Lecture 10: Price discrimination Part II

Transcription

1 Lecture 10: Price discrimination Part II EC 105. Industrial Organization. Matt Shum HSS, California Institute of Technology EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 1 / 30

2 Price discrimination is endemic! In this lecture we consider various examples of pricing practices which can be interpreted as (typically second-degree) price discrimination Consider real-world examples where firm sets different prices for different groups of customers 1 Airline tickets 2 Student vs. Adult tickets 3 Journal subscriptions: personal vs. institutional 4 Senior discounts 5 Food at airports, sporting events, concerts All of these examples correspond to inverse elasticity principle: customers with more elastic demand get lower prices. Most real-world cases of price discrimination are more subtle however! EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 2 / 30

3 Example: nonlinear pricing Charging customers a price depending on the quantity purchased (electricity, telephone service) Typically exhibit quantity discounts: Consumers who buy more are paying less per unit Makes sense if demand more elastic at higher quantities Examples: Starbucks: 12 oz (25 yuan), 16oz (30 yuan), 20oz (35 yuan) 12 oz: 2.08/oz; 16oz: 1.875/oz; 20oz: 1.75/oz Buy one, get second at 50% buy 5 for the price of 4 Internet examples: subscriptions vs. a la carte Online newspapers Movie streaming (Netflix) Uber: $99 for pool rides for a month EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 3 / 30

4 Example: Dynamic pricing Dynamic pricing: pricing which changes over time Examples: Airline tickets Music concert Los Angeles Olympics (2028) Typically, price is highest at beginning, then falls Makes sense if earlier customers have less elastic demand For many internet products (gaming apps): freemium pricing free to play Upgrades cost money Way for firm to discriminate between casual users (elastic demand) vs loyal/serious users (inelastic demand) EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 4 / 30

5 Example: Market segmentation Sometimes market institutions can help firms price discriminate by sorting consumers by their valuations This is called market segmentation. One example are secondary markets (markets for used goods) Consumers with high preference for quality buy new goods Consumer with lower preference for quality buy used goods. Existence of secondary markets allows new good producers to raise prices and get higher profits Very relevant in e-commerce: used good markets proliferate online Amazon marketplace, Taobao, etc. Similar logic for leasing/sharing economy: Airbnb Ability to share good may stimulate demand for the good Gaming: existence of markets for in-game accessories ( gold-mining ) can increase player interest in the game EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 5 / 30

6 The Basic Model A firm produces a single good at marginal cost c. Consumers receive utility V (q) T (q) if they purchase a quantity q and utility 0 otherwise. Two cases: 1. { 1, 2} 2. [, ] EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 6 / 30

7 The Two Type Case The Two Type Case Monopolist offers two bundles (we assume the monopolist serves both types; λ is sufficiently large): (q 1, T 1), directed at type 1 consumers (in proportion λ), and (q 2, T 2), directed at type 2 consumers (in proportion 1 λ). The monopolist s profit is Π m = λ(t 1 cq 1 ) + (1 λ)(t 2 cq 2 ) Monopolist faces two types of constraints. The individual rationality constraint for type (IR()) requires that consumers of type are willing to buy. Since 2 consumers can always buy the 1 bundle, the relevant IR constraint is IR( 2): 1V (q 1) T 1 0 (1) EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 7 / 30

8 The Two Type Case The incentive compatibility constraint for type (IC()) requires that consumers of type prefer the bundle designed for them rather than that designed for type The relevant IC constraint is that of the high-valuation consumers, IC( 2 ): 2 V (q 2 ) T 2 2 V (q 1 ) T 1 (2) In fact, we will proceed ignoring IC( 1 ) and then show that the solution of the subconstrained problem satisfies it. We thus solve the problem: max Π m s.t. (1) and (2) EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 8 / 30

9 The Two Type Case Since IR( 1 ) will hold with equality at the optimum, we can rewrite (2) as T 2 2 V (q 2 ) [ 2 V (q 1 ) T 1 ] = 2 V (q 2 ) ( 2 1 )V (q 1 ) T 1 can be chosen to appropriate the type- 1 surplus entirely, but T 2 must leave some net surplus to the type 2 consumers, because they can always buy the bundle (q 1, T 1 ) and have net surplus 2 V (q 1 ) T 1 = ( 2 1 )V (q 1 ) EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 9 / 30

10 The Two Type Case Substituting this into the objective function, the monopolist solves the following unconstrained problem max q1,q 2 λ( 1 V (q 1 ) cq 1 ) + (1 λ)[ 2 V (q 2 ) cq 2 ( 2 1 )V (q 1 )] First Order Conditions are: 1 V (q 1 ) = [ c 1 1 λ λ ] (3) 2 V (q 2 ) = c (4) It follows from (4) that the quantity purchased by the high value consumers is socially optimal (marginal utility equal marginal cost). No distortion at the top and from (3) that the quantity consumed by the low-demand consumers is socially suboptimal: 1V (q 1) > c and their consumption is distorted downwards. quantity degradation EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 10 / 30

11 The Two Type Case It remains to check that the low demand consumers do not want to choose the high demand consumers bundle. Because they have zero surplus, we require that 0 1 V (q 2 ) T 2. But this condition is equivalent to which is satisfied 0 ( 2 1 )[V (q 2 ) V (q 1 )], Bundle 2, although offering higher quantity, is too expensive for the low types. EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 11 / 30

12 The Two Type Case Intuition Monopolist attempt to extract the high demand consumers large surplus faces threat of personal arbitrage: High demand consumer can consume the low-demand consumers bundle if his own bundle does not generate enough surplus. To relax this personal arbitrage constraint, the monopolist offers a relatively low consumption to the low demand consumers. OK because typically high demand consumers suffer more from a reduction in consumption than low demand ones (single crossing property). Since low demand consumers are not tempted to exercise personal arbitrage, no distortion at the top (recall welfare gains can be captured by the monopolist through an increase in T 2. EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 12 / 30

13 Now we consider a mathematical generalization. Let be distributed with density f () (and CDF F ) on an interval [, ]. Monopolist offers a nonlinear tariff T (q). A consumer with type purchases q() and pays T (q()). Monopolist s aggregate profit (across all consumer types) is: Π m = [T (q()) cq()]f ()d The monopolist maximizes his profit subject to two types of constraints EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 13 / 30

14 IR constraints For all, V (q()) T (q()) 0 As before, it suffices that IR () holds: V (q()) T (q()) 0 (5) If (5) holds, any type can realize a nonegative surplus consuming s bundle: V (q()) T (q()) ( )V (q()) 0 EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 14 / 30

15 IC constraints should not consume the bundle designed for ( ) IC(): for all, : U() U(, ) = V (q()) T (q()) V (q( )) T (q( )) U(, ) (6) These constraints are not very tractable in this form. However, we can show that it suffices to require that the ICs are satisfied locally ; i.e., a necessary and sufficient condition for = argmax U(, ) = V (q( )) T (q( )) is given by the FOC (evaluated at the true type ): V (q()) = T (q()) (7) This says that a small increase in the quantity consumed by they type consumer generates a marginal surplus V (q()) equal to the marginal payment T (q()). Thus, the consumer does not want to modify the quantity at the margin. EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 15 / 30

16 Now from the IC U() = max V (q( )) T (q( )) Using the envelope theorem (red part =0): U() Thus we can write (use U() = 0 ) U() = = V (q())+q () [V (q()) T (q())] U(t) dt + U() = t V (q(t))dt EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 16 / 30

17 Note that consumer s utility grows with at a rate that increases with q(). This is important for the derivation of the optimal quantity function, as it implies that higher quantities differentiate different types more, in that the utility differentials are higher. Since leaving a surplus to the consumer is costly to the monopolist (recall T (q()) = V (q()) U()), the monopolist will tend to reduce U and to do so, will induce (most) consumers to consume a suboptimal quantity. Intuition from the previous equations is that optimality will imply a bigger distortion for low- consumers. EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 17 / 30

18 Since T (q()) = V (q()) U() = V (q()) V (q(t))dt, we can write ( ) Π m = V (q()) V (q(t))dt cq() f ()d (8) EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 18 / 30

19 Recall: Integration by Parts If u and v are continuous functions on [a, b] that are differentiable on (a, b), and if u and v are integrable on [a, b], then b a u(x)v (x)dx + b a u (x)v(x)dx = u(b)v(b) u(a)v(a) Let g = uv. Then g =uv + vu. By the fundamental theorem of calculus, b a g = g(b) g(a). Then and the result follows. b a g (x)dx = u(b)v(b) u(a)v(a), EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 19 / 30

20 Now, integrating by parts, with f () = F () and V (q(t))dt = G(), [ ] V (q(t))dt f ()d is equal to V (q())d 0 V (q())f ()d = V (q())[1 F ()]d Then going back to (8) EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 20 / 30

21 We can write as Π m = Π m = ( V (q()) V (q(t))dt cq() ) f ()d ([V (q()) cq()]f () V (q())[1 F ()]) d Now max Π m w.r.t. q( ) requires that the term under the integral be maximized w.r.t. q() for all, yielding: [V (q()) c]f () V (q())[1 F ()] = 0 or equivalently V (q()) = c + [1 F ()] V (q()) (9) f () EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 21 / 30

22 V (q()) = c + [1 F ()] V (q()) > c f () Thus marginal willingness to pay for the good V (q()) exceeds the marginal cost c for all but the highest value consumer = Quantity degradation (for < ) No distortion at the top (MU = MC) for EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 22 / 30

23 Moreover, for this specification of preferences, we can get a simple expression for the price-cost margin. Let T (q) p(q) denote the marginal price when the consumer is consuming q units. From consumer optimization T (q()) = V (q()) V (q()) = T (q()) Substituting in (9), which I write again here V (q()) = c + [1 F ()] V (q()), f () = p(q()) we have: p(q()) c p(q()) = [1 F ()] f () (10) EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 23 / 30

24 f () We will assume that the hazard rate of the distribution of types, [1 F ()], is increasing in (common assumption, satisfied by a variety of distributions). We can rewrite (9) as: ( Or letting Γ() ( ) [1 F ()] V (q()) = c f () ) [1 F ()] f (), simply as: Γ()V (q()) = c, where Γ () > 0 by our increasing-hazard-rate assumption EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 24 / 30

25 Γ()V (q()) = c Then totally differentiating (with variables q() and ), we obtain: dq() d using V concave and Γ () > 0 = Γ () Γ() V (q()) V (q()) > 0, Thus q () > 0: q() increases with. (Higher types get higher quantity) EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 25 / 30

26 Now consider the price-cost margin. Recall p(q()) c p(q()) = [1 F ()] f () = 1 Γ() The derivative of the RHS with respect to is: 1 1 [ ] 2 [Γ() + Γ ()] < 0 Thus p c c decreases with consumer type,and therefore with output EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 26 / 30

27 Finally, recall T (q) = p(q). Hence Thus T (q) is concave. As a result: T (q) = dp dq = dp/d dq/d = ( ) (+) < 0 Average price per unit T (q)/q decreases with q (Maskin and Riley s quantity discount result). Because a concave function is the lower envelope of its tangents, the optimal nonlinear payment schedule can also be implemented by offering a menu of two part tariffs (where the monopolist lets the consumer choose among the continuum of two-part tariffs) EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 27 / 30

28 IC constraints We argued that it was enough to require that the ICs are satisfied locally ; i.e., that a necessary and sufficient condition for = argmax U(, ) = V (q( ), ) T (q( )) is given by the FOC (evaluated at the true type ): V (q(), ) q = T (q()) This is because of the single crossing property, 2 V (q(),) q > 0 EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 28 / 30

29 The Inverse Elasticity Rule Again Decompose the aggregate demand function into independent demands for marginal units of consumption. Fix a quantity q and consider the demand for the q th unit of consumption. By definition, the unit has price p. The proportion of consumers willing to buy the unit is D q (p) 1 F ( q(p)), where q(p) denotes the type of consumer who is indifferent between buying and not buying the q th unit at price p: q(p)v (q) = p (11) The demand for the q-th unit is independent of the demand for the qth unit for q q (due to no income effects). We can thus apply the inverse elasticity rule. EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 29 / 30

30 The Inverse Elasticity Rule Again The optimal price for the qth unit is given by p c p = dp D q dd q p However and from (11) We thus obtain dd q dp p c p = f ( q(p)) d q(p) dp dq(p) q(p) = dp p = 1 F ( q(p)) q(p)f ( q(p)) which is equation (10), thus unifying the theories of second degree and third degree price discrimination EC 105. Industrial Organization. (Matt Shum HSS, CaliforniaLecture Institute10: of Price Technology) discrimination Part II 30 / 30