# 15. Firms, and monopoly Varian, Chapters 23, 24, and 25.

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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 Our approach today p(q)
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 = MC p(q) + qp’(q) = c’(q) Revenue, R(q) = p(q)q Cost Marginal cost Revenue from extra unit sold Revenue lost on all sales due to price fall

Perfect competition (many firms)
Max p(q)q – c(q) Perfect competition: p(q) = p p=MR = MC p + 0 = c’(q) Revenue, R(q) = p(q)q Cost Marginal cost Revenue from extra unit sold Firm is too small to affect price

Pricing in the short run
Perfect competition: p = 20 c(q) = q+0.1q2 Find the firm’s profit-maximizing q Monopolist p(q)= q c(q) = q+0.1q2 Find the firm’s profit-maximizing q

Cost function definitions
c(q) = q+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 Solve for q ATC MC q

How many firms will there be?
p Perfect competition Knowing q P = MC(q) Q=D(P) #firms = Q/q ATC MC D(p) q

The long run outcome Perfect competition: D(P) = 600 - 20P
c(q) = q+0.1q2 Find the long run q Find the long run price, and # of firms Natural monopoly: D(P) = P c(q) = q+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 ATC 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 linear, so is marginal revenue P = A – Bq MR = A – 2Bq pm Profit MC Demand MR qm Optimal quantity set by monopolist

Inefficiency of monopoly
Dead weight loss 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 q q Elasticity > 1 Elasticity = 1
Constant elasticity of demand Elasticity < 1 q q p = q -e p = a - bq

Monopolist’s decision
Natural monopoly: D(P) = P c(q) = q+0.1q2 What q will monopolist choose? What is their profit? What is elasticity of demand at this price/quantity?

Monopoly in an Edgeworth box
Can we understand the inefficiency of monopoly in terms of our Edgeworth box analysis? Recall: the market got the right prices… …but monopolist doesn’t

“Market”-determined prices
Person B wBx y wBy Contract curve Endowment wAy Person A x wAx

Suppose A could choose a price
Person B wBx y wBy B’s price offer curve Endowment wAy Person A x wAx

A chooses a point off the contract curve
Person B wBx y Competitive equilibrium wBy A’s best bundle B’s price offer curve Endowment wAy Person A x wAx

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

If you buy x units, you pay a total of T + cx
First degree PD Outcome is Pareto efficient Consumer earns no consumer surplus Profit of fully discriminating monopolist MC = c Profit of non- discriminating monopolist Demand xm x* Alternative pricing mechanism: If you buy x units, you pay a total of T + cx Entry fee

Entry fees in the Edgeworth box
Person B wBx y A’s best bundle with PD is on the contract curve Entry fee paid by B to A wBy B’s price offer curve Endowment A’s best without PD wAy Person A x wAx Recall: The 2nd fundamental theorem

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 MC = c Demand Demand x*A x*B Consumer A 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 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….
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 But then A would choose option B She gets surplus V from option B, and 0 from option A Monopolist gets profit U A’s demand B’s demand TB V W U MC x*B x*A x

R RA RB x x*B x*A Option A Option B is better than option A
for person A RA RB Option B x x*B x*A

The monopolist can do a little better….
A’s demand 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 B’s demand V W U 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 still willing to take option A’’ over option B’’ Profit up by DW-DU A’s demand DW B’s demand V W U MC DU x*B x*A x’’B 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 Loss from lost sales to B-types from further decreasing x+B V Stop when = W U MC x*B x*A x x+B

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 MC A B A B x+B x*A x x+B=0 x*A x

2nd degree price discrimination
High type: DH(P) = 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) = 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 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

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

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