Multiproduct rms, nonlinear pricing and price discrimination Lecture notes
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1 Multiproduct rms, nonlinear pricing and price discrimination Lecture notes Estelle Cantillon November 29, 2007
2 1 Introduction In many industries, rms produce several products. This raises a new set of positive (pricing, product positioning, economies of scope) and normative (welfare distortion, mergers, unbundling) questions. We will distinguish between environments where consumers consume a single unit of a product and environments when they consume several products and/or several units of the same product. Unit demand consumers Pricing Product positioning Economies of scope Multi-unit demand consumers Demand Pricing Economies of scope
3 2 Unit demand consumers Even when rms are multiproduct, as long as consumers only buy one unit, the estimation of demand can proceed as before. What changes is rm behavior. 2.1 Pricing Suppose monopolist produces two products with demand D j (p 1 ; p 2 ); j = 1; 2: Each product is produced under a separate constant marginal cost technology. Optimal prices solve max p 1 ;p 2 (p 1 c 1 )D 1 (p 1 ; p 2 ) + (p 2 c 2 )D 2 (p 1 ; p 2 )
4 The FOCs can be expressed in matrix notations as " p1 c 1 p 2 c 2 # = " D1 D 2 # Let D ij j : Solving for p 1 c 1 ; we get: p 1 c 1 = D 1D 22 D 2 D 21 D 11 D 22 D 12 D 21 = D 1 + D D 12 D D 22 D 2 D 21 D 11 D 22 D 12 D 21 Prices are higher when D 12 > 0 (products are substitutes - which comes directly from demands based on the discrete choice models we have seen so far). They depend on both own and cross price elasticities. Appl n: Leslie (2005) Leslie asks whether a Broadway theater sets prices for its di erent seat categories optimally for a speci c play "Seven Guitars". Setting corresponds to multiproduct monopolist facing unit-demand consumers.
5 Demand speci cation distinguishes between regular tickets, tickets bought with a coupon and tickets bought on the same day at a booth. The outside good is all other Broadway shows. Utilities from each alternative is given by: U ijt = 8 >< >: it q j ( 1 y 2 i p jt ) j = h; m; l it q h ( 1 y 2 i p ct ) coupons it q h (( 1 y 2 i p bt 1 y i 2 ) tickets bought at booth ( 1 y 2 i p o ) o outside good where y i is the income of individual i ( 1 y 2 i is the budget for entertainment), it is individual i s taste for the show at time t (assumed to be distributed according to an exponential exp(x t ): The model also includes a probability of receiving a coupon in any period (if not, then alternative c is not available), and the fact that some categories may ll up and thus are not available when consumers asks for it. Conditional on a value for parameters, on a sequence over consumers and on draws over coupons, we can easily partition the set of (y i ; it ) according to which alternative is chosen and thus predict market shares (the distribution of it is assumed
6 and the distribution of y i is known)! Simulated MLE, result is estimated demand q jt (p t ; X t ; b ) Counterfactual: max fpht ;p mt ;p lt ;p bt ;p ct g p jt q jt (p t ; :): Explores several scenarios P P t j6=o
7 2.2 Product positioning So far, we have considered how the monopolist should optimize its pricing structure, given the set of products (with xed attributes) that it sells. We now endogenize the products. Suppose the monopolist faces two types of consumers, characterized by their taste for quality (quantity), 2 f L ; H g; 0 < L < H (the proportion of high types is ): Types are private information. Monopolist faces a cost of 1 2 q2 for producing at quality q: Consumers get utility q p from consuming a good of quality q and paying p for it. First best benchmark: Qualities served to each type solves max q q 2 q2 for 2 f L ; H g : ql F B = L ; qh F B = H: First best can be achieved when types are known to all through rst degree (perfect) price discrimination. Monopolist sets prices such that p i = i qi F B : 1
8 In practice types are not observed so the monopolist solves (conditional on "full coverage") 1 max (p q L ;q H ;p L ;p H H 2 q2 H ) + (1 )(p L subject to H q H p H H q L p L and L q L p L q2 L ) This is the standard second degree price discrimination problem (screening problem). The solution is qh = H; ql = L elsewhere) 1 (no distortion at the top, downward distortion p = H q > c = q q H +q L 2 (price di erences are not aligned with marginal costs di erences) Total welfare is lower but pro ts are higher.
9 Extension 1: Types are continuously distributed according to a cdf F (and quality can take continuous values too) Suppose preferences take the form u(q; ) P (q). Given any price function P (q); q() = arg max q u(q; ) P (q); yielding a utility level v() = max q u(q; ) P (q): The monopolist solves max q(:) Z (P (q() C(q()) df () = Z (u(q(); ) C(q() v()) df () under the usual incentive compatibility and individual rationality constraints which can be shown to be equivalent to q(:) monotonically increasing, v 0 () = u (q(); ) > 0 and v() 0: After integration by parts (substituting v 0 ) we get max q(:) Z u(q(); ) C(q() 1 F () u (q(); ) f() This expression can be maximized point-wise, yielding as FOC u q (q(); ) = C q (q() + 1! F () u q (q(); ) f() df ()
10 Under a single crossing condition u q > 0 the second order condition q(:) increasing is satis ed. We have as before no distortion at the top and downward distortion in the quality elsewhere. Extension 2: Types are multi-dimensional Issues now are whether monopolist has enough instruments and to what extent those multiple dimensions can be collapsed into a "su cient statistics". Also which ones are observable and thus amenable to "personalized pricing" (third degree price discrimination) and which ones aren t. See Rochet and Stole (2002) as an example. Extension 3: Market is an oligopoly Consider again the model above with several rms and consumer preferences given by u j (q j ; ) P j (q j ); j = 1; :::; n: Given P j (:); the indirect utility of consumer from rm j s o ering is given by v j () = max qj u j (q j ; ) P j (q j ): Thus from rm j s perspective, consumer s best alternative is v j () = maxf0; v 1 (); :::; v j 1 (); v j+1 (); :::g
11 The best response by rm j is the same as the best response by a monopolist which faces a consumer with utility u j (q; ) P j (q) and an outside option v j (): The fact that the outside option depends on types implies that the identity of marginal consumer is harder to pin down ex-ante. Extra structure is needed in general (Stole, 2005). Horizontal di erentiation (Spulber, 1989) Suppose for example the source of heterogeneity among consumers is location. Let measure the distance of a consumer to rm 1 located at 0 on the hotelling line. Assume further that u 1 (q; ) < 0 < u 2 (q; ) and that u1 q (q; ) < 0 < u2 q (q; ): Consider rm 2 s problem, taking P 1 (q) as xed. v 2 () = maxf0; max q u 1 (q; ) P 1 (q)g: By the envelope thm, v 0 2 () = u1 (q(); ) < 0: Let be the marginal consumer (the consumer indifferent between the o er from rm 1 and the o er from rm 2). Firm 1 s market is [0; ] and rm 2 s market is [ ; 1]: Thus we can separately determine the allocation of quality and market shares. Price competition is entirely focussed on the marginal consumer (each rm acts as a second-degree price discriminating monopolist on its market when it determines q()). Here competition lower prices but does not a ect quality distortion - relative to the case where both rms would be
12 under single ownership. Vertical heterogeneity. Issue now is to introduce some "friction" to avoid perfect Bertrand competition. Possibilities include cost asymmetries (Stole, 1995), precommitment to qualities ex-ante (Champsaur and Rochet, 1989), or private information about costs (Biglaiser and Mezzetti, 2000).
13 Application: McManus (2007) Looks at evidence of second degree price discrimination and quality distortion in the specialty co ee market around the University of Virginia. Demand speci cation allows for a vertical dimension of preference for products and an horizontal dimension of preference due to location. In each period t (portion of the day), each consumer i located at location l makes a purchase (j is product, x is product line (drip, regular, specialty)): ( xi q x U ijlt = j + p j + D jl + j + " ijlt for j 2 J t X t + " iolt outside good (X t contains period dummies) where q j is the size, D jl is distance from consumer s location and j is unobserved quality. Assuming a lognormal distribution for xi and a logit distribution for "; and given the known distribution of locations, one can then compute expected market shares for each product during period t as a function of the value of parameters (also knows total population) (SMLE)
14 Combines demand estimates with costs information to check several predictions of the theory: "No distortion at the top": at the largest size, the marginal bene t of an extra ounce should be equal to the extra cost. Marginal bene t of increase the size of product j by one ounce is computed as MB j () = P l is computed on the basis of cost data. P R t x xi q ( x 1) j xi jj : Marginal cost Role of outside good, and substitution across product categories.
15 Explains di erence between expressos and specialty co ee by the fact that expressos are lower margin substitutes for specialty co ees, and low and at distortion for drip co ee by fact that competes most ercely with outside good.
16 2.3 Economies of scope When rms produce several products a natural question that arises concern the existence and importance of economies of scope. Let Y 2 R 2 a vector of outputs (two products). Let P 1 Y the projection of Y on the x axis and P 2 Y the projection of Y on the y axis. Let C(Y ) be the cost to produce output Y: De nition 1: There are economies of scope in region A if C(Y ) C(P 1 Y ) + C(P 2 Y ) for all Y 2 A such that P 1 Y; P 2 Y 2 A This is especially relevant in the context of mergers (synergy and thus e ciency motivation for mergers versus market power motivation) and in the context of regulation (unbundling, dismantling of regulated monopolies).
17 Non-identi cation result In the NEIO, we have the FOC condition for optimal prices to infer rms marginal costs. A common assumption of the pricing model is that rms have constant marginal costs. Can we identify else than marginal costs (under the assumption that products are technology independent and marginal costs are constant)? Consider the pro t function of a rm with cost function C(D 1 (p 1 ; p 2 ); D 2 (p 1 ; p 2 )) : For simplicity, suppose the rm produces two goods, and rewrite its optimization problem as max p p 1 ;p 1 D 1 (p 1 ; p 2 ) + p 2 D 2 (p 1 ; p 2 ) C(D 1 (p 1 ; p 2 ); D 2 (p 1 ; p 2 )) 2 For given observed optimal prices, the FOCs yields two nonlinear equations where C q1 (D 1 (p 1 ; p 2 ); D 2 (p 1 ; p 2 )) and C q2 (D 1 (p 1 ; p 2 ); D 2 (p 1 ; p 2 )): This is too little to infer the cost function unless costs are indeed independent across products and marginal costs are constant. The issue here is that we cannot in general infer information of greater dimension than the information we observe. Additional sources of variations (observed prices under di erent demand conditions) can help identify
18 slopes of the cost function at di erent levels but will usually not be able to cover the whole range of costs.
19 Cost function estimation Idea here is to use observed cost data as a function of outputs to evaluate presence of cost synergies. Issue raised by de nition 1 is that would ideally need to observe cost for individual products (i.e. observe single product rms). In practice this is rarely the case so that researchers typically rely on parametric assumption on cost function to estimate those costs. This is an issue and in practice di erent papers have reached di erent conclusions re the existence of economies of scope in the local and national telephony in the US following the break-up of AT&T in 1984 and the separation of local from long distance services. Beresteanu (2005) proposes an alternative non parametric test that relies on a weaker condition that "economies of scoe", "cost complementarity". De nition 2: C exhibits cost complementarities on A is C is submodular in A; i.e. if for any Y 2 A; Y 0 > 0 such that Y + Y 0 ; Y + P 1 Y and Y + P 2 Y 0 2 A; C(Y + Y 0 ) C(Y ) [C(Y + P 1 Y 0 ) C(Y )] + [C(Y + P 2 Y 0 ) C(Y )]
20 Advantage is that data requirement is smaller (you do not need to observe single product rms). Applies method to data on telephony and concludes that there exist cost complementarities between local and long distance telephony.
21 3 Multi-unit demand consumers 3.1 Demand Empirical models of consumer level demand that we have seen so far, assume that each consumer consumes at most one unit of a single product in every observation period. In order to study rm behavior in the presence of multi-unit demand consumers, we now relax this assumption. As we do so, the main concern will be to understand the data requirement for estimating demand that exibly incorporates substitution and complementarity patterns across goods Consumers may buy single units of several goods Suppose there are two goods A and B: Income not spent on A or B is spent to purchase a continuous compositive commodity. Utilities from each options are given
22 by: u A = A p A + A u B = B p B + B u AB = u A + u B + u 0 = 0 (normalization) where " [ A # ; " B ] 0 are#! unobserved product qualities distributed according to a normal 0 1 N ; and is de ned as (u 0 1 AB u A ) (u B u 0 ). De nition (Samuelson, 1974): Two goods are substitutes if the cross-price elasticity of compensated demand is positive. The two goods are complements otherwise.
23 In this simple setting, there is no wealth e ect and the quality of substitutes or complements depends on the cross-elasticity of demand. Moreover it can be shown that cross-elasticities are positive (goods A and B are substitutes) i < 0; goods are complements if > 0, and independent otherwise. The model has 5 parameters: A ; B ; ; and : The question is whether these parameters can be separately identi ed in the data (the concern is that observing consumers choose both A and B, or none, could be due to either a high value for or a high value of ): Without variations in the data, we only observe three independent moment conditions: P A ; P B and P AB : So the model is not identi ed. Identi cation can come from di erent sources of variation in the data: (1) variation in prices or explanatory variables entering A or B but not both (for example, a cross-section), (2) panel data and a parametric assumption according to which jt = j + " jt where " jt is i.i.d. across time and time. See Gentskow (2007) for a discussion and an application to online and o ine news. In principle, demand for such a exible model could rely on only aggregate data as long as we can observe the proportion of consumers buying both goods.
24 3.1.2 Consumers have multi-unit demand (Hendel, 1999 and Dubé, 2004) Suppose there is a single good but consumers may want to purchase several units of that good. The success or failure of any pricing scheme by the rm will depend on how individual demand varies with quantities. Without strong functional form assumptions, it should be clear that how individual demand varies with quantities cannot be identi ed from aggregate data. Idea is to subdivide each agent (or rm) f as the aggregation of J f independent decision makers making a decision for a speci c task j: The task involves the acquisdition of a single unit of a product (or the outside good). The utility from the choice of option k for task j of rm f is given by U fjk = X kfj + kj p k + " fjk The model allows for preferences for a computer brand at the rm level but also allows a rm to buy computers of di erent brands for the di erent tasks (observable task characteristics include divisions for example). However, the model rules out any other interactions among purchases for di erent tasks.
25 Let x fjk = 1 if rm f chooses product k for task j: Suppose that the number of tasks within a rm is distributed according to F J (:jd f ; ) (which depends on parameters and rm characteristics D f ): The predicted number of computers bought by rm j is equal to 2 J 1X 6 X f X 4 J f =0 j=0 k 3 7 x fjk (; X) 5 Pr(J f jd f ; ) Hendel (1999) applies some extension of this model to the demand for computers by the corporate sector. Dubé (2004) applies this idea to Carbonated soft drinks. The output of the empirical analysis is an individual demand for the products that allows the purchase of several units of the same products and the purchase of di erent products.
26 3.2 Pricing When consumers buy several goods or several units of the same goods, rms can generally do better than linear prices. We say that a rm price discriminates when the prices it charges for consecutive units of the same good or across several of its products cannot be explained by marginal costs variations Bundling Bundling refers to the practice of selling several products together for a di erent price than component products (in case of mixed bundling) or sell the products only as a bundle (pure bundling). Bundling is a form of (second degree) price discrimination that allows rms to extract more surplus from consumers. Adams and Yellen (1976) provided the basic intuition for why bundling may raise pro ts. Suppose a monopolist produces two goods, both a zero marginal costs. There are
27 two types of consumers. Type 1 have valuation 3 for good 1 and valuation 4 for good 2, type 2 have valuation 5 for good 1 and valuation 2.5 for good 2. p 1 p 2 p 12 pro t Opt. linear prices Opt. bundling McAfee, McMillan, Whinston (1989) derived a su cient condition for mixed bundling to be superior to linear prices (graphical intuition). More on this, next class.
28 3.2.2 Two-part tari s and other nonlinear price schedules Suppose consumers get utility V (q) T when they consume q units of the product and pay T for it (they get zero otherwise). Assume that V 0 > 0 and V 00 < 0: Suppose further that is privately observed by consumers and can take two values L ; H with c < L < H (and that low types have proba ): At any linear price p; the demand from types is given by arg max q V (q) pq: The FOC V 0 (q) = p condition implicitly de nes a downward sloping demand function D (p): Aggregate demand is given by D(p) = D L (p) + (1 )D H (p): Under linear pricing, the rm chooses its price to maximize (p c)d(p): With a two-part tari, for any given p; the highest pro t the rm achieve is when it sets the xed component equal to S L (p) = max q L V (q) pq (assuming that S L (p) > S H (p)): The optimal price in a two part tari is then given as the price that maximizes S L (p) + (p c)d(p):
29 Pro ts under a two-part tari are higher, prices are lower (applications: telephone, gas, electricity, taxi, amusement park, membership yield discount on cultural events,...). Two-part tari s can be generalized to multi-part tari s or more generally non linear prices. It is easy to reinterpret quality in section 2.2. as quantity and thus view the model as one of optimal non-linear prices. By de nition D(q; ) = u q (q; ) and thus the FOC can be rewritten as u q (q(); ) = C q (q()) + 1 F () u q (q(); ) f() or D(q(); ) C q (q()) = 1 F () D (q(); ) f() P 0 (q()) P 0 (q()) C q (q() = 1 F () D f() D = 1 (q(); )
30 Wilson (1993) shows that the monopolist can very close to the pro t from optimal non linear prices just with a 3 or 4 part tari s Pricing in competitive environments As before competition reduces distortion. special cases. Existing results are mainly derived in
31 3.3 Economies of scope In the presence of bundling we usually richer price information (prices on bundles or prices for di erent quantitites that help us identify cost synergies. Consider the context of procurement auctions (multi-products, with buyer interested in several products) we can use the FOCs for optimal prices to infer marginal costs for component items: max (p p 1 ;p 2 ;p 1 c 1 )D 1 (p 1 ; p 2 ; p 12 )+(p 2 c 2 )D 2 (p 1 ; p 2 ; p 12 )+(p 12 c 12 )D 12 (p 1 ; p 2 ; p 12 ) If Jacobian is invertible, c = p + rd 1 D p 1 c 1 p 2 c 2 p 12 c = D 1 D 2 D Cantillon and Pesendorfer (2006) show that Jacobian is indeed invertible if D j (p 1 ; p 2 ; p 12 ) > 0 for all j (otherwise bounds).
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