Industrial Organization Lecture 4: Discrimination

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1 Industrial Organization Lecture 4: Discrimination Marie-Laure Allain Ecole Polytechnique January 14, 2015 Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

2 Introduction So far, we have assumed that prices were uniform : unit price, for perfect competition as well as for monopolies / oligopolies One good = one price All consumers pay the same price, irrespective of their valuation (ex: rationing) In practice, different units of a same good can be sold at different prices to the same customer or to different ones: discrimination Example: Different classes (airlines) or packages - different prices for different services Different prices for the same class too: plane tickets prices depends on many factors (when you buy the ticket, flexibility, etc.). Quantity rebates ( the second half price : same product sold at different prices for the same consumer), fidelity rebates Supermarket chains charge higher prices in richer areas. Special prices for young, retirees, large families Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

3 Introduction Debate on price discrimination: law versus economics Law: discrimination has a negative flavour: does not seem fair a priori Contrast with universal service for telecoms, postal services, electricity Reflecting cost difference is not discriminatory (e.g. delivered prices p + tx) Economics: discrimination = personalisation Other things equal, A firm benefits from offering tailored prices Customers: some win, some lose, and the overall effect is ambiguous (efficiency / redistribution) Strategic interaction: discrimination can increase competition Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

4 Introduction Three types of discrimination (Pigou, 1920) First degree price discrimination: perfect discrimination A specific price for each unit (for the same or different customers): p ih (q ih ) or T i (q i ). Scond degree price discrimination: self-selection Same menu for all customers, with options tailored to specific needs {p i, q i } i or T (q) (example: two-part tariffs) Third degree price discrimination: based on observable characteristics Age, location, family, etc: a tariff for each group (movie theater, railway services, etc.): {p g } g (example: discount for students) Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

5 Price discrimination: outline First degree discrimination Third degree discrimination Second degree discrimination Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

6 First degree price discrimination Perfect discrimination : Supposes perfect information about customers Different prices for different customers Different prices for different units sold to same customer (not very realistic, but useful benchmark) Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

7 First degree price discrimination Example: consider a monopoly Consumer willing to pay u i for i th unit Constant unit cost c Benchmark: no discrimination Uniform price p Trade-off markup / volume : (p c) p = 1 ε Monopoly: p > c inefficient as W < W Welfare is not maximal, but it is shared between firm and consumers. Cf next figure: due to uniform pricing, consumers with high valuation have a positive surplus. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

8 First degree price discrimination No discrimination: welfare is shared. P Consumers surplus Monopoly profit «Dead weight loss» (welfare loss) p c = c q c q Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

9 First degree price discrimination Assume perfect dicrimination is possible: Maximal price for each unit: p i = u i All units such that u i > c are traded Trade is efficient: W = W But consumer surplus is zero: the whole welfare is seized by the firm. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

10 First degree price discrimination Discrimination: welfare is maximal, but seized by the monopoly. P Monopoly profit = welfare p c = c q c q Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

11 First degree price discrimination Remark: implementation Quantity q Utility: U(q), U 0, U 0 Production cost C(q) Efficient trade: q : max q U(q) C(q) U (q ) = C (q )(= p ) Nonlinear tariff T (q) = U(q) Or two-part tariff: T (q) = S(p ) + p q ( sell the technology ) Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

12 Price discrimination: outline First degree discrimination Third degree discrimination Second degree discrimination Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

13 Third degree price discrimination Supposes limited information about customers Identify customer segment More realistic: necessitates accessible information, e.g. student card, ID card (age), etc. In practice, sellers have to prevent arbitrage from buyers who would buy on the cheapest segment to resell to the high-price segment. Prevent distributors from selling actively outside their territories, and limit exports: quotas, dual pricing for domestic/exports (medicines), etc. Example of country-based prices: cars, pharmaceutical products Easier to prevent arbitrage for services (resell a meal at a restaurant?) Guarantees become void if sale outside country of origin Assumes also that the firms cannot discriminate within a group. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

14 Third degree price discrimination Example Framework A monopolist faces a constant unit cost c n consumers groups, demand for each group D i (p), i {1,..., n} No discrimination (refresher): price p for aggregate demand D = Di i Optimal pricing: p c = 1 p ε(p) with ε(p) = D (p)p D(p) the elasticity of demand on total market. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

15 Third degree price discrimination Example (contd) Discrimination: the monopolist maximizes its profit p i D i (p i ) c i i D i (p i ) yields the price p i for group i: p i c p i = 1 ε i (p i ) where ε i (p i ) = D i (p i )p i D i (p i ) is the elasticity of demand on market i. Note that ε(p) = D i (p) D(p) ε i(p i ). i Optimal pricing implies that the monopolist should charge more in markets with the lower elasticity of demand. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

16 Third degree price discrimination Welfare analysis: property For a given total output, not discriminating is more efficient: n n max U i (q i ) C(Q) st. q i = Q i=1 i=1 U i (q i ) = p : FOC yield i {1,..., n 1}, U i (q i) = U 1 (q 1). By definition of the demand function, p = U 1 (q 1). Price differences thus yield inefficient outcomes: U i (qi) = pi Implication: discrimination can increase total surplus only if it increases volume of trade Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

17 Third degree price discrimination Illustration Two markets, sizes N 1 and N 2, reservation prices v 1 and v 2 > v 1 Discrimination: p i = v i No discrimination: the firm can Either serve all consumers at price p 1 = v 1 : yields a profit π 1 = (N 1 + N 2 )(v 1 c); Or serve only market 2 at price p 2 = v 2 : yields a profit π 2 = N 2 (v 2 c). p p v 2 π 1 v 2 π 2 v 1 v 1 c c q q Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

18 Third degree price discrimination if π 1 > π 2, first option prevails; then forbidding discrimination: Does not affect total surplus; But benefits consumers in the second market. If instead π 1 < π 2, forbidding discrimination: Leads firm to withdraw from first market; Total surplus decreases: no consumer benefits and the firm loses. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

19 Third degree price discrimination Impact on competition? Discrimination tends to intensify competition: No discrimination: average sensitivity of residual demand Discrimination: more direct competition for each group Illustration: spatial differentiation à la Hotelling Two firms 1 and 2, located at the two ends of a street of length L Same constant unit cost c. Unit transportation cost t Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

20 Third degree price discrimination Benchmark: No discrimination - price free on board p i A consumer located at distance x from firm i(j i) will buy from i if: p i + tx < p j + t(l x) Demands thus are: D 1 = L 2 (p 2 p 1 ) 2t and D 2 = L D 1 Maximizing profit π i = (p i c)d i yields the best-response p i = tl+c+p j 2, hence the equilibrium prices: p 1 = p 2 = c + tl Consumer x thus buys from the closer shop and pays Lt + c + tmin{x, L x} Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

21 Third degree price discrimination Discrimination: delivered price p i (x) Actually, the good is not necessarily delivered, but the transport cost is subsidized by the firm. Homogenous good (delivered to same address), consumer x compares the two prices: similar to Bertrand competition; But the cost of delivery is not the same: c + tx for firm 1 and c + t(l x) for firm 2. As in Bertrand with asymmetric costs, the closer firm wins at price just below its competitor s price: Consumer x thus buys from the closer shop and pays Lt + c tmin{x, L x} Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

22 Third degree price discrimination c + 3Lt 2 without discrimina-on c + Lt c c + Lt 2 with discrimina-on 0 L 2 L Figure : Prices paid by consumers. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

23 Third degree price discrimination Conclusion: total prices are lower when firms compete in personalized delivered prices Remark: Firms are better off absent discrimination But each firm benefits from offering personalized prices Asymmetric regulation A unilateral move to discrimination can intensify competition Illustration: meeting competition defence for dominant firm Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

24 Price discrimination: outline First degree discrimination Third degree discrimination Second degree discrimination Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

25 Second degree price discrimination Firm Does not identify consumer type Can still discriminate by offering a menu of options Different customers choose different packages: Price, quantity, quality, services Each customer has more information about its preferences than the firm Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

26 Second degree price discrimination Tariff-based discrimination (quantity dimension): e.g. menu of two-part tariffs yields a concave, non-linear tariff (progressive rebate) T 1 (q) = F 1 +p 1 q T 0 (q) = p 0 q T 2 (q) = F 2 +p 2 q T(q) Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

27 Example Second degree price discrimination Consumers, gross surplus: U = θu(q) with u > 0 and u < 0. θ is the intensity of preference With two part tariff T = pq + F, Consumer s variable surplus is S(p, θ) = θu(q) pq Consumer s utility is θu(q) T (Inverse) demand is defined by q = D(p, θ) p = θu (q). Benchmark: perfect discrimination (personalized tariffs) Consumer θ s variable surplus is maximum for p = c: defines q such that S (c, θ) = θu(q ) cq The optimal tariff for consumer θ is T (q, θ) = S (c, θ) + }{{} cq }{{} fixed part variable part The firm then seizes the whole surplus of consumer θ at its maximum: π = T (q, θ) cq = S (c, θ). What if tariff must be the same for all consumers? Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

28 Second degree price discrimination Uniform tariff: simplified model Assume two types of consumers: low with θ 1 and high θ 2 > θ 1, in proportions λ 1 and λ 2. With two part tariff T = pq + F, Utility normalized to zero for outside option not buying Optimal two-part tariff Maximize λ 1 [F + (p c)d(p, θ 1 )] + λ 2 [F + (p c)d(p, θ 2 )] Subject to participation constraint: F S(p, θ 1 )( S(p, θ 2 )). Yields F = S(p, θ 1 ) with p maximizing λ 1 [S(p, θ 1 ) + (p c)d(p, θ 1 )] + λ 2 [S(p, θ 1 ) + (p c)d(p, θ 2 )] = W (p, θ 1 ) + λ 2 (p c) [D(p, θ 2 ) D(p, θ 1 )] }{{} >0 Without the second term, the firm would choose efficient price p = c; The second term drives the price above marginal cost: p > c; However, since it recovers some surplus through fixed fee, p < p m (θ 2 ). Profit is however less than with perfect discrimination. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

29 Second degree price discrimination A non-linear tariff can do better: offer two options (t 1, q 1 ) and (t 2, q 2 ) so as to max λ 1(t 1 cq 1 ) + λ 2 (t 2 cq 2 ) (q 1,t 1 ),(q 2,t 2 ) s.t.θ 1 u(q 1 ) t 1 0 (PC 1 ) θ 2 u(q 2 ) t 2 0 (PC 2 ) θ 1 u(q 1 ) t 1 θ 1 u(q 2 ) t 2 (IC 1 ) θ 2 u(q 2 ) t 2 θ 2 u(q 1 ) t 1 (IC 2 ) (PC i ) are consumer i s participation constraint: higher utility if they accept their offer than no offer; (IC i ) are consumer i s incentive constraint: higher utility if they accept their offer than the offer designed for the other type of customers; Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

30 Second degree price discrimination IC constraints ensure self-selection of the customers: the main idea behind second-degree price discrimination. Remark: revealed preferences argument: incentive constraints can be rewritten as θ 2 [u(q 2 ) u(q 1 )] t 2 t 1 θ 1 [u(q 2 ) u(q 1 )] which implies that q 2 q 1 (as 0 < θ 1 < θ 2 ). Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

31 Second degree price discrimination The participation constraint of consumers of type 1 is thus The more restrictive of the two participation constraints (as θ 1 u(q 1 ) θ 2 u(q 2 )); Necessarily binding (otherwise, could uniformly increase both tariffs): implies t 1 = θ 1 u(q 1 ). Furthermore, the more restrictive incentive constraint is that of type 2 ( high ) consumers: Assume (IC 2 ) is binding: then t 2 t 1 = θ 2 [u(q 2 ) u(q 1 )] (IC 1 ) is satisfied too: t 2 t 1 θ 1 [u(q 2 ) u(q 1 )] Yet (IC 2 ) must be binding, otherwise can increase both tariffs; Implies t 2 = θ 2 u(q 2 ) (θ 2 θ 1 )u(q 1 ). Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

32 Second degree price discrimination A firm can thus Implement any profile q 2 q 1 At the maximal prices t 1 = θ 1 u(q 1 ) t 2 = θ 2 u(q 2 ) (θ 2 θ 1 )u(q 1 ). The programme of the firm thus amounts to max λ 1[θ 1 u(q 1 ) cq 1 ] + λ 2 [θ 2 u(q 2 ) (θ 2 θ 1 )u(q 1 ) cq 2 ] (q 1,q 2 ) = λ 1 [θ 1 u(q 1 ) λ 2 λ 1 (θ 2 θ 1 )u(q 1 ) cq 1 ] + λ 2 [θ 2 u(q 2 ) cq 2 ] Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

33 Second degree price discrimination The previous program yields the FOC: FOC q1 : [θ 1 λ 2 λ 1 (θ 2 θ 1 )]u (q 1 ) = c FOC q2 : θ 2 u (q 2 ) = c For large consumers: efficient quantity θ 2 u (q 2 ) = c For small consumers: quantity is lower than efficient level: [θ 1 λ 2 λ 1 (θ 2 θ 1 )]u (q 1 ) = c The solution is such that q 2 > q 1 Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

34 Second degree price discrimination Intuition: reducing quantity offered to small consumers Relaxes large users incentive constraint Allows the firm to extract more surplus from them Remark: The optimal menu differs from the one obtained with single two-part tariff Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

35 Second degree price discrimination More general model: continuum of consumers θ distributed over [θ 1, θ 2 ] according to density f (θ) and cumulative distribution function F (θ). Intuitively, consumer is excluded if θ is close to 0 let ˆθ denote the (endogenous) threshold characterizing the first consumer served. The programme of the firm becomes: + max q(.),t(.) ˆθ [t(θ) cq(θ)]f (θ)dθ s.t. θ, θu(q(θ)) t(θ) 0 (P θ ) θu(q(θ)) t(θ) θu(q(θ )) t(θ ) θ (I θ ) Note: as before, the incentive constraints imply that the quantity must increase with θ. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

36 Second degree price discrimination Defining the rent left to the consumer r θu t, we rewrite the programme as: max q(.),t(.) + ˆθ [θu(q(θ)) cq(θ) r(θ)]f (θ)dθ s.t. θ, r(θ) 0 (P θ ) r(θ) r(θ ) + (θ θ )u(q(θ )) θ (I θ ) The incentive constraint moreover implies r (θ) = d [θu(q(θ)) t(θ)] dθ = d dθ [max θ {θu(q(θ )) t(θ )}] = u(q(θ)) > 0 (using the envelope theorem) The rent r thus increases in θ: the binding participation constraint is that of ˆθ: r(ˆθ) = 0. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

37 Second degree price discrimination The rent can thus be rewritten as r(θ) = θ ˆθ r (x)dx = θ ˆθ u(q(x))dx Any increasing quantity profile q(θ) together with the rent profile r(θ) = θˆθ u(q(x))dx can be implemented using the non linear tariff t(θ) = θu(q(θ)) r(θ): Consider the incentive constraint I θ. The rent of a consumer of type θ who would choose a contract designed for a θ would be: V (θ, θ ) = θu(q(θ )) t(θ ) = (θ θ )u(q(θ )) + r(θ ) = (θ θ )u(q(θ )) + θ ˆθ u(q(x))dx Thus dv dθ = (θ θ )u (q(θ ))q (θ ) is positive if θ < θ and negative if θ > θ: consumer θ thus chooses the right contract (θ = θ). Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

38 Second degree price discrimination The expected rent over all consumers is thus equal to + ˆθ r(θ)f (θ)dθ = (integration by parts) = = + θ ˆθ [ ˆθ + + ˆθ + ˆθ [ θ u(q(x))dx]f (θ)dθ f (x)dx]u(q(θ))dθ [1 F (θ)]u(q(θ))dθ. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

39 Second degree price discrimination The objective of the firms can thus be rewritten as max q(.) + ˆθ max q(.) {[θu(q(θ)) cq(θ)]f (θ) [1 F (θ)]u(q(θ))}dθ + ˆθ [(θ 1 F (θ) f (θ) 1 F (θ) f (θ) s.t. θ, q (θ) 0 )u(q(θ)) cq(θ)]f (θ)dθ Assuming that h(θ) θ increases with θ, the solution is implicitely given by maximizing over q(.) inside the integral: h(θ)u (q) = c. As h(θ) < θ, the quantity is below the efficient level. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

40 Second degree price discrimination The total price paid by the consumer θ is t(θ) = θu(q(θ)) r(θ) Can be implemented via a non-linear tariff t = T (q), such that t(θ) = T (q(θ)) The marginal price T (q) satisfies T (q (θ)) = θu (q (θ)) = c + l(θ)u (q (θ)) where l(θ) 1 F (θ) f (θ) represents the likelihood ratio. Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

41 Second degree price discrimination When the likelihood ratio decreases with θ (as is the case for most usual distributions): The marginal price T (q) decreases with θ (since l(θ) decrease, and u < 0), The non-linear tariff T (q) is therefore concave Marie-Laure Allain (Ecole Polytechnique) IO 4: Discrimination January 14, / 41

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