04/21/23

産業組織研究会

Matching as SignalMatching as Signal

Kyushu UniversityKyushu University

Nobuaki HoriNobuaki Hori

2

Introduction

• Motivation– Why is ranking of universities (vertically

and/or horizontally) stable, (in some cases,) irrelevantly to the quality of research or instruction ?

• ⇒Matching + Signaling– Matching between workers and schools– Names of universities work as “signals”

3

Introduction

• Signaling with a continuum of differential types– Mailath (1987): direct choice of vertical signal

(education) – This model : matching 　→　 Pooling is robust

• Matching between the workers and univ.– Epple, Romano and Sieg (2006), Akabayashi a

nd Naoi (2008)↑Univ. are intrinsically differentiated.– This model: potentially identical

4

Introduction

• Matching tournament– Cole, Mailath and Postlewaite (1992, 95, 98,

2001), Hopkins(2006) : matching is goal.– This model: matching is instrument.

5

Main results

• Various types of ranking are realized as a self-enforcing belief

• While potentially identical, highly ranked universities can enjoy the status by charging higher tuition fees.

• When matching is assortative, a rent absorption of tuition fee implements efficient level of educational performance.

6

ModelModel

• Workers (=students)Workers (=students)– heterogeneous in innate abilities, heterogeneous in innate abilities, aa

• Educational performance, Educational performance, ee– workplace productivityworkplace productivity– cost of performancecost of performance

• AssumptionsAssumptions

min max[ , ] ~ ( ), ( ) ia a a F a F a i

( , )iy e a

( , )ic e a

2( , ) ( , ) ( , )0, 0, 0i i iy e a y e a y e a

e a e a

2( , ) ( , ) ( , )0, 0, 0i i ic e a c e a c e a

e a e a

7

Model

• Utility functionUtility function

• • But But ee and and aa are not observable are not observable

⇒ ⇒ Incentives for Incentives for SignalingSignaling• Matches between workers and Matches between workers and

universities work as a signaling universities work as a signaling devicedevice.

, (productivity) (wage)e a y w

( , )

(wage) (cost of effort) (tuition fee)

u w c e a p

8

Timing

1. Universities simultaneously set tuition fees p(j)

↓↓

2. Students and universities make one-to-one matches, bargaining over educational performances e

↓↓

3. Workers are paid w(s) in the labor market.

9

ModelModel

• UniversitiesUniversities– Continuous, indexed byContinuous, indexed by– ⇒ ⇒ Positive measure of students can Positive measure of students can

not go to universitiesnot go to universities

• Lexicographic Preference (1)p (2)e

• Interpretation of educational performanceInterpretation of educational performance1.1. educational service offered by universitieseducational service offered by universities

2.2. score of achievement test prior to score of achievement test prior to entranceentrance

0 0[ ,1], 0 1 j J

00

10

Model

• Assumptions for the labor marketAssumptions for the labor market– A worker’s info.A worker’s info.

– Perfect competitionPerfect competition

(educational performance) not observable

(innate ability) not observable

(educational t observablrack) e

e

a

s

0[ ,1] s J

( ) [ | ]w s E y s

11

Equilibrium ConceptsEquilibrium Concepts

1.1. Perfect Bayesian Nash eq.Perfect Bayesian Nash eq.

2.2. Stable matchingStable matching

* * *( ( )) ( ( )) ( ( ( )), )

( ) ( ) ( ( ), ) for any i i i i

i

w s a p s a c e s a a

w s p s c e s a s J

*( ) : equilibrium of

( ) : necessary to get record i is a s a

e s e s

12

Image of self-enforcing rank

• Universities are ranked by Universities are ranked by – ⇒ ⇒ Matching is “assortative”Matching is “assortative”

( )w j

( )w j ( )e j

( )a j

13

A social belief

• J (set of university) could have some pooling sub-ranges,

0 1 1 2 1[ , ),[ , ), [ ,1), 1, 2, , M M

0 1 1where 1 M

14

A social belief

• : The probability that the innate ability of the graduate is higher than that of . – When the both belong to a same sub-

range

– Otherwise

Pr[ ( ) ( )]a j a j j

j

Pr[ ( ) ( )] Pr[ ( ) ( )] 1/ 2a j a j a j a j

, Pr[ ( ) ( )] 1j j a j a j For

15

The social belief (Image M=2)

z

1

1

1l

3l

4l

2l

1l 2l 3l 4l(agents)i

(universities)j

z

16

Matching stage

• is given.• stable matching

* * *( ( )) ( ( )) ( ( ( )), )

( ) ( ) ( ( ), ) for any i i i i

i

w s a p s a c e s a a

w s p s c e s a s J

( )p j

* equilibrium of ( ) :

necessary to get record ( ) : ii s as a

e se s

17

Matching stage

Lemma 1Lemma 1 If an equilibrium contains a If an equilibrium contains a pooling range for any pair pooling range for any pair universities which belong to it,universities which belong to it,

• : equilibrium performance of : equilibrium performance of aa• : equilibrium payment of : equilibrium payment of aa

Lemma 2 Lemma 2 In equilibrium,In equilibrium,

( ) ( ), ( ) ( ), ( ) ( )p j p j e j e j w j w j

( )a( )a

( )a a is weakly increasing in

18

Matching stage

• rational expectationsrational expectations

• In equilibrium,

1*1( ( )) ( ( ), ) ( ) (

m

m

a

i i m maw s a y a a dF a

)

*( ( )) ( )e s a a

*( ( )) ( )p s a a

19

Image of stable match (separating)

2

12 1 1( , ) ( ) / ( ) ( )

m

m

a

m m may e a F a

( , )iu s i a

( , )iu s i a ( , )iu s i a

e1( )ma( )ma1( )ma

11 1( , ) ( ) / ( ) ( )

m

m

a

m m may e a F a

20

Boundary conditions

• Initial boundary condition

• Boundary condition for two ranges

0

min

0 0 0 0 0

0

(0, ) ( )( ( ), ) ( ) ( ( ), )

a

ay a dF a

y a a a c a a

1

1

111 1

1

( ( ), ) ( )( ) ( ( ), )

( ( ), ) ( )( ) ( ( ), )

m

mmm m m

m m

m

mmkm km km

m m

y a a dF aa c a a

y a a dF aa c a a

21

Image of stable match (separating)

1

1

( , ) ( )( )

m

m

a

am

m m

y e a dF aa

1( , ) m mu s a

e( )ma1( )ma

1( , ) m mu s a

1( )ma

( )

( ), m

m

a

a m m

recursively determi is

from

ned

22

Fee setting stage

Definition

Lemma 3

1

1

ˆIf ( ) ( , ) for any ,

ˆa same tuition fee which implements ( ) ( , )

can be best responces for the universities on the range .

m m m

m m m

a e m m

a e

m

1

1

1

ˆ( , )

arg max ( , ) ( ) / ( , )

m m

m

m m mm

e

y e a dF a c e a

23

Graphical image

1

1( ( ), ) ( ) /

m

m m m mmEy y a a dF a

1( )ma 1ˆ( , )m me

*( )ma

1

1

( )

( )

m

m

w

a

( )m mEy a

e

24

Equilibrium performances

1

Proposition 1

ˆ( ) ( , ) can be

a perfect Bayesian Nash equilibrium.=m m ma e

0 1 min 0ˆ( ) 0 ( , ), [ , ) a e a a a In equilibrium,

for

0 0 1 1 2ˆ ˆ( ) ( ) ( ) a e e

( ) 0ma m recursively determine for are

by adopting

d

Lemma

25

Equilibrium performances

0amina maxa1a 2a 4a3a

( )a*( )e a

a

e

26

Tuition fees

1

0

0

min

0 1

0 0 1 01 0

0

ˆ( ( , ), ) ( )ˆ( ) ( ( , ), )

(0, ) ( )

a

a

a

a

y e a dF aa c e a

y a dF a

Proposition 2

Equilibrium tuition fee schedule is given by

(initial condition)

27

1

1

1

1

11

1

11

( ) ( )

ˆ( ( , ), ) ( )ˆ( ( , ), )

ˆ( ( , ), ) ( )ˆ( ( , ), )

m

m

m

m

m m

a

m mam m m

m m

a

m mam m m

m m

a a

y e a dF ac e a

y e a dF ac e a

and

28

Equilibrium performances

0amina maxa1a 2a 4a3a

( )a

a

p