PRICE CAPS IN MULTI-PRICE MARKETS
Oren Bar-Gill
Harvard Law School
UCL conference on
Behavioral IO and Consumer Protection
October 17-18, 2014
Introduction
§
Multi-dimensional pricing is common in consumer markets.
Examples: Credit cards, Mortgages, Cellphones, Air Travel.
§
Lawmakers might become concerned that a specific pricedimension is excessively high and decide to regulate it.
Examples:
CARD Act cap on late fees.
Dodd-Frank Act cap on prepayment penalties in mortgage contracts.
Usury ceilings.
Some courts used the Penalty Doctrine to cap ETFs in cellphone contracts.
EU caps on roaming fees and international calling rates.
Singapore Telecommunications Act (2000): Cap on price that hotels can charge
for international phone calls.
• Banning certain fees (e.g., the credit card no activity fee). - A price cap of
zero.
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1
Introduction
§
What are the positive effects of these price caps?
• Unintended consequences: Adjustment in the unregulated
price dimensions.
§
What are the normative effects of these price caps?
• Do they help consumers?
• Do they increase welfare?
§
Answers depend on:
• Reason for pre-regulation pricing
- Type (and direction) of misperception:
misperception vs. price misperception.
• Market structure
utility
2
Preview of Results
§ The nature and scope of welfare-enhancing regulation depends
on both the type and direction of misperception.
• Greater scope for welfare-enhancing price caps with price
underestimation and utility overestimation.
§ A monopolist is less likely to increase the unregulated price in
response to a cap on the regulated price.
• Less unintended consequences.
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The Literature
§
§
§
Markets with multi-dimensional products, and multidimensional prices, have been studied in the IO literature:
• Standard two-part tariff pricing.
• Products with an aftermarket – for parts or service. See,
e.g., Farrell and Klemperer (2007) and Farrell (2008).
Multi-dimensional pricing has been studied in the behavioral
IO literature. See, e.g., DellaVigna and Malmendier (2004),
Gabaix and Laibson (2006), Grubb (2009), Heidhues and
Koszegi (2010), Spiegler (2011), Heidhues, Koszegi and
Murooka (2012), Armstrong and Vickers (2012).
Most of the IO, and behavioral IO, literature does not study
price caps.
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The Literature
§
Important exceptions:
DellaVigna and Malmendier (2004)
• Briefly discuss the potential welfare benefits of price
regulation.
• Study naiveté about time preferences – related to utility
misperception.
• I consider different types (and directions) of misperception
and compare their positive and normative implications.
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The Literature
§
Important exceptions:
Heidhues and Koszegi (2010)
• Analyze the welfare implications of legal restrictions on
pricing in credit markets.
• Study naiveté about time preferences – related to utility
misperception.
• Focus on competition. Don’t study the effects of market
power.
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The Literature
§
Important exceptions:
Armstrong and Vickers (2012)
• Model
appears to cover both utility and price
misperception, but in a way that masks the positive and
normative differences between them.
• Focus on competition. Don’t study the effects of market
power.
• Allows for heterogeneity in consumer misperception,
which I don’t.
- Price cap increases welfare by limiting the crosssubsidization of sophisticated consumers by less
sophisticated consumers.
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The Literature
§
Bar-Gill and Bubb (2012)
• Each price is incurred only once.
- No incentive effects, beyond the purchase decision.
• Seller’s cost structure: One-dimensional, fixed cost.
• Linear demand.
• Price underestimation.
§
See also: Agarwal et al (2013)
• Model similar to the one in Bar-Gill and Bubb (2012).
- Also focus on price underestimation.
• More general treatment of market power.
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Framework of Analysis
Product
Two-dimensional product (𝑋, 𝑌)
Consumer chooses consumption levels (𝑥, 𝑦)
X is a binary dimension: 𝑥 ∈ 0,1
If 𝑥 = 1, the consumer chooses a Y-dimension use level 𝑦 ∈ 𝑅!
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Framework of Analysis
Product
X and Y are two dimensions of a single product
Or: X and Y are effectively bundled
All sellers offer X and Y together; no seller offers only X
(or only Y).
- X and Y are difficult to separate, or
- Bundling is profitable for efficiency or behavioral reasons.
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Framework of Analysis
Seller’s Costs
X-dimension: Fixed cost 𝑐!
Y-dimension: Per-unit cost 𝑐!
Total cost for a consumer who decided to purchase the product:
𝐶 𝑐! , 𝑐! = 𝑐! + 𝑦𝑐!
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Framework of Analysis
Value to Consumers
Base-value v
– Distributed across consumers according to the CDF 𝐹(𝑣)
Use-value 𝑢(𝑦)
– Varies with y, but in a manner common to all consumers
– Assume: 𝑢! 𝑦 > 0 , 𝑢′′ 𝑦 < 0
Total value: 𝑣 + 𝑢(𝑦) – Separability drives some of the results.
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Framework of Analysis
Seller’s Decisions
Sets prices:
- Base price: 𝑝!
- Per-use price: 𝑝!
Total price: 𝑃 𝑝! , 𝑝! = 𝑝! + 𝑦𝑝!
Seller’s profit per-product purchased:
π 𝑝! , 𝑝! = 𝑃 𝑝! , 𝑝! − 𝐶 𝑐! , 𝑐!
= 𝑝! − 𝑐! + 𝑦 𝑝! 𝑝! − 𝑐!
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Framework of Analysis
Seller’s Decisions
Assumption: 𝜋 𝑝! , 𝑝! is increasing in 𝑝! .
[Otherwise a cap on 𝑝! would have an indeterminate effect on the perunit profit and, therefore, on 𝑝! .]
Seller’s total profit:
Π 𝑝! , 𝑝! = π 𝑝! , 𝑝! ∙ 𝐷 𝑝! , 𝑝!
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Framework of Analysis
Consumer’s Decisions
Use Decision
max 𝑢 𝑦 − 𝑦𝑝!
!
- FOC: 𝑢! 𝑦 = 𝑝! è 𝑦 = 𝑦 𝑝!
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Framework of Analysis
Consumer Misperception
Utility misperception: 𝑢 𝑦 = 𝛿𝑢 𝑦 , where 𝛿 ∈ 0, ∞
Price misperception: 𝑝! = 𝛿𝑝! , where 𝛿 ∈ 0, ∞
Misperceptions apply only ex ante.
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Framework of Analysis
Anticipated Use Level
Utility misperception: max! 𝛿𝑢 𝑦 − 𝑦𝑝!
- FOC: 𝛿𝑢! 𝑦 = 𝑝! è 𝑦 = 𝑦 𝑝! ; 𝛿
Price misperception: max! 𝑢 𝑦 − 𝑦𝛿𝑝!
- FOC: 𝑢! 𝑦 = 𝛿𝑝! è 𝑦 = 𝑦 𝑝! ; 𝛿
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Framework of Analysis
Purchase Decision
Actual (net) value:
𝑉 𝑣, 𝑝! , 𝑝! = 𝑣 + 𝑢(𝑦) − 𝑝! + 𝑦𝑝!
Perceived (net) value:
𝑉 𝑣, 𝑝! , 𝑝! ; 𝛿 = 𝑣 + 𝑢(𝑦) − 𝑝! + 𝑦𝑝!
- Utility misperception: 𝑢 𝑦 = 𝛿𝑢 𝑦 , 𝑝! = 𝑝!
- Price misperception: 𝑢 𝑦 = 𝑢 𝑦 , 𝑝! = 𝛿𝑝!
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Framework of Analysis
Purchase Decision
The consumer will purchase iff 𝑉 𝑣, 𝑝! , 𝑝! ; 𝛿 > 0.
𝑣 > 𝑣 𝑝! , 𝑝! ; 𝛿 ≡ 𝑝! + 𝑦𝑝! − 𝑢 𝑦 Assuming a unit mass of consumers, the demand for the product is:
𝐷 𝑝! , 𝑝! ; 𝛿 = 1 − 𝐹 𝑣 𝑝! , 𝑝! ; 𝛿
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Framework of Analysis
Purchase Decision
Perceived overall consumer surplus:
!
𝑆 𝑝! , 𝑝! ; 𝛿 =
! !! ,!! ;!
𝑉 𝑣, 𝑝! , 𝑝! ; 𝛿 𝑓 𝑣 𝑑𝑣
Actual overall consumer surplus:
𝑆 𝑝! , 𝑝! ; 𝛿 =
!
! !! ,!! ;!
𝑉 𝑣, 𝑝! , 𝑝! 𝑓 𝑣 𝑑𝑣
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Framework of Analysis
Social Welfare
The Social Welfare Function
𝑊 𝑝! , 𝑝! ; 𝛿 =
!
=
! !! ,!! ;!
𝑣 + 𝑢 𝑦 𝑝!
− 𝑐! + 𝑦 𝑝! 𝑐!
𝑓 𝑣 𝑑𝑣
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Framework of Analysis
The First-Best Optimum
Optimal use level: max! 𝑢 𝑦 − 𝑦𝑐!
- FOC: 𝑢! 𝑦 ∗ = 𝑐! è 𝑦 ∗ = 𝑦 𝑐!
Optimal purchases:
𝑣 > 𝑣 ∗ = 𝑣 𝑐! , 𝑐! = 𝑐! + 𝑦 𝑐! ∙ 𝑐! − 𝑢 𝑦 𝑐!
𝐷∗ 𝑐! , 𝑐! = 1 − 𝐹 𝑣 𝑐! , 𝑐!
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Framework of Analysis
The First-Best Optimum
Social welfare at the first-best optimum:
𝑊 ∗ = 𝑊 𝑐! , 𝑐! ; 1 =
!
=
! !! ,!!
𝑣 + 𝑢 𝑦 𝑐!
− 𝑐! + 𝑦 𝑐! 𝑐!
𝑓 𝑣 𝑑𝑣
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Framework of Analysis
The Law
Price cap on the per-use price:
𝑝! ≤ 𝑝!
Or: Price cap on the base price:
𝑝! ≤ 𝑝!
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The Second-Best Optimum
General
Second-best optimum =
Maximum consumer surplus, given the misperception:
max!! ,!! 𝑆 𝑝! , 𝑝! ; 𝛿 s.t. Π 𝑝! , 𝑝! = 0
Or: Maximum welfare, given the misperception:
max!! ,!! 𝑊 𝑝! , 𝑝! ; 𝛿 s.t. Π 𝑝! , 𝑝! ≥ 0
Note: Π 𝑝! , 𝑝! = 0 implies 𝑊 𝑝! , 𝑝! ; 𝛿 = 𝑆 𝑝! , 𝑝! ; 𝛿
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The Second-Best Optimum
Utility Misperception - Underestimation
𝑝!∗ (𝛿 )
𝑐!
𝑐!
𝑝!∗ (𝛿 )
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The Second-Best Optimum
Utility Misperception - Overestimation
𝑝!∗ (𝛿 )
𝑐!
𝑐!
𝑝!∗ (𝛿 )
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The Second-Best Optimum
Price Misperception - Underestimation
𝑝!∗ (𝛿 )
𝑐!
𝑐!
𝑝!∗ (𝛿 )
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The Second-Best Optimum
Price Misperception - Overestimation
𝑝!∗ (𝛿 )
𝑐!
𝑐!
𝑝!∗ (𝛿 )
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Competition
General
Seller solves:
max!! ,!! 𝑆 𝑝! , 𝑝! ; 𝛿 s.t. Π 𝑝! , 𝑝! = 0
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Competition
General
Competition max!! ,!! 𝑆 𝑝! , 𝑝! ; 𝛿 s.t. Π 𝑝! , 𝑝! = 0
Second best optimum –
max!! ,!! 𝑊 𝑝! , 𝑝! ; 𝛿 s.t. Π 𝑝! , 𝑝! = 0
Source of inefficiency: Maximizing perceived surplus, rather than
actual surplus.
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Competition
Comparison: The Object (and Direction) of Misperception
WelfareEnhancing
Price Cap
𝑝!! (𝛿 )
𝑝!! (𝛿 )
𝑝!∗ (𝛿 )
𝑐!
Utility
Underestimation
WelfareEnhancing
Price Cap
𝑐!
𝑝!∗ (𝛿 )
Price
Underestimation
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Competition
Comparison: The Object (and Direction) of Misperception
WelfareEnhancing
Price Cap
𝑝!! (𝛿 )
𝑝!! (𝛿 )
𝑝!∗ (𝛿 )
𝑐!
Price
Overestimation
WelfareEnhancing
Price Cap
𝑐!
𝑝!∗ (𝛿 )
Utility
Overestimation
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Monopoly
General
A monopolistic seller maximizes its profit function:
Π 𝑝! , 𝑝! = 𝜋 𝑝! , 𝑝! ∙ 1 − 𝐹 𝑣 𝑝! , 𝑝! ; 𝛿
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Monopoly
General
Per-use price: 𝑝!! = 𝑝!!
§ Follows from the separability of the X and Y dimensions in
this model.
Base price:
§ Monopoly: 𝑝! = 𝑐! − 𝑦 𝑝! 𝑝! − 𝑐! +
!!! ! !! ,!!
! ! !! ,!!
§ Competition: 𝑝! = 𝑐! − 𝑦 𝑝! 𝑝! − 𝑐!
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Monopoly
Utility Underestimation and Price Overestimation
§ Inadequately low demand
§ High base price reduces demand further
§ Monopoly pricing reduces welfare
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Monopoly
Utility Overestimation and Price Underestimation
§ Excessively high demand in a competitive market
§ High base price avoids purchases that generate a social loss,
but might also deter purchases that generate a social gain
§ Welfare effect of monopoly pricing is indeterminate
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Monopoly
Effect of a Cap, 𝑝! , on the Unregulated Price, 𝑝!
§ Less (upward) adjustment of base price
◦ Competition: Sellers must raise the base price
◦ Monopoly: A monopolist may decide to absorb the lost
profit from the price cap, rather than suffer the reduction in
demand that would result from an increase in the
unregulated, salient price.
è Increases welfare, at least when the cap responds to
utility underestimation.
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Extensions
§ Misperception on Multiple Dimensions
§ Multiple Misperceptions on a Single Dimension
§ Beyond Price Caps
§ Multi-Dimensional Quality and Quality Floors
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Extensions
§
Misperception on Multiple Dimensions
• The analysis will hold, if we assume that the X dimension,
instead of being free from misperception, is subject to the
same type (and direction) of misperception as the Y
dimension but at a lesser degree.
• The benefit from a price cap on 𝑝! will disappear, if sellers
can easily find a third dimension, Z, which is subject to a
misperception of equal magnitude.
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Extensions
§
Multiple Misperceptions on a Single Dimension
What happens if dimension Y is subject, simultaneously, to
utility misperception (under- or over-estimation) and price
misperception (under- or over-estimation)?
Multiple misperceptions can either increase or decrease the
scope for a welfare-increasing price cap.
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Extensions
§
Beyond Price Caps
While I focus on price caps, lawmakers can, and do, restrict
prices in other ways:
• Restricting sellers’ ability to reprice (CARD Act).
• Changing defaults / Requiring explicit opt-in.
Examples: Credit card overlimit fees, Overdraft protection.
• Mandating disclosure of a specific fee
- Trigger an “outrage constraint”
- APR disclosure; what fees are included in the “finance
charge” definition.
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Extensions
§
Multi-Dimensional Quality and Quality Floors
• Multiple quality dimensions
• Lawmakers intervene on a single dimension
- Restrict sellers ability to disclaim implied warranties
- FDA / CPSC regulate certain quality dimensions
- Unconscionability doctrine polices the dispute resolution
mechanism
- Consumer protection laws mandate minimum cancellation or
withdrawal rights
• Similar misperceptions
- Quality misperception corresponds to price misperception
- Utility misperception affects price and quality in a similar
way
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