Lecture 6: Price discrimination II (Nonlinear Pricing)

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1 Lecture 6: Price discrimination II (Nonlinear Pricing) EC 105. Industrial Organization. Fall 2011 Matt Shum HSS, California Institute of Technology November 14, 2012 EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

2 Outline Outline 1 The Two Type Case 2 Empirical calibration: Cable TV Markets 3 The Continuum of Types Case C 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

3 The Basic Model Outline 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}, 1 < 2; lo and hi types 2. [, ] EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

4 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. IR( 1 ) : 1 V (q 1 ) T 1 0; IR( 2 ) : 2 V (q 2 ) T 2 0. (1) Since 2 consumers can always buy the 1 bundle, only IR( 1) is relevant EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

5 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 IC( 2 ) : 2 V (q 2 ) T 2 2 V (q 1 ) T 1 (2) IC( 1 ) : 1 V (q 1 ) T 1 1 V (q 2 ) T 2 (3) The relevant IC constraint is that of the high-valuation consumers, IC( 2 ). Assume this for now. 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. IC( 2 ) and IR( 1 ) EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

6 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. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

7 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 ] (4) 2 V (q 2 ) = c (5) It follows from (5) that the quantity purchased by the high value consumers is socially optimal (marginal utility equal marginal cost) (absence of distortion at the top), and from (4) that the quantity consumed by the low-demand consumers is socially suboptimal: 1V (q 1) > c and their consumption is distorted downwards EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

8 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 )], EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

9 The Two Type Case 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 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. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

10 Regulation in Cable The Two Television Type Case 185 The theory described in the previous section applies also to the case of con- Figure 1. Quality degradation with two types adapted from Maskin and Riley (1984) indifference curves 2.1. Continuous for U(q) Types C(q) but Discrete Qualities EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

11 Empirical calibration: Cable TV Markets 194 The Journal of LAW& ECONOMICS Total Prices/Channels Table 3 Preliminary Evidence of Quality Degradation Three-Good Markets Two-Good Markets Mean Difference Mean Difference p 3 /channel 1.06 (.03).04 (.01) 1.10 (.03).61 (.06) p 2 /channel 1.10 (.03).13 (.02) 1.72 (.07)... p 1 /channel 1.23 (.04)... N Note. Reported are the average price per channel for each offered cable service. Channels include all top 40 satellite channels and, for the lowest quality service, all major broadcast networks. Ratios are formed with total price and total channels. Values in the Difference columns are the difference in price per channel in that row and the row that follows. The cable system in one two-good market included no satellite or broadcast networks in its lowest quality service. Standard errors are in parentheses. mand (Crawford 2000). 23 As such, in this paper, we directly measure cable service quality by exploiting the restrictions of a theoretical model of optimal price and quality choice to recover quality measures consistent with the observed prices EC 105. Industrial andorganization. marketfall shares (MattWe ShumLecture then HSS, California 6: Price relate discrimination Institute these of Technology) II to (Nonlinear thepricing) networks available November to14, determine / 30

12 Empirical Regulation calibration: Cable intv Cable Markets Television 201 EC 105. Industrial market Organization. share from Fall 2011 Table (Matt1. Shum The Lecture HSS, WTP California 6: Pricefor discrimination Institute the of cut Technology) II types, (Nonlinear those Pricing) consumers just November 14, / 30 Table 5 Recovered Parameter Values and Implied Qualities Variable Three-Good Markets Two-Good Markets One-Good Markets Net type distribution: f f f f t t t Qualities: q q q % Degradation: (t3 q 3)/t (t2 q 2)/t (t1 q 1)/t Price/quality ratio q 3/p q 2/p q 1/p N Note. Parameters of net type distribution are obtained using the procedure in Section 4.2. Quality measures are calculated using equation (12). Percentage of degradation evaluated at cut types is defined as the marginal type just inclined to purchase that quality.

13 The Continuum of Types Case Insights from previous example generalize to case where there are an infinite number of types Now 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 profit is Π m = [T (q()) cq()]f ()d The monopolist maximizes his profit subject to two types of constraints EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

14 IR constraints The Continuum of Types Case For all, V (q()) T (q()) 0 As before, it suffices that IR () holds: V (q()) T (q()) 0 (6) If (6) holds, any type can realize a nonegative surplus consuming s bundle: V (q()) T (q()) ( )V (q()) 0 EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 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(, ) (7) 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()) (8) 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. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

16 Equation (8) can be used to characterize the payment function once the quantity function q() is known. Assume for now that q() is strictly monotonic in (we ll show this later). Letting α( ) = q 1 ( ), i.e., α(q()) =, then (8) is T (q) = α(q)v (q) Now from the IC U() = max V (q( )) T (q( )) Using the envelope theorem, U() Thus we can write (use U() = 0 ) U() = = V (q()) U(t) dt + U() = t V (q(t))dt EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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 (9) EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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 (9) EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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()) (10) f () EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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 = EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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 price of an extra unit when the consumer already consumes q units. From consumer optimization T (q()) = V (q()) V (q()) = T (q()) Substituting in (10), 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 () (11) EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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 (10) 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. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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() Thus q () > 0, q() increases with V (q()) V (q()) > 0, EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

28 IC constraints The Continuum of Types Case 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. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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 (12) 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. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 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 (12) 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 (11), thus unifying the theories of second degree and third degree price discrimination EC 105. Industrial Organization. Fall 2011 (Matt ShumLecture HSS, California 6: Price discrimination Institute of Technology) II (Nonlinear Pricing) November 14, / 30

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