Download Firm Objectives Profit Second Order Condition Graphical Presentation

Professor Jay Bhattacharya
Spring 2001
Firm Objectives
Profit
• Cost minimization: Given a fixed output
level (without any story about how this
output is determined) firms choose the
minimum cost combination of inputs.
• Profit maximization: Firms choose that
level of output that yields the highest level
of profits.
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Econ 11--Lecture 13
• Profit = Revenue – Cost
• Last class, we analyzed costs in detail.
– The cost minimization problem produces a
function C(Q), which represents minimum
costs given output Q.
• Revenue is output multiplied by the price at
which that output sells—R(Q) = PQ.
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Profit Maximization—Choosing Output
• If a firm produces at all, it will produce an
amount such that MR = MC.
– If the extra revenue generated from producing 1
extra unit of output (MR) exceeds the
additional cost of producing that unit, then the
firm can increase its profit by expanding output
by 1 unit.
d Π dR dC
dR dC
=
−
=0 Þ
=
dQ dQ dQ
dQ dQ
• Interpretation: To maximize profits, set
marginal revenue (dR/dQ) equal to marginal
cost (dC/dQ).
Econ 11--Lecture 13
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Second Order Condition
C,R
• The SOC is important because of “S” shaped cost
curves.
• Also, if price falls below AVC, the firm (if it
produced positive amounts of output) would earn
a loss. Instead, it should go out of business
Econ 11--Lecture 13
Econ 11--Lecture 13
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Graphical Presentation
d 2 Π d 2 R d 2C
=
−
<0
dQ 2 dQ 2 dQ 2
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Supply: How much will firms produce?
• max Π = R(Q) – C(Q)
• First order condition:
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Econ 11--Lecture 13
slope = P
R=PQ
C(Q)
Q**
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Π
R(Q)-C(Q)
0
Q* Q
Econ 11--Lecture 13
Q**
Q*
Q
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Professor Jay Bhattacharya
Spring 2001
Nicholson Example Problem
Lump-Sum Tax
• Would a lump-sum profits tax affect the
profit maximizing quantity of output? How
about a proportional tax on profits? How
about a tax assessed on each unit of output?
• Profit maximization under a lump-sum
profits tax: (T is the lump-sum tax)
Π = PQ − C ( Q ) − T
• First order condition is exactly the same:
d Π dR dC
=
−
=0
dQ dQ dQ
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Econ 11--Lecture 13
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Lump-Sum Tax (continued)
Spring 2001
• Under a proportional tax on profits (say, t)
the firm’s problem is:
Π = ( PQ − C ( Q ) ) (1 − t )
R=PQ
• The first order condition is:
R=PQ-T
C(Q)
Econ 11--Lecture 13
d Π æ dR dC ö
dR dC
=ç
−
=
÷ (1 − t ) = 0 Þ
dQ è dQ dQ ø
dQ dQ
Q* Q
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Proportional Tax (continued)
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Econ 11--Lecture 13
Π = PQ (1 − t ) − C ( Q )
R=PQ(1-t)
Econ 11--Lecture 13
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• Under a tax (say, t) on output the firm’s
problem is:
R=PQ
C(Q)
Econ 11--Lecture 13
Tax on Output
• The firm makes less profits, but the
marginal and limit conditions are the same:
C,R
If the firm
produces positive
Q without the tax,
it produces positive
Q with the tax.
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Proportional Tax on Profits
• However, it may be optimal for firms to go
out of business if the lump sum tax is high
enough: C,R
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Econ 11--Lecture 13
(1-t)C(Q)
Q* Q
• The first order condition is:
d Π dR
dC
dR
dC
=
(1 − t ) − = 0 Þ (1 − t ) =
dQ dQ
dQ
dQ
dQ
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Spring 2001
Econ 11--Lecture 13
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Professor Jay Bhattacharya
Spring 2001
Tax on Output (continued)
Prices and Industry Structure
• The price that firms can charge will depend upon
the structure of the industry.
• This tax distorts the firm’s FOC—the
optimal Q differs from Q*:
– In a competitive industry, firms cannot affect prices by
cutting back on (or increasing) output.
– In a monopolistic or oligopolistic industry, changes in
output affect market price.
C,R
The firm is more
likely to go out of
business with this
risky tax scheme.
R=PQ
• In general, output prices that firm faces will be a
function of the firm’s output:
C(Q)
– P = P(Q)
– This is exactly the (inverse) market demand function.
Qnew Q*
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Q
Econ 11--Lecture 13
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Supply Curve
Competitive Supply
P
• In a competitive industry, firms take prices as
fixed.
– If firms charge prices higher than marginal costs, they
lose all their business.
– If firms charge prices lower than marginal costs, they
make negative profits.
C′(Q )
• For price taking firms, dR/dQ = P
• The first order condition is P = MC = dC/dQ
– The firm will choose output at a point where price is
equal to marginal cost.
Q
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Econ 11--Lecture 13
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How Does Supply Vary with
Input Prices
P
For example, an
increase in wages
shifts supply back
since it increases
marginal costs.
Econ 11--Lecture 13
Econ 11--Lecture 13
Q
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Monopoly Supply
• An increase in marginal cost = a shift in the
supply curve
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• Monopolists do not sit back and take prices.
• They manipulate prices by curtailing output below
competitive levels.
– For monopolists, prices are a function of quantity
produced (the inverse market demand curve).
• However, like firms in a competitive industry,
monopolists still maximize profits. The FOC is:
dC
dR dP (Q )
Q + P (Q ) =
=
dQ
dQ
dQ
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Econ 11--Lecture 13
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Professor Jay Bhattacharya
Spring 2001
Monopoly Supply (Graphical)
Nicholson Example Problem #2
P
dR/dQ—Marginal
Revenue curve
C′(Q )
P(Q) — Demand curve
Area in box =
Total revenue
Q(monopoly) Q(competitive)
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Econ 11--Lecture 13
Q
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Example #2 Solved
Econ 11--Lecture 13
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• Inverting two demand functions yields:
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p1 = − q1 + 50
2
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max Π = q æ − 1 q + 50 ö + q æ − 1 q + 25 ö − (q + q )2
÷ 2ç
÷
1ç
1
2
1
2
q1 , q2
è 2
ø
è 4
ø
Econ 11--Lecture 13
1
p2 = − q2 + 25
4
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Example #2 Solution (IV)
• Plugging the inverse demand functions back
into the profit function gives the firm’s
maximization problem in terms of q1 and q2:
Econ 11--Lecture 13
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• The firm chooses q1 and q2 to maximize
profits:
max Π = q p + q p − (q + q )2
1 1
2 2
1
2
q1 , q2
Example #2 Solution (III)
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Econ 11--Lecture 13
Example #2 Solution (continued)
• Let q1 be the amount sold in Australia at
price p1.
• Let q2 be the amount sold in Lapland at
price p2.
• The firm’s total revenues from the two
locations equal p1q1 + p2q2.
• The firm’s total costs of producing q1 + q2
are 0.25(q1 + q2)2.
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• Universal Widget produces high-quality widgets
at its plant in Gulch, Nevada, for sale throughout
the world. The cost function for total widget
production (q) is given by TC = 0.25q2. Widgets
are demanded only in Australia (where demand is
q = 100 – 2p), and Lapland (where demand is q =
100 – 4p). If Universal Widget can control q in
each market, how many should it sell in each
location in order to maximize profits? What price
will be charged in each location?
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• The first order conditions for the firm’s
problem and their solution are:
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1
− q1 − q2 + 50 = 0
2
2
1
− q1 − q2 + 25 = 0
2
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q1 = 30 Þ p1 = 35
q 2 = 10 Þ p2 = 22.5
Econ 11--Lecture 13
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Professor Jay Bhattacharya
Spring 2001
Producer Surplus
3 Ways to Measure PS
• Producer surplus = Revenue - Variable Cost
• Profit = Revenue - Total Cost
• Producer surplus = Profit if no fixed costs
or if in the long-run
• AKA ‘operating profit’
• PS = Revenue - Variable Cost
• PS = Area where Price > MC
• PS = Area to the left of the supply curve
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Producer Surplus
P
P = C ′(Q )
• The industry supply curve is the horizontal
sum of the existing firms’ supply curves
Econ 11--Lecture 13
P
P
Q1
Q
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Industry Supply
Market
price line
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Q
P
Q2
Econ 11--Lecture 13
Q
Q1+Q2
Q
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