competitive nonlinear pricing for signals

Introduction Model Monopoly Duopoly Conclusion

Competitive Nonlinear Pricing for Signals

Zhuoran Lu

School of Management, Fudan University

April 12, 2019


Motivation

In many vertical markets, signaling cost can be manipulated

I Education: the tuition scheme is chosen by the university.

I Luxury good: the price schedule is chosen by the retailer.

I Advertising: the tariff is chosen by the media company.

Lu (2018) studies monopolistic nonlinear pricing for signals

I Monopolistic seller with type-independent participation.

I Price transparency leads to higher price and lower quantity.

This paper studies competitive nonlinear pricing for signals

I Solves the optimal pricing for different market structures.

I Assesses the effects of pricing transparency on signaling.


Overview

ModelI Sellers choose price schedules for a good with intrinsic value.

I Buyer chooses how much to purchase as a signal to receivers.

I Buyer has a two-dimensional preference over the good.

In the monopoly case

I If receivers observe prices, there is either downward distortionin allocation, or full efficiency under some certain conditions.

I If receivers do not observe prices, allocation is more dispersed.

In the duopoly case

I Competition between sellers results in larger quantities ofsignals and higher market coverage than under monopoly.


Literature

Classic signaling and screening models

I Spence (1973), Mussa & Rosen (1978), Maskin & Riley (1984).

Monopolistic nonlinear pricing for signals

I Rayo (2013), Lu (2018).

Competitive nonlinear pricing for non-signals

I Gilbert & Matutes (1993), Villas-Boas & Schmidt-Mohr (1999),

Armstrong & Vickers (2001), Rochet & Stole (2002), Ellison

(2005), Yang & Ye (2008), Ye & Zhang (2017).

Multi-dimensional screening models

I McAfee & McMillan (1988), Armstrong (1996), Rochet & Chone

(1998), Armstrong & Rochet (1999), Thanassoulis (2004).


Model


Model

Players and actions

I Two schools post tuition schemes Ti(z), i = 1, 2.

I A worker has a two-dimensional preference over education:

Vertical type θ ∼ U [0, 1]; horizontal type di ∼ U [0, 12 ].

I Worker chooses at most one school to attend;

Obtains productivity Q(z, θ) = γθz + z, where γ > 0.

I A competitive labor market posts wages Wi(z), i = 1, 2.

InformationI Worker’s type (θ, {di}) is privately known.

I Worker’s education choice (i, z) is publicly observed.

I In the observed case, employers observe both Ti(z).

I In the unobserved case, employers observe neither of Ti(z).


Worker (𝜃, 𝑑1)

School 1

School 2

𝑑1

𝑑2

A Duopoly Education Market


Payoffs

I School i’s expected profit Πi = E[Ti(z(θ))].

I Worker’s gross utility by attending school i

Vi(z, θ) = Wi(z)− Ti(z)− C(z, θ),

where C(z, θ) = z2/2 + (1− θ)z is the effort cost.

I Worker obtains zero utility if purchases no education.

I Surplus: S(z, θ) = (γ + 1)θz − z2/2; zfb(θ) = (γ + 1)θ.

Worker’s problem

I Worker incurs a transport cost kdi ≥ 0 if attends school i.

I Worker chooses i ∈ {∅, 1, 2} and z to maximize net utility

Ui(z, θ, di) = Vi(z, θ)− kdi.


A Direct Mechanism

Timing

I School i offers a contract {zi(θ), Ti(θ)} to worker.

I Labor market posts Wi(z) based on the observability of offers.

Market belief is correct if Wi(z) = E[Q(z, θ)|zi(θ)].I Worker reports a type θ to only the school he attends.

School i’s problem

I In the observed case, max. Πi s.t. IC, IR and correct belief.

I In the unobserved case, max. Πi s.t. IC and IR.

I Focus on the symmetric school-optimal separating equilibrium.


Preliminaries

Lemma 1.In both cases, an allocation {z(θ), V (θ)} is IC if

(i) z(θ) is non-decreasing.

(ii) Define θ0 := inf{θ|z(θ) > 0}, then for all θ > θ0,

V (θ) = V (θ0) +

∫ θ

θ0

−Cθ(z(s), s)ds = V (θ0) +

∫ θ

θ0

z(s)ds.

Market shareI School i’s market share for type θ is twice

si(θ) := min

{Vi(θ)

k,1

4+Vi(θ)− V−i(θ)

2k

}.

I The market segment switching type θ1 is given by

V1(θ1) + V2(θ1) =k

2.


A Bertrand-Spence Benchmark

I Suppose k = 0, Bertrand competition leads to Ti(z) ≡ 0.

I The model is translated to a Spence’s signaling game.

I In the least-cost separating equilibrium, education levels are

zs(θ) = (2γ + 1)θ ≥ (γ + 1)θ = zfb(θ).

I Define the signaling intensity as

zs(θ)− zfb(θ)zfb(θ)

=γ

γ + 1.

I The greater γ is, the more intense signaling activity is.


Over-Investment in Education

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.6

1.2

1.8

2.4

3

Assumption: γ = 1 and k = 0


Monopoly


The Observed Case

I A monopolist maximizes the joint profit of both schools.

Monopolist’s problem

I The monopolist solves

maxz(θ)

∫ θ1

θ0

[S(z, θ)− V (θ)]V (θ)

kdθ︸︷︷︸

Phase I: partially covered range

+

∫ 1

θ1

[S(z, θ)− V (θ)]1

4dθ︸︷︷︸

Phase II: fully covered range

s.t. V ′(θ) = z(θ), z′(θ) ≥ 0, V (θ1) =k

4.

I If θ0 ∈ (0, 1], then V (θ0) = 0; otherwise, V (θ0) is free.


Solving Monopolist’s Problem

I Define the Hamiltonian for each phase as

Phase I: H1 = [S(z, θ)− V (θ)]V (θ)

k+ λz.

Phase II: H2 = [S(z, θ)− V (θ)]1

4+ λz.

I Applying the Maximum Principle, for each phase j = 1, 2,

∂Hj

∂z= 0 and λ(θ) = −∂Hj

∂Vwith λ(1) = 0.

I Solving Phase I yields an ODE

(γ + 3)V − V V − V 2

2= 0. (1)

I Solving Phase II yields z(θ) = (γ + 2)θ − 1 for θ ∈ [θ1, 1].


Determining cutoff types

I Suppose θ0 > 0, then from (1), for θ ∈ [θ0, θ1),

V (θ) =γ + 3

4(θ − θ0)2, z(θ) =

γ + 3

2(θ − θ0).

I Applying smooth pasting,

θ0 =1

γ + 2− γ + 1

2(γ + 2)

√k

γ + 3, (2)

θ1 =1

γ + 2+

γ + 3

2(γ + 2)

√k

γ + 3. (3)

I Thus, θ0 > 0 if k < 4(γ+3)(γ+1)2

; θ1 < 1 if k < 4(γ+1)2

(γ+3) .

I If θ0 > 0 and θ1 = 1, then θ0 is given by λ(1) = 0.


Optimal Contract — Low Signaling Intensity

Proposition 1.

When γ ≤ 1, equilibrium exists. The optimal contract satisfies:

I For k ∈(

0, 4(γ+1)2

(γ+3)

],

zmo(θ) =

{γ+32 (θ − θmo

0 ) if θmo0 ≤ θ < θmo

1

(γ + 2)θ − 1 if θmo1 ≤ θ ≤ 1,

where θmo0 and θmo

1 are given by (2) and (3), respectively.

I For k > 4(γ+1)2

(γ+3) ,

zmo(θ) =γ + 3

2

(θ − θmo

0

)if θmo

0 ≤ θ ≤ 1.

where θmo0 = 1−γ

γ+3 ≥ 0.


Welfare Implications

Corollary 1.

I When γ < 1, θmo0 > 0 and zmo(θ) < zfb(θ) on [0, 1).

I When γ = 1 and k ≥ 4(γ+1)2

(γ+3) , zmo(θ) = zfb(θ) on [0, 1].

IdeaI Signaling can mitigate screening distortion (Lu 2018).

I Market penetration also induces higher education supply.

I When γ = 1 and k is big, the two forces offset screening.


Signaling Intensity and Horizontal Differentiation

(a) γ = 0.5 and k = 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.3

0.6

0.9

1.2

1.5

(b) γ = 0.5 and k = 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.3

0.6

0.9

1.2

1.5



(a) γ = 1 and k = 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.4

0.8

1.2

1.6

2

(b) γ = 1 and k = 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.4

0.8

1.2

1.6

2


Optimal Contract — High Signaling Intensity

Vertical market is partially covered

I When γ > 1, if k ≤ 4(γ+3)(γ+1)2

, θmo0 ≥ 0 and θmo

1 < 1.

I The optimal contract is characterized by Proposition 1.

Vertical market is fully covered

I When γ > 1, if k > 4(γ+3)(γ+1)2

, θmo0 = 0, meaning λ(0) = 0.

I Solving Phase I is equivalent to solving the program

(γ + 3)V − V V − V 2

2= 0.

s.t. V (0) = 0, V (θ1) =k

4, V (θ1) = (γ + 2)θ1 − 1.

I Solving Phase II follows exactly the previous steps.


Optimal Contract — High Signaling Intensity

Proposition 2.

When γ > 1, equilibrium exists. The optimal contract satisfies:

I For k ∈(

0, 4(γ+3)(γ+1)2

],

zmo(θ) =

{γ+32 (θ − θmo

0 ) if θmo0 ≤ θ < θmo

1

(γ + 2)θ − 1 if θmo1 ≤ θ ≤ 1,

where θmo0 and θmo


I For k > 4(γ+1)2

(γ+3) ,

zmo(θ) =

{γ+32 θ if 0 ≤ θ < θmo

1

(γ + 2)θ − 1 if θmo1 ≤ θ ≤ 1,

where θmo1 =

√kγ+3 .



(a) γ = 1.5 and k = 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

2

2.5

(b) γ = 1.5 and k = 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.5

1

1.5

2

2.5


The Effects of Horizontal Differentiation

Corollary 2.

For k < k := min{

4(γ+1)2

(γ+3) ,4(γ+3)(γ+1)2

}, as k increases:

(i) V mo(θ) increases on (θmo0 , 1].

(ii) zmo(θ) increases on (θmo0 , θmo

1 ]; remains the same on (θmo1 , 1].

(iii) The market coverage [θmo0 , 1] extends, but [θmo

1 , 1] shrinks.

IdeaI Monopolist provides more rents to gain market share.

I This can be achieved by more education or larger coverage.

I The optimal allocation requires a balance between these two.


The Unobserved Case

Monopolist’s problem

I Given W (z), the monopolist solves

maxz(θ)

∫ θ1

θ0

[W (z)− C(z, θ)− V (θ)]V (θ)

kdθ︸︷︷︸

Phase I: partially covered range

+

∫ 1

θ1

[W (z)− C(z, θ)− V (θ)]1

4dθ︸︷︷︸

Phase II: fully covered range

s.t. V ′(θ) = z(θ), z′(θ) ≥ 0, V (θ1) =k

4.

I If θ0 ∈ (0, 1], then V (θ0) = 0; otherwise, V (θ0) is free.

I In equilibrium, W (z) = E[Q(z, θ)|z(θ)] using Bayes’ rule.


Solving Monopolist’s Problem


Phase I: H1 ≡ [W (z)− C(z, θ)− V (θ)]V (θ)

k+ λz.

Phase II: H2 ≡ [W (z)− C(z, θ)− V (θ)]1

4+ λz.

I In equilibrium, W ′(z) = Qz(z, θ) +Qθ(z, θ) · θ′(z).

I Solving Phase I yields an ODE

(2γ + 3)V − γV V...V

V 2− V V +

γV 2

V− V 2

2= 0. (4)

I Solving Phase II with the desired initial condition yields

W (z) =γ

2(γ + 1)z2 +

2(γ + 1)

γ + 2z.

I This implies that z(θ) = 2(γ + 1)(θ − 1γ+2) for θ ∈ [θ1, 1].


Determining cutoff types

I Suppose θ0 > 0, then from (4), for θ ∈ [θ0, θ1),

V (θ) =4γ + 3

4(θ − θ0)2, z(θ) =

4γ + 3

2(θ − θ0).

I Applying smooth pasting,

θ0 =1

γ + 2− 1

4(γ + 1)

√k

4γ + 3, (5)

θ1 =1

γ + 2+

4γ + 3

4(γ + 1)

√k

4γ + 3. (6)

I Thus, θ0 > 0 if k < 16(γ+1)2(4γ+3)(γ+2)2

; θ1 < 1 if k < 16(γ+1)4

(4γ+3)(γ+2)2.

I If θ0 > 0 and θ1 = 1, then θ0 is given by λ(1) = 0.


Optimal Contract

Proposition 3.

For all γ > 0, equilibrium exists. The optimal contract satisfies:

I For k ∈(

0, 16(γ+1)4

(4γ+3)(γ+2)2

],

zmu(θ) =

4γ+32 (θ − θmu

0 ) if θmu0 ≤ θ < θmu

1

2(γ + 1)(θ − 1γ+2) if θmu

1 ≤ θ ≤ 1,

where θmu0 and θmu


I For k > 16(γ+1)4

(4γ+3)(γ+2)2,

zmu(θ) =4γ + 3

2(θ − θmu

0 ) if θmu0 ≤ θ ≤ 1.

where θmu0 = 1

2γ+3 .


Education Levels are More Dispersed

Proposition 4.

(i) For all γ, k > 0, θmu0 > θmo

0 and θmu1 < θmo

1 .

(ii) There is a cutoff θ ∈ (θmu0 , θmu

1 ), such that zmu(θ) < zmo(θ)on (θmo

0 , θ); zmu(θ) > zmo(θ) on (θ, 1]. The length of theinterval (θmo

0 , θ) is increasing in k, and vanishes as k → 0.

IdeaI Education demand is more elastic due to signal jamming.

I Monopolist supplies more education in fully covered range.

I But it is not as profitable in partially covered range due to IC.


Comparison of Education Levels

(a) γ = 1, k = 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.00

0.53

1.07

1.60

2.13

2.67

(b) γ = 1, k = 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.00

0.53

1.07

1.60

2.13

2.67


Consumer Surplus and Tuition

Proposition 5.

(i) There is a cutoff θ ∈ (θmu0 , 1), such that V mu(θ) < V mo(θ)

on (θmo0 , θ); V mu(θ) > V mo(θ) on (θ, 1]. The length of the

interval (θmo0 , θ) is increasing in k, and vanishes as k → 0.

(ii) There is a cutoff z ∈ (0, zfb(1)), such that Tmu(z) > Tmo(z)on (0, z); Tmu(z) > Tmo(z) on (z, zfb(1)]. The length of theinterval (0, z) is increasing in k, and vanishes as k → 0.

Implication

I A lower-type who is close to either school benefits more fromthe increase in horizontal differentiation in the observed case.


Duopoly


The Observed Case

I From now on, assume that k < k so that θmo0 > 0.

School’s problem

I Given the other school’s contract, school i solves

maxzi(θ)

∫ θ1

θ0

[S(z, θ)− Vi(θ)]Vi(θ)

kdθ︸︷︷︸

Phase I: local monopoly range

+

∫ 1

θ1

[S(z, θ)− Vi(θ)][

1


2k

]dθ︸︷︷︸

Phase II: competition range

s.t. V ′(θ) = z(θ), z′(θ) ≥ 0, V (θ1) =k

4.

I If θ0 ∈ (0, 1], then Vi(θ0) = 0; otherwise, Vi(θ0) is free.


Solving School’s Problem


Phase I: H1 = [S(z, θ)− Vi(θ)]Vi(θ)

k+ λz.

Phase II: H2 = [S(z, θ)− Vi(θ)][

1


2k

]+ λz.

I Solving Phase I follows exactly the same steps of monopoly.

I Solving Phase II is equivalent to solving the program

V = γ + 2− 2

k

[(γ + 1)θV − V − V 2

2

](7)

s.t. V (θ1) =k

4, V (θ1) =

√(γ + 3)k

2, V (1) = γ + 1.


Optimal Contract

Proposition 6.

In the observed case, the duopoly optimal contract satisfies:

zdo(θ) =

{γ+32 (θ − θdo0 ) if θdo0 ≤ θ < θdo1V do(θ) if θdo1 ≤ θ ≤ 1,

where V do(θ) and θdo1 are given by (7), and θdo0 = θdo1 −√

kγ+3 .

Equilibrium Discontinuity

I Equilibrium is discontinuous at k = 0.

I It is always optimal to exclude sufficiently low types if k > 0.


Education Levels and Market Coverage

Proposition 7.

If k < k, then θdo0 < θmo0 and zdo(θ) > zmo(θ) for θ ∈ (θdo0 , 1). In

contrast to the monopoly case, more worker types (in terms ofboth vertical and horizontal type) receive education, and eachparticipating type obtains higher net utility.

IdeaI Competition leads to more education in competition range.

I This relaxes IC, thus lower types also receive more education.


The Effects of Competition on Education Levels

(a) γ = 1 and k = 1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.4

0.8

1.2

1.6

2

(b) γ = 1 and k = 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.4

0.8

1.2

1.6

2


The Effects of Competition on Market Coverage

(a) γ = 1 and k = 1.5

0 0.05 0.1 0.15 0.2 0.25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

covered market segment

uncovered market segment

(b) γ = 1 and k = 3

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1




The Unobserved Case

School’s problem

I Given W (z) and the other school’s contract, school i solves

maxzi(θ)

∫ θ1

θ0

[W (z)− C(z, θ)− Vi(θ)]Vi(θ)

kdθ︸︷︷︸

Phase I: local monopoly range

+

∫ 1

θ1

[W (z)− C(z, θ)− Vi(θ)][

1


2k

]dθ︸︷︷︸

Phase I: competition range

s.t. V ′(θ) = z(θ), z′(θ) ≥ 0, V (θ1) =k

4.

I If θ0 ∈ (0, 1], then Vi(θ0) = 0; otherwise, Vi(θ0) is free.

I In equilibrium, W (z) = E[Q(z, θ)|z(θ)] using Bayes’ rule.


Solving School’s Problem


Phase I: H1 = [W (z)− C(z, θ)− Vi(θ)]Vi(θ)

k+ λz.

Phase II: H2 = [W (z)− C(z, θ)− Vi(θ)][

1


2k

]+ λz.

I Solving Phase I follows exactly the same steps of monopoly.

I Solving Phase II is equivalent to solving the program

...V =

V 2

γV

2(γ + 1)− 2

k

[(γ + 1)θV − V − V 2

2

] (8)

s.t. V (θ1) =k

4, V (θ1) =

√(4γ + 3)k

2,V (1)[V (1)− γ]

V (1)= γ + 1.


Optimal Contract

Proposition 8.

In the unobserved case, the duopoly optimal contract satisfies:

zdu(θ) =

{4γ+32 (θ − θdu0 ) if θdu0 ≤ θ < θdu1

V du(θ) if θdu1 ≤ θ ≤ 1,

where V du(θ) and θdu1 are given by (8), and θdu0 = θdu1 −√

k4γ+3 .

Equilibrium Discontinuity

I Equilibrium is discontinuous at k = 0.

I It is always optimal to exclude sufficiently low types if k > 0.


The Effects of Competition on Education Levels

(a) γ = 1 and k = 3.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.00

0.58

1.16

1.75

2.33

2.91

(b) γ = 1 and k = 3.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.00

0.57

1.14

1.70

2.27

2.84


The Effects of Competition on Market Coverage

(a) γ = 1 and k = 3.5

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



(b) γ = 1 and k = 3.8

0 0.05 0.1 0.15 0.2 0.250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1




Summary

In this paper

I We studied competitive nonlinear pricing for signals.

I We solved the optimal pricing for different market structures.

Main resultsI In the monopoly observed case, there is either downward

distortion in quantity, or full efficiency, depending on thedegrees of signaling intensity and horizontal differentiation.

I In the monopoly unobserved case, signal quantities are moredispersed due to signal jamming and market penetration.

I Market competition results in a higher market coverage andlarger quantities for both the observed and unobserved case.

competitive nonlinear pricing for signals

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