Problem Set 8
Due 2:00PM Sep. 13
1. A firm has cost function C(y) = y 2 + y + 4, where y is the number of units of output it
produces. Let p denote the market price for a unit of output, which the firm assumes to be
invariant to its choice of output.
(i) Write down the firm’s profit function
Solution: π(y) = py − (y 2 + y + 4)
(ii) At what level of output is profit maximized?
Solution: The FOC is π 0 (y) = 0, or π 0 (y) = p − 2y + 1 = 0, or y =
2. A monopolist faces inverse demand function p(q) = 720 − 41 q 3 +
function c(q) = 540q.
19 2
3 q
p+1
2 .
− 54q, and has cost
(i) Write down the firm’s profit function
Solution: Profit is revenue minus cost, or π(q) = p(q)q − c(q). Therefore, we have
1
19
π(q) = 720q − q 4 + q 3 − 54q 2 − 540q
4
3
(1)
(ii) Show that the profit function has three stationary points, equal to q = 3, q = 6, and
q = 10.
Solution: We have
π 0 (q) = 180 − q 3 + 19q 2 − 108q
(2)
which is polynomial of degree 3, with three stationary points. You can substitute in the
values of q to verify that these are indeed stationary points.
(iii) Which of these points are minima and maxima?
Solution: We take the second derivative of π(q) to obtain π 00 (q) = −3q 2 + 38q − 108.
Substituting in for each value of q that is a stationary point, we have π 00 (3) = −21,
π 00 (6) = 12, and π 00 (10) = −28. Thus, q = 3 and q = 10 are local maxima.
1
(iv) What level of output should the firm choose?
Solution: It has to be one of q = 3 or q = 10. We can check directly by substituting
in for each value of q, so we get π(3) = 204.75 and π(10) = 233.33, so q = 10 is the
optimal choice.
3. Find a value x ∈ R that maximizes the function f (x) = x(1−x)
. Now find the value of x ∈ R
2
that maximizes the function f (x) = x(1 − x). Explain the relationship between the solutions.
Solution: Take first order conditions for each problem, and you see that the solution is
x = 21 . This a solution for both problems since the first problem is just the second divided
by a constant, which doesn’t affect the value of x that maximizes the expression.
4. For which value of x ∈ [1, 3] is the function x2 (1 − x)2 maximized. Hint: Draw the picture.
Solution: The solution is x = 3. Using FOCs will not help you, here!
5. For what values of x ∈ R is the function f (x) = 2x3 − 3x2 globally maximized?
Solution: f 0 (x) = 0 for x = 0 and x = 1, and f 00 (0) = −6, so the point 0 is a local maximum.
In fact, global maxima do not exist for this problem, since limx→∞ f (x) = ∞.
6. Prove the following claim: Let f (x) defined for x ∈ R be a differentiable function. Then, f (x)
is (strictly) convex if and only if −f (x) is (strictly) concave. Hint: go back to the differentiable
characterization of convexity and concavity given in the lecture notes!
Solution: If f (x) is convex, then f 00 (x) ≥ 0. But then, −f 00 (x) ≤ 0, which implies that
−f (x) is concave. Similarly, if f (x) is concave, then f 00 (x) ≤ 0, so that −f 00 (x) ≥ 0, which
implies that −f (x) is convex.
7. Suppose a firm has revenue function R(y), and differentiable convex cost function C(y), where
y is the level of output produced by the firm.
(i) Suppose that the firm operates in a competitive market, so that the price of good y in
the market can be written as p, i.e. it is not a function of y. Show that if R(y) = py,
the firm’s optimal choice of output equates price with marginal cost.
Solution: We have π(y) = R(y)−C(y), and the FOC is R0 (y) = C 0 (y). Since R0 (y) = p,
this yields p = C 0 (y), as we are supposed to show.
(ii) Now, suppose that the firm is a monopolist, and that the market price is a differentiable
function of output, p(y). Assume that p0 (y) < 0.
2
(a) Interpret this assumption.
Solution: The monopolist is the sole seller of good y, so the quantity he produces
has an effect on market price. In particular, the assumption p0 (y) < 0 formalizes
the idea that the more the monopolist produces, the lower the market price will be.
(b) Show that the monopolist’s optimal choice of output is lower than the optimal choice
of output of the competitive firm.
Solution: The FOC for the monopolist is R0 (y) = C 0 (y), or p0 (y) + p(y) = C 0 (y),
or p(y) = C 0 (y) − p(y)y. Since p0 (y) < 0 and y ≥ 0. this implies that p(y) ≥ C 0 (y),
which implies that the monopolist’s solution sets a strictly higher price than the
competitive firm’s. Since price is strictly decreasing in output, the monopolist must
be producing lower output.
3