Nonlinear Pricing

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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

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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

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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.

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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.

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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.

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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.

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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.

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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.

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Taxes

• How should Crown Casino be taxed?

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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.

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Graphical Representation
Three-Part
Linear
Two-Part
P(q)q
Fixed-Fee
Block
Nonlinear
q
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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

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Game Theory

• What can game theory tell us about the form of
  nonlinear prices?

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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

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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) lt P(q1) P(q2))

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Exercise

• Pricing Stephen King over the Internet

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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

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Participation Constraint

• Consumer is utility function, ui(x) gives
  value for a quantity x. Let ui(0) 0.
• Participation constraint says that the consumers
  expenditure, P(x)x cannot exceed
  willingness-to-pay.
• That is, ui(x) P(x)x

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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.

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Two Types

• High demand (H) customers and low demand (L)
  customers with
• Condition The single crossing property

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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?

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Participation
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Incentive Compatibility
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Combine and Rearrange
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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

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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

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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)

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Profit Maximisation

• What should the quantities be to induce
  truth-telling?
• Firms 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.

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Graphical Treatment
WTP
PL A PH A B C
B
But H-type prefers L-deal so needs to set PH A
C
A
C
Quantity
xL
xH
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Incentive to reduce amount to L-type
WTP
PL A PH A C
B
Gains
A
C
Quantity
xL
xH
Loses
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Optimal Bundles
WTP
PL A PH A C D
B
A
C
D
Quantity
xL
xH
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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.

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Other Characteristics

• Quantity is not the only dimension to price in a
  nonlinear fashion
• Quality versioning
• Contract conditions can service be interrupted?

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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