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Profit maximization by firms

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Profit maximization by firms

ECO61

Udayan Roy

Fall 2008

- 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

- A firm’s profit, P, is equal to its revenue R less its cost C
- P = 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,
- P = R(Q) – C(Q) = P(Q)Q – C(Q)

9-3

- 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

9-4

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.

9-6

- 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 DQ marginal units sold, measured on a per unit basis

9-9

- An increase in sales quantity (DQ) changes revenue in two ways:
- Firm sells DQ additional units of output, each at a price of P(Q). This is the output expansion effect: PDQ
- 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: QDP

9-10

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

Output expansion effect: PQ

9-11

- The output expansion effect is PDQ
- The price reduction effect is QDP
- Therefore the additional revenue per unit of additional output is MR = (PDQ + QDP)/DQ = P + QDP/DQ
- When demand is negatively sloped, DP/DQ < 0. So, MR < P.
- When demand is horizontal, DP/DQ = 0. So, MR = P.

9-12

- 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 P

- Step 2: Shut-Down Rule
- Check whether the quantity from Step 1 yields higher profit than shutting down

9-14

- 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

Marginal cost

Profit-maximizing price

E

B

profit

Average cost

Average

D

C

cost

Demand

Marginal revenue

QMAX

Costs and

Revenue

Quantity

0

- Recall that profit = (P - AC) Q
- Therefore, the firm will stay in business as long as price (P) is greater than average cost (AC).

Average total cost

Demand

Costs and

Revenue

Quantity

0

The firm maximizes

profit by producing

the quantity at which

MC

marginal cost equals

marginal revenue.

MC

2

AC

=

=

=

=

P

MR

MR

P

AR

MR

1

2

MC

1

Q

Q

Q

1

MAX

2

Costs

and

Revenue

Quantity

0

MC

AC

Costs

and

Revenue

ACmin

=

=

P

AR

MR

Quantity

0

- 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

- Face a perfectly horizontal demand curve
- 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.

9-21

- 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