Question 4 (21 points)
Martin Marvel publishes comic books. The only inputs he needs are superhero character ideas (a form of
capital) (K) and hours of cartoonists' labor (L). His production function for comic books (q) is
q = f(K, L) = 1.5 \cdot K^{1/2} \cdot L^{3/4}
Assume that superhero character ideas can be purchased for $27 each and the hourly wage of cartoonists'
labor is $18.
a. (0 points) For practice (using the skills you developed in the previous problem set) you can verify that in
the long run Martin's labor demand is $L(q) = q^{4/5}$, capital demand is $K(q) = q^{4/5}$, and his long run cost
function is $C(q) = 30q^{4/5}$.
b. (3 points) Determine Martin's long run average cost function, $AC(q)$ and long run marginal cost function,
$MC(q)$. Does the production process exhibit economies, diseconomies, or constant economies of scale?
c. (4 points) Suppose that in the short run Martin is stuck with exactly 324 superhero character ideas, but
is able to hire as much labor as he wishes. What is Martin's short run total cost function, $SC(q)$?
d. (2 points) What are Martin's short run average cost function, $SAC(q)$, and short run marginal cost
function, $SMC(q)$?
e. (2 points) Use Martin's long run demand for capital from part (a) to determine the level of output for
which the long run demand for capital is exactly equal to the short run fixed level of capital in part (c).
f. (4 points) Compare $AC(q)$ and $SAC(q)$ at the output you found in (e). In addition, compare $MC(q)$ and
$SMC(q)$ at this output. Interpret your findings.
g. (6 points) Provide a single plot showing four clearly labeled curves: $AC(q)$, $MC(q)$, $SAC(q)$, and $SMC(q)$,
with $0 \le q \le 12,000$ on the x-axis and $0 \le \$/q \le 10$ on the y-axis. I encourage you to use a free
online graphing calculator (like Desmos) to plot these functions and include the electronic plot as your
answer. Alternatively, you can sketch the curves by hand, but they must have the correct shapes (increasing,
decreasing, U-shaped, etc.). Be sure that your curves are consistent with your insights from part (f). Hint:
This picture should look similar to the envelope concept discussed in class.
