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 + 3 X 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 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
270 Exercises
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 fi nding 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).
Sample Solution