1Pricing, Monopolists, and DiscriminationISyE 6230Single-Product Monopolist¡ Demand is q=D(p); Inverse demand function p = P(q)l Assume D’(p) < 0 and Revenue concave¡ C(q) is cost of producing q unitsl C’(q) > 0¡ Monopolistl maxp[pD(p) – C(D(p))]2Single-Product Monopolist¡ First order conditions:l pm– C’(D(pm))=-D(pm)/D’(pm)l Or (pm-C’)/pm= 1/ε (1)¡ Where ε = -D’pm/D is demand elasticity at the monopoly price pml Lerner Index = relative “markup”¡ (Equivalently, if qm´ D(pm),l MR(qm) ´ P(qm)+P’(qm)qm= C’(qm) )¡ Implications?Definitions¡ Consumer surplus:l What customers would pay in excess of what they already spend¡ Total surplus = consumer + supplier¡ Social welfare = consumer surplus + firm’s profit¡ Dead-weight loss:l Reduction in benefit due to inefficient allocation of resources3Elasticity¡ Given a function B = f(A…)¡ Elasticity of B with respect to A (eB,A)=% change in B / % change in A=(MB/B)/(MA/A) = (∂B/∂A)*(A/B)Elasticity¡ Price Elasticity of Demand (eQ,P)l eQ,P= (∂Q/∂P)*(P/Q)l 3 regions:¡ eQ,P< -1 elastic¡ eQ,P= -1 unit elastic¡ eQ,P> -1 inelastic¡ If we raise price by 1%, then what happens to demand?4Elasticity¡ Income Elasticity of Demandl eQ,I= (∂Q/∂I)*(I/Q)¡ If eQ,I=2.0 for auto, then a 10% increase in income will lead to what? Elasticity¡ Cross-Price Elasticityl How does quantity demand change with respect to price change in another good?l eQi,Pj= (∂Qi/∂Pj)*(Pj/Qi)l eQi,Pj> 0 à i and j are substitutesl eQi,Pj< 0 à i and j are complementsl Examples?5Representative Income & Price Elasticities¡ Necessities versus luxury items?¡ Price elasticies?-1.21.2Housing: ownership-0.181Housing: rent-0.541.06Gasoline-1.290.71Charitable Giving-0.20.22Medical Services-1.140.61Electricity-1.23Auto-0.210.28FoodPrice ElasticityIncome ElasticityItemLinear Demand Curve¡ Linear demand àQ = a + bP (b < 0)¡ eQ,P=(∂Q/∂P)*(P/Q) =¡ Implications?PQba/b6Alternative Demand Curve¡ Suppose that Q = aPb, b <0l eQ,P= (∂Q/∂P)*(P/Q) = (baPb-1)(P/(aPb))l We also have log Q = log a + b log P¡ Generalized version:l X = aPxbPycIdl log X = log a + b log Px+ c log Py+ d log Il eX,Px=b, ex,Py=c, ex,I=dEstimating Demand Curves¡ Run experimentationl Change prices and see how demand changesl Issues:¡ Do you have representative customers?¡ Can you deal with confounding factors?¡ Use historical datal Past salesl Past economic datal Issue: ¡ Can you account for confounding issues (e.g., growth, income changes)?7Multi-Product Monopolist¡ Produces goods i = 1,…,n¡ Charges prices p=(p1,…,pn)¡ Sells quantities q=(q1,…,qn) where qi= Di(p) is the demand for good i¡ Cost of producing is C(q1,…,qn)Multi-Product Monopolist¡ Maximizesl ∑i=1npiDi(p) –C(D1(p),…,Dn(p)) à¡ If C(q1,…,qn) = ∑i=1nCi(qi), and let Ri´ piDi, then algebra gives us:¡ where εii´-(∂Di/∂pi)(pi//Di) & εij´ –(∂Dj/∂pi)(pi/Dj)(2) i allfor ∑∑∂∂∂∂=∂∂+∂∂+≠ jijjijijjiiiipDqCpDppDpD )'(1'∑≠−−=−ijiiiijjjjiiiiiRDCppCpεεε8Multi-Product Monopolist¡ Multi-product monopolist compared to before? ¡ Goods that are substitutes (εij< 0)l (An increase in piraises demand for good j)l RHS side is now bigger than before, so Lerner exceeds inverse of its own elasticity¡ Goods that are complements (εij> 0)l (A decrease in piraises the demand for good j)l RHS side is now smaller than before, so Lerner index is less than inverse of own elasticity of demandl Could even sell some goods below marginal cost (à Lerner index < 0) so as to raise demand for other goods!Price discrimination¡ Same commodity sold at different prices to different consumersl Entrance ticket to Disneylandl (But not always…freight charges may not be!)¡ These price differences cannot be explained by the difference in the marginal cost of making the goods available for the various consumers¡ An economic good is defined not only according to its physical properties but also to the space, time, and state of the world at which it is available for consumption (Debreu 1959)9Necessary for Price-Discrimination¡ Firm must have some market power¡ Firm controls the sale of its productsl Secondary markets make this more difficult¡ Consumers should have heterogeneous utilities from the goodl I.e., different price elasticities of demand First-degree (Perfect) Price Discrimination¡ Seller charges each individual consumer his reservation price¡ Seller must know how much the consumer is willing to pay¡ Examples?10Second-degree discrimination¡ Firm makes n separate prices and offers them (all) to each group (who purchase at a price smaller or equal to their reservation pricesl 1st, 2nd, or 3rdrailway tickets¡ Other ways to achieve this are through quantity discounts and two-part tariffs¡ Each consumer “self-selects” by their own choiceThird-degree discrimination¡ Seller separates consumers into groups and offers the monopoly price to each class¡ Uses a direct signal about demand¡ Examples?11Third-degree Example¡ Firm can distinguish customer segmentsl P1(q1) = 100-q1l P2(q2) = 50 – q2¡ C(q) = cq where c=$0PQ1001005050Q1= 100-p1or P1=100-q1Q2= 50-p2or P2=50-q2Third-degree Example¡ If monopolist cannot price discriminate on type:PQ15010010050Q = 100-p or P=100-qQ=Q1+Q2= 150-2p or P=75-q/212Third-degree Example¡ Uniform price in the market:l If sell with uniform price to both customer types¡ maxppD(p) = p(150-2p) = 150p-2p2¡ à pm= 75/2, qm= 75, Πm= 2812.5l If sell with uniform price to only p=100-q customers¡ à pm= 50, qm= 50, Πm= 2500¡ Suppose can price discriminatel p1= 50, q1= 50, Π1= 2500l p2= 25, q2= 25, Π2= 625l à Πtotal= 3725¡ Can monopolist ever be worse under 3rddegree PD?Third-degree Multi-market¡ Monopolist charges linear tariff for each market, l {p1, …,pi, …,pm}l {q1=D1(p1),…qi,…,qm}l Q=∑i=1mDi(pi)¡ Monopolistl maxp[∑i=1mpiDi(pi) – C(∑i=1mDi(pi))]l à (pi-C’(q))/pi= 1/εil Implications?13Second-degree Discrimination¡ Monopolist cannot tell customers apart but can offer different bundles¡ Needs to make sure that consumer does not choose a bundle directed towards another customer¡ Examples?¡ How would he do this?Second-degree Discrimination¡ T(q) = A + pq is two-part tariff¡ Customer i pays Tiand consumes qi¡ Customer utility is θiV(q) – Tiif they pay Tiand consume q units; 0 otherwise l V(0) = 0, V’(q) > 0, V”(q) < 0¡ λ is probability customer is