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)
• Differentiate wrt q and set equal to zero:
MR = MCp(q) + qp’(q) = c’(q)
Revenue, R(q) = p(q)q Cost
Revenue fromextra unit sold Revenue lost on all
sales due to price fall
Marginalcost
Perfect competition (many firms)
Max p(q)q – c(q)
• Perfect competition: p(q) = p
p=MR = MCp + 0 = c’(q)
Revenue, R(q) = p(q)q Cost
Revenue fromextra unit sold Firm is too small
to affect price
Marginalcost
Perfect competition: • p = 20• c(q) =
62.5+10q+0.1q2
• Find the firm’s profit-maximizing q
Pricing in the short run
Monopolist• p(q)= 50 - 0.1q• c(q) =
62.5+10q+0.1q2
• Find the firm’s profit-maximizing q
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
Cost function definitions
How many firms will there be?
Perfect competition• In long run,
competition forces profits to 0– P = ATC(q)– P = MC(q)– C’(q) = C(q)/q
• Solve for q
q
pATC
MC
How many firms will there be?
Perfect competition• Knowing q
– P = MC(q)– Q=D(P)– #firms = Q/q
q
pATC
MC
D(p)
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
The long run outcome
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
• D(p)<q at p where MC=ATC
• Happens when fixed cost high relative to– marginal cost– inverse demand
• Fixed cost can only be covered by p>MC
q
p
ATC
MCD(p)
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 linear, so is marginal revenue
• P = A – Bq• MR = A – 2Bq
MC
DemandMR
Optimal quantity set by monopolist
pm
qm
Profit
Inefficiency of monopoly
MC
DemandMR
pm
qm q*
Dead weight loss
Mark-up overMarginal cost
(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
q
p
Constant elasticityof demand
q
p
Elasticity > 1
Elasticity < 1
Elasticity = 1
p = q -e p = a - bq
Natural monopoly:• D(P) = 600 - 20P• c(q) =
640+10q+0.1q2
Monopolist’s decision
• 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
• Alternative pricing mechanism:
If you buy x units, you pay a total of T + cx
MC = c
Demand
Profit of non-discriminatingmonopolist
Profit of fullydiscriminatingmonopolist
• Outcome isPareto efficient
• Consumer earnsno consumersurplus
Entry feex*xm
With more than one consumer...
MC = c
Demand
Profit from consumer A
Consumer A Consumer B
MC = c
Demand
Profit from consumer B
….charge a different entry fee to each….but the same marginal price
x*Bx*A
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
RA = TA + cx
Consumer B
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….• Best pricing policy:
Offer two options:
Option A: x*A for $(U+V+W)+cx*A
Option B: x*B for $U+cx*B
• But then A would choose option B– She gets surplus V from option
B, and 0 from option A– Monopolist gets profit U
x
A’s demand
MC
U
V
W
x*B x*A
TA
TBB’sdemand
x
R
x*B x*A
RB
RA
Option A
Option B
Option B is betterthan option Afor person A
The monopolist can do a little better….
• Option A’:
x*A for $(U+W)+cx*A
• A will be happy to take this offer– She gets a surplus of V– Monopolist gets profit
U+W
x
A’s demand
B’sdemand
MC
U
V
W
x*B x*A
…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 still willing to take option A’’ over option B’’
• Profit up by DW-DUDU
DW
x’’B x
A’s demand
MC
U
V
W
x*B x*A
B’sdemand
…and the best it can do is?
• Stop when =W
x+B
Gain from higher fees paid byA-types from further decreasing x+
B
Loss from lost sales to B-typesfrom further decreasing x+
B
x
A’s demand
MC
U
x*B x*A
B’sdemand V
Should the monopolist bother selling to low-demand consumers?
x+B xx*A
AB
MC
Going further, you lose moreon the B-types than you gainon the A-types
x+B=0 xx*A
AB
MC
Going all the way to zero, you lose less on the B-types thanyou gain on the A-types
YES: Sell to B-types NO: Sell only to A-types
High type:• DH(P) = 100 - P
Low type:• DH(P) = 70 – P
• MC=10
2nd degree price discrimination
• What bundles should the monopolist offer?
• At what prices?
High type:• DH(P) = 100 - P
Low type:• DH(P) = X – P
• MC=10
2nd degree price discrimination
• 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– Peak-load pricing
• 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
Assume marginal cost of production is zero
Consumer type Word processor Spreadsheet
Type A 120 100
Type B 100 120
Selling strategies
Sell separately
• Highest price to sell 2 word processors is 100• Highest for spreadsheet is 100
• Sell two of each, for profit of 400
Bundle
• Can sell a bundle to each consumer for 220
• Total profit is 440
• Dispersion of prices falls with bundling