(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 refl ected 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 S rich and S poor ,

remembering that X poor and X rich 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 fi nd

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 ( S poor + S rich )/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 coeffi cient of X poor in the above formula had been equal to 2 and the coeffi cient of

X rich 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) S poor + S rich . How should the

grant money be allocated to maximize this function? (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)