Chapter 9

Profit Maximization

Profit-MaximizingPrices and Quantities

A firm’s profit, , is equal to its revenue R less its cost C = R – C

Maximizing profitFirm’s revenue, R(Q) = P(Q)QFirm’s cost of production, C(Q)

Overall,= R(Q) – C(Q) = P(Q)Q – C(Q)

Demand function:

Qd=D(P)

Inverse demand function:P=P(Qd)

it shows how much the firm must charge to sell any given Q

Profit-Maximization: An Example

Noah and Naomi face weekly inverse demand function P(Q) = 200-Q for their garden benches

Weekly cost function is C(Q)=Q2

Suppose they produce in batches of 10To maximize profit, they need to find the

production level with the greatest difference between revenue and cost

Q P R C ∏0 200 0 0 0

10 190 1900 100 1800

20 180 3600 400 3200

30 170 5100 900 4200

40 160 6400 1600 4800

50 150 7500 2500 5000

60 140 8400 3600 4800

70 130 9100 4900 4200

80 120 9600 6400 3200

90 110 9900 8100 1800

100 100 10000 10000 0

Q P R C ∏0 200 0 0 0

10 190 1900 100 1800

20 180 3600 400 3200

30 170 5100 900 4200

40 160 6400 1600 4800

50 150 7500 2500 500060 140 8400 3600 4800

70 130 9100 4900 4200

80 120 9600 6400 3200

90 110 9900 8100 1800

100 100 10000 10000 0

Figure 9.2: A Profit-Maximization Example

Marginal Revenue

Marginal Revenue: the extra revenue produced by the Q marginal units sold, measured on a per unit basis

Q P R C ∏0 200 0 0 0

10 190 1900 100 1800

20 180 3600 400 3200

30 170 5100 900 4200

40 160 6400 1600 4800

50 150 7500 2500 500060 140 8400 3600 4800

70 130 9100 4900 4200

80 120 9600 6400 3200

90 110 9900 8100 1800

100 100 10000 10000 0

Marginal Revenue and Price

An increase in sales quantity (Q) changes revenue in two ways

Firm sells Q additional units of output, each at a price of P(Q), the output expansion effect

Firm also has to lower price as dictated by the demand curve; reduces revenue earned from the original (Q-Q) units of output, the price reduction effect

Price-taking firm faces a horizontal demand curve and is not subject to the price reduction effect 9-7

Figure 9.4: Marginal Revenue and Price

P-Taker

Firm’s extra R from selling more Q

Figure 9.4: Marginal Revenue and Price

Firm’s extra R from selling more Q= A-B

A

B

Price-Taker firm: MR=D curve since MR=P

Downward-sloping demand curve: MR=P when sales =0 and MR<P elsewhere.

Figure 9.4: Marginal Revenue and Price

MR

MR=

Sample Problem 1 (9.1):

If the demand function for Noah and Naomi’s garden benches is Qd = D(P) = 1,000/P1/2, what is their inverse demand function?

Profit-Maximizing Sales Quantity

Two-step procedure for finding the profit-maximizing sales quantity

Step 1: Quantity RuleIdentify positive sales quantities at which MR=MCIf more than one, find one with highest

Step 2: Shut-Down RuleCheck whether the quantity from Step 1 yields

higher profit than shutting down

Supply Decisions

Price takers are firms that can sell as much as they want at some price P but nothing at any higher price Face a perfectly horizontal demand curve Firms in perfectly competitive markets, e.g. MR = P for price takers

Use P=MC in the quantity rule to find the profit-maximizing sales quantity for a price-taking firm

Shut-Down Rule: If P>ACmin, the best positive sales quantity maximizes profit. If P<ACmin, shutting down maximizes profit. If P=ACmin, then both shutting down and the best positive sales

quantity yield zero profit, which is the best the firm can do.

Figure 9.6: Profit-Maximizing Quantity of a Price-Taking Firm

The best choice: P=MC

Supply Function of aPrice-Taking Firm

A firm’s supply function shows how much it wants to sell at each possible price: Quantity supplied = S(Price)

To find a firm’s supply function, apply the quantity and shut-down rulesAt each price above ACmin, the firm’s profit-

maximizing quantity is positive and satisfies P=MCAt each price below ACmin, the firm supplies nothingWhen price equals ACmin, the firm is indifferent

between producing nothing and producing at its efficient scale

Figure 9.7: Supply Curve of a Price-Taking Firm

Figure 9.9: Law of Supply

Law of Supply: when market price increases, the profit-maximizing sales quantity for a price-taking firm never decreases

Change in Input Price and the Supply Function

How does a change in an input price affect a firm’s supply function?

Increase in price of an input that raises the per unit cost of productionAC, MC curves shift upSupply curve shifts up

Increase in an unavoidable fixed costAC shifts upwardMC unaffectedSupply curve does not shift

Figure 9.10: Change in Input Price and the Supply Function

Figure 9.11: Change in Avoidable Fixed Cost

Short-Run versusLong-Run Supply

Firm’s marginal and average costs may differ in the long and short run

This affects firm response over time to a change in the price it faces for its product

Suppose the price rises suddenly and remains at that new high level

Use the quantity and shut-down rules to analyze the long-run and short-run effects of the price increase on the firm’s output

Figure 9.13(a): Quantity Rule

Firm’s best positive quantity:Q*

SR in short run

Q*LR in long run, a

larger amount

Figure 9.13(b): Shut-Down Rule

New price is above the avoidable short-run average cost at Q*

SR and the long-run average cost at Q*

LR

Firm prefers to operate in both the short and long run

Producer Surplus

A firm’s producer surplus equals its revenue less its avoidable costs = producer surplus – sunk costRepresented by the area between firm’s price level

and the supply curveCommon application: investigate welfare

implications of various policiesCan focus on producer surplus instead of profit

because the policies can’t have any effects on sunk costs

Figure 9.16: Producer Surplus

Sample Problem 2 (9.8)

Suppose Dan’s cost of making a pizza is C(Q) = 4Q + Q2/40), and his marginal cost is MC = 4 + (Q/20). Dan is a price taker. What is Dan’s supply function? What if Dan has an avoidable fixed cost of $10?