Chapter 9 Profit Maximization

Profit-Maximizing Prices and Quantities A firm’s profit, , is equal to its revenue R less its cost C  = R – C Maximizing profit Firm’s revenue, R(Q) = P(Q)Q Firm’s cost of production, C(Q) Overall,  = R(Q) – C(Q) = P(Q)Q – C(Q)

Demand function: Q d =D(P) Inverse demand function: P=P(Q d ) 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)=Q 2 Suppose they produce in batches of 10 To maximize profit, they need to find the production level with the greatest difference between revenue and cost

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QPRC∏

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

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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 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 Q d = D(P) = 1,000/P 1/2, what is their inverse demand function?

Profit-Maximizing Sales Quantity Two-step procedure for finding the profit- maximizing sales quantity Step 1: Quantity Rule Identify positive sales quantities at which MR=MC If more than one, find one with highest  Step 2: Shut-Down Rule Check 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>AC min, the best positive sales quantity maximizes profit. If P<AC min, shutting down maximizes profit. If P=AC min, 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 a Price-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 rules At each price above AC min, the firm’s profit- maximizing quantity is positive and satisfies P=MC At each price below AC min, the firm supplies nothing When price equals AC min, 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 production AC, MC curves shift up Supply curve shifts up Increase in an unavoidable fixed cost AC shifts upward MC unaffected Supply 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 versus Long-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 cost Represented by the area between firm’s price level and the supply curve Common application: investigate welfare implications of various policies Can 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 + Q 2 /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?