Research suggests that successful performance on exams requires preparation (that is, studying - L) and rest (that is, sleep - S). Neither by itself produces good exam grades, but in the right combination they maximize your exam performance. We can then model your exam grades as emerging from a production process that takes hours of studying and hours of sleep as inputs. Suppose this production process has increasing returns to scale.
a) On a graph with hours of sleep on the horizontal axis and hours of studying on the vertical, illustrate an isoquant that represents a particular exam performance level qA
b) Suppose you are always willing to pay $20 to get back an hour of sleep and $5 to get back an hour of studying. Illustrate on your graph the least cost way to get to the exam grade qA.
c) Suppose a new caffeine/ginseng drink comes on the market, and you find it makes you twice as productive when you study. What in your graph will change? Suppose that the production technology described above can be captured by the production function q = 40LS, where q is your exam grade, L is the number of hours spent studying, and S is the number of hours spent sleeping. MPL = 40S and MPK =40L.
d) Prove that this production process indeed have increasing returns to scale.
e) What is the equation of isoquant?
f) What is the equation for a slope of an isoquant? What is this called? What does it indicate?
g) Set up the cost minimization problem and solve for the conditional studying and sleeping demands as functions of pL(what you are willing to pay to get back an hour of studying) ,pS (what you are willing to pay to get back an hour of sleep), and q (your exam performance).
h) Discuss the demand functions you derived in g). Is sleep and studying normal inputs? What happens to optimal amount sleep and pS increases? What happens to optimal amount of studying as pL increases?
i) Now, assume that you are always willing to pay $20 to get back an hour of sleep and $5 to get back an hour of studying. What is your “optimal production plan”?
j) Derive the cost function and simplify the function as much as you can.
k) Continue with total cost function derived in part j) and derive the average cost. Are marginal and average cost curves for this problem upward or downward sloping? Explain. What is the relationship between MC and AC? Draw the MC and AC.
l) What is your “optimal production plan” if you wish to reach the exam performance of 160?
m) What is the cost of this 160- score performance? Now suppose your instructor clearly tells you that you have to study for 5 hours to get 160-score performance. Reconsider the “short-run” problem where studying is fixed at L .
n) What is your “optimal production plan”? What is your “optimal production plan” if you wish to reach the exam performance of 160? Draw this short run solution along with long run solution in one graph.
o) What is the short-run cost function? What is the short-run marginal cost function? Average cost function?
p) Compare cost functions from part o) with cost functions from part k).