Transcript Profit maximization by firms

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
– 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
• 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)
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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
P  R C  PQ C
P  (200  Q )  Q  Q 2
P  200Q  Q 2  Q 2  200Q  2Q 2
P  2(100Q  Q 2 )  2(50 2  50 2  100Q  Q 2 )
P  2(50 2  [50 2  100Q  Q 2 ])
P  2(50 2  [50 2  2  50  Q  Q 2 ])
P  2(50 2  [50  Q ]2 )
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
9-6
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 DQ
marginal units sold, measured on a per unit basis
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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
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Figure 9.4: Marginal Revenue and Price
Price reduction
effect of output
expansion: QDP.
Non-existent
when demand is
horizontal
Output expansion effect: PDQ
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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.
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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 P
• 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
Costs and
Revenue
Marginal cost
ProfitE
maximizing
price
B
profit
Average
cost D
Average cost
C
Demand
Marginal revenue
0
QMAX
Quantity
Profit: downward-sloping demand of pricesetting 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
Costs and
Revenue
Average total cost
Demand
0
Quantity
Profit Maximization: horizontal demand for a price taking firm
Costs
and
Revenue
The firm maximizes
profit by producing
the quantity at which
marginal cost equals
marginal revenue.
MC
MC2
AC
P = MR1 = MR2
P = AR = MR
MC1
0
Q1
QMAX
Q2
Quantity
Shut down because P < AC at all Q: horizontal demand for a price
taking firm
Costs
and
Revenue
MC
AC
ACmin
P = AR = MR
0
Quantity
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