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Price Discrimination in Competitive Settings
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Price Discrimination in Competitive Settings
Slide 1
• Price discrimination is well understood for monopoly, ...
• ... but imperfect competition is most common economic setting.
• Goal of lectures is to explain what we know about these
interactions.
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Several questions arise.
• How is consumer surplus, profit, social welfare, and the
distribution of prices affected by an increase in competition
given that firms practice some degree of price discrimination?
Slide 2
• How is consumer surplus, profit and welfare affected when
firms practice some degree of price discrimination given the
presence of imperfect competition?
• In what situations will competing firms choose to price
discriminate when allowed by law? (Non-cooperatively or
cooperatively?) Does price discrimination encourage too much
or too little entry?
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Necessary conditions for Price Discrimination
Slide 3
• Market power (short-run)
• Distinguishable market segments (segmenting device)
• Enforceable market segments (no arbitrage)
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Definitions of Price Discrimination
Standard: Three Degrees of Price Discrimination (Pigou)
Slide 4
• First degree. Extract entire consumer surplus
• Second degree. Approximation to first degree; or more
recently, a revised notion of nonlinear pricing.
• Third degree. Pricing conditional on observable and
verifiable consumer characteristics.
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More Useful Definitions of Price Discrimination
Slide 5
Two relevant dimensions:
1. Direct versus indirect
2. Interpersonal versus intrapersonal
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Direct versus Indirect:
Slide 6
Easiest (conceptually and practically) to think of two types of price
discrimination: direct and indirect.
• Direct price discrimination: e.g., third-degree,
location-based, time of day, etc.
• Indirect price discrimination: e.g., nonlinear pricing,
quality variations, “second-degree,” etc.
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Interpersonal versus intrapersonal:
(see, Armstrong and Vickers, 2002)
• Interpersonal: Price discrimination meant to segment across
consumers
Slide 7
• Intrapersonal: Price discrimination that applies to different
units purchased by the same consumer.
For example, varying prices across heterogeneous consumers is
interpersonal price discrimination (of the direct kind), while
offering a set of homogenous consumers the same nonlinear price
(e.g., two-part tariff) is intrapersonal.
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Roadmap of Lectures
Lecture 1: Direct (3rd-degree) interpersonal price discrimination
Slide 8
Lecture 2: Indirect (“2nd-degree”) interpersonal price
discrimination
Lecture 3: Application: Demand under Uncertainty
N.B.: The topic of intrapersonal price discrimination is examined in
Armstrong and Vickers (2002) and the Handbook of I. O. chapter.
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Why should competition matter?
Index consumer heterogeneity by θ and let q represent a vector of
consumption. With quasi-linear preferences, utility is
u(q, θ) + w.
Slide 9
Suppose each firm j 6= i offers a price schedule of Tj (qj , s(θ)),
where s(θ) is an observable statistic stochastically related to θ.
N.B.: this incorporates most forms of price discrimination.
Then, the consumer’s indirect utility relative to firm i is
X
Tj (qj , θ).
v i (qi , θ) = max u(qi +q−i , θ) − P min
q−i
{
j6=i
qj =q−i ,}
j6=i
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v i (qi , θ) = max u(qi +q−i , θ) −
q−i
{
P
min
j6=i
qj =q−i ,}
X
Tj (qj , s(θ)).
j6=i
This looks as though profit maximization in oligopoly is formally
equivalent to monopoly with consumer preferences v i (qi , θ).
Two problems from competition
Slide 10
1. What may be natural to assume about the fundamental
consumer preferences (e.g., ordered elasticities by θ or
single-crossing property) is no longer natural under
competition. For example,
uqi ,θ (qi , θ) ≥ 0 6⇔ vqi i ,θ (qi , θ) ≥ 0.
2. Best-response function of oligopolist is similar to optimization
of monopolist, but fixed point is embedded in {v i (qi , θ)}N
i=1 .
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Lecture I: Direct Price Discrimination
Recall that direct price discrimination arises when firms condition
prices on some observable characteristic. Hence, in our setting in
which consumer preferences are
u(q, θ) + w,
Slide 11
a price scheme, p(θ) which depends upon θ would be a clear case.
More generally, only statistics s which covary with θ are
observable: e.g. f (θ|s). In this case, p(s) is the price scheme and
aggregate demand for the segment with characteristic s is
Z
q(p, s) = N (s) q(p, θ)f (θ|s)dθ,
where q(p, θ) is the exact demand function for type θ (that is,
p = uq (q(p, θ), θ)) and N (s) is the number of buyers with s.
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A General Welfare Analysis
• We begin by noting that there is a connection between welfare
changes from price discrimination and aggregate output
changes from price discrimination.
Slide 12
• Following an argument very similar to that used by Varian
(1985) in the case of monopoly, we can put bounds on the
change in social welfare when moving from one pricing regime
to another.
• Result: Increased aggregate output is a necessary condition
for welfare to increase when going from a uniform pricing
regime to a price discrimination regime.
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Welfare Analysis (continued):
• Consider a market with M distinguishable consumer segments
and J competing firms. Define a product by firm and segment,
so there are n = M J products.
– Let p0 and p1 be the equilibrium n-tuple price vectors
across these markets under two different regimes.
Slide 13
– Let q0 and q1 represent the associated consumer demands
at these prices.
– Let c0 and c1 represent the aggregate associated costs of
production of these outputs by the firms.
• Suppose the consumer preferences are quasi-linear in wealth.
Then there exists an indirect utility function, v(p) which
measures the sum of the consumer surpluses given p.
• Similarly, let π(p) ≡ (p − c)0 q represent the sum of all firms’
profits.
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Welfare Analysis (continued):
• Let ∆ represent the change due to moving from regime 0 to 1.
• Thus,
∆π = π(p1 ) − π(p0 ) = q01 p1 − q00 p0 − ∆c.
Slide 14
• Because indirect utility functions are convex in prices we have
q01 (p0 − p1 ) ≥ ∆v ≥ q00 (p1 − p0 ).
• Noting that ∆W = ∆π + ∆v,
p0 ∆q − ∆c ≥ ∆W ≥ p1 ∆q − ∆c.
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Welfare Analysis (continued):
p0 ∆q − ∆c ≥ ∆W ≥ p1 ∆q − ∆c.
• Let p0 = (pu , . . . , pu ), a uniform price vector, and
p1 = (p1 , . . . , pn ) is a price discriminating vector. And let
marginal costs be constant at c per unit.
• Then, we have the following useful welfare bounds:
Slide 15
(pu − c)
n
X
∆qi ≥ ∆W ≥
i=1
n
X
(pi − c)∆qi .
i=1
• Hence, a necessary condition for welfare to improve under
price discrimination is that aggregate output increases.
[intuition?]
• A sufficient condition for welfare improvement is that the
change in output valued at the new profit margins is positive.
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Welfare Analysis (continued):
Slide 16
• Result: when evaluating the impact of price discrimination on
social welfare under competition, we are naturally interested in
aggregate output effects across markets.
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A Simple Model of Horizontal Heterogeneity
Many of results that we know can be illustrated in a simple
Hotelling model of spatial price discrimination. We begin here.
A Simple Horizontal Heterogeneity Model (ala Hotelling):
Slide 17
• Consumers are uniformly located on a linear market, [0, 1],
with θ denoting their location. Firms(plants) are located at
each endpoint and produce at cost c.
• Consumer preferences:
v − τ θ − pl if buy from left firm, and
v − τ (1 − θ) − pr if buy from right firm.
• v represents base value, and τ is differentiation measure.
Assume that v is sufficiently large for market coverage.
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Remarks on Setting:
• Direct price discrimination corresponds to pi (θ); that is, price
is conditioned on location.
Slide 18
• Firms have reverse orderings over customer segments. This
horizontal heterogeneity is critical for the effects of
competition. (The opposite setting with vertical
heterogeneity will generate different results in many
dimensions.)
• v is sufficiently large (specifically, v ≥ c + 23 τ ) so that the
market is covered by both multi-plant monopolist and
duopolists. Hence, industry demand elasticity is effectively
zero, while cross-price elasticity is not. Thus, there is no
welfare improvement from competition. More complex model
could address this issue.
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Uniform pricing benchmark:
Slide 19
• Uniform-pricing multi-plant Monopoly: If v is sufficiently
large such that the market is covered, the optimal price is
easily seen to be
1
p = v − τ,
2
and total profits across plants are found to be
1
π = v − c − τ.
2
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Uniform pricing benchmark (continued):
• Uniform-pricing duopoly: The marginal consumer, θ, who
is indifferent between the two duopolists offers is defined by
v − τ θ − pl = v − τ (1 − θ) − pr .
Slide 20
Duopolists choose prices simultaneously to maximize
1 p−i − pi
(pi − c)
+
.
2
2τ
As is well known, the Nash equilibrium is
pl = pr = c + τ,
and total industry profits are
π=
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1
τ.
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Direct discrimination:
Slide 21
• Discriminating multi-plant Monopoly: For sufficiently
large v, a monopolist will extract all fo the consumer surplus
by choosing
pl (θ) = v − τ θ and pr (θ) = v − τ (1 − θ).
Total profits are
1
π = v − c − τ.
4
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Direct discrimination (continued):
• Discriminating duopoly: Following arguments in Lederer
and Hurter (1986), in equilibrium distant j firm offers pj (θ) = c
and close firm attracts consumer θ with a price of
Slide 22
pi (θ) = c + τ (1 − 2θ).
The symmetric equilibrium profit level for each duopolist is
Z 12
1
π=
τ (1 − z)dz = τ.
4
0
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Remarks on Horizontal Model
Slide 23
• Social welfare is not improved by competition (relative to
monopoly) because all consumers purchase from nearest firm.
(This is due to market coverage.) Nor does price discrimination
have any welfare effects.
• Profits for a monopolist increase with price discrimination
relative to uniform pricing; profits for duopolists decrease with
price discrimination relative to uniform pricing.
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Remarks on Horizontal Model (continued)
• Price dispersion under price discrimination. Monopoly
prices range (over domain θ) over a smaller interval relative to
duopoly.
1
pm (θ) ∈ [v − τ, v] versus pd (θ) ∈ [c, c + τ ].
2
Slide 24
Particularly relevant considering empirical evidence (e.g.,
Borenstein and Rose (1994)).
• In first stage of two-stage duopoly game in which firms can
commit to either uniform pricing or price discrimination, it is a
dominant strategy to use price discrimination (even though is
toughens competition in the second stage). See e.g., Thisse and
Vives (1988).
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A Simple Model of Vertical Heterogeneity
Slide 25
• The simple linear-city market model developed above had the
strong market segment of one firm be the weak market segment
of the other. (Reverse ordering over segments)
• We now consider the case where both firms agree as to the
strong (more inelastic) market segment and the weak (more
elastic) segment. Symmetric ordering over segments. (See
Holmes (1989) and Borenstein (1985).)
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Model of Vertical Heterogeneity (continued)
• Consider a setting with two firms, j = A, B and two
distinguishable consumer segments, i = 1, 2.
Slide 26
B
• Demand functions are symmetric: qij (pA
i , pi ). When prices are
B
equal within customer segments, pA
i = pi = pi , it is useful to
consider the function qi (p) ≡ qiA (p, p) ≡ qiB (p, p).
• Define the market elasticity for segment i at equal prices as
p 0
εm
q (p),
i (p) = −
qi (p) i
• Define the firm elasticity for segment i at equal prices as
εfi (p)
p 0
p ∂qiA (p, p)
c
=−
qi (p) +
= εm
i (p) + εi (p),
b
qi (p)
qi (p) ∂pi
where εci (p) > 0 is the cross-price elasticity of demand.
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Model of Vertical Heterogeneity (continued)
Given that
c
εfi (p) = εm
i (p) + εi (p),
• a monopolist will choose prices across markets such that
1
pi − c
= m
,
pi
εi (pi )
Slide 27
• but non-cooperative duopolists (in a symmetric price
equilibrium) will set prices such that
pi − c
1
= m
.
pi
εi (pi ) + εci (pi )
• Competition (relative to monopoly) raises consumer surplus
and lowers prices and profits under price discrimination.
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Model of Vertical Heterogeneity (continued)
• For output effects, we need a bit more notation. Define the
marginal profit from an increase in price in market i, given
B
prices are initially equal at p = pA
i = pi .
∂πi (p) ≡
Slide 28
qiA (p)
∂qiA (p, p)
+ (p − c)
.
∂pA
i
We assume ∂πi (p) is decreasing in p.
• Equilibrium conditions:
∂πi (p∗i ) = 0, i = 1, 2
∂π1 (p∗u ) + ∂π2 (p∗u ) = 0
(discrimination),
(uniform pricing).
• Let i = 2 be the strong market. Then ∂πi (p) decreasing implies
that p∗2 > p∗u > p∗1 .
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Model of Vertical Heterogeneity (continued)
• Suppose that regulation requires |p2 − p1 | ≤ r.
• If this constraint binds, then p2 = p1 + r, and discriminating
firms will choose p1 such that
∂π1 (p∗1 ) + ∂π2 (p∗1 + r) = 0,
Slide 29
yielding solutions parametric in r: p∗1 (r) and p∗2 (r) = p∗1 (r) + r.
• Total output can now be parameterized in r:
Q(r) = q1 (p∗1 (r)) + q2 (p∗1 (r) + r).
• r = 0 corresponds to uniform pricing. If Q(r) increases in r,
then aggregate output increases under price discrimination.
• If Q(r) decreases in r, then aggregate output (and necessarily
social welfare) decreases under price discrimination.
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Model of Vertical Heterogeneity (continued)
• Condition for Q0 (r) > 0 (i.e., necessary condition for welfare to
increase):
Slide 30
(p2 − c) d
2q20 (p2 ) dp2
∂q2a (p2 , p2 )
∂pa2
a
(p1 − c) d
∂q1 (p1 , p1 )
− 0
2q1 (p1 ) dp1
∂pa1
c
ε1 (p1 )
εc2 (p2 )
+ m
−
> 0.
ε1 (p1 ) εm
2 (p2 )
• First term is Robinson’s (1930) adjusted-concavity condition.
With linear demand curves, it is zero.
• Second term is entirely novel, due to competition.
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Model of Vertical Heterogeneity (continued)
Hence, assuming linear demand, Q0 (r) > 0 iff
εm
εc1 (p1 )
1 (p1 )
>
εm
εc2 (p2 )
2 (p2 )
Slide 31
The is referred to as the elasticity-ratio condition (ERC).
• Interpretation: Price discrimination increases output if
inter-segment elasticity ratio is greater with respect to the
outside option than with respect to the rival good.
• For example, at monopoly outcome, right hand side is 1. Here,
price discrimination increases output if strong market is
relatively more competitive than weak market.
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Model of Vertical Heterogeneity (continued)
Again, restricting attention to linear demand systems, we have
Slide 32
• Result 1: Output increases under price discrimination iff
elasticity-ratio condition (ERC) is satisfied. Hence, ERC is a
necessary condition for welfare improvement.
• Result 2: ERC implies profits increase from discrimination.
(See Holmes, 1989.) Compare to result in the horizontal setting
where profits always decrease.
When ERC fails, welfare and output must decrease
(applying Result 1). Also, profits may rise or fall as shown by
examples in Holmes (1989).
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Model of Vertical Heterogeneity (continued)
Slide 33
• Result 3: Regardless of ERC, price discrimination under
competition causes prices to rise in some markets and fall in
others: p∗1 < p∗u < p∗2 .
In the horizontal setting (i.e., asymmetric firm preferences
over segments), Result 3 does not hold. E.g., all prices
decreased in our simple horizontal heterogeneity model:
pu = c + τ (uniform pricing) versus p(θ) = c + τ − 2θτ
(discriminatory pricing). Corts (1998) generalizes this idea.
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Application: Private restrictions on discrimination
• We have already seen that competing firms may sometimes
benefit from private restrictions on price discrimination.
Slide 34
• Policy question: would social welfare be harmed by allowing
firms to write contracts restricting price discrimination among
themselves?
• Winter (1997) tackles this question. Consider previous vertical
heterogeneity setting but allow restrictions of the form
|p2 − p1 | ≤ r, where r is agreed to by firms before the
price-setting game.
• Real example from U.S.: firms collectively agreed not to double
manufacturer coupons.
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Application: Private restrictions (continued)
• Answer depends entirely upon the adjusted-concavity condition
and the elasticity-ratio condition.
• If the adjusted-concavity condition is dominant, then there is a
wedge between private and social benefits.
Slide 35
For example, if relative market and cross-price elasticities
are equal, then firms will prefer r > 0 iff social welfare is
reduced. A prohibition on such restrictions is socially efficient.
• Suppose that demand is linear, so adjusted-concavities are
equal across markets. Winter (1997) (and Holmes (1989))
shows restriction raises profit only if ERC fails. [intuition?]
• But when ERC fails, price discrimination reduces output and
social welfare. Hence, private incentives are aligned with social
objectives.
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Application: Entry and Monopolistic Competition
• Throughout welfare analysis, two distortions have appeared:
inefficiently low aggregate output and mis-allocations among
consumers.
Slide 36
• With the possibility of costly entry, a third welfare concern
emerges – the possibility of too much or too little entry.
• Katz (1984). Consider a world of informed and uninformed
consumers (e.g., Salop and Stiglitz (1977)). Uninformed
consumers are assumed to have larger (inelastic) demands
relative to informed consumers. [NB: any vertically
heterogeneous, two-segment, direct discrimination model will
suffice for our purposes.]
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Application: Monopolistic Competition (continued)
• Under uniform pricing with a small proportion of uninformed
consumers, all firms offer a price equal to minimum average
cost, and first-best efficiency emerges.
Slide 37
• Now, consider same world but with price discrimination. (Here,
the price discrimination takes the form of firms offering two
products to the different consumers at two different prices.)
Long-run equilibrium requires that average price equals
average cost. But starting at the socially efficient number of
firms, profits will be positive, implying too much entry.
• Result: Price discrimination can generate too much entry in a
world of monopolistic competition.
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Application: Entry Deterrence
• Consider a monopolist serving two geographic markets.
Suppose that in one market, an entrant emerges.
Slide 38
• The monopolist’s best response in this case is to reduce price in
the attacked market to accommodate entry.
• Policy issue: by banning price discrimination across markets,
monopolist has less incentive to reduce price in the attacked
market. Hence, entry is more profitable for the entrant.
• Result: Allowing price discrimination may reduce socially
beneficial entry.
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Application: Entry Deterrence
Slide 39
• Compare and contrast this with the previous result on
monopolistic competition: price discrimination by symmetric
firms selling to all customer segments causes too much entry;
when some firms cannot sell to all customer segments (i.e., they
can enter only one segment), price discrimination reduces entry.
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Slide 40
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Lecture II: Indirect Price Discrimination
There are at least two interesting dimensions to consider:
Slide 41
• Exclusive versus common agency: In equilibrium, does a
consumer buy from only one firm (each buyer is“exclusive” to
one firm) or from multiple firms (multiple firms have same
buyer in “common”.)
Under exclusive agency, competitive effects occur over the
value of the second-best alternative (mathematically, the
participation constraint). Under common agency, competitive
effects occur on the margin of purchase (mathematically, the
incentive constraints).
• Vertical versus horizontal heterogeneity: Is the ordering
over the relevant consumer segments shared by firms (vertical),
or are they in conflict (horizontal).
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Lecture II: Indirect Price Discrimination
Let’s be more precise here about what is mathematically meant by
horizontal and vertical heterogeneity. Suppose consumer
preferences for firm j’s product of quality q is
uj (qj , θ) − Tj (qj ),
Slide 42
j = 1, 2.
• Horizontal Heterogeneity:
u1qθ (q1 , θ) ≥ 0 ≥ u2qθ (q2 , θ),
u1θ (q1 , θ) > 0 > u2θ (q2 , θ).
• Vertical heterogeneity:
ujqθ (qj , θ) ≥ 0, j = 1, 2;
ujθ (qj , θ) > 0, j = 1, 2.
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Review of monopoly model
• Consumer preferences.
u(q, θ) + w,
– (i) u(0, θ) = 0, which is without loss of generality if the
consumer’s outside option is independent of θ;
Slide 43
– (ii) uθ (q, θ) > 0, which implies that consumer surplus is
always increasing in type; and
– (iii) the single-crossing property that
uqθ (q, θ) > 0.
which guarantees that demand curves are ordered by θ.
• Consumer heterogeneity. θ ∈ [θ0 , θ1 ], with CDF F (θ) and
density f (θ) > 0.
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Review of the monopoly model (continued)
• Firm preferences. Convex costs of production are C(q) per
consumer, with C(0) = Cq (0) = 0.
Slide 44
• Firm chooses price schedule, T (q) to maximize
Z θ1
[T (q(θ)) − C(q(θ))]f (θ)dθ,
θ0
subject to
q(θ) ∈ arg max u(q, θ) − T (q) ∀θ
q
v(θ) ≡ max u(q, θ) − T (q) ≥ 0 ∀θ
q
incentive constraints,
participation constraints.
• Note also that v(θ) ≡ u(q(θ), θ) − T (q(θ)) by definition.
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Review of the monopoly model (continued)
• Following Mirrlees’s taxation paper, we can perform a trick
using an indirect utility function to simplify the firm’s problem.
Slide 45
• Profit is the difference between total surplus and consumer
surplus. Hence, we can rewrite the objective function above as
Eθ [π] = Eθ [u(q(θ), θ) − C(q(θ)) − v(θ)] ,
• Because v(θ) ≡ u(q(θ), θ) − T (q(θ)) is the maximum surplus a
consumer of type θ can obtain over all q, the envelope theorem
provides that
v 0 (θ) = uθ (q(θ), θ).
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Review of the monopoly model (continued)
• But using integration by parts, we know that
Z θ1
Z θ1
θ1
v(θ)f (θ)dθ = v(θ)[F (θ) − 1]|θ=θ0 −
v 0 (θ)[F (θ) − 1]dθ.
θ0
Slide 46
θ0
Simplifying both sides using our result for v 0 (θ), we have
1 − F (θ)
Eθ [v(θ)] = v(θ0 ) + Eθ
uθ (q(θ), θ) .
f (θ)
• Hence, the objective function of the firm can restated entirely
in terms of q(θ) and the utility of the lowest type consumer,
v(θ0 ).
1 − F (θ)
Eθ u(q(θ), θ) − C(q(θ)) −
uθ (q(θ), θ) − v(θ0 ) ,
f (θ)
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Review of the monopoly model (continued)
• Revised program. Choose {q(θ), v(θ0 )} to maximize
1 − F (θ)
Eθ u(q(θ), θ) − C(q(θ)) −
uθ (q(θ), θ) − v(θ0 ) ,
f (θ)
subject to
Slide 47
q(θ) ∈ arg max u(q, θ) − T (q) ∀θ
q
v(θ) ≥ 0 ∀θ
incentive constraints,
participation constraints.
• Clearly, we need to simplify the constraints too.
• The participation constraint easily simplifies when we recall
that v 0 (θ) = uθ (q(θ), θ) > 0. Hence,
v(θ0 ) ≥ 0
⇐⇒
v(θ) ≥ 0 ∀θ.
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Review of the monopoly model (continued)
• Simplifying the incentive constraints.
Slide 48
– Because of the single-crossing property, q(θ) must be
nondecreasing. It is well known that for any q(θ) that is
nondecreasing, there exists a unique price schedule, T (q),
such that q(θ) ∈ arg maxq u(q, θ) − T (q) and
T (q(θ0 )) = u(q(θ0 ), θ0 ) − v(θ0 ).
– Hence, we can replace the incentive constraints above with
simply the requirement that q(θ) be nondecreasing.
– For intuition, suppose q is smooth and strictly increasing.
Here, T (q) is constructed by solving the consumer’s FOC:
T 0 (q(θ)) = uq (q(θ), θ),
subject to T (θ0 ) = u(q(θ0 ), θ0 ) − v(θ0 ).
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– If the consumer’s objective function, using this solution for
T (q), is strictly concave, then we are finished.
– To this end, totally differentiate the consumer’s FOC with
respect to θ:
T 00 (q(θ))q 0 (θ) = uqq (q(θ), θ)q 0 (θ) + uqθ (q(θ), θ).
Slide 49
Simplifying for the SOC, we have
uqq (q(θ), θ) − T 00 (q(θ)) = −
uqθ (q(θ), θ)
< 0.
q 0 (θ)
The righthand side is negative because of the single-crossing
property and the fact that q(θ) is increasing.
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Review of the monopoly model (continued)
• Simplified program. Choose {q(θ), v(θ0 )} to maximize
1 − F (θ)
Eθ u(q(θ), θ) − C(q(θ)) −
uθ (q(θ), θ) − v(θ0 ) ,
f (θ)
Slide 50
subject to q(θ) nondecreasing v(θ0 ) ≥ 0.
• Ignore for now the condition that q(θ) be nondecreasing.
– In this case, the firm should set v(θ0 ) as low as possible
without violating the participation constraint, so v(θ0 ) = 0.
– Although the firm is maximizing over a function, q(θ), the
maximum can be found by choosing q(θ) to maximize the
integrand for each value of θ.
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Review of the monopoly model (continued)
Slide 51
– This pointwise optimization requires that for each θ, q(θ)
satisfies
uq (q(θ), θ) − Cq (q(θ)) =
1 − F (θ)
uqθ (q(θ), θ).
f (θ)
This equation is crucial in our analysis of indirect price
discrimination.
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Review of the monopoly model (continued)
• Can we ignore the monotonicity of q(θ)? Consider the
integrand in the firm’s objective function:
Λ(q, θ) ≡ u(q, θ) − C(q) − uθ (q, θ) − v(θ0 ).
Slide 52
The optimal q(θ) is determined by the equation Λq (q(θ), θ) = 0.
Therfore,
Λqθ (q(θ), θ)
q 0 (θ) = −
.
Λqq (q(θ), θ)
Providing that Λ(q, θ) is quasi-concave in q (an implicit
assumption), then
q 0 (θ) ≥ 0 ⇐⇒ Λqθ (q(θ), θ) ≥ 0.
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Review of the monopoly model (continued)
• What sort of assumptions guarantee Λqθ (q, θ) ≥ 0?
Slide 53
Answer: A sufficient set of assumptions is that
– u(q, θ) is quadratic in (q, θ), and
– (1 − F (θ))/f (θ) is nonincreasing.
• Finally, note that once q(θ) is determined, v(θ) can be
determined via integration, and finally T (q) can be determined
by the identity: T = u − v.
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Review of the monopoly model (continued)
• Back to the firm’s choice of q(θ). What does it mean?
Slide 54
uq (q(θ), θ) − Cq (q(θ)) =
1 − F (θ)
uqθ (q(θ), θ) ≥ 0.
f (θ)
• Note that consumption is below the socially efficient level for
all θ < θ1 because the righthand side is positive. For the
consumer with θ = θ1 , consumption is socially efficient.
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Review of the monopoly model (continued)
Slide 55
• Interpretation 1: the lefthand side is the marginal social benefit
of an extra unit of production. The righthand side is the
marginal cost to the firm in terms of a shift of rents from the
firm to the consumer. These are referred to as information
rents.
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Review of the monopoly model (continued)
• Interpretation 2. Rewrite the condition:
[uq (q(θ), θ) − Cq (q(θ)]f (θ) = [1 − F (θ)]uqθ (q(θ), θ) ≥ 0.
Slide 56
– Suppose the firm raises q(θ) by a small increment by
lowering the marginal price of the q(θ).
– The lefthand side is the extra surplus the firm generates
multiplied by the probability that this particular consumer
type is present to purchase. The righthand side is the
additional consumer surplus that must be given to all higher
types who would have purchased at the higher price,
multiplied by their relative probability in the consumer
population.
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One-dimensional models (continued)
Slide 57
Example of Monopoly: u(q, θ) = (θ + 1)q and θ ∈ [0, 1].
q f b (θ) = θ + 1 first-best full-information allocation.
θ + 1 − q m (θ) = (1 − θ), second-best monopoly allocation.
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quality
2
qfb ( t )
1.5
Slide 58
1
qm ( t )
0.5
type
0.2
&
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0.6
0.8
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Review of the monopoly model (continued)
This is the classic monopoly pricing tradeoff where,
M R(q) = P (q) + qP 0 (q),
Slide 59
and profit maximization requires
P (q) − M C(q) = −qP 0 (q) ≥ 0.
The only subtlety is that the present analysis is this tradeoff
applied to every marginal unit of consumption.
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Review of the monopoly model (continued)
Slide 60
• Final note. Suppose that uθ (q, θ) < 0 and uqθ (q, θ) < 0, instead
of the reverse. Then our solution changes trivially.
– v(θ1 ) = and all lower types receive surplus;
– the condition for q(θ) becomes
[uq (q(θ), θ) − Cq (q(θ)]f (θ) = −F (θ)uqθ (q(θ), θ) ≥ 0.
• This reformulation will be useful in what follows.
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Exclusive Sales: Buying from one firm
• In all models of exclusive sales, the competition enters the
mathematics through the participation constraint which now
depends upon type. A consumer buys from firm 1, for example,
only if
Slide 61
v1 (θ) ≥ max{v2 (θ), 0}.
This is a setting of type-dependent participation. (See
Lewis-Sappington, et al. for this problem in general.)
• In equilibrium, typically some set of θ’s consumers go to one
firm, and the other θ’s go to the other firm. How the set of θ is
partitioned is determined by the competition and the nature of
consumer preferences.
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Exclusive Sales (contiunued)
One-dimensional models.
Slide 62
• There is a tension. Most models of price discrimination and
competition suggest using both horizontal and vertical
dimensions at the same time.
• Unfortunately, multi-dimensional models of self-selection are
very difficult to solve. (Rochet and Stole, 2002).
• Solution has been for many papers to examine one-dimensional
models. We’ll survey a few here.
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One-dimensional models of exclusive sales
Slide 63
• Model 1. Spulber (1989), Stole (1995). Consider a duopoly,
j = 1, 2. Heterogeneity is of the horizontal type:

vq − τ θq if consumer from firm j = 1
u(q, θ) =
vq − τ (1 − θ)q if consumer from firm j = 2.
Here, v is a known value; private information is over θ.
• If there were no competition and the firm on the “left” (j = 1)
was a monopolist, she would choose q(θ) to satisfy:
v − τ θ − C 0 (q1 (θ)) =
F (θ)
τ.
f (θ)
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One-dimensional models (continued)
• Note that the envelope condition implies that v10 (θ) > 0 and
v20 (θ) < 0. Recall
v1 (θ) ≥ max{v2 (θ), 0}.
Slide 64
Hence, the participation constraint for firm 1 will bind at only
one point.
• With competition, buyers near the previous monopolist-left
firm continue to consume the same amount as before.
1
F (θ)
v − τ θ − C 0 (q1 (θ)) =
τ, ∀θ ∈ [0, ),
f (θ)
2
v − τ (1 − θ) − C 0 (q(θ)) =
1 − F (θ)
1
τ, ∀θ ∈ ( , 1].
f (θ)
2
• Allocations are same as monopoly’s over the purchase region.
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One-dimensional models (continued)
Slide 65
• Effect of competition is to lower prices uniformly to all
consumer types, and to transfer most distorted consumers
toward a closer alternative.
• Can put this market on a unit circle (ala Salop) and consider
entry effects. Not much gas been determined, unfortunately.
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One-dimensional models (continued)
Example of Model 1: v = 3, C(q) = 12 q 2 and τ = 1.
3
Slide 66
2.5
2
1.5
0.2
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0.6
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One-dimensional models (continued)
Slide 67
• Model 2. Stole (1995). Consider a duopoly, j = 1, 2.
Heterogeneity is of the vertical type:

α θq if consumer from firm j = 1
1
u(q, θ) =
α2 θq if consumer from firm j = 2.
Here, each αj is a known value; private information is over θ.
• Without loss of generality, assume that α1 ≥ α2 . Note that this
is a case where there is differentiation between the goods: good
1 is superior.
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One-dimensional models (continued)
• Result: In equilibrium, if α1 > α2 , all consumers buy from
firm 1.
Slide 68
• If α1 = α2 , firms offer pricing at cost and consumers are
indifferent. Profits are zero.
• If there were no competition and the firm on the “left” (j = 1)
was a monopolist, she would choose q(θ) to satisfy:
α1 θ − C 0 (q1 (θ)) =
&
1 − F (θ)
.
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One-dimensional models (continued)
Slide 69
• With competition, offering the monopoly scheme will induce
firm 2 to skim off low-θ consumers. In equilibrium, firm 2 offers
to sell at cost, but firm 1 must reduce the distortion to the
lower interval of buyers.
• The full solution requires control-theoretic techniques.
• Example of Model 2: α1 = 0.60, α2 = 0.40, C(q) = 12 q 2 and
θ ∈ [1, 2].
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One-dimensional models (continued)
Quality
1.2
1
0.8
Slide 70
0.6
0.4
0.2
1.2
1.4
1.6
1.8
2
ν
Lower constraint moves toward first-best output as α1 becomes
close to α2 .
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One-dimensional models (continued)
What do we learn from these sorts of models?
Slide 71
• Competition leads to greater consumer surplus, lower profits,
greater social welfare (fixing number of firms).
• How competition impacts the market depends fundamentally
on the model.
• Many other varieties that one could consider. What is robust?
Sensible?
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Multi-dimensional models
• Rochet and Stole (2001). Consider two-dimensional preferences:
u(q, θ1 , θ2,j ) = θ1 q − θ2 .
Slide 72
• The assumption is that the consumer’s marginal value of
quality is θ1 for all firms, but a different θ2,j is drawn for each
firm.
• Hence, the first parameter measures vertical heterogeneity; the
second parameter measures horizontal heterogeneity.
• Assume that θ1 has CDF F (θ1 ) and density f (θ1 ); θ2,j has
CDF Gj (θ2,j ) and density gj (θ2,j ).
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Multi-dimensional models (continued)
• First, consider a monopoly version of this model in which the
firm offers T (q). Suppose that the support of θ2 is unbounded.
• It turns out there is a simple way setup the maximization
program: define
v1 (θ1 ) = max θ1 q − T (q).
Slide 73
q
A consumer will only show to buy up if v1 (θ1 ) ≥ θ2 . Hence,
there is not relevant participation constraint. Rather, a firm
will obtain participation with probability G(v1 (θ1 )).
• As such, the firm maximizes
G(v1 (θ1 ))(θ1 q − C(q) − v1 (θ1 ),
subject to the standard incentive constraints.
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Multi-dimensional models (continued)
• This turns out to be a messy problem to solve. Formally, you
obtain a boundary-value problem. Still, we can say some things
about the monopoly solution:
Slide 74
– The resulting quality allocation lies between the first-best,
q f b (θ), and the solution which would have arrived if θ2 ≡ 0.
Hence, uncertainty about participation lowers distortions.
– Why? Because there is a marginal cost to reducing
consumer surplus in terms of lost market share. The
distortions were used to reduce consumer surplus; hence,
they are less valuable.
– Also, there is typically pooling or efficiency for the lowest
types.
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Multi-dimensional models (continued)
Slide 75
Monopoly Example: C(q) = 21 q 2 , θ1 ∈ [4, 5] and θ2 ∈ [0, ∞) with
exponential distribution parameter σ.
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Multi-dimensional models (continued)
Now let’s think about competition. Imagine θ2,j represents distance
to firm j. And let {v1 (θ1 ), . . . , vn (θ1 )} represent the vector of
indirect utilities derived from each firms nonlinear pricing schedule.
Slide 77
• In a semi-logit specification, we would have market share
determined by
Gk (vk , θ1 ) =
1 + exp vσk
exp vσk
f (θ1 ).
P
1)
+ i6=k exp vi (θ
σ
• In a Hotelling duopoly we have:
1
Gj (vj , θ1 ) = G min vj , (1 + vj − vi (θ1 ))
f (θ1 ).
2
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Multi-dimensional models (continued)
Results
• If market is uncovered, solution looks similar to monopoly.
Downward distortions, but less great as when market share
does not matter.
Slide 78
• If market is fully covered (everyone buys from some firm), then
an equilibrium is cost-plus-fee pricing:
Tj (q) = C(q) + Kj .
Hence, allocations are efficient, but firms make profits off
horizontal component.
• Similar structure to preferences used in empirical-structural IO
literature: e.g., Berry, Levinson and Pakes (1994).
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Common Sales: Buying from multiple firms
Here, in equilibrium consumers purchase from both duopolists.
(Perhaps makes most sense in regulation setting.) Competition
enters through incentive constraints.
One-dimensional models.
Slide 79
• Consider two firms, each of which offer nonlinear price schedule
of Tj (qj ). The buyer consumes q1 of good 1 from firm 1, and q2
of good 2 from firm 2. Consumer utility is
u(q1 , q2 , θ) − T1 (q1 ) − T2 (q2 ).
We assume identical single-crossing assumptions as in the case
of monopoly: uqj ,θ > 0 and uθ > 0. Here, we consider vertical
heterogeneity, but it does not matter as it did in the
exclusive-buying settings. (See Mezzetti (1997).)
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Common sales (continued)
• Suppose that you are firm 1 and that q2 is fixed. Then you
would behave as a monopolist and choose q1 (θ) to solve
uq1 (q1 (θ), q2 , θ) − Cq (q1 (θ)) =
Slide 80
1 − F (θ)
uq1 θ (q1 (θ), q2 , θ).
f (θ)
• In reality, however, the choice of q2 will depend upon the offer
of T1 (q) if u12 6= 0.
• Take firm 2’s pricing schedule as fixed: T2 (q2 ). We can then
define the consumer’s best-response function given (q1 , θ):
q̂2 (q1 , θ) = arg max u(q1 , q, θ) − T2 (q).
q∈Q2
• Note that q̂2 is increasing in θ given the assumed
single-crossing property.
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• From firm 1’s point of view, it faces a consumer with utility
Û 1 (q1 , θ) = u(q1 , q̂2 (q1 , θ), θ) − T2 (q̂2 (q1 , θ)).
• Suppose that q̂2 is smooth and Û satisfies a single-crossing
property in (q1 , θ). (This is a source of difficulty for uq1 q2 < 0.
Slide 81
• Then firm 1’s optimal price-discriminating solution is to choose
q1 (θ) to satisfy
Ûq11 (q1 (θ), θ) − Cq (q1 (θ)) =
1 − F (θ) 1
Ûq1 θ (q1 (θ), θ).
f (θ)
• Using the equilibrium condition that q2 (θ) = q̂2 (q1 (θ), θ) and
applying the envelope theorem to replace the derivatives of Û 1
with derivatives of q̂2 , we obtain:
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Common sales (continued)
uq1 (q1 , q2 , θ) − Cq (q1 ) =
1 − F (θ)
∂ q̂2 (q1 , θ)
uq1 θ (q1 , q2 , θ) + uq2 θ (q1 , q2 , θ)
.
f (θ)
∂q1
Slide 82
• Comparing, we see that the presence of a duopolist introduces
a second strategic term. If the goods are substitutes, uq1 q2 < 0,
competition leads to less distortion; if the goods are
complements,uq1 q2 > 0, there is a greater distortion.
• Hence, the result in common agency is very similar to what we
have known since Cournot (1838): collusion/merger is socially
good for complements and socially bad for substitutes.
• A similar result arises for horizontal heterogeneity.
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Common sales (continued)
Slide 83
• Multidimensional models also confirm these results in the
tractable settings where they have been explored.
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Slide 84
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Lecture III: Application to Demand Uncertainty
• Previous lectures have focused on settings in which distinct
customer segments consumed distinct allocations.
Heterogeneity was between consumers; the underlying
distribution was known.
Slide 85
• But sometimes underlying distribution is also unknown. For
example, f (θ|s) where s is an aggregate demand state. Now,
there is aggregate uncertainty.
• With simultaneous reporting, a grand mechanism could learn s
from the underlying type reports; thus, we could condition on
common variation, as in McAfee-Reny, et al.
• With variable unit costs (e.g., capacity constraints), cost
externalities suggest using a grand mechanism. Wilson (1993),
Harris-Raviv (1989) for peak-load capacity auction models.
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Unit Demand Models: Monopoly (continued)
• But suppose that there are limitations on the mechanism you
can offer: you must post all units with their associated prices
before observing any choices. There is no role for an auctioneer.
Slide 86
• That is, customers choose from the set of goods and prices you
have posted, and you cannot adjust your prices in response.
• All is not lost, however. You may optimally post only a few
items at the lowest price, so if aggregate demand is high, price
will naturally rise as these units stock out. Hence, you can
make prices respond indirectly to aggregate demand.
• We consider two classes of models in this regard: (1) models
with no interesting segmentation over θ, only s; (2) and models
with both.
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Unit Demand Models: Monopoly (continued)
• Suppose that a consumer values a single unit of consumption;
this value is θ, drawn from a distribution, f (θ|s), conditioned
on a state of aggregate demand, s = 1, . . . , S.
• Hence, the per-consumer demand curve in state s, is
D(p, s) = 1 − F (p|s).
Slide 87
• We will frequently work with inverse market demand, using the
notation for price as a function of market demand, q,
Ps (q),
for the demand price, conditioned on state s.
• The firm posts a number of units, q(p) at each price p.
• Let Q(p) be the total number of units offered at price p or less.
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Unit Demand Models: Monopoly (continued)
• NB: this is not a nonlinear price schedule. It is a list of
“buckets”. It is closer to a supply function than a nonlinear
price schedule.
Slide 88
• The firm produces at marginal cost, c, and has ex ante capacity
cost k per unit.
• Rationing. In all of these models with multiple prices, we need
a story of allocation.
– Efficient (parallel) rationing.
– Proportional rationing.
– Inefficient rationing.
– Walrasian auctioneer? Makes sense only in model of supply
function equilibria. See Klemperer and Meyer (1989).
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Unit Demand Models: Monopoly (continued)
• Case 1: Efficient Rationing and Monopoly.
Slide 89
• Suppose that there are 2 states: P1 (q) and P2 (q) are the
demand functions with P2 (q) > P1 (q). P1 (0) > c + k.
• We use q1 to represent output in low-demand state and
Q2 = q1 + q2 to represent output in high-demand state. Here,
q2 represents additional output consumed in state s = 2.
• Thought experiment. Suppose that s is not aggregate demand
uncertainty, but consumer heterogeneity. What are optimal
prices for given q1 and q2 . This is familiar from lecture 2.
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Unit Demand Models: Monopoly (continued)
• A monopolist facing two consumer types, and wishing to sell q1
the low type and q1 + q2 to the high type, will choose
p1 = ABCD and p2 = ABCD + DEF G. BCEH is surplus left
to high-demand customer types.
Slide 91
• Now, return to the aggregate demand uncertainty setting.
Think of there being a continuum of unit-demand consumers
tracing out the demand curves in each state.
• Under efficient rationing, the highest-demand consumers arrive
first in either state, and so they will purchase the lowest-priced
items first.
• Because of this, within the first demand state, the best that
can be accomplished is a uniform price per unit. Why? This
implies that p1 = CD for exactly q1 units.
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Unit Demand Models: Monopoly (continued)
• Now consider the s = 2 state. As before,only a uniform price
can be implemented for the additional units q2 . Moreover, we
need p1 ≤ p2 .
Slide 92
• The highest demand consumers in s = 2 will buy the units
priced at p1 . Then, they will buy the residual units at p2 . This
implies the highest price which sells q2 additional units is
p2 = F G. Because there is no way to screen within state with
unit demands, consumers obtain more consumer surplus than
in our thought experiment.
• Nonetheless, monopoly price setting with aggregate demand
uncertainty and efficient rationing looks very similar to
nonlinear pricing. Indeed, it is nonlinear pricing with
additional price schedule restrictions.
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Unit Demand Models: Monopoly (continued)
• Choice of outputs, q1 and q2 (together with p1 and p2 ).
• The monopolist solves:
max (P1 (q1 ) − c)q1 + f2 (P2 (q1 + q2 ) − c)q2 − k(q1 + q2 ),
{q1 ,q2 }
Slide 93
subject to P1 (q1 ) ≤ P2 (q1 + q2 ), where fs is the probability of s.
• The first-order conditions (ignoring the price constraint) are
1
P1 (q1 ) 1 −
= c + k − f2 P20 (q1 + q2 )q2 ,
ε1 (q1 )
1
k
=c+ .
P2 (q1 + q2 ) 1 −
ε2 (q1 )
f2
• Very similar to nonlinear pricing model with consumer
heterogeneity rather than aggregate demand heterogeneity.
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Unit Demand Models: Monopoly (continued)
• Case 2: Proportional Rationing and Monopoly.
Slide 94
• Perhaps surprisingly, profit is higher for the monopolist under
proportional rationing: low-demand consumers buy up the
low-priced items to raise the price for the high-demand
consumers.
• In the high-demand state, residual demand is
q1
D(p2 ) 1 −
.
D2 (p1 )
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Unit Demand Models: Monopoly (continued)
• In the high-demand state, residual demand is
q1
D(p2 ) 1 −
.
D2 (p1 )
Slide 95
• Thus, the optimal program of hte monopolist is
k
D1 (p1 )
max (p1 −c−k)D1 (p1 )+f2 p2 − c −
D2 (p2 ) 1 −
,
f2
D2 (p1 )
{p1 ,p2 }
subject to p2 ≥ p1 .
• Given resulting prices, quantities are determined recursively:
1
q1 = D(p1 ) and q2 = D(p2 )(1 − D2q(p
).
1)
• With k = 0, monotonicity constraint is slack iff
state-contingent monopoly price is increasing in s.
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Unit Demand Models: Perfect Competition
• Model is first due to Prescott (1975)
Slide 96
• Define qi (p) as the output posted at prices p by firm i. Qi (p) is
the total output firm i sells at prices p or lower.
P
P
• q(p) ≡ i qi (p) and Q(p) ≡ i Qi (p).
• Suppose that s = 1, . . . , S. In free-entry perfectly competitive
model, prices are determined by zero-profit condition.
Rationing rules determine output at those prices.
• To see this, suppose that given equilibrium distribution of
supply, units offered at price p sell in states s = s0 , . . . , S.
(Because demand is ordered, this will always be an upper
interval.)
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Unit Demand Models: Competition (continued)
• Suppose that given equilibrium distribution of supply, units
offered at price p sell in states s = s0 , . . . , S.
Slide 97
• The probability a given firm sells these p-priced units is
1 − F (s0 ). The zero-profit condition, is there fore,
p(s) = c +
k
,
1 − F (s − 1)
where p(1) = c + k and q(1) is always sold.
• Independent of the rationing rule, we have determined the
zero-profit prices {p(1), . . . , p(S)}.
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Unit Demand Models: Competition (continued)
Slide 98
• Rationing rules and output. Taking our prices as given, we
know that firms must supply enough output in each state so as
to clear the output market. This depends upon the the residual
demand function which depends upon the rationing rule:
– Efficient rationing. The residual demand at a price p(s)
given cumulative output purchased at lower prices,
Q(p(s − 1)), is Ds (p(s)) − Q(p(s − 1)). Hence, equilibrium
requires that
q(p(s)) = max{Ds (p(s)) − Q(p(s − 1)), 0},
with q(p(1)) = D1 (c + k).
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Price Discrimination in Competitive Settings
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Unit Demand Models: Competition (continued)
Slide 99
– Proportional rationing. In a world of proportional
rationing, residual demand at p(s) is


s−1
X
q(j) 
.
RDs (p(s)) = Ds (p(s)) 1 −
Dj (p(j))
j=1
Equilibrium requires this to be zero in each state, which
provides a recursive relationship determining the quantities
offered at each spot price.
– Unlike efficient rationing, ex post misallocations can arise,
so the perfectly competitive outcome need not be Pareto
efficient.
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Unit Demand Models: Monopoly v. Competition
• Example: Suppose that s = 1, 2, D(p, s) = s(4 − p) and
c = k = 1.
• Monoply with fully-flexible pricing: p(1) = 3 and p(2) = 27 .
• Monopoly with efficient rationing: p(1) =
Slide 100
46
15
and p(2) =
49
15 .
• Monopoly with proportional rationing: p(1) = 3 and p(2) = 72 .
• Competition: p(1) = c + k = 2 and p(2) = c +
k
f2
= 3.
• Conclusions: In the example (and in general, Dana (1999)),
dispersion is greater with perfect competition.
• Empirical evidence from Borenstein and Rose (1994). Indirect
price discrimination or aggregate demand uncertainty and price
rigidities?
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Unit Demand Models: Competition (continued)
Slide 101
• Oligopoly.
• Dana (1999) has also extended the polar cases of monopoly and
perfect competition to oligopoly within a class of proportional
rationing and multiplicative demand uncertainty.
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Models of More Complex Demand
• Previous models have considered unit demands. This means
that the firm(s) cannot screen between consumers with
differing demand intensities. (Proportional rationing does this
somewhat already.)
Slide 102
• Now, consider a class of models where the firms can screen on θ
as well as s. We’ll use Dana (1999b) to illustrate this kind of
model.
• Consider an airline offering two flights at distinct times, A and
B.
• There are two demand states, s = a, b with equal probability.
Demand state s = a implies that flight A is in high demand
and B is in low demand; State s = b is the reverse.
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Dana (1999b) model (continued)
• Preferences.
Slide 103
– The high-demand flight has N1 consumers who prefer it; the
low demand flight has only N2 < N1 .
– These preferences are not rigid, however. Consumers of type
θ obtain v from flying their ideal flight and v − θ from the
other flight. v is common across consumers, θ is distributed
according to CDF F (θ) on [0, θ̄].
– To be precise, then, each consumer’s type is a pair: {y, θ},
where y is the preferred flight, either A or B.
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Dana (1999b) model (continued)
• Perfect Competition. In equilibrium, zero profits requires
that pL = c + k and pH = c + 2k.
• N2 units of the low demand flight always sells at pL = c + k.
Slide 104
• Consumers will fly their less preferred flight if θ < pH − pL = k.
Hence, under proportional rationing,
qL = N2 + F (k)(N1 − qL ) and q2 = (1 − F (k))(N1 − qL ),
or after substitution,
qL = N2 + (N1 − N2 )F (k)/(1 + F (k)),
qH = (N1 − N2 )(1 − F (k))/(1 + F (k)).
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Dana (1999b) model (continued)
• Monopoly.
• A monopolist will always choose pH = v (why?) and choose
∆p = pH − pL = v − pL to maximize profits.
• Consumers with θ < ∆p will fly their less preferred flight.
Slide 105
• Solving for the monopolist’s objective function:
F (∆p)
(θ − ∆p − c − k)
max 2 N2 + (N1 − N2 )
∆p
1 − F (∆p)
1 − F (∆p)
(θ − c − 2k).
+ (N1 − N2 )
1 + F (∆p)
The first-order condition for this objective is
F (∆p)
F 0 (∆p)(N1 − N2 )
N2 + (N1 − N2 )
+
(k−∆p) = 0.
1 − F (∆p)
(1 + F (∆p))2
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Dana (1999b) model (continued)
• To repeat, the FOC is
F (∆p)
F 0 (∆p)(N1 − N2 )
N2 + (N1 − N2 )
+
(k−∆p) = 0.
1 − F (∆p)
(1 + F (∆p))2
• When evaluated at ∆p = 0, this is simply
Slide 106
−N2 + F 0 (0)(N1 − N2 )k.
If this is positive (i.e., k is important), then ∆p > 0.
• The first-order condition is negative at ∆p = k, so
∆p < k.
• The price differential helps shift demand into the low-demand
flight, but less than socially efficient. ∆p∗ < k also implies that
the price dispersion is smaller under monopoly.
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Dana (1999b) model (continued)
• Research frontiers?
Slide 107
• Consider placing rigid nonlinear price schedules rather than
price-quantity buckets in the case of monopoly. Question about
which IC constraints arise. How does competition alter this
more general nonlinear pricing setting.
• Compare to fully-flexible pricing with learning. General
interaction of learning, price discrimination via nonlinear
pricing, and segmenting over aggregate demand states. When
are rigid prices optimal given the learning setting?
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