### Perfect competition and monopoly

```15. Firms, and monopoly
Varian, Chapters 23, 24, and 25
The firm
• The goal of a firm is to maximize profits
• Taking as given
– Necessary inputs
– Costs of inputs
– Price they can charge for a given quantity
• We will ignore inputs for this course (Econ
102, or I/O will cover this)
Standard theory
• Intuition
– Firm chooses a price, p, at which to sell, in
order to maximize profits
• Our approach today
– The firm chooses a quantity, q, to sell
– Inverse demand function is given
p(q)
Firm decision in the short run
Max p(q)q – c(q)
Revenue, R(q) = p(q)q
Cost
• Differentiate wrt q and set equal to zero:
MR = MC
p(q) + qp’(q) = c’(q)
Revenue from
extra unit sold
Revenue lost on all
sales due to price fall
Marginal
cost
Perfect competition (many firms)
Max p(q)q – c(q)
Revenue, R(q) = p(q)q
Cost
• Perfect competition: p(q) = p
p=MR = MC
p + 0 = c’(q)
Revenue from
extra unit sold
Firm is too small
to affect price
Marginal
cost
Pricing in the short run
Perfect competition:
• p = 20
• c(q) =
62.5+10q+0.1q2
• Find the firm’s
profit-maximizing q
Monopolist
• p(q)= 50 - 0.1q
• c(q) =
62.5+10q+0.1q2
• Find the firm’s profitmaximizing q
Cost function definitions
c(q) = 62.5+10q+0.1q2
• Fixed cost: the part of the cost function
that does not depend on q
• Variable cost: the part of the cost function
that does depend on q
• Total cost: FC+VC
• Average total cost: (FC+VC)/q=c(q)/q
How many firms will there be?
p
Perfect competition
• In long run,
competition forces
profits to 0
– P = ATC(q)
– P = MC(q)
– C’(q) = C(q)/q
ATC
MC
• Solve for q
q
How many firms will there be?
p
Perfect competition
• Knowing q
ATC
– P = MC(q)
– Q=D(P)
– #firms = Q/q
MC
D(p)
q
The long run outcome
Perfect competition:
• D(P) = 600 - 20P
• c(q) =
62.5+10q+0.1q2
• Find the long run q
• Find the long run
price, and # of
firms
Natural monopoly:
• D(P) = 600 - 20P
• c(q) =
640+10q+0.1q2
• What is q when
MC=ATC?
• How many firms will
there be?
Natural monopoly
p
• D(p)<q at p where
MC=ATC
• Happens when fixed
cost high relative to
ATC
– marginal cost
– inverse demand
• Fixed cost can only
be covered by p>MC
MC
D(p)
q
Monopolist
• Natural monopolies
– Electricity
– Telephones
– Software?
• Monopoly can also be by government
protection
– Patented drugs
• Imposed with violence
– Snow-shovel contracts in Montreal
Monopolist
• No competition
• Monopolist free to choose price
– MR(q) no longer constant p
– Single price: set MR(q) = MC(q)
• More elaborate pricing schemes to follow
– Price discrimination
Monopoly pricing
(no price discrimination)
• Note:
When
demand is
pm
linear, so is
marginal
revenue
• P = A – Bq
• MR = A – 2Bq
Profit
MC
MR
qm
Optimal quantity set by monopolist
Demand
Inefficiency of monopoly
pm
Mark-up over
Marginal cost
MC
Demand
MR
qm
q*
(Price) elasticity of demand
• The elasticity of demand measures the
percent change in demand per percent
change in price:
e = -(dq/q) / (dp/p)
= -(p/q)*(dq/dp) < 0
Optimal mark-up formula
p(q) + qp’(q) = c’(q)
can be rearranged to make:
p = MC / (1 – 1/|e|)
This can be rearranged to yield:
(p – MC)/MC = 1 / (|e| - 1) > 0
Demand elasticity
p
p
Elasticity > 1
Constant elasticity
of demand
Elasticity = 1
Elasticity < 1
p = q -e
q
q
p = a - bq
Monopolist’s decision
Natural monopoly:
• D(P) = 600 - 20P
• c(q) =
640+10q+0.1q2
• What q will
monopolist choose?
• What is their profit?
• What is elasticity of
demand at this
price/quantity?
Price discrimination
• Idea is to charge a different price for
different units of the good sold
• What does “different units” mean
• Purchased by different people
– E.g., children, students, pensioners, military
• Different amounts purchased by a given
person
– E.g., quantity discounts, entrance fees, etc.
Three degrees of discrimination
• First degree PD
– Each consumer can be charged a different
price for each unit she buys
• Second degree PD
– Prices can change with quantity purchased,
but all consumers face the same schedule
• Third degree PD
– Prices can’t vary with quantity, but can differ
across consumers
First degree PD
• Outcome is
Pareto efficient
Profit of fully
discriminating
monopolist
• Consumer earns
no consumer
surplus
MC = c
Profit of nondiscriminating
monopolist
Demand
xm
x*
Entry fee
• Alternative pricing mechanism:
If you buy x units, you pay a total of T + cx
With more than one consumer...
….charge a different entry fee to each
….but the same marginal price
Profit from consumer A
Profit from consumer B
MC = c
Demand
x*A
Consumer A
MC = c
Demand
x*B
Consumer B
Entry fees as “two-part-tariffs”
• Let A’s consumer surplus be TA and let
B’s be TB .
• Monopolist sets a pair of price schedules:
Consumer A
Consumer B
RA = TA + cx
RB = TB + cx
Entry fees
Price per unit = c
Second degree PD
• Suppose again there are two types of
people – A-types and B-types
• Half is A-type, half B-type
• …but now we cannot tell who is who
• Can the monopolist still capture some of
the consumer surplus? Yes - airlines
• All of it? No
A problem of information….
TA
• Best pricing policy:
Offer two options:
Option A: x*A for \$(U+V+W)+cx*A
Option B: x*B for \$U+cx*B TB
• But then A would choose
option B
A’s demand
B’s
demand
V
W
U
MC
x*B
– She gets surplus V from option
B, and 0 from option A
– Monopolist gets profit U
x*A
x
R
Option A
Option B is better
than option A
for person A
RA
RB
Option B
x*B
x*A
x
The monopolist can do a little
better….
A’s demand
• Option A’:
x*A for \$(U+W)+cx*A
B’s
demand
• A will be happy to take
this offer
V
W
U
– She gets a surplus of V
– Monopolist gets profit
U+W
MC
x*B
x*A
x
…but it can do even better
• Option A’’:
x*A for \$(U+W+DW)+cx*A
• Option B’’
x’’B for \$(U-DU)+cx’’B
A’s demand
DW
B’s
demand
V
MC
• A still willing to take
option A’’ over option B’’
• Profit up by DW-DU
W
U
DU
x’’B x*B
x*A
x
…and the best it can do is?
Gain from higher fees paid by
A-types from further decreasing x+B
A’s demand
B’s
demand V
Loss from lost sales to B-types
from further decreasing x+B
• Stop when
W
U
MC
x+ B
x*B
x*A
x
=
Should the monopolist bother
selling to low-demand consumers?
Going further, you lose more
on the B-types than you gain
on the A-types
Going all the way to zero, you
lose less on the B-types than
you gain on the A-types
YES: Sell to B-types
NO: Sell only to A-types
MC
A
B
x+B
MC
x*A
A
B
x
x+B=0
x*A
x
2nd degree price discrimination
High type:
• DH(P) = 100 - P
Low type:
• DH(P) = 70 – P
• MC=10
• What bundles
should the
monopolist offer?
• At what prices?
2nd degree price discrimination
High type:
• DH(P) = 100 - P
Low type:
• DH(P) = X – P
• MC=10
• For what value of X
will the monopolist
not sell to low
types?
Outcome
B-types
• They buy less than the Pareto efficient
quantity: x+B < x*B
• They earn zero consumer surplus
A-types
• They buy the Pareto optimal amount, x*A
• They earn positive consumer surplusFN
– this is always what they could earn if they
pretended to be B-types
FN: Whenever x+B >0
Third degree price discrimination
• Monopolist faces demand in two markets,
A and B
• Suppose marginal cost is constant, c
• Then the monopolist just sets prices so
that
pA = c / (1 – 1/|eA|)
pB = c / (1 – 1/|eB|)
Some problems
• Non-constant marginal cost?
– Replace c above with c’(xA+xB)
• What if demands are inter-dependent?
– E.g., xA(pA,pB) and xB(pB,pA)
• Applications
• A: Riding the metro in rush-hour
• B: Riding off-peak
– Children’s and adults’ ticket prices
Bundling
• Suppose a monopolist sells two (or more)
goods
• It might want to sell them together – that
is, in a “bundle”
• E.g.s
– Software – Word, PowerPoint, Excel
– Magazine subscriptions
Software example
Two types of consumer who have different
valuations over two goods
Type A
120
100
Type B
100
120
Assume marginal cost of production is zero
Selling strategies
Sell separately
Bundle
• Highest price to sell 2
word processors is
100
• Highest for