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Nonlinear Pricing

Linear Prices. Suppose you operate a service or sell a good.A customer may purchase a quantity, q.You can charge a price based on that quantity: I.e., a function, P(q).A linear price has P(q) = p. Examples of Nonlinear Pricing. Regulated IndustriesPer unit price depends on quantityTaxes depend

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Nonlinear Pricing

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    1. Nonlinear Pricing How to use price innovation to increase profits?

    2. Linear Prices Suppose you operate a service or sell a good. A customer may purchase a quantity, q. You can charge a price based on that quantity: I.e., a function, P(q). A linear price has P(q) = p

    3. Examples of Nonlinear Pricing Regulated Industries Per unit price depends on quantity Taxes depend on income Unregulated Industries Quantity discounts Rebates Take-or-pay contracts Credits towards subsequent purchases

    4. Railroads and Transport Railroad tariffs specify charges based on the weight, volume, and distance of each shipment. For instance, discounts on the charge per mile per hundredweight are offered for full-car shipments and for long-distance shipments. In other transport industries such as trucking, airlines, and parcel delivery the rates depend also on the speed of delivery or the time of the day, week, or season.

    5. Electricity Electricity tariffs specify energy charges based on the total kilo-Watt hours used in the billing period, as well as demand charges based on the peak power load during the year. Lower rates apply to successive blocks of kilo-Watt hours and in some cases the demand charges are also divided into blocks. Energy rates for most industrial customers are further differentiated by the time of use, as between peak and offpeak periods.

    6. Telecommunications Telephone companies offer a variety of tariffs for measured toll service and WATS lines. Each tariff provides the least-cost service for a particular range of traffic volumes. Rates are also differentiated by distance and time of use. Mobile phone contracts have a take-or-pay quality.

    7. Airlines Airline fares allow “frequent flier” credits toward free tickets based on accumulated mileage. The retail value of a free ticket increases sharply with the number of miles used to acquire it. Further discounts are offered for advance purchase, noncancellation, round trip, weekend stays, and duration.

    8. Rental Agreements Rental rates for durable equipment and space,such as vehicles and parking lots, are lower if the duration is longer. Rates for rental cars are also differentiated by the size of the vehicle and the time of use.

    9. Advertising Newspaper and magazine advertising rates are based on the size and placement of the insertion, the total number of lines of advertising space purchased by the customer during the year, and in some cases the annual dollar billings.

    10. Taxes How should Crown Casino be taxed?

    11. Classifying Nonlinear Prices Block-declining or tapered tariffs: marginal prices decline in successive increments as purchases increase. Two-part tariff: comprises an initial fixed fee and a per-unit price for units purchased.

    12. Graphical Representation

    13. Finding the Optimal Price Using linear pricing Cannot be a truth telling mechanism No customer has an incentive to report they are a high willingness-to-pay type Outcome: pool all customers together! All customers treated equally

    14. Game Theory What can game theory tell us about the form of nonlinear prices?

    15. Strategic Consumers Given a nonlinear price schedule, P(q); will choose q to maximise their own payoff, u(q) – P(q)q Firm wants to choose P(q) to maximise its profits:

    16. Possibility of Resale If resale is possible, some consumers may benefit from arbitrage E.g., for quantity discounts, consumer may buy a large order and sell to non-bulk purchasers (long-distance telephone transmission) E.g., increasing schedules, open multiple accounts and purchase small amount from each Expensive resale will still constrain tariffs If bulk resale is excluded, can exclude multiple accounts by ensuring price schedule is not increasing (e.g., P(q1 + q2) < P(q1) + P(q2))

    17. Exercise Pricing Stephen King over the Internet

    18. Revelation Principle Can always find a truth telling mechanism with the same outcome A truth telling mechanism will cause customers to be separated by their type. Must satisfy constraints: Participation: wants to purchase a positive quantity Incentive Compatibility: wants to tell the truth

    19. Participation Constraint Consumer i’s utility function, ui(x); gives ‘value’ for a quantity x. Let ui(0) = 0. Participation constraint says that the consumer’s expenditure, P(x)x cannot exceed willingness-to-pay. That is, ui(x) = P(x)x

    20. Incentive Compatibility What per unit price, p, would be required to have the consumer purchase x. Maximise ui(x) – p.x Same as (inverse) demand function.

    21. Two Types High demand (H) customers and low demand (L) customers with: Condition: The ‘single crossing property’

    22. Truth-telling Mechanism Mechanism: if consumer says they are a high-type, then let them consume xH for a total expenditure of PH; otherwise, they can consume xL for total expenditure of PL. What does (PH, xH) and (PL, xL) have to be to get consumers to tell the truth?

    23. Participation

    24. Incentive Compatibility

    25. Combine and Rearrange

    26. Which Constraints will Bind? Suppose that (H-types earn no surplus) Then But this implies Which violates the L-type PC. So we must conclude that

    27. Which Constraints will Bind? Suppose that (L-types are given a ‘rent’) Then, as H-type IC is binding But violates SCP. So it must be that above condition not binding and instead that:

    28. Price Properties Low demand customers earn no surplus (charged their WTP) High demand customers earn a surplus and are charged a price just sufficient to induce them to tell the truth (I.e., purchase xH rather than xL)

    29. Profit Maximisation What should the quantities be to induce truth-telling? Firm’s profit is PH – c. xH + PL – c. xL Maximised subject to previous constraints Which gives L-type has (marginal) value that exceeds MC and consumes an inefficiently small amount but H-type has (marginal) value equal to marginal cost.

    30. Graphical Treatment

    31. Incentive to reduce amount to L-type

    32. Optimal Bundles

    33. Generalisation Intuition: if high demand consumer pays a marginal price greater than marginal cost, then firm could lower price to them by a small amount and induce them to buy more. This would be profitable since price exceeded marginal cost on those sales. Would not affect other sales since they are optimised at lower values of consumption.

    34. Other Characteristics Quantity is not the only dimension to price in a nonlinear fashion Quality: versioning Contract conditions: can service be interrupted?

    35. Summary Nonlinear prices can increase profits and efficiency when firms cannot tell customers’ WTP Strategic customers constrain the form of the non-linear price General principle: Extract rents from ‘low’ types Price to ‘high’ types to encourage truthful revelation Can be applied to characteristics other than quantity

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