# 3 Price Discrimination

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1 Joe Chen 26 3 Price Discrimination There is no universally accepted definition for price discrimination (PD). In most cases, you may consider PD as: producers sell two units of the same physical good at different prices (, either to the same consumer or to different consumers.) Note that: There is no PD if the price difference reflects the costs of serving; One cannot infer PD does not happen when differentiated products are sold to different consumers. It is useful to classify, according to Pigou (1920), price discriminations by first, second, and third degree PD: First degree (1 ) PD: producers capture the whole consumer surplus; Second degree (2 ) PD: in the case of incomplete information, producers use selfselection devices to extract consumer surplus; Third degree (3 ) PD: there exist direct signals about demands, and producers use this signal to price-discriminate.

2 Joe Chen First degree (Perfect) PD One necessary condition for the 1 PD is complete information: producers know the demands. Given this, how can a monopolistic producer engage in 1 PD? Consider a market with n identical consumers and (market) demand function D(p). Denote the inverse demand as: p = P (q). The monopoly can offer a tariff T (q) such that: T (q) = Z q Z0 q 0 P (x)dx/n P (x)dx/n where >0, andq c is the competitive market demand. if q = q c /n otherwise, Consider an affine pricing schedule (or a two-part tariff): T (q) =A + pq. Note that when A =0, T (q) is a linear tariff. LetS c = S c /n + p c q T (q) = 0 where p c is the competitive market demand. Z q c 0 if q>0 if q =0, [P (x) p c ]dx, andoffer: When consumers have different demands: S i c T (q) = + pc q 0 if q>0 if q =0. No that 1 PD has the same price and quantity as in perfect competition.

3 Joe Chen Third degree (multimarket) PD Based on some direct signal (exogenous information), a producer can divide consumers into m groups. It is then as if there are m independent markets. Note that for the markets to be truly independent, we need to assume: No arbitrage between groups; The producer cannot price discriminate within the group. Hence the monopolistic producer charges a linear tariff for each group. That is, the monopoly solves: X m ³X max p m id i (p i ) C D i(p i ). p 1,p 2,...,p m i=1 i=1 Recall the multiproduct monopoly s problem (independent demands, separable costs), we know that p i are determined according to the inverse elasticity rule: where ε i = D 0 i (p i)p i /D i (p i ). p i C 0 (q) p i = 1 ε i, Some examples: Low-price discount to firsttimemagazinesubscribers; Student and senior citizen discounts; Legal and medical service bills; Goods in poorcountries that do not reflect transportation costs and import taxes, etc. Compared to uniform pricing in all markets, 3 PD makes the monopoly and consumers in high-elasticity markets better off, while it makes consumers in the lowelasticity markets worse off. How about total social surplus? Let C q i = ³X ³X i c, and consumer surplus S i (p i )= R p D i(x)dx. We can write the change i q i

4 Joe Chen 29 in social surplus when monopoly practices 3 PD with {p 1,...,p m } and uniform price with p: W = X [S i(p i ) S i (p)] + X (p i c)q i X (p c)q i i i i. Note that: S i (p i ) S i (p)+si(p)(p 0 i p), since: S 00 i ( ) = D0 i ( ) > 0, ors( ) is convex. Hence, S i (p i ) S i (p) q i [p i c (p c)], or, S i (p i ) S i (p)+(p i c)q i (p c)q i (p i c)q i (p i c)q i =(p i c) q i. Hence, we have the lower bound for the social surplus change: W X i (p i c) q i. Similarly, S i (p) S i (p i )+S 0 i(p i )(p p i ), following the same reasoning, we have the upper bound: W X i (p c) q i. Thus, a necessary condition for the 3 PD to be preferred socially is that it raises total output. The elimination of the 3 PD can be dangerous if it reduces outputs significantly (for example, it leads to the closure of markets). Sometimes it can be a win-win situation (Pareto improvement) to allow firms to price-discriminate.

5 Joe Chen Second degree PD (arbitrage and screening) Note that the absence of direct signals empowers consumers with the ability to engage in personal arbitrage. Perfect PD is, in general, not possible. However, even when there is no exogenous information (direct signals) for the monopolistic producer to tell the consumers apart, a monopoly can still extract some consumer surplus. Let s first go through a common way of 2 PD the two-part tariff Two-part tariffs A two-part tariff, T (q) =A + pg, offers a (continuum) menu of bundles {T,q} located on a straight line. Uniform pricing is a special case of two-part tariff when A =0. Suppose the consumers have the following preferences: θv (q) T if she pays T for quantity q U = 0 otherwise; where: V (0) = 0, V 0 ( ) > 0, andv 00 ( ) < 0. LetV (q) = 1 (1 q)2 2 and there be two groups of consumers: one with taste parameter θ 1 and the other taste parameter θ 2. Let the fraction of consumers with θ = θ 1 be λ, andθ 1 <θ 2. The monopolistic producer has a constant marginal cost c<θ 1. Solving the consumer s problem: we have: and consumer surplus S i (p) except for T as: max.θ i V (q) (A + pq), q q i = D i (p) =1 p/θ i, S i (p) =θ i V (D i (p)) pd i (p) = (θ i p) 2. 2θ i The aggregate demand at price p can be expressed as: D(p)=λD 1 (p)+(1 λ)d 2 (p) =1 p[λ/θ 1 +(1 λ)/θ 2 ] 1 p/ e θ.

6 Joe Chen 31 Perfect PD: charging a price p 1 = c and A i = (θ i p) 2 2θ i.henceprofit Π 1 : Π 1 = λ (θ 1 c) 2 2θ 1 +(1 λ) (θ 2 c) 2 2θ 2. This is the highest profit that a monopoly can get; Welfare is optimal. Monopoly pricing (linear tariff): the monopoly solves: max. (p c)d(p). p And, the monopoly price is: p 2 =(c + e θ)/2. The monopoly may want to give up consumers with lower θ, andservesonlyθ 2 consumers. This cannot be optimal if: or, D 1 ( c + θ 2 ) 0, 2 θ 1 c + θ 2. 2 Let s assume this inequality holds. Two-part tariff: A = S 1 (p). Now, the monopoly solves: max.s 1 (p)+(p c)d(p). p And, the monopoly price is: p 3 = cθ 1 /(2θ 1 e θ). Compare these results, we have: Π 1 Π 3 Π 2, and (under the assumption: θ 1 (c + θ 2 )/2), c = p 1 <p 3 <p 2. Note again, ignoring the redistributive concerns, welfare (social surplus) is higher under the two-part tariff than that of the linear one, because: the marginal price is lower.

7 Joe Chen 32 In general, for any linear tariff T (q) =pq with p>c, there exists a two-part tariff et (q) = A e + epq such that, if consumers are offered the choice between T and T e,both types of consumers as well as the monopoly are made better off. As it turns out it is tricky to demonstrate this algebraically. We ll get into the idea in just a minute. For now, a figure helps a lot... Consider c<ep <p,anda e =(p ep)d 2 (p). Type-θ 1 consumers prefer the linear tariff, whiletype-θ 2 consumers choose the two-part tariff. The change in the monopoly s profit (fromtype-θ 2 consumers) is: (1 λ)[ A e + epd 2 (ep) pd 2 (p) c(d 2 (ep) D 2 (p))] =(1 λ)[ep(d 2 (ep) D 2 (p)) c(d 2 (ep) D 2 (p))] =(1 λ)(ep c)(d 2 (ep) D 2 (p)) > Tie-in Sales and 2 PD Consider a consumer who enjoys one unit of basic (tying) good together with q complementary (tied) goods, and derive surplus: θv (q) T (q). The producer produces the basic good at cost c 0, and the complementary good at cost c per unit. Note that c 0 becomes a fixed cost per customer served. It plays no role in the pricing decision of the complementary good. Suppose tie-in sales are not allowed, and there are competitive firms willing to provide the complementary good at price c. If the producer serves both types of customers, it sets the price of the basic good to S 1 (c), thesurplusofthetype-θ 1 consumers. If tie-in sales are allowed, we have already shown that the producer would charge a price p>cfor the complementary good, and S 1 (p) <S 1 (c) for the basic good. But note that: here, the tie-in sale reduces welfare as long as the producer serves both types of consumers. But the prohibition of tied-in sales may result in only type-θ 2 consumers

8 Joe Chen 33 are served (if (1 λ)s 2 (c) S 1 (c)). In this case, tie-in sales are not necessary detrimental Non-linear tariffs Again, studying a figure helps us to draw the following conclusions: 1. Type-θ 1 (low-demand) consumers derive no net surplus; type-θ 2 (high-demand) consumers derive a positive surplus; 2. The incentive compatibility (personal-arbitrage) constraint for the high-demand consumers are binding; 3. The high-demand consumers purchase the social optimal quantity, q 2 = D 2 (c); the low-demand consumers purchase a suboptimal quantity, q 1 < D 1 (c). Thisisthe so-called absence of distortion at the top. The monopoly s profit is: Π m = λ(t 1 cq 1 )+(1 λ)(t 2 cq 2 ). It chooses two bundles: (T 1,q 1 ) and (T 2,q 2 ) to maximize Π m. There are two kinds of constraints: IR: θ 1 V (q 1 ) T 1 0. The IR (individual rationality) constraints require the consumers to buy. Note that if the IR constraint for type-θ 1 consumers is satisfied, type-θ 2 consumers are automatically willing to buy, because: they can always choose to buy q 1 at T 1 and get net surplus θ 2 V (q 1 ) T 1 > 0. IC: θ 2 V (q 2 ) T 2 θ 2 V (q 1 ) T 1. The IC (incentive compatibility) constraints are used to prevent personal arbitrage (type-θ 2 consumers choose the bundle designed for type-θ 1 consumers). Since the idea is to induce type-θ 2 consumers to reveal their high demands, thus, the IC constraint for type-θ 1 consumers is most likely irrelevant (not binding). We can proceed taking into account only type-θ 2 consumers IC constraint and check later if the IC constraint for type-θ 1 consumers is satisfied.

9 Joe Chen 34 When Π m is maximized, it must be the case that both constraints (the IR of type-θ 1 and the IC of type-θ 2 ) are binding; i.e., T 1 = θ 1 V (q 1 ),andθ 2 V (q 2 ) T 2 = θ 2 V (q 1 ) T 1. Hence, T 2 = θ 2 V (q 2 ) θ 2 V (q 1 )+T 1 = θ 2 V (q 2 ) (θ 2 θ 1 )V (q 1 ). Observe that: Type-θ 1 consumers surplus is entirely appropriated; Note also, T 2 <θ 2 V (q 2 ) so that type-θ 2 consumers enjoy net surplus: (θ 2 θ 1 )V (q 1 ) > 0. Using T 1 and T 2, we can write down the monopoly s problem as: max λ[θ 1V (q 1 ) cq 1 ]+(1 λ)[θ 2 V (q 2 ) (θ 2 θ 1 )V (q 1 ) cq 2 ]. (q 1,q 2 ) The FOCs are: ³ θ 1 V 0 (q 1 )=c/ θ 2 V 0 (q 2 )=c. 1 1 λ λ θ 2 θ 1 θ 1 ; Observe that: 0 < 1 1 λ θ 2 θ 1 λ θ 1 < 1 when: λ ( θ 2 θ 1 θ 2, 1). Hence, as long as both types of consumers are served (λ is not too small), θ 1 V 0 (q 1 ) >c(the quantity consumed by type-θ 1 consumers is suboptimal). Note also: the FOCs V 0 (q 1 ) >V 0 (q 2 ),sincev ( ) is concave, we have q 1 <q 2. Finally, check the IR constraint for type-θ 1 consumers. θ 1 V (q 2 ) T 2 = θ 1 V (q 2 ) θ 2 V (q 2 )+(θ 2 θ 1 )V (q 1 )=(θ 2 θ 1 )[V (q 1 ) V (q 2 )] < 0=θ 1 V (q 1 ) T 1.

10 Joe Chen Quality Discrimination So far, we consider the monopoly discriminates among consumers with different tastes by offering different quantities of the same good at different prices. The same analysis applies to the situation that the monopoly discriminates among consumers with different tastes for quality by offering different qualities at different prices. Consumers have preference U = θs p (if buying), where: s is the quality, and p = p(s). The producer produces s with cost c(s), where:c( ) is increasing and convex. Let q c(s): thecostofqualitys; Let V (q) c 1 (q) =s: the quality obtained at cost q. Then, U = θv (q) p(v (q)) = θv (q) ep(q), where: ep(q) p(v (q)). Moreover, by construction, the monopoly s cost function is linear in q. Therefore, it should be clear now that quality discrimination is identical to the above analysis. device. The monopolistic producer uses lower-quality goods as a market-segmentation

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