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Profit maximization by firms. ECO61 Udayan Roy Fall 2008. Revenues and costs. A firm’s costs (C) were discussed in the previous chapter A firm’s revenue is R = P  Q

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Profit maximization by firms l.jpg

Profit maximization by firms

ECO61

Udayan Roy

Fall 2008


Revenues and costs l.jpg

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 maximizing prices and quantities l.jpg

Profit-MaximizingPrices and Quantities

  • 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


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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

9-4


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Profit-Maximization: An Example

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 l.jpg

Figure 9.2: A Profit-Maximization Example

9-6


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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 l.jpg

Example


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Marginal Revenue

  • 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


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Marginal Revenue and Price

  • 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


Figure 9 4 marginal revenue and price l.jpg

Figure 9.4: Marginal Revenue and Price

Price reduction effect of output expansion: QP. Non-existent when demand is horizontal

Output expansion effect: PQ

9-11


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Marginal Revenue and Price

  • 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


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Demand and marginal revenue


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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 P

  • Step 2: Shut-Down Rule

    • Check whether the quantity from Step 1 yields higher profit than shutting down

9-14


Profit l.jpg

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


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Marginal cost

Profit-maximizing price

E

B

profit

Average cost

Average

D

C

cost

Demand

Marginal revenue

QMAX

Profit: downward-sloping demand of price-setting firm

Costs and

Revenue

Quantity

0


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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 l.jpg

Average total cost

Demand

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

Costs and

Revenue

Quantity

0


Profit maximization horizontal demand for a price taking firm l.jpg

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

Profit Maximization: horizontal demand for a price taking firm

Costs

and

Revenue

Quantity

0


Shut down because p ac at all q horizontal demand for a price taking firm l.jpg

MC

AC

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

Costs

and

Revenue

ACmin

=

=

P

AR

MR

Quantity

0


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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

      • 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.

9-21


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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


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