Chapter 9

Profit Maximization

McGraw-Hill/Irwin Copyright © 2008 by The McGraw-Hill Companies, Inc. All Rights Reserved.

Main Topics

Profit-maximizing quantities and pricesMarginal revenue, marginal cost, and

profit maximizationSupply decisions by price-taking firmsShort-run versus long-run supplyProducer surplus

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Profit-MaximizingPrices and Quantities

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

Maximizing profit is another example of finding a best choice by balancing benefits and costsBenefit of selling output is firm’s revenue, R(Q) =

P(Q)QCost of selling that quantity is the firm’s cost of

production, C(Q)Overall,

= R(Q) – C(Q) = P(Q)Q – C(Q)

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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

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Figure 9.2: A Profit-Maximization Example

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Marginal Revenue

In general marginal benefit must equal marginal cost at a decision-maker’s best choice whenever a small increase or decrease in her action is possible

Here the firm’s marginal benefit is its marginal revenue: the extra revenue produced by the Q marginal units sold, measured on a per unit basis

Q

QQRQR

Q

RMR

)()(

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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

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Figure 9.4: Marginal Revenue and Price

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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

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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.

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

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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

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Figure 9.7: Supply Curve of a Price-Taking Firm

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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

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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

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Figure 9.10: Change in Input Price and the Supply Function

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Figure 9.11: Change in Avoidable Fixed Cost

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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

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Figure 9.13(a): Quantity Rule

Firm’s best positive quantity:Q*

SR in short run

Q*LR in long run, a

larger amount

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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

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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

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Figure 9.16: Producer Surplus

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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?