Transcript p(y)

Monopoly
垄断

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
A monopolized market has a single seller.
The monopolist’s demand curve is the
(downward sloping) market demand curve.
So the monopolist can alter the market price by
adjusting its output level.
$/output unit
p(y)
Higher output y causes a
lower market price, p(y).
Output Level, y
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A legal fiat; e.g. US Postal Service
A patent; e.g. a new drug
Sole ownership of a resource; e.g. a toll highway
Formation of a cartel; e.g. OPEC
Large economies of scale; e.g. local utility
companies.

Suppose that the monopolist seeks to maximize its
economic profit,
( y)  p( y)y  c( y).

What output level y* maximizes profit?
( y)  p( y)y  c( y).
At the profit-maximizing output level y*
d( y) d
dc( y)

0
p( y)y 
dy
dy
dy
so, for y = y*,
d
dc( y)
.
p( y)y 
dy
dy


At the profit-maximizing output level the slopes
of the revenue and total cost
curves are equal;
MR(y*) = MC(y*).
In a monopoly market, P>MR, how to prove?
Marginal revenue is the rate-of-change of
revenue as the output level y increases;
d
dp( y)
MR( y) 
.
p( y)y  p( y)  y
dy
dy
dp(y)/dy is the slope of the market inverse
demand function so dp(y)/dy < 0. Therefore
dp( y)
MR( y)  p( y)  y
 p( y)
dy
for y > 0.
E.g. if p(y) = a - by then
R(y) = p(y)y = ay - by2
and so
MR(y) = a - 2by < a - by = p(y) for y > 0.
E.g. if p(y) = a - by then
R(y) = p(y)y = ay - by2
and so
MR(y) = a - 2by < a - by = p(y) for y > 0.
a
p(y) = a - by
a/2b
a/b y
MR(y) = a - 2by
Marginal cost is the rate-of-change of total
cost as the output level y increases;
dc( y)
MC( y) 
.
dy
E.g. if c(y) = F + ay + by2 then
MC( y)  a  2by.
$
c(y) = F + ay + by2
F
$/output unit
y
MC(y) = a + 2by
a
y
At the profit-maximizing output level y*,
MR(y*) = MC(y*). So if p(y) = a - by and if
c(y) = F + ay + by2 then
MR( y*)  a  2by*  a  2by*  MC( y*)
and the profit-maximizing output level is
aa
y* 
2(b  b )
causing the market price to be
aa
p( y*)  a  by*  a  b
.
2(b  b )
$/output unit
a
p(y) = a - by
MC(y) = a + 2by
a
y
MR(y) = a - 2by
$/output unit
a
p(y) = a - by
MC(y) = a + 2by
a
y* 
aa
2(b  b )
y
MR(y) = a - 2by
$/output unit
a
p(y) = a - by
p( y*) 
aa
ab
2(b  b )
MC(y) = a + 2by
a
y* 
aa
2(b  b )
y
MR(y) = a - 2by


Proposition: A monopolist will never choose to
operate where the demand curve is inelastic.
Q: How to prove it?
d
dp( y)
MR( y) 
p( y)y  p( y)  y
dy
dy
y dp( y) 

 p( y) 1 
.

 p( y) dy 
d
dp( y)
MR( y) 
p( y)y  p( y)  y
dy
dy
y dp( y) 

 p( y) 1 
.

 p( y) dy 
Own-price elasticity of demand is
p( y) dy
1


so MR( y)  p( y) 1  .
  
y dp( y)
1

MR( y)  p( y) 1   .
 
Suppose the monopolist’s marginal cost of
production is constant, at $k/output unit.
For a profit-maximum
1

MR( y*)  p( y*) 1    k which is
k


p( y*) 
.
1
1

p( y*) 
k
1
1

.
•E.g. if  = -3 then p(y*) = 3k/2,
•and if  = -2 then p(y*) = 2k.
•So as  rises towards -1 the monopolist
alters its output level to make the market
price of its product to rise.
 1
Notice that, since MR ( y*)  p( y*)1    k ,
 
1
1

p( y*) 1    0  1   0



1
 1    1.
That is,

So a profit-maximizing monopolist always
selects an output level for which market
demand is own-price elastic.


Markup pricing: Output price is the marginal cost
of production plus a “markup.”
How big is a monopolist’s markup and how does it
change with the own-price elasticity of demand?
1

p( y*) 1    k



k
p( y*) 

1 1 
1

k
is the monopolist’s price. The markup is
k
k
p( y*)  k 
k  
.
1 
1 
•E.g. if  = -3 then the markup is k/2,
•and if  = -2 then the markup is k.
•The markup rises as the own-price
elasticity of demand rises towards -1.

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Two kinds of taxes:
---Profit tax
---Quantity tax

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A profits tax levied at rate t reduces profit from
(y*) to (1-t)(y*).
Q: How is after-tax profit, (1-t)(y*), maximized?
A: By maximizing before-tax profit, (y*).
So a profits tax has no effect on the monopolist’s
choices of output level, output price, or demands for
inputs.
I.e. the profits tax is a neutral tax (中性税).

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A quantity tax of $t/output unit raises the
marginal cost of production by $t.
So the tax reduces the profit-maximizing output
level, causes the market price to rise, and input
demands to fall.
The quantity tax is distortionary（扭曲）.
$/output unit
p(y)
p(y*)
MC(y)
y
y*
MR(y)
$/output unit
p(y)
MC(y) + t
p(y*)
t
MC(y)
y
y*
MR(y)
$/output unit
p(y)
p(yt)
p(y*)
MC(y) + t
t
MC(y)
y
yt y*
MR(y)
$/output unit
p(y)
p(yt)
p(y*)
The quantity tax causes a drop
in the output level, a rise in the
output’s price and a decline in
demand for inputs.
MC(y) + t
t
MC(y)
y
yt y*
MR(y)
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
p=a-by
MR=a-2by
With tax, MC=c+t
Profit maximization: a-2by=c+t
y=(a-c-t)/2b
p(y)=a-by=a-(a-c-t)/2
dp/dt=1/2
The monopolist passes on half of the tax.



Can a monopolist “pass” all of a $t quantity tax to
the consumers?
Suppose the marginal cost of production is
constant at $k/output unit.
With no tax, the monopolist’s price is
k
p( y*) 
.
1 

The tax increases marginal cost to $(k+t)/output
unit, changing the profit-maximizing price to
(k  t ) 
p( y ) 
.
1 by
 buyers is
The amount of the tax paid
t

p( yt )  p( y*).
(k  t ) 
k
t
p( y )  p( y*) 


1 
1  1 
t
•is the amount of the tax passed on to
buyers.
•E.g. if  = -2, the amount of
the tax passed on is 2t.
•Because  < -1（to make MR positive）,
 /1) > 1 and so the
monopolist passes on to consumers more
than the tax!



Social welfare=consumer surplus+ producer
surplus
A market is Pareto efficient if it achieves the
maximum possible total gains-to-trade.
Otherwise a market is Pareto inefficient.
$/output unit
The efficient output level
ye satisfies p(y) = MC(y).
p(y)
MC(y)
p(ye)
ye
y
$/output unit
The efficient output level
ye satisfies p(y) = MC(y).
p(y)
CS
MC(y)
p(ye)
ye
y
$/output unit
The efficient output level
ye satisfies p(y) = MC(y).
p(y)
CS
p(ye)
MC(y)
PS
ye
y
$/output unit
p(y)
CS
p(ye)
The efficient output level
ye satisfies p(y) = MC(y).
Total gains-to-trade is
maximized.
MC(y)
PS
ye
y
$/output unit
p(y)
p(y*)
MC(y)
y
y*
MR(y)
$/output unit
p(y)
p(y*)
CS
MC(y)
y
y*
MR(y)
$/output unit
p(y)
p(y*)
CS
MC(y)
PS
y
y*
MR(y)
$/output unit
p(y)
p(y*)
CS
MC(y)
PS
y
y*
MR(y)
$/output unit
p(y)
p(y*)
CS
MC(y)
PS
y
y*
MR(y)
$/output unit
p(y)
p(y*)
CS
PS
MC(y*+1) < p(y*+1) so both
seller and buyer could gain
if the (y*+1)th unit of output
was produced. Hence the
MC(y) market
is Pareto inefficient.
y
y*
MR(y)
$/output unit Deadweight loss（额外净损失）
measures the gains-to-trade
p(y) Not achieved by the market.
p(y*)
MC(y)
DWL
y
y*
MR(y)
The monopolist produces
$/output unit
less than the efficient
quantity, making the
p(y)
market price exceed the
efficient market
p(y*)
MC(y)
price.
e
DWL
p(y )
y*
y
ye
MR(y)

A natural monopoly arises when the firm’s
technology has economies-of-scale （规模经济）
large enough for it to supply the whole market at a
lower average total production cost than is possible
with more than one firm in the market.
$/output unit
ATC(y)
p(y)
MC(y)
y
$/output unit
ATC(y)
p(y)
p(y*)
MC(y)
y*
MR(y)
y


A natural monopoly deters entry by threatening
predatory pricing （掠夺性定价）against an
entrant.
A predatory price is a low price set by the
incumbent firm when an entrant appears, causing
the entrant’s economic profits to be negative and
inducing its exit.

E.g. suppose an entrant initially captures onequarter of the market, leaving the incumbent firm
the other three-quarters.
$/output unit
ATC(y)
p(y), total demand = DI + DE
DE
DI
MC(y)
y
$/output unit
ATC(y)
An entrant can undercut the
incumbent’s price p(y*) but ...
p(y), total demand = DI + DE
DE
p(y*)
pE
DI
MC(y)
y
$/output unit
An entrant can undercut the
ATC(y)
incumbent’s price p(y*) but
p(y), total demand = DI + DE
the incumbent can then
DE
lower its price as far
p(y*)
as pI, forcing
DI
pE
the entrant
to exit.
pI
MC(y)
y

Like any profit-maximizing monopolist, the natural
monopolist causes a deadweight loss.
$/output unit
ATC(y)
p(y)
p(y*)
MC(y)
y*
MR(y)
y
$/output unit
ATC(y)
p(y)
Profit-max: MR(y) = MC(y)
Efficiency: p = MC(y)
p(y*)
p(ye)
MC(y)
y*
MR(y)
ye y
$/output unit
ATC(y)
p(y)
Profit-max: MR(y) = MC(y)
Efficiency: p = MC(y)
p(y*)
DWL
p(ye)
MC(y)
y*
MR(y)
ye y


Why not command that a natural monopoly
produce the efficient amount of output?
Then the deadweight loss will be zero, won’t it?
$/output unit
At the efficient output
level ye, ATC(ye) > p(ye)
ATC(y)
p(y)
ATC(ye)
p(ye)
MC(y)
MR(y)
ye y
$/output unit
ATC(y)
p(y)
ATC(ye)
p(ye)
At the efficient output
level ye, ATC(ye) > p(ye)
so the firm makes an
economic loss.
MC(y)
Economic loss
MR(y)
ye y


So a natural monopoly cannot be forced to use
marginal cost pricing. Doing so makes the firm
exit, destroying both the market and any gains-totrade.
Regulatory schemes can induce the natural
monopolist to produce the efficient output level
without exiting.

Average cost pricing
2nd best solution
 Difficulty: How to measure costs?


Government-ownership
$/output unit
ATC(y)
p(y)
P(y)=AC(y)
MC(y)
PAC
MR(y)
yAC
y

Underlying technology


Market size

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Minimum efficient scale----output level that
minimizes average cost.
Openness
Collusion

Cartel---- several different firms in an industry
collude to reduce output and increase price.
p
AC
Demand
p*
MES
y
AC
Demand
p*
MES
y

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What causes monopoly
Profit-maximizing choices of monopoly
Markup pricing
Taxing a monopoly
Inefficiency of monopoly
Natural monopoly (自然垄断)