Download Profit maximization

By: Brian Murphy


Given a function for cost with respect to
quantity produced by a firm and market
demand with respect to price set by the firm,
find the price for a manufactured good that
will optimize profits for the firm.
Key variables:





p = price of manufactured good
Q = quantity manufactured
Q(p) = market demand function
C(Q) = cost function for manufacturing process
Π(Q) = profit function = R(Q) – C(Q)

Given cost and demand function:
Take market demand function and solve for p in
terms of Q to get inverse market demand (p(Q)).
 Calculate Revenue function (R(Q) = p(Q)*Q
 Find marginal revenue function MR(Q) = dR(Q)/dQ
 Find marginal cost function MC(Q) = dC(Q)/dQ
 Set MR = MC and solve for optimal quantity Q*.
 Plug Q* into p(Q) to get profit maximizing price p*.
 Plug Q* into Π(Q) to calculate profit for p*.

A firm faces the following market demand:
Q(p) = 27.5 -0.5p
and the following costs:
C(Q) = 100 – 5Q + Q2
What price should the firm set to maximize
profits?
Find inverse market demand:
Take Q(p) = 27.5 – 0.5p
0.5p = 27.5 –Q
p(Q) = 55 – 2Q
Find revenue function:
R(Q) = p(Q) * Q = 55Q – 2Q2
Find marginal revenue function:
MR(Q) = dR(Q)/dQ = 55 – 4Q
Find marginal cost function:
C(Q) = 100 – 5Q + Q2
MC(Q) = -5 + 2Q
Set marginal revenue equal to marginal cost:
MC(Q) = MR(Q) -> 55 – 4Q = -5+2Q
6Q = 60 -> Q* = 10.
Plug Q* into p(Q):
p(Q) = 55 – 2Q, p(Q*) = 35 = p*.
Calculate profit function:
Π(Q) = R(Q) – C(Q) = 55Q – 2Q2 -100 +5Q – Q2
= 60Q – 3Q2 -100
With Q* = 10
Π(Q*) = $200 = maximized profit.