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