Introduction

World

Optimal Choice

Small Model

Full Model

University Competition, Grading Standards and Grade Inflation Sergey V. Popov1

Dan Bernhardt2

1 Centre

for Advanced Studies Higher School of Economics 2 Department

of Economics University of Illinois

6 July 2011

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Grading Policies Are Not Bestowed Upon

1967 1999 2005

Harvard U 2.8 3.42 3.45

U Illinois 2.77 3.12 3.19

S. Rojstaczer, GradeInflation.com: Grades definitely rise. Students get smarter? Grade inflation?..

GPAs go up faster in better universities.

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Universities Can Adjust Policies

Bar & Zussman (2011): registered republican professors grade differently from registered democrat. Bagues, Labini and Zinovyeva (2008): Italian universities respond to funding shocks by changing grading policies.

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

What Other People Do?

Yang & Yip (2003): grade inflation leads to people with good grades and people with bad grades earn same product. Dubey & Geanakoplos (2010): discrete grading makes sense if you want students to exercise effort. Ostrovsky & Schwarz (2010): optimal information revelation might require to give out same grades to people of different abilities. Zubrickas (2010): optimal grading schedule to milk out effort involves revelation in the middle and giving the same grade at the right tail.

Introduction

World

Optimal Choice

Small Model

Full Model

Results Preview

Grading standards are lower in better universities. Social planner sets higher grading standards in better universities. Grading standards go down faster in better universities.

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

The World We study the market of fresh alumni. Students attend two kinds of universities — H and I. University type represents the academic ability distribution of its student. We use a continuum of universities to model that the labor market effect of one university is negligible.

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

The World We study the market of fresh alumni. Students attend two kinds of universities — H and I. University type represents the academic ability distribution of its student. We use a continuum of universities to model that the labor market effect of one university is negligible.

Students are getting employed in two kinds of jobs — good and bad jobs. Employers use grades and interviews to give wages. Universities understand the effect of their own grading on the placement.

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

Universities

There are two kinds of universities — H and I. Universities choose grading policies as academic ability cutoff for “A” grade, θˆu , u ∈ {H , I }.

Not A 0

0.2

A

θ

I

0.4

0.6

0.8

θ

Universities maximize the total wage received by alumni.

1

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

Students

Students are characterized by academic and social abilities,

¯ × [µ, µ (θ, µ) ∈ [θ, θ] ¯ ]. Social ability µ is independent of academic ability θ, µ ∼ G(·). Academic ability θ of students of university u has distribution Fu (·).

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

Students

Students are characterized by academic and social abilities,

¯ × [µ, µ (θ, µ) ∈ [θ, θ] ¯ ]. Social ability µ is independent of academic ability θ, µ ∼ G(·). Academic ability θ of students of university u has distribution Fu (·). There is α of H students and 1 − α of I students. FH has a better upper tail than FI , for every point where one cuts the upper tail.

Introduction

World

Optimal Choice

Small Model

Jobs

There are two types of jobs — good and bad.

Full Model

Conclusion

Introduction

World

Optimal Choice

Small Model

Jobs

There are two types of jobs — good and bad. Good jobs have technology S θµ. Bad jobs have technology sθµ, S > s ≥ 0.

Full Model

Conclusion

Introduction

World

Optimal Choice

Small Model

Jobs

There are two types of jobs — good and bad. Good jobs have technology S θµ. Bad jobs have technology sθµ, S > s ≥ 0. There is a measure Γ of good jobs.

Full Model

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Jobs

There are two types of jobs — good and bad. Good jobs have technology S θµ. Bad jobs have technology sθµ, S > s ≥ 0. There is a measure Γ of good jobs. Good jobs pay W , bad jobs pay w, W > w > 0.

Conclusion

Employment

Hire these for good jobs Hire these too

µ

social skill cutoff for B students =K/E(θ|B)

social skill cutoff for A students =K/E(θ|A) These get A These get B grading standard θ

K is chosen so that total quantity of students employed on good jobs is Γ.

Introduction

World

Optimal Choice

Small Model

Motivation

Social planner: Maximizes total output.

Full Model

Conclusion

Introduction

World

Optimal Choice

Small Model

Motivation

Social planner: Maximizes total output. ... equivalent to maximizing output on good jobs.

Full Model

Conclusion

Introduction

World

Optimal Choice

Small Model

Motivation

Social planner: Maximizes total output. ... equivalent to maximizing output on good jobs. Universities: Universities maximize their total alumni wage.

Full Model

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Motivation

Social planner: Maximizes total output. ... equivalent to maximizing output on good jobs. Universities: Universities maximize their total alumni wage. ... equivalent to maximizing employment on good jobs.

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Motivation

Social planner: Maximizes total output. ... equivalent to maximizing output on good jobs. Universities: Universities maximize their total alumni wage. ... equivalent to maximizing employment on good jobs. Universities realize how they affect employers.

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Social Planner’s Problem Total productivity of IA

Total productivity of HA

max α θˆH ,θˆI

α

subject to

Z |

Z

µ ¯ µ ˆHA

µ ¯ µ ˆHB

Z

}|

θ¯ θˆH

{

µθdFH dG +(1 − α)

θˆH

Z

Z

µ ˆHA

µ ¯ µ ˆHB

Z

Z

µ ˆIA µ ¯

θˆH

dFH dG + (1 − α)

dFH dG + (1 − α)

Z

Z

}|

θ¯

{

µθdFI dG +

θˆI θˆI

µθdFI dG {z }

θ

Total productivity of HB

θ¯

θˆH θ

µ ¯

µθdFH dG +(1 − α) µ ˆ {z } | IB

θ

µ ¯

z Z

Z

Total productivity of HB

α

α

z Z

Z

Z

µ ¯ µ ˆIA µ ¯

µ ˆIB

Z

Z

θ¯ θˆI

dFI dG+

θˆI

dFI dG = Γ θ

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

University u’s Problem

max θˆ

Z

µ ¯ µ ˆuA

Z

θ¯ θˆu

dFu dG +

Z

µ ¯ µ ˆuB

Z

θˆu

dFu dG θ

subject to

µ ˆuA E [θ|u , θ > θˆu ] ≥ K , µ ˆuB E [θ|u , θ < θˆu ] ≥ K .

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

Equilibrium

Labor market works as described. Universities choose standards as best responses to each other.

Proposition Equilibrium exists. It is unique with respect to who gets employed on good jobs.

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

“A” Students Are Always Employed

Proposition University’s best response involves acquiring a good job for a positive mass of students. That means every equilibrium implies hiring some HA and some IA guys for good jobs.

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

Simple Model: µ ¯=µ=1

This assumption means that interviews do not matter. If the amount of good jobs is too small, then equilibrium will have to include only HA and IA students. Interior equilibrium condition when only A students get a good job: E [θ|H , θ > θˆH∗ ] = E [θ|I , θ > θˆI∗ ].

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

Equilibirum and Social Planner

Proposition Social planner picks equal grading standard.

Proposition When the measure of good jobs is not large, the grading standard in H is lower than in I.

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

Equilibirum and Social Planner

Proposition Social planner picks equal grading standard.

Proposition When the measure of good jobs is not large, the grading standard in H is lower than in I.

Corollary There are too many H students employed on a good job, and too few I students in equilibrium.

First-Best vs Equilibrium

1 0.9 First best 0.8 0.7

θH

0.6

too many H

0.5

EIAθ=EHAθ Equilibrium

θ =θ H

0.4

I

Capacity 0.3

too little I

0.2 0.1 0

0

0.2

0.4

0.6

0.8

θI

Note: Γ = 0.25, α = 0.5. fH (x ) = 2x, fI (x ) = 2 − 2x.

1

Introduction

World

Optimal Choice

Small Model

Full Model

Conclusion

Linear Density Family

f (x | b ) = 1 −

b + bx , x ∈ [0, 1] 2

Proposition When only “A” students are getting a good job, grading standards go down faster in H than in I.

Introduction

World

Optimal Choice

Small Model

Full Model

Limited Heterogeneity in Social Skills

No “B” students get good jobs. Also, some “A” students with low social skills may not get a good job.

Proposition When µ2 g (µ) is increasing, in equilibrium H universities have lower grading standards and lower social skill cutoff for hiring at good jobs.

θˆ = K





1 − G(ˆ µu ) = R (ˆ µ) − µ ˆu g (ˆ µu )ˆ µ2u 1

ˆ =K µ ˆE [θ|θ > θ]

Conclusion

Equilibrium

1 0.9 0.8 0.7 Choice of I

θ

0.6 Market clearing condition for I Market clearing condition for H Optimality condition

0.5 0.4 0.3

Choice of H

0.2 0.1 0

0

0.2

0.4

0.6

0.8

1

µ

¯ = [0, 1]. µ is uniform on [0, 1], fH (x ) = 2x, fI (x ) = 2 − 2x, [θ, θ]

Introduction

World

Optimal Choice

Small Model

Full Model

Social Planner

Proposition Social planner’s choice is interior when Γ is small enough.

Proposition Social planner: θˆHP > θˆIP , µ ˆP ˆP HA < µ IA .

Conclusion

Social Planner 1 0.9 0.8 Choice for I 0.7

θ

0.6 0.5 Choice for H

0.4 0.3 0.2

Market clearing condition for I Market clearing condition for H Social planner’s choice

0.1 0

0

0.2

0.4

0.6

0.8

µ ¯ = [0, 1]. µ is uniform on [0, 1], fH (x ) = 2x, fI (x ) = 2 − 2x, [θ, θ]

1

Introduction

World

Optimal Choice

Small Model

Full Model

Better in µ Dimension

Some people argue that students with better µ have better chances at getting to better schools, so H university should have FH ≻C FI and GH ≻C GI .

Proposition When Fu = Fu ′ and Gu ′ ≻C Gu , θu∗ < θu∗′ .

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Better in µ Dimension

Some people argue that students with better µ have better chances at getting to better schools, so H university should have FH ≻C FI and GH ≻C GI .

Proposition When Fu = Fu ′ and Gu ′ ≻C Gu , θu∗ < θu∗′ .  h i µ∗u )  θˆ∗ = K 1∗ − 1 1−Gu (ˆ ∗ ∗ 2 u µ ˆu gu (ˆ µu ) (ˆ µu )  µ ˆ∗u = E [θ|uK,θ>θˆ∗ ] u

Conclusion

Better School Better G 1 Market clearing condition for both types Optimality condition 1 Optimality condition 2

0.9 0.8 0.7

θ

0.6 0.5 Choice of U’

0.4 0.3 0.2 0.1 0

0

0.2

0.4

0.6

0.8

1

µ

Figure: Equilibrium outcomes. µ is uniformly distributed on [0, 1] for U ′ , and density of the social skill distribution is 2µ for U; fU (θ) = fU ′ (θ) = 2θ , ¯ = [ 0, 1] . [θ, θ]

Introduction

World

Optimal Choice

Small Model

Full Model

Better in Both Dimensions

Corollary When both µ2 gH (µ) and µ2 gI (µ) are increasing, and FH ≻C FI and GH ≻C GI , θH∗ < θI∗ .

Conclusion

Introduction

World

Optimal Choice

Small Model

Full Model

Better in Both Dimensions

Corollary When both µ2 gH (µ) and µ2 gI (µ) are increasing, and FH ≻C FI and GH ≻C GI , θH∗ < θI∗ . To prove, consider a type H ′ , with FH ′ = FH and GH ′ = GI .

Conclusion

Better F and Better G 1 Market clearing condition for F

I

0.9

Market clearing condition for F

H

0.8

Optimality condition for G

I

0.7

Optimality condition for G

H

Choice of I

θ

0.6 0.5 Choice of H’

0.4 0.3

Choice of H

0.2 0.1 0

0

0.2

0.4

0.6

0.8

1

µ

Figure: Equilibrium outcomes. µ is uniformly distributed on [0, 1] for H ′ and I, and density of the social skill distribution is 2µ for H; ¯ = [ 0, 1] . fH ′ (θ) = fH (θ) = 2θ , fI (θ) = 2 − 2θ , [θ, θ]

Introduction

World

Optimal Choice

Small Model

Full Model

Summing Up

Socially optimal standards are more demanding to H. Equilibrium might result in H being less demanding to students than I for a big class of g (·). Grade inflation can be caused by: increase in Γ — then eventually H will become much stricter than I. increasing gap between FH (·) and FI (·).

Model predicts faster grade inflation in better universities.

Conclusion