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Price Discrimination and Monopoly: Nonlinear Pricing

Price Discrimination and Monopoly: Nonlinear Pricing. Introduction. Annual subscriptions generally cost less in total than one-off purchases Buying in bulk usually offers a price discount these are price discrimination reflecting quantity discounts

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Price Discrimination and Monopoly: Nonlinear Pricing

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  1. Price Discrimination and Monopoly: Nonlinear Pricing

  2. Introduction • Annual subscriptions generally cost less in total than one-off purchases • Buying in bulk usually offers a price discount • these are price discrimination reflecting quantity discounts • prices are nonlinear, with the unit price dependent upon the quantity bought • allows pricing nearer to willingness to pay • so should be more profitable than third-degree price discrimination • How to design such pricing schemes? • depends upon the information available to the seller about buyers • distinguish first-degree (personalized) and second-degree (menu) pricing

  3. First-degree price discrimination 2 • Monopolist can charge maximum price that each consumer is willing to pay • Extracts all consumer surplus • Since profit is now total surplus, find that first-degree price discrimination is efficient

  4. First-degree price discrimination 3 • Suppose that you own five antique cars • Market research shows there are collectors of different types • keenest is willing to pay $10,000 for a car, second keenest $8,000, third keenest $6,000, fourth keenest $4,000, fifth keenest $2,000 • sell the first car at $10,000 • sell the second car at $8,000 • sell the third car to at $6,000 and so on • total revenue $30,000 • Contrast with linear pricing: all cars sold at the same price • set a price of $6,000 • sell three cars • total revenue $18,000

  5. First-degree price discrimination 4 • First-degree price discrimination is highly profitable but requires • detailed information • ability to avoid arbitrage • Leads to the efficient choice of output: since price equals marginal revenue and MR = MC • no value-creating exchanges are missed

  6. First-degree price discrimination 5 • The information requirements appear to be insurmountable • but not in particular cases • tax accountants, doctors, students applying to private universities • No arbitrage is less restrictive but potentially a problem • But there are pricing schemes that will achieve the same outcome • non-linear prices • two-part pricing as a particular example of non-linear prices • charge a quantity-independent fee (membership?) plus a per unit usage charge • block pricing is another • bundle total charge and quantity in a package

  7. Two-part pricing • Jazz club serves two types of customer • Old: demand for entry plus Qodrinks is P = Vo – Qo • Young: demand for entry plus Qydrinks is P = Vy – Qy • Equal numbers of each type • Assume that Vo > Vy: Old are willing to pay more than Young • Cost of operating the jazz club C(Q) = F + cQ • Demand and costs are all in daily units

  8. Two-part pricing 2 • Suppose that the jazz club owner applies “traditional” linear pricing: free entry and a set price for drinks • aggregate demand is Q = Qo + Qy = (Vo + Vy) – 2P • invert to give: P = (Vo + Vy)/2 – Q/2 • MR is then MR = (Vo + Vy)/2 – Q • equate MR and MC, where MC = c and solve for Q to give • QU = (Vo + Vy)/2 – c • substitute into aggregate demand to give the equilibrium price • PU = (Vo + Vy)/4 + c/2 • each Old consumer buys Qo = (3Vo – Vy)/4 – c/2 drinks • each Young consumer buys Qy = (3Vy – Vo)/4 – c/2 drinks • A surplus profit from each pair of Old and Young is U =(PU-c)QU=(1/8)*(Vo + Vy – 2c)2

  9. Two-part pricing 3 • If there are n customers of each type per evening, the jazz-club owner’s profit, U is: U = nU –F = =(n/8)*(Vo + Vy – 2c)2 – F.

  10. (a) OldCustomers (b)YoungCustomers (c) Old/Young Pair of Customers Price Price Price V a V o o e V y b d f V +V i c h o y + g 4 2 k j c MC MR V +V V + V V o y o V - c y Quantity Quantity Quantity o y 2 Two part pricing 4 This example can be illustrated as follows: Linear pricing leaves each type of consumer with consumer surplus

  11. Two part pricing 5 • Jazz club owner can do better than this • Consumer surplus at the uniform linear price is: • Old: CSo = (Vo – PU).Qo/2 = (Qo)2/2 • Young: CSy = (Vy – PU).Qy/2 = (Qy)2/2 • So charge an entry fee (just less than): • Eo = CSoto each Old customer and Ey = CSy to each Young customer • check IDs to implement this policy • each type will still be willing to frequent the club and buy the equilibrium number of drinks • So this increases profit by Eo for each Old and Ey for each Young customer

  12. Two part pricing 6 • The jazz club can do even better • reduce the price per drink • this increases consumer surplus • but the additional consumer surplus can be extracted through a higher entry fee • Consider the best that the jazz club owner can do with respect to each type of consumer

  13. Two-Part Pricing 7 Using two-part pricing increases the monopolist’s profit $/unit Vi The entry charge converts consumer surplus into profit Set the unit price equal to marginal cost This gives consumer surplus of (Vi - c)2/2 c MC MR Set the entry charge to (Vi - c)2/2 Vi Vi - c Quantity Profit from each pair of Old and Young now d = [(Vo – c)2 + (Vy – c)2]/2 - F

  14. Block pricing • There is another pricing method that the club owner can apply • offer a package of “Entry plus X drinks for $Y” • To maximize profit apply two rules • set the quantity offered to each consumer type equal to the amount that type would buy at price equal to marginal cost • set the total charge for each consumer type to the total willingness to pay for the relevant quantity • Return to the example:

  15. Block pricing 2 Old Young $ $ Willingness to pay of each Old customer Willingness to pay of each Young customer Vo Quantity supplied to each Old customer Quantity supplied to each Young customer Vy c MC c MC Qo Vo Qy Vy Quantity Quantity WTPo = (Vo – c)2/2 + (Vo – c)c = (Vo2 – c2)/2 WTPy = (Vy – c)2/2 + (Vy – c)c = (Vy2 – c2)/2

  16. Block pricing 3 • How to implement this policy? • card at the door • give customers the requisite number of tokens that are exchanged for drinks

  17. A final comment • One final point • average price that is paid by an Old customer = (Vo2 – c2)/2(Vo – c) = (Vo + c)/2 • average price paid by a Young customer = (Vy2 – c2)/2(Vo – c) = (Vy + c)/2 • identical to the third-degree price discrimination (linear) prices • but the profit outcome is much better with first-degree price discrimination. Why? • consumer equates MC of last unit bought with marginal benefit • with linear pricing MC = AC (= average price) • with first-degree price discrimination MC of last unit bought is less than AC (= average price) so more is bought

  18. Second-degree price discrimination • What if the seller cannot distinguish between buyers? • perhaps they differ in income (unobservable) • Then the type of price discrimination just discussed is impossible • High-income buyer will pretend to be a low-income buyer • to avoid the high entry price • to pay the smaller total charge • Take a specific example • Ph = 16 – Qh • Pl = 12 – Ql • MC = 4

  19. Second-degree price discrimination 2 • First-degree price discrimination requires: • High Income: entry fee $72 and $4 per drink or entry plus 12 drinks for a total charge of $120 • Low Income: entry fee $32 and $4 per drink or entry plus 8 drinks for total charge of $64 • This will not work • high income types get no consumer surplus from the package designed for them but get consumer surplus from the other package • so they will pretend to be low income even if this limits the number of drinks they can buy • Need to design a “menu” of offerings targeted at the two types

  20. Second-degree price discrimination 3 • The seller has to compromise • Design a pricing scheme that makes buyers • reveal their true types • self-select the quantity/price package designed for them • Essence of second-degree price discrimination • It is “like” first-degree price discrimination • the seller knows that there are buyers of different types • but the seller is not able to identify the different types • A two-part tariff is ineffective • allows deception by buyers • Use quantity discounting

  21. 4 4 MC MC Low income consumers will not buy the ($88, 12) package since they are willing to pay only $72 for 12 drinks Second degree price discrimination 4 These packages exhibit quantity discounting: high- income pay $7.33 per unit and low-income pay $8 This is the incentive compatibility constraint High-income Low-Income So any other package offered to high-income consumers must offer at least $32 consumer surplus So will the high- income consumers: because the ($64, 8) package gives them $32 consumer surplus The low-demand consumers will be willing to buy this ($64, 8) package So they can be offered a package of ($88, 12) (since $120 - 32 = 88) and they will buy this $ High income consumers are willing to pay up to $120 for entry plus 12 drinks if no other package is available Profit from each high- income consumer is $40 ($88 - 12 x $4) $ And profit from each low-income consumer is $32 ($64 - 8x$4) 16 Offer the low-income consumers a package of entry plus 8 drinks for $64 12 $32 8 $32 $32 $40 $32 $64 $8 $24 $16 $32 $32 $8 8 12 16 8 12 Quantity Quantity

  22. 4 4 MC MC Second degree price discrimination 5 The monopolist does better by reducing the number of units offered to low-income consumers since this allows him to increase the charge to high-income consumers A high-income consumer will pay up to $87.50 for entry and 7 drinks High-Income So buying the ($59.50, 7) package gives him $28 consumer surplus Can the club- owner do even better than this? Suppose each low-income consumer is offered 7 drinks So entry plus 12 drinks can be sold for $92 ($120 - 28 = $92) Each consumer will pay up to $59.50 for entry and 7 drinks Low-Income $ $ Profit from each ($92, 12) package is $44: an increase of $4 per consumer 16 Yes! Reduce the number of units offered to each low-income consumer Profit from each ($59.50, 7) package is $31.50: a reduction of $0.50 per consumer 12 $28 $87.50 $31.50 $44 $92 $59.50 $28 $48 $28 7 12 16 8 12 7 Quantity Quantity

  23. Second-degree price discrimination 6 • Will the monopolist always want to supply both types of consumer? • There are cases where it is better to supply only high-demand types • high-class restaurants • golf and country clubs • Take our example again • suppose that there are Nl low-income consumers • and Nh high-income consumers

  24. Second-degree price discrimination 7 • Suppose both types of consumer are served • two packages are offered ($57.50, 7) aimed at low-income and ($92, 12) aimed at high-income • profit is $31.50xNl + $44xNh • Now suppose only high-income consumers are served • then a ($120, 12) package can be offered • profit is $72xNh • Is it profitable to serve both types? • Only if $31.50xNl+ $44xNh > $72xNh 31.50Nl> 28Nh This requires that Nh 31.50 < = 1.125 Nl 28 There should not be “too high” a fraction of high-demand consumers

  25. Second-degree price discrimination 8 • Characteristics of second-degree price discrimination • extract all consumer surplus from the lowest-demand group • leave some consumer surplus for other groups • the incentive compatibility constraint • offer less than the socially efficient quantity to all groups other than the highest-demand group • offer quantity-discounting • Second-degree price discrimination converts consumer surplus into profit less effectively than first-degree • Some consumer surplus is left “on the table” in order to induce high-demand groups to buy large quantities

  26. Non-linear pricing and welfare • Non-linear price discrimination raises profit • Does it increase social welfare? • suppose that inverse demand of consumer group i is P = Pi(Q) • marginal cost is constant at MC = c • suppose quantity offered to consumer group i is Qi • total surplus – consumer surplus plus profit –is the area between the inverse demand and marginal cost up to quantity Qi Price Demand Total Surplus c MC Qi Qi(c) Quantity

  27. Price Demand Total Surplus c MC Qi Qi(c) Quantity Non-linear pricing and welfare 2 • Pricing policy affects • distribution of surplus • output of the firm • First is welfare neutral • Second affects welfare • Does it increase social welfare? • Price discrimination increases social welfare of group i if it increases quantity supplied to group i Q’i

  28. Price Demand Total Surplus c MC Qi Qi(c) Quantity Non-linear pricing and welfare 3 • First-degree price discrimination always increases social welfare • extracts all consumer surplus • but generates socially optimal output • output to group i is Qi(c) • this exceeds output with uniform (non-discriminatory) pricing

  29. Non-linear pricing and welfare 4 Low demand offered less than the socially optimal quantity Price • Menu pricing is less straightforward • suppose that there are two markets • low demand • high demand PU L • Uniform price is PU MC • Menu pricing gives quantities Q1s, Q2s Qls QlU Quantity Price • Welfare loss is greater than L High demand offered the socially optimal quantity • Welfare gain is less than G PU G MC QhU Quantity Qhs

  30. Price PU L MC Qls QlU Quantity Price PU G MC QhU Quantity Qhs Non-linear pricing and welfare 5 • It follows that ΔW < G – L = (PU – MC)ΔQ1 + (PU – MC)ΔQ2 = (PU – MC)(ΔQ1 + ΔQ2) • A necessary condition for second-degree price discrimination to increase social welfare is that it increases total output • “Like” third-degree price discrimination • But second-degree price discrimination is more likely to increase output

  31. The incentive compatibility constraint • Any offer made to high demand consumers must offer them as much consumer surplus as they would get from an offer designed for low-demand consumers. • This is a common phenomenon • performance bonuses must encourage effort • insurance policies need large deductibles to deter cheating • piece rates in factories have to be accompanied by strict quality inspection • encouragement to buy in bulk must offer a price discount

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