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Profit maximization

Profit maximization. By: Brian Murphy. Scenario. 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:

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Profit maximization

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  1. Profit maximization By: Brian Murphy

  2. Scenario • 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)

  3. Procedure • 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*.

  4. Example 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?

  5. Example (cont’d.) 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

  6. Example (cont’d.) 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*.

  7. Example (cont’d.) 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.

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