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:
S poor = 40 + X poor
S rich = 45 + 3X rich
(a) Plot the above relationships and interpret the differences between the slopes and intercepts in intuitive terms. Do you think that the difference in the "productivity" of additional educational spending between rich and poor reflected in the above formulas is realistic? (To answer, you might focus on the differences in home life for the groups and differences in the availability of extra-curricular enrichment activities.)
(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 transformation curve is a straight line. (Hint: You should be able to find the transformation curve solely by locating its endpoints.)
(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 maximize its overall average test score, which equals (Spoor + Srich)/2.
How should it allocate the grant money? Find the answer by using a diagram that extends the iso-crime line approach from the chapter. (You will not get full credit for finding the answer by trial-and-error number crunching, although you are welcome to include such numbers along with your diagram). Using the results of (a), explain why your answer comes out the way it does.
(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. How should the grant money be allocated to maximize this function(1/5)S poor + (S rich)^(1/2)? (here, you can crunch some numbers, or use calculus if you like). Show the solution in your plot and contrast it to that from (c).