Consider the problem you face as a student as you determine how much to study for an exam by modeling yourself as a “producer of an exam score” between 0 and 100.
A: Suppose that the marginal payoff to studying for the initial hours you study increases but that this marginal payoff eventually declines as you study more.
(a) Illustrate, on a graph with “hours studying for the exam” as an input on the horizontal axis and “exam score” (ranging from0 to 100) as an output on the vertical axis, what your production frontier will look like.
(b) Now suppose that your tastes over leisure time (i.e. non-study time) and exam scores satisfies the usual five assumptions about tastes that we outlined in Chapter 4. What will your producer tastes look like. (Be careful to recognize that the producer picture has “hours studying” and not leisure hours on the horizontal axis.)
(c) Combining your production frontier with graphs of your indifference curves, illustrate the optimal number of hours you will study.
(d) Suppose that you and your friend differ in that your friend’s marginal rate of substitution at every possible “production plan” is shallower than yours. Who will do better on the exam?
(e) Notice that the same model can be applied to anything we do where the amount of effort is an input and how well we perform a task is the output. As we were growing up, adults often told us: “Anything worth doing is worth doing well.” Is that really true?
B: Now suppose that you and your friends Larry and Daryl each face the same “production technology” x = 3ℓ2 −0.2ℓ3 where x is the exam grade and ℓ is the number of hours of studying. Suppose further that each of you has tastes that can be captured by the utility function u (ℓ, x) = x −αℓ.
(a) Calculate your optimal hours of studying as a function of α.
(b) Suppose the values for α are 7, 10, and 13 for you, Larry and Daryl respectively. How much time will each of you study?
(c) What exam grades will each of you get?
(d) If each of you had 10 hours available that you could have used to study for the exam, could you each have made a 100? If so, why didn’t you?