in the rich neighborhood and one in the poor neighborhood. The schools are equal in size and currently have equal budgets. The city receives $10 million in federal grant money that can be used to supplement the budgets of the two schools, which are initially identical. For each school, the average score on a standardized achievement test depends on how many dollars are allocated to the school. Letting S denote the average test score and X denote additional spending in millions of dollars, the relationships between scores and additional spending for the two schools are as follows: Spoor = 40 + Xpoor , Srich = 45 + 3 Xrich.
A) Plot the above relationships and interpret the differences between the slopes and intercepts in intuitive terms. Do you think that the dif-ference in the " productivity " of additional educational spending between rich and poor refl ected in the above formulas is realistic?
B) Derive and plot the community ’ s transformation curve between Srich and Spoor , remembering that Xpoor and Xrich must sum to 10. Because of the linear relationship between S and X for each group, the transfor-mation curve is a straight line.
C) Find and plot the test scores that would result if the city divided the grant money equally between the schools.
D) Find and plot the test scores that would result if the city allocated the grant money to equalize the scores across schools.
E) Finally, consider the case in which the community’s goal is to maxi- mize its overall average test score, which equals (Spoor + Srich)/2 how should it allocate the grant money?
F) What if the coefficient of Xpoor in the above formula had been equal to 2 and the coefficient of Xrich had been equal to 1.5? Without drawing any diagrams or doing any computations, you should be able to tell how the community would allocate the grant money if its goal were to maximize the average score. What allocation would it choose?
G) Suppose the community’s social welfare function is (1/5)Spoor + Srich. How should the grant money be allocated to maximize this func- tion?