Chapter 11: Competitive Markets Profit Maximization Example Profit Maximization at Beau Apparel: An Illustration

Profit Maximization at Beau Apparel Beau Apparel, Inc., is a clothing manufacturer that produces moderately priced men’s shirts. Beau Apparel is a price-taker Beau Apparel is one of many firms that produce a fairly homogeneous product, and none of the firms in this moderate- price shirt market engages in any significant advertising. To choose the quantity that maximizes its profits, Beau Apparel needs estimates of the market price and its costs

Price and Cost Forecasts In December 2010, the manager of Beau Apparel prepares the firm’s production plan for the first quarter of 2011. The Marketing/Forecasting Division forecasts a 2011 price of $15 The manager of Beau Apparel estimates a cubic equation for short-run cost AVC = 20 – 0.003Q + 0.00000025Q2 the coefficients from the AVC function are used to determine the short-run marginal cost function, SMC = a + 2bQ + 3cQ2 SMC = 20 – 0.006Q + 0.00000075Q2 Now that the firm knows the price and estimates of AVC and MC: Should the firm produce or shut down? If it produces, what is the profit maximizing quantity?

The Shutdown Decision To answer the shut down question, the manager determines the quantity that minimizes AVC and the value of AVC at that quantity. It was previously determined that AVC = 20 – 0.003Q + 0.00000025Q2. The quantity where AVC is minimized is –b/(2c) Min Q = -(-0.003)/(2 * 0.00000025) = 6000 Now substitute the 6000 output level back into the AVC equation. Min AVC = 20 – (0.003*6000) + (0.00000025*60002) = $11 AVC is minimized at $11 while producing 6000 units.

The Shutdown Decision Cont. The manager now compares this minimum AVC with the forecasted price of $15. Since $15 > $11, the firm should produce and not shutdown.

The Output Decision To maximize profits or minimize loss, marginal revenue (price for a price-taker) should equal marginal cost. P = SMC = 20 – 0.006Q + 0.00000075Q2 15 = 20 – 0.006Q + 0.00000075Q2 0 = 5 – 0.006Q + 0.00000075Q2 Since this equation cannot be factored algebraically, it must be solved using the quadratic formula. Q = (-(-0.006) ± √(0.0062 – 4*5*0.00000075)) ÷ (2*0.00000075) Q = (0.006 ± 0.004583) ÷ 0.0000015 There are two solutions: Q = 945 and Q = 7,055

The Output Decision Cont. Calculate AVC for each quantity, Q = 945 and Q = 7,055 AVC945 = 20 – (0.003*945) + (0.00000025*9452) = $17.39 AVC7,055 = 20 – (0.003*7055) + (0.00000025*70552) = $11.28 Compare the price to AVC for each quantity. At Q = 945, price = $15 < $17 = AVC. The manager would not produce here, since the AVC is higher than the price. At Q = 7,055, price = $15 > $ 11.28 = AVC. The manager will produce 7,055 units, because price is greater than AVC. Now that the manager has determined the profit maximizing quantity, he calculates the profit or loss.

Computing Total Profit or Loss Remember TR = P*Q TVC = AVC*Q TC = TVC + TFC = (AVC*Q) + TFC TP = TR – TC = (P*Q) – [(AVC*Q) + TFC] The manager expects TFC to be $30,000. For P = $15, we have already calculated that AVC = $11.28 and Q = 7,055 units. TP = (15*7055) – (11.28*7055) -30,000 TP = -$3,755 Even though Beau is experiencing a loss at $15, they should continue to produce, since the loss of -$3,755 is much less than the $30,000 in fixed costs that would still have to be paid even if production stopped. AVC ≤ P < ATC