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

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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)

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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)p(q)

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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 Revenue from extra unit sold Revenue lost on all sales due to price fall Marginal cost

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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 Revenue from extra unit sold Firm is too small to affect price Marginal cost

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Perfect competition: p = 20 c(q) = 62.5+10q+0.1q 2 Find the firm’s profit-maximizing q Pricing in the short run Monopolist p(q)= 50 - 0.1q c(q) = 62.5+10q+0.1q 2 Find the firm’s profit- maximizing q

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c(q) = 62.5+10q+0.1q 2 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

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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 p ATC MC

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

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Perfect competition: D(P) = 600 - 20P c(q) = 62.5+10q+0.1q 2 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+0.1q 2 What is q when MC=ATC? How many firms will there be?

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Natural monopoly D(p)MC q p ATC MC D(p)

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Monopolist Natural monopolies –Electricity –Telephones –Software? Monopoly can also be by government protection –Patented drugs Imposed with violence –Snow-shovel contracts in Montreal

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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

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Monopoly pricing (no price discrimination) Note: When demand is linear, so is marginal revenue P = A – Bq MR = A – 2Bq MC Demand MR Optimal quantity set by monopolist pmpm qmqm Profit

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Inefficiency of monopoly MC Demand MR pmpm qmqm q* Dead weight loss Mark-up over Marginal cost

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(Price) elasticity of demand The elasticity of demand measures the percent change in demand per percent change in price: = -(dq/q) / (dp/p) = -(p/q )*( dq/dp) < 0

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Optimal mark-up formula p(q) + qp’(q) = c’(q) can be rearranged to make: p = MC / (1 – 1/| |) This can be rearranged to yield: (p – MC)/MC = 1 / (| | - 1) > 0

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Demand elasticity q p Constant elasticity of demand q p Elasticity > 1 Elasticity < 1 Elasticity = 1 p = q - p = a - bq

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Natural monopoly: D(P) = 600 - 20P c(q) = 640+10q+0.1q 2 Monopolist’s decision What q will monopolist choose? What is their profit? What is elasticity of demand at this price/quantity?

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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

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“Market”-determined prices Person A x y Person B BxBx AyAy Endowment ByBy AxAx Contract curve

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Suppose A could choose a price Person A x y Person B BxBx AyAy Endowment ByBy AxAx B’s price offer curve

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A chooses a point off the contract curve Person A x y Person B BxBx AyAy Endowment ByBy AxAx B’s price offer curve A’s best bundle Competitive equilibrium

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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.

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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

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

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

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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* B x* A

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Entry fees as “two-part-tariffs” Let A’s consumer surplus be T A and let B’s be T B. Monopolist sets a pair of price schedules: Consumer A R A = T A + cx Consumer B R B = T B + cx Entry fees Price per unit = c

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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

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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 TATA TBTB B’s demand

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x R x* B x* A RBRB RARA Option A Option B Option B is better than option A for person A

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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’s demand MC U V W x* B x* A

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…but it can do even better Option A’’: x* A for $(U+W+ W)+cx* A Option B’’ x’’ B for $(U- U)+cx’’ B A still willing to take option A’’ over option B’’ Profit up by W- U UU WW x’’ B x A’s demand MC U V W x* B x* A B’s demand

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…and the best it can do is? Stop when = W x+Bx+B Gain from higher fees paid by A-types from further decreasing x + B Loss from lost sales to B-types from further decreasing x + B x A’s demand MC U x* B x* A B’s demand V

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Should the monopolist bother selling to low-demand consumers? x+Bx+B x x* A A B MC Going further, you lose more on the B-types than you gain on the A-types x + B =0 x x* A A B MC 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

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High type: D H (P) = 100 - P Low type: D H (P) = 70 – P MC=10 2 nd degree price discrimination What bundles should the monopolist offer? At what prices?

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High type: D H (P) = 100 - P Low type: D H (P) = X – P MC=10 2 nd degree price discrimination For what value of X will the monopolist not sell to low types?

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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 surplus FN –this is always what they could earn if they pretended to be B-types FN: Whenever x + B >0

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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 p A = c / (1 – 1/| |) p B = c / (1 – 1/| |)

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Some problems Non-constant marginal cost? –Replace c above with c’(x A +x B ) What if demands are inter-dependent? –E.g., x A (p A,p B ) and x B (p B,p A ) Applications –Peak-load pricing A: Riding the metro in rush-hour B: Riding off-peak –Children’s and adults’ ticket prices

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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

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Software example Two types of consumer who have different valuations over two goods Assume marginal cost of production is zero Consumer typeWord processorSpreadsheet Type A120100 Type B100120

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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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