Profit Maximization Example

Profit Maximization at Beau Apparel: An Illustration

Chapter 11: Competitive Markets

Profit Maximization at Beau ApparelBeau Apparel, Inc., is a clothing

manufacturer that produces moderately priced men’s shirts.

Beau Apparel is a price-takerBeau 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 ForecastsIn 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:1.Should the firm produce or shut down?2.If it produces, what is the profit maximizing quantity?

The Shutdown DecisionTo answer the shut down question, the manager

determinesthe 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)

= $11AVC 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 DecisionTo 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) ÷ 0.0000015There are two solutions: Q = 945 and Q = 7,055

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The Output Decision Cont.Calculate AVC for each quantity, Q = 945 and Q =

7,055 AVC945 = 20 – (0.003*945) + (0.00000025*9452) =

$17.39AVC7,055 = 20 – (0.003*7055) + (0.00000025*70552) =

$11.28Compare 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 LossRemember

TR = P*QTVC = AVC*QTC = 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 = -$3,755Even 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