Profit Maximization in Mathematical Economics: Practice Problems and Solutions

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Note: Here, I will present solve problems typical of those offered in a mathematical economics or advanced microeconomics course. The problems were originally compiled by Dr. Charles N. Steele and are reprinted with his generous permission. The solutions to the problems are my own work and not necessarily the only way to solve the problems.

Problem 1. Suppose a firm faces a demand curve for its product P = a - bQ, and the firm's costs of production and marketing are C(Q) = cQ + d, where P is price, Q is quantity, and a, b, c, and d are positive constants. Find the following:

a. The formula for profit Π in terms of Q.

b. The first order condition (FOC) for maximum profit.

c. The second order condition (SOC) for maximum profit.

Solution 1a. Π = TR - TC = PQ - C(Q) = aQ - bQ² - cQ - d = Π = - bQ² + (a-c)Q - d

Solution 1b. FOC: dΠ/dQ = -2bQ + (a-c) ≡ 0. Thus, -2bQ = -(a-c) and Q = (a-c)/2b.

Solution 1c. SOC: d²Π/dQ² = -2b < 0, since it is given that b > 0. Thus, Q = (a-c)/2b is a maximum.

Problem 2. Suppose the firm faces a demand curve for its product P = 32 - 2Q, and the firm's costs of production and marketing are C(Q) = 2Q². Find the following.

a. The formula for profit Π in terms of Q.

b. The FOC and SOC for maximum total revenue.

c. The price and quantity that maximize total revenue, and the corresponding value of total revenue.

d. The FOC and SOC for maximum profit.

e. The price and quantity that maximize profit, and the corresponding value of profit.

f. What would the competitive price and quantity be, assuming C(Q) = 2Q² represented the industry cost function?

Solution 2a. Π = TR - TC = PQ - C(Q) = 32Q - 2Q² - 2Q² = Π = 32Q - 4Q²

Solution 2b. TR = 32Q - 2Q²

FOC: d[TR]/dQ = 32 - 4Q ≡ 0. Thus, Q = 8.

SOC: d²[TR]/dQ² = -4 < 0. Thus, Q = 8 is a maximum.

Solution 2c. The quantity that maximizes total revenue is Q = 8, according to the first and second-order conditions in Solution 2b. The price that maximizes total revenue is

32 - 2*8 = P = 16. Total revenue at this level is PQ = 16*8 = TR =128. We note that AVC here is 2Q = 2*8 = 16, so price is at least equal to average variable cost.

Solution 2d. FOC: dΠ/dQ = 32 - 8Q = 0. Thus, Q = 4.

SOC:d²Π/dQ² = -8 < 0. Thus, Q = 4 is a maximum.

Solution 2e. The quantity that maximizes profit is Q = 4, according to the first and second-order conditions in Solution 2d. The price that maximizes profit is

32 - 2*4 = P = 24. Total profit at this level is 32*4 - 4*4² = Π = 64.

Here, 24 > 16, so P > AVC, and it is optimal for the firm to produce Q = 4.

Solution 2f. The firm will produce at P = MC, where P = 32 - 2Q. TC = 2Q², so MC = 4Q. Thus, 32 - 2Q = 4Q. Thus, 32 = 6Q and Q = 32/6 = Q = 16/3. P = 32 - 2(16/3) = P = 64/3

See Mr. Stolyarov's complete list of offerings of Mathematical Economics Problems and Solutions.