Profit maximization by firms

ECO61Muhammad Farhan Javed

Revenues and costs

• A firm’s costs (C) were discussed in the previous chapter

• A firm’s revenue is R = P Q– Where P is the price charged by the firm for the

commodity it sells and Q is the quantity of the firm’s output that people buy

– We discussed the link between price and quantity consumed – the demand curve – earlier

• Now it is time to bring revenues and costs together to study a firm’s behavior

Profit-MaximizingPrices and Quantities

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

• We assume that a firm’s actions are aimed at maximizing profit

• Maximizing profit is another example of finding a best choice by balancing benefits and costs– Benefit of selling output is firm’s revenue, R(Q) = P(Q)Q– Cost 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 10• To maximize profit, they need to find the

production level with the greatest difference between revenue and cost

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Profit-Maximization: An Example

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Note that [50 – Q]2 is always a positive number. Therefore, to maximize profit one must minimize [50 – Q]2. Therefore, to maximize profit, Noah and Naomi must produce Q = 50 units. This is their profit-maximizing output.

When Q = 50, π = 2 502 = 5000. this is the biggest profit Noah and Naomi can achieve.

Figure 9.2: A Profit-Maximization Example

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Choice requires balance at the margin

• 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

Example

Marginal Revenue• 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

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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). This is the output expansion effect: PQ– Firm also has to lower price as dictated by the demand

curve; reduces revenue earned from the original Q units of output. This is the price reduction effect: QP

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Figure 9.4: Marginal Revenue and Price

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Output expansion effect: PQ

Price reduction effect of output expansion: QP. Non-existent when demand is horizontal

Marginal Revenue and Price• The output expansion effect is PQ• The price reduction effect is QP• Therefore the additional revenue per unit of

additional output is MR = (PQ + QP)/Q = P + QP/Q

• When demand is negatively sloped, P/Q < 0. So, MR < P.

• When demand is horizontal, P/Q = 0. So, MR = P.

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Demand and marginal revenue

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

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Profit

• Profit equals total revenue minus total costs.– Profit = R – C– Profit/Q = R/Q – C/Q– Profit = (R/Q - C/Q) Q– Profit = (PQ/Q - C/Q) Q– Profit = (P - AC) Q

Profit: downward-sloping demand of price-setting firm

profit

Averagecost

Quantity

Profit-maximizing price

QMAX0

Costs andRevenue

Demand

Marginal cost

Marginal revenue

Average cost

B

C

E

D

Profit: downward-sloping demand of price-setting firm

• Recall that profit = (P - AC) Q• Therefore, the firm will stay in business as

long as price (P) is greater than average cost (AC).

Shut down because P < AC at all Q: downward-sloping demand of price-setting firm

Quantity0

Costs andRevenue

Demand

Average total cost

Profit Maximization: horizontal demand for a price taking firm

Quantity0

Costsand

Revenue

MC

AC

MC1

Q1

MC2

Q2

The firm maximizesprofit by producing the quantity at whichmarginal cost equalsmarginal revenue.

QMAX

P = MR1 = MR2 P = AR = MR

Shut down because P < AC at all Q: horizontal demand for a price taking firm

Quantity0

Costsand

Revenue

MC

AC

P = AR = MR

ACmin

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

• not subject to the price reduction effect

– 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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Price determination

• We have seen how the price is determined in the case of price setting firms that have downward sloping demand curves

• But how is the price that price taking firms use to guide their production determined?– For now think of it as determined by trial and error.

Pick a random price. See what quantity is demanded by buyers and what quantity is supplied by producers. Keep trying different prices whenever the two quantities are unequal

– The market equilibrium price is the price at which the quantities supplied and demanded are equal