Download Chapter 9

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:
Qd=D(P)
Inverse demand function:
P=P(Qd)
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)=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
Q
P
R
C
∏
0
200
0
0
0
10
190
1900
100
1800
20
180
3600
400
3200
30
170
5100
900
4200
40
160
6400
1600
4800
50
150
7500
2500
5000
60
140
8400
3600
4800
70
130
9100
4900
4200
80
120
9600
6400
3200
90
110
9900
8100
1800
100
100
10000
10000
0
Q
P
R
C
∏
0
200
0
0
0
10
190
1900
100
1800
20
180
3600
400
3200
30
170
5100
900
4200
40
160
6400
1600
4800
50
150
7500
2500
5000
60
140
8400
3600
4800
70
130
9100
4900
4200
80
120
9600
6400
3200
90
110
9900
8100
1800
100
100
10000
10000
0
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
Q
P
R
C
∏
0
200
0
0
0
10
190
1900
100
1800
20
180
3600
400
3200
30
170
5100
900
4200
40
160
6400
1600
4800
50
150
7500
2500
5000
60
140
8400
3600
4800
70
130
9100
4900
4200
80
120
9600
6400
3200
90
110
9900
8100
1800
100
100
10000
10000
0
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
9-7
effect
Figure 9.4: Marginal Revenue and Price
Firm’s extra R from
selling more Q
P-Taker
Figure 9.4: Marginal Revenue and Price
B
A
Firm’s extra R from
selling more Q= 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 Qd = D(P) =
1,000/P1/2, what is their inverse demand
function?
Profit-Maximizing Sales Quantity
Two-step procedure for finding the profitmaximizing 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 profitmaximizing 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.
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 ACmin, the firm’s profitmaximizing quantity is positive and satisfies P=MC
At each price below ACmin, the firm supplies nothing
When price equals ACmin, 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 shortrun average cost at
Q*SR and the longrun 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 + Q2/40), 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?