Optimal Timing of College Subsidies
Enrollment, Graduation and the Skill Premium

Kazushige Matsuda
Princeton University

May 23, 20190 / 33

Motivation

The skill premium has been rising in the US from 50% in 1980 to 90% now.

1 / 33

Motivation

The skill premium has been rising in the US from 50% in 1980 to 90% now.
Policies which increase the supply of college educated labor would dampen the skill
premium (“Race between Technology and Education”, Goldin and Katz (2008).).

1 / 33

Motivation

The skill premium has been rising in the US from 50% in 1980 to 90% now.
Policies which increase the supply of college educated labor would dampen the skill
premium (“Race between Technology and Education”, Goldin and Katz (2008).).
Existing literature often equates enrollment with graduation.

1 / 33

Motivation

The skill premium has been rising in the US from 50% in 1980 to 90% now.
Policies which increase the supply of college educated labor would dampen the skill
premium (“Race between Technology and Education”, Goldin and Katz (2008).).
Existing literature often equates enrollment with graduation.
But almost half of the college enrollees in the US drop out.

1 / 33

Motivation

The skill premium has been rising in the US from 50% in 1980 to 90% now.
Policies which increase the supply of college educated labor would dampen the skill
premium (“Race between Technology and Education”, Goldin and Katz (2008).).
Existing literature often equates enrollment with graduation.
But almost half of the college enrollees in the US drop out.
It is important to understand how policy can aﬀect graduation.

1 / 33

This paper

I consider a new college subsidy scheme whose amount varies with year of college,i.e., freshmen, sophomores, etc...: year-dependent subsidies

2 / 33

This paper

I consider a new college subsidy scheme whose amount varies with year of college,i.e., freshmen, sophomores, etc...: year-dependent subsidies
Year-dependent subsidies will have diﬀerential impacts on enrollment & graduation.

2 / 33

This paper

I consider a new college subsidy scheme whose amount varies with year of college,i.e., freshmen, sophomores, etc...: year-dependent subsidies
Year-dependent subsidies will have diﬀerential impacts on enrollment & graduation.
◮

Individuals enroll based on high school ability.

◮

Some college enrollees learn their college ability is low and drop out.

2 / 33

This paper

I consider a new college subsidy scheme whose amount varies with year of college,i.e., freshmen, sophomores, etc...: year-dependent subsidies
Year-dependent subsidies will have diﬀerential impacts on enrollment & graduation.
◮

Individuals enroll based on high school ability.

◮

Some college enrollees learn their college ability is low and drop out.

◮

Back-loaded subsidies would decrease enrollment and increase graduation.

2 / 33

This paper

I consider a new college subsidy scheme whose amount varies with year of college,i.e., freshmen, sophomores, etc...: year-dependent subsidies
Year-dependent subsidies will have diﬀerential impacts on enrollment & graduation.
◮

Individuals enroll based on high school ability.

◮

Some college enrollees learn their college ability is low and drop out.

◮

Back-loaded subsidies would decrease enrollment and increase graduation.

◮

Front-loaded subsidies, vice versa

2 / 33

This paper

I consider a new college subsidy scheme whose amount varies with year of college,i.e., freshmen, sophomores, etc...: year-dependent subsidies
Year-dependent subsidies will have diﬀerential impacts on enrollment & graduation.
◮

Individuals enroll based on high school ability.

◮

Some college enrollees learn their college ability is low and drop out.

◮

Back-loaded subsidies would decrease enrollment and increase graduation.

◮

Front-loaded subsidies, vice versa

Questions: What timing of subsidies will maximize the number of college graduatesand social welfare?

2 / 33

What I do

A quantitative heterogeneous agent overlapping generation lifecycle model with◮borrowing constraints◮

endogenous enrollment/graduation decisions.

3 / 33

What I do

A quantitative heterogeneous agent overlapping generation lifecycle model with◮borrowing constraints◮

endogenous enrollment/graduation decisions.

Examine the eﬀect of year-dependent subsidies on graduation, the skill premium,
and welfare.

3 / 33

What I do

A quantitative heterogeneous agent overlapping generation lifecycle model with◮borrowing constraints◮

endogenous enrollment/graduation decisions.

Examine the eﬀect of year-dependent subsidies on graduation, the skill premium,
and welfare.
◮

Focus on the relative sizes across years in college (slope).

3 / 33

What I do

A quantitative heterogeneous agent overlapping generation lifecycle model with◮borrowing constraints◮

endogenous enrollment/graduation decisions.

Examine the eﬀect of year-dependent subsidies on graduation, the skill premium,
and welfare.
◮

Focus on the relative sizes across years in college (slope).

◮

I fix the total budget of college subsidies from now on.

3 / 33

Outline

1

Introduction

2

Model

3

Calibration

4

Results

5

Conclusion

3 / 33

Demography

I focus on steady state from now on.

4 / 33

Demography

I focus on steady state from now on.
OLG: each individual has one oﬀspring living with them until independence.

4 / 33

Demography

I focus on steady state from now on.
OLG: each individual has one oﬀspring living with them until independence.
New independent individuals make an enrollment decision.

4 / 33

Demography

I focus on steady state from now on.
OLG: each individual has one oﬀspring living with them until independence.
New independent individuals make an enrollment decision.
One period is two years and completing college requires 2 periods.

4 / 33

Demography

I focus on steady state from now on.
OLG: each individual has one oﬀspring living with them until independence.
New independent individuals make an enrollment decision.
One period is two years and completing college requires 2 periods.
At the beginning of the 2nd period of college, enrollees make a dropout decision.

4 / 33

Demography

I focus on steady state from now on.
OLG: each individual has one oﬀspring living with them until independence.
New independent individuals make an enrollment decision.
One period is two years and completing college requires 2 periods.
At the beginning of the 2nd period of college, enrollees make a dropout decision.
Once an individual finishes their schooling, they will be high school graduates(e = HS), college dropouts (CD), or college graduates (CG ).

4 / 33

Demography

I focus on steady state from now on.
OLG: each individual has one oﬀspring living with them until independence.
New independent individuals make an enrollment decision.
One period is two years and completing college requires 2 periods.
At the beginning of the 2nd period of college, enrollees make a dropout decision.
Once an individual finishes their schooling, they will be high school graduates(e = HS), college dropouts (CD), or college graduates (CG ).
After that, they face a standard life cycle problem with income risk, incomplete
markets for insurance, and borrowing constraints.

4 / 33

Timeline

5 / 33

Preferences: three parts

The lifetime utility is the sum of the following three parts:
1

The expected discounted sum
E1

J
󰁛
j=1

β̃j u(cj , ℓj ) where u(c, ℓ) =(c µ ℓ1−µ )1−γ
1−γ

where cj denotes consumption and ℓj is leisure at age j.

6 / 33

Preferences: three parts

The lifetime utility is the sum of the following three parts:
1

The expected discounted sum
E1J󰁛β̃j u(cj , ℓj ) where u(c, ℓ) =j=1

(c µ ℓ1−µ )1−γ
1−γ

where cj denotes consumption and ℓj is leisure at age j.
2

Expected utility of college attendance:
λj (θc , φ) = λ + λθ θc + λφj φCollege utility depends on college ability θc and college taste φ.

6 / 33

Preferences: three parts

The lifetime utility is the sum of the following three parts:
1

The expected discounted sum
E1J󰁛β̃j u(cj , ℓj ) where u(c, ℓ) =j=1

(c µ ℓ1−µ )1−γ
1−γ

where cj denotes consumption and ℓj is leisure at age j.
2

Expected utility of college attendance:
λj (θc , φ) = λ + λθ θc + λφj φCollege utility depends on college ability θc and college taste φ.

3

Parental altruism: They enjoy their children’s lifetime utility with a weight ν.

6 / 33

Key Factors of Educational Decisions
1

Common factors◮

Tuition pe , subsidies s1 (q), s2 (q), and credit limits Ac1 , Ac2 .

◮The price of eﬀective labor w CG , w CD , w HS .

7 / 33

Key Factors of Educational Decisions
1

2

Common factors◮

Tuition pe , subsidies s1 (q), s2 (q), and credit limits Ac1 , Ac2 .

◮The price of eﬀective labor w CG , w CD , w HS .

Idiosyncratic factors: enrollment◮

Initial asset a, family income q, and taste φ ∼ N(0, 1)◮

CD
High school ability θh : signal of θc and labor productivity εHSj (θh , η), εj (θh , η).

◮Idiosyncratic transitory productivity η ∼ ΠHS (·)

7 / 33

Key Factors of Educational Decisions
1

2

3

Common factors◮

Tuition pe , subsidies s1 (q), s2 (q), and credit limits Ac1 , Ac2 .

◮The price of eﬀective labor w CG , w CD , w HS .

Idiosyncratic factors: enrollment◮

Initial asset a, family income q, and taste φ ∼ N(0, 1)◮

CD
High school ability θh : signal of θc and labor productivity εHSj (θh , η), εj (θh , η).

◮Idiosyncratic transitory productivity η ∼ ΠHS (·)

Idiosyncratic factors: graduation◮

Realized college ability θc aﬀects college utility λj (θc , φ) and εCGj (θc , η)◮Idiosyncratic transitory productivity η ∼ ΠCD (·)
7 / 33

Education stage: Enrollment

V0 (a, θh , η, q, φ) = max[V1c (a, θh , η, q, φ), V1 (a, HS, θh , η)]
󰁿
󰁾󰁽
󰂀 󰁿
󰁾󰁽
󰂀
enrolling

not enrolling

Individuals only observe θh but not θc before the enrollment decision.

8 / 33

Education stage: Enrollment

V0 (a, θh , η, q, φ) = max[V1c (a, θh , η, q, φ), V1 (a, HS, θh , η)]
󰁿
󰁾󰁽
󰂀 󰁿
󰁾󰁽
󰂀
enrolling

not enrolling

Individuals only observe θh but not θc before the enrollment decision.
College ability is correlated with high school ability.
θc = θh + 󰂃c and 󰂃c ∼ N(0, σc2 )8 / 33

Education stage: Enrollment

V0 (a, θh , η, q, φ) = max[V1c (a, θh , η, q, φ), V1 (a, HS, θh , η)]
󰁿
󰁾󰁽
󰂀 󰁿
󰁾󰁽
󰂀
enrolling

not enrolling

Individuals only observe θh but not θc before the enrollment decision.
College ability is correlated with high school ability.
θc = θh + 󰂃c and 󰂃c ∼ N(0, σc2 )I assume enrollees are overoptimistic on college abilities.
θc = µc (θh ) + θh + 󰂃c and 󰂃c ∼ N(0, σc2 ), (Perceived law of motion)󰁿 󰁾󰁽 󰂀󰁿 󰁾󰁽 󰂀
bias

whereactual abilityµc (θh ) = µc0 + µc1 θh

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Education stage: First half of college
V1c (a, θh , η, q, φ) = maxu(c, 1 − h − h̄) + Eθc |θh λ1 (θc , φ)′c,h,a ,y+ βEθc |θh Eη′ max[V2c (a′ , θc , η ′ , q, φ), V2 (ã(a′ ), CD, θh , η ′ )]
󰁿
󰁾󰁽
󰂀 󰁿
󰁾󰁽
󰂀
continue

subject todropout

c + a′ + pe = a + y + s1 (q) − T (c, a, y )′cy = w HS εHS
1 (θh , η)h, a ≥ −A1 c ≥ 0, 0 ≤ h ≤ 1 − h̄θc = θh + µc (θh ) + 󰂃c , 󰂃c ∼ N(0, σc2 ), η ′ ∼ ΠCD
They can work as high school graduates.

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Education stage: First half of college
V1c (a, θh , η, q, φ) = maxu(c, 1 − h − h̄) + Eθc |θh λ1 (θc , φ)′c,h,a ,y+ βEθc |θh Eη′ max[V2c (a′ , θc , η ′ , q, φ), V2 (ã(a′ ), CD, θh , η ′ )]
󰁿
󰁾󰁽
󰂀 󰁿
󰁾󰁽
󰂀
continue

subject todropout

c + a′ + pe = a + y + s1 (q) − T (c, a, y )′cy = w HS εHS
1 (θh , η)h, a ≥ −A1 c ≥ 0, 0 ≤ h ≤ 1 − h̄θc = θh + µc (θh ) + 󰂃c , 󰂃c ∼ N(0, σc2 ), η ′ ∼ ΠCD
They can work as high school graduates.
Going to college requires a fraction h̄ of time.

9 / 33

Education stage: First half of college
V1c (a, θh , η, q, φ) = maxu(c, 1 − h − h̄) + Eθc |θh λ1 (θc , φ)′c,h,a ,y+ βEθc |θh Eη′ max[V2c (a′ , θc , η ′ , q, φ), V2 (ã(a′ ), CD, θh , η ′ )]
󰁿
󰁾󰁽
󰂀 󰁿
󰁾󰁽
󰂀
continue

subject todropout

c + a′ + pe = a + y + s1 (q) − T (c, a, y )′cy = w HS εHS
1 (θh , η)h, a ≥ −A1 c ≥ 0, 0 ≤ h ≤ 1 − h̄θc = θh + µc (θh ) + 󰂃c , 󰂃c ∼ N(0, σc2 ), η ′ ∼ ΠCD
They can work as high school graduates.
Going to college requires a fraction h̄ of time.
At the beginning of j = 2, they observe θc and η ′ and make a dropout decision.

ã

9 / 33

Education stage: Second half of college

V2c (a, θc , η, q, φ) = maxu(c, 1 − h − h̄) + λ2 (θc , φ) + βEη′ V3 (ã(a′ ), CG , θc , η)′c,h,a ,ysubject to
c + a′ + pe − s2 (q) − y + T (c, a, y ) =󰀫(1 + r )a(1 + r s )a

if a ≥ 0
if a < 0′c′CGy = w CD εCD
2 (θc , η)h, a ≥ −A2 c ≥ 0, 0 ≤ h ≤ 1 − h̄, η ∼ ΠThey can work as college dropouts.

10 / 33

Education stage: Second half of college

V2c (a, θc , η, q, φ) = maxu(c, 1 − h − h̄) + λ2 (θc , φ) + βEη′ V3 (ã(a′ ), CG , θc , η)′c,h,a ,ysubject to
c + a′ + pe − s2 (q) − y + T (c, a, y ) =󰀫(1 + r )a(1 + r s )a

if a ≥ 0
if a < 0′c′CGy = w CD εCD
2 (θc , η)h, a ≥ −A2 c ≥ 0, 0 ≤ h ≤ 1 − h̄, η ∼ ΠThey can work as college dropouts.
At the end of the period, one completes college and draws η ′ from ΠCG .
Financial Market

10 / 33

After the education stage

Individuals face a standard lifecycle problem with borrowing limit Ae .

Working Stage

11 / 33

After the education stage

Individuals face a standard lifecycle problem with borrowing limit Ae .

Working StageAt jb , individuals transfer asset to oﬀspring after observing their high school ability.
Tranfer

11 / 33

After the education stage

Individuals face a standard lifecycle problem with borrowing limit Ae .

Working StageAt jb , individuals transfer asset to oﬀspring after observing their high school ability.
Tranfer

I assume retirees oﬀer no labor, receive pension p(e, θ), and have no access to loans.
Retirement Stage

11 / 33

Goods Sector

A representative firm produces final good from capital K and aggregate labor H:
Y = F (K , H) = K α H 1−α12 / 33

Goods Sector

A representative firm produces final good from capital K and aggregate labor H:
Y = F (K , H) = K α H 1−α

H is composed of two skills: skilled labor H S and unskilled labor H U :1

H = (a(H S )ρ + (1 − a)(H U )ρ ) ρ1where 1−ρis the elasticity of substitution.

12 / 33

Goods Sector

A representative firm produces final good from capital K and aggregate labor H:
Y = F (K , H) = K α H 1−α

H is composed of two skills: skilled labor H S and unskilled labor H U :1

H = (a(H S )ρ + (1 − a)(H U )ρ ) ρ1where 1−ρis the elasticity of substitution.
◮

College graduates work as skilled labor: w CG = w S◮

High school graduates and college dropouts work as unskilled labor:w HS = w CD = w U Share of skilled labor by college dropout

12 / 33

College

A representative college requires κ units of skilled labor to provide education.
pe E − w S κE
where E is the measure of college enrollees and pe is tuition.

13 / 33

College

A representative college requires κ units of skilled labor to provide education.
pe E − w S κE
where E is the measure of college enrollees and pe is tuition.
I assume colleges are competitive and there is free entry: pe = w S κ

13 / 33

Government
The government collects tax T (c, a, y ) and spend the revenues on◮college subsidies
Ge =

󰁛󰁝

j=1,2

Sjc

sj (q)dµcj

14 / 33

Government
The government collects tax T (c, a, y ) and spend the revenues on◮college subsidies
Ge =

󰁛󰁝

j=1,2

◮

Sjc

sj (q)dµcj

other government consumption
Gc = gY

14 / 33

Government
The government collects tax T (c, a, y ) and spend the revenues on◮college subsidies
Ge =

󰁛󰁝

j=1,2

◮

Sjc

sj (q)dµcj

other government consumption
Gc = gY◮

retirement benefits

J 󰁝
󰁛
j=jr

p(e, θ)dµj
Sj

14 / 33

Government
The government collects tax T (c, a, y ) and spend the revenues on◮college subsidies
Ge =

󰁛󰁝

j=1,2

◮

Sjc

sj (q)dµcj

other government consumption
Gc = gY◮

retirement benefits

J 󰁝
󰁛
j=jr

p(e, θ)dµj
Sj

The tax function is assumed to be
T (c, a, y ) = τc c + τk ra1a≥0 + τl y − dYNGovernment Budget
14 / 33

Outline

1

Introduction

2

Model

3

Calibration

4

Results

5

Conclusion

14 / 33

Calibration Strategy

There are two sets of parameters:

15 / 33

Calibration Strategy

There are two sets of parameters:
Those estimated outside of the model or fixed based on the literature.

15 / 33

Calibration Strategy

There are two sets of parameters:
Those estimated outside of the model or fixed based on the literature.
The remaining parameters to match moments given the first set of parameter values.

15 / 33

Labor Productivity

I assume labor productivity
ln 󰂃ej (θ, η) = ln 󰂃e + ln ψje + 󰂃eθ θ + ln η

16 / 33

Labor Productivity

I assume labor productivity
ln 󰂃ej (θ, η) = ln 󰂃e + ln ψje + 󰂃eθ θ + ln η◮

where ψje is the age profile of workers at age j estimated from PSID.

PSID

16 / 33

Labor Productivity

I assume labor productivity
ln 󰂃ej (θ, η) = ln 󰂃e + ln ψje + 󰂃eθ θ + ln η◮

where ψje is the age profile of workers at age j estimated from PSID.

After filtering out age eﬀects, I regress hourly wages on ln AFQT80.

PSID

Ability

16 / 33

Labor Productivity

I assume labor productivity
ln 󰂃ej (θ, η) = ln 󰂃e + ln ψje + 󰂃eθ θ + ln η◮

where ψje is the age profile of workers at age j estimated from PSID.

After filtering out age eﬀects, I regress hourly wages on ln AFQT80.

log AFQT

HS

CD

CG

.61
(.32)

.74
(.32)

1.31
(.24)

PSID

Ability

16 / 33

Transitory Labor Productivity Process

I assume πηe (η ′ |η) is a two-state Markov chain approximating

Markov Chainln η ′ = ρe ln η + 󰂃eη , 󰂃eη ∼ N(0, σηe2 )17 / 33

Transitory Labor Productivity Process

I assume πηe (η ′ |η) is a two-state Markov chain approximating

Markov Chainln η ′ = ρe ln η + 󰂃eη , 󰂃eη ∼ N(0, σηe2 )Minimum Distance Estimator separately for each education level.

ρe
σηe2

HS

CD

CG

0.94
0.017

0.95
0.021

0.95
0.025

17 / 33

Intergenerational Ability Transmission

New independent individuals draw their high school abilities θh′ .
θh′ = m + mθ θ + 󰂃θ , 󰂃θ ∼ N(0, σh2 )I regressed children’s ability on parents’ ability to get mθ = 0.46.

18 / 33

Subsidies and Loansqfamily income

subsidies to students

subsidies to colleges

total s̄(q)12
3- $30,000$30,000 - $80,000$80,000 -$2,820$668$143$10,477$10,477$10,477$13,297$11,145$10,620

The government subsidizes the education sector $10, 477 in the data.
In the model, students receive all subsidies but pay the full cost of education.

19 / 33

Subsidies and Loansqfamily income

subsidies to students

subsidies to colleges

total s̄(q)12
3- $30,000$30,000 - $80,000$80,000 -$2,820$668$143$10,477$10,477$10,477$13,297$11,145$10,620

The government subsidizes the education sector $10, 477 in the data.
In the model, students receive all subsidies but pay the full cost of education.
In the current system, college subsidies are constant and s1 (q) = s2 (q) = s̄(q).

19 / 33

Subsidies and Loansqfamily income

subsidies to students

subsidies to colleges

total s̄(q)12
3- $30,000$30,000 - $80,000$80,000 -$2,820$668$143$10,477$10,477$10,477$13,297$11,145$10,620

The government subsidizes the education sector $10, 477 in the data.
In the model, students receive all subsidies but pay the full cost of education.
In the current system, college subsidies are constant and s1 (q) = s2 (q) = s̄(q).
Students’ interest rate is the prime rate plus ιs = 2.3%, annual.

19 / 33

Subsidies and Loansqfamily income

subsidies to students

subsidies to colleges

total s̄(q)12
3- $30,000$30,000 - $80,000$80,000 -$2,820$668$143$10,477$10,477$10,477$13,297$11,145$10,620

The government subsidizes the education sector $10, 477 in the data.
In the model, students receive all subsidies but pay the full cost of education.
In the current system, college subsidies are constant and s1 (q) = s2 (q) = s̄(q).
Students’ interest rate is the prime rate plus ιs = 2.3%, annual.
The loan limit for the first half Ac1 is $6,125 (= $2, 625 + $3, 500) from Staﬀord loan.
The loan limit for the second half Ac2 is $23,000.

19 / 33

The Remaining Parameters
Parameter

Description

Value

µ0c
µ1c

college ability bias intercept
college ability bias slope
college utility intercept
college utility slope
first period college taste
second half college taste
productivity of skilled laborproductivity of CDs.d. of college ability
education costconsumption share of preferencetime discount rate
altruismlump-sum transfer ratio
borrowing wedge (r − = r + ι)intergenerational ability transmission intercept
intergenerational ability transmission s.d.

0.190
-0.409
-23.2
241
64.1
41.3
0.457
1.02
0.340
0.226
0.418
0.938
0.09480.125
18.0%-0.0471
0.171

λ
λθ
λφ1
λφ2
aS
󰂃CD
σc
κ
µ
β
v
d
ι
m
σh
Parameters

20 / 33

Matched Moments

MomentModel

TargetEnrollment rate of ability quartile
Graduation rate of ability quartile
Enrollment rate of family income quartile
Graduation rate of family income quartile
Skill premium for CG
Skill premium for CD
Expected/Actual graduation rate −1
Education cost/mean income at 48
Hours of work
K /YTransfer/mean income at 48
Log pre-tax/post-tax income
Borrowers
Mean of AFQT
Standard deviation of AFQT

(figure)(figure)(figure)(figure)
90.8%
19.6%
0.431
0.320
33.8%
1.298
67.0%
61.2%
6.59%
-0.0135
0.217(figure)(figure)(figure)(figure)
90.2%
19.9%
0.433
0.33
33.3%
1.325
66%
61%
6.3%
0
0.213

21 / 33

Optimism

A survey shows that students believe that there is an 86% chance of graduating
while only 60% graduate

22 / 33

Optimism

A survey shows that students believe that there is an 86% chance of graduating
while only 60% graduate
To match this fact, the calibrated µ0c is positive and◮the bias for the mean ability is 48% of the standard deviation of college ability.

22 / 33

Optimism

A survey shows that students believe that there is an 86% chance of graduating
while only 60% graduate
To match this fact, the calibrated µ0c is positive and◮the bias for the mean ability is 48% of the standard deviation of college ability.

Low ability students are more optimistic (µ1c < 0), which is consistent with data.

22 / 33

Model Fit

Figure: Enrollment rates

Figure: Graduation rates

23 / 33

Validation 1: Partial Equilibrium Eﬀect of Year-Invariant subsidies

I simulate the partial equilibrium response of enrollment to an $1,000 increase insubsidies for all the enrollees evenly.

24 / 33

Validation 1: Partial Equilibrium Eﬀect of Year-Invariant subsidies

I simulate the partial equilibrium response of enrollment to an $1,000 increase insubsidies for all the enrollees evenly.
◮

The additional subsidies are given to only one generation.

24 / 33

Validation 1: Partial Equilibrium Eﬀect of Year-Invariant subsidies

I simulate the partial equilibrium response of enrollment to an $1,000 increase insubsidies for all the enrollees evenly.
◮

The additional subsidies are given to only one generation.

◮

All the prices and the distribution of initial state are fixed at the current level.

24 / 33

Validation 1: Partial Equilibrium Eﬀect of Year-Invariant subsidies

I simulate the partial equilibrium response of enrollment to an $1,000 increase insubsidies for all the enrollees evenly.
◮

The additional subsidies are given to only one generation.

◮

All the prices and the distribution of initial state are fixed at the current level.

The aggregate enrollment rate of the aﬀected generation increases by 1.05
percentage points in the simulation, which is broadly in the range.

24 / 33

Validation 1: Partial Equilibrium Eﬀect of Year-Invariant subsidies

I simulate the partial equilibrium response of enrollment to an $1,000 increase insubsidies for all the enrollees evenly.
◮

The additional subsidies are given to only one generation.

◮

All the prices and the distribution of initial state are fixed at the current level.

The aggregate enrollment rate of the aﬀected generation increases by 1.05
percentage points in the simulation, which is broadly in the range.
◮

The fraction of college graduates increases by 0.45 percentage points.

24 / 33

Validation 1: Partial Equilibrium Eﬀect of Year-Invariant subsidies

I simulate the partial equilibrium response of enrollment to an $1,000 increase insubsidies for all the enrollees evenly.
◮

The additional subsidies are given to only one generation.

◮

All the prices and the distribution of initial state are fixed at the current level.

The aggregate enrollment rate of the aﬀected generation increases by 1.05
percentage points in the simulation, which is broadly in the range.
◮

The fraction of college graduates increases by 0.45 percentage points.

◮

The fraction of college dropouts increases by 0.60 percentage points.

24 / 33

Validation 2: Sluggish increase in college graduates

In the US, the number of college graduates increased sluggishly despite the increasein the skill premium.

25 / 33

Validation 2: Sluggish increase in college graduates

In the US, the number of college graduates increased sluggishly despite the increasein the skill premium.
Derive the two steady states’ aS and 󰂃CD imitating 1980 and 2000 skill premiums.

25 / 33

Validation 2: Sluggish increase in college graduates

In the US, the number of college graduates increased sluggishly despite the increasein the skill premium.
Derive the two steady states’ aS and 󰂃CD imitating 1980 and 2000 skill premiums.

Compare the changes of the numbers of college graduates and dropouts with data.
1980

2000

change (model)change (data)college graduate premium
college dropout premium

46.2%
12.1%

90.9%
19.6%

44.7pp
7.5pp

43.2pp7.4pp

share of college graduatesshare of college dropouts

28.0%
42.8%

32.9%
41.3%

4.9pp
-1.5pp4.98pp2.41pp25 / 33

Outline

1

Introduction

2

Model

3

Calibration

4

Results

5

Conclusion

25 / 33

Main Exercises

Exercise 1: Increase overall spending without changing the structure of subsidies,financed by increased tax on labor income.

26 / 33

Main Exercises

Exercise 1: Increase overall spending without changing the structure of subsidies,financed by increased tax on labor income.
Exercise 2: Keep total spending fixed but choose subsidies by year (year-dependent
subsidies) to maximize the number of college graduates in steady state.

26 / 33

Main Exercises

Exercise 1: Increase overall spending without changing the structure of subsidies,financed by increased tax on labor income.
Exercise 2: Keep total spending fixed but choose subsidies by year (year-dependent
subsidies) to maximize the number of college graduates in steady state.
Exercise 3: Keep total spending fixed and choose subsidies to maximize welfare insteady state.

26 / 33

Exercise 1: Year Invariant Subsidies

Ge

0.75 ḠeḠe

1.5Ḡe2Ḡeenrollment rateshare of college graduatesskill premium

72.7%
32.1%
95.0%

74.2%
32.9%
90.9%

77.2%
34.2%
82.8%

77.8%
35.0%
78.3%

26 / 33

Exercise 2: Year Dependent Subsidies That Maximize College Graduatesmax

g1 ,g2 ,τℓsubject to
g1

󰁝

S1c

󰁝

S2CG

s̄(q)dµc1 + g2

󰁝

dµCG
2

S2c

s̄(q)dµc2 = Geand the government budget constraint where sj (q) = gj s̄(q).

27 / 33

Exercise 2: Year Dependent Subsidies That Maximize College Graduatesmax

g1 ,g2 ,τℓsubject to
g1

󰁝

S1c

󰁝

S2CG

s̄(q)dµc1 + g2

󰁝

dµCG
2

S2c

s̄(q)dµc2 = Geand the government budget constraint where sj (q) = gj s̄(q).
sj (q)

year-invariant Ḡeyear-dependent Ḡes1 (1)
s1 (2)
s1 (3)s2 (1)s2 (2)
s2 (3)$13,599$11,447$10,922$13,599$11,447$10,922$4$4$3$42,436$35,720$34,082

Back-loaded
27 / 33

Exercise 2: Year Dependent Subsidies That Maximize College Graduatesyear-invariant/dependent

invariant Ḡedependent Ḡeinvariant 1.5Ḡeenrollment rateshare of college graduatesskill premium

74.2%
32.9%
90.9%

68.7%
34.5%
82.6%

77.2%
34.2%
82.8%

Share of college graduates increases more than increasing the total budget by 50%.
Skill premium decreases more than increasing the total budget by 50%.
Enrollment decreases.

28 / 33

Mechanism

In the current system, increasing enrollment encourages more people who are more
likely to drop out.

29 / 33

Mechanism

In the current system, increasing enrollment encourages more people who are more
likely to drop out.
The enrollment margin is not so important from the perspective of getting people tograduate.

29 / 33

Mechanism

In the current system, increasing enrollment encourages more people who are more
likely to drop out.
The enrollment margin is not so important from the perspective of getting people tograduate.
It is easier to create incentives for the marginal dropout to finish than to create
incentives for the marginal non-enrollee to enroll and finish.

29 / 33

Mechanism

In the current system, increasing enrollment encourages more people who are more
likely to drop out.
The enrollment margin is not so important from the perspective of getting people tograduate.
It is easier to create incentives for the marginal dropout to finish than to create
incentives for the marginal non-enrollee to enroll and finish.
Decreasing subsidies for the first period serves mainly to discourage people who areunlikely to graduate from enrolling.

29 / 33

Mechanism

In the current system, increasing enrollment encourages more people who are more
likely to drop out.
The enrollment margin is not so important from the perspective of getting people tograduate.
It is easier to create incentives for the marginal dropout to finish than to create
incentives for the marginal non-enrollee to enroll and finish.
Decreasing subsidies for the first period serves mainly to discourage people who areunlikely to graduate from enrolling.
The higher subsidies for the second period encourages marginal dropouts to finish.

29 / 33

Mechanism

In the current system, increasing enrollment encourages more people who are more
likely to drop out.
The enrollment margin is not so important from the perspective of getting people tograduate.
It is easier to create incentives for the marginal dropout to finish than to create
incentives for the marginal non-enrollee to enroll and finish.
Decreasing subsidies for the first period serves mainly to discourage people who areunlikely to graduate from enrolling.
The higher subsidies for the second period encourages marginal dropouts to finish.
In addition, we can shift subsidies away from college dropouts to college graduates.

29 / 33

Exercise 3: Year Dependent Subsidies That Maximize Welfare of Newborns
󰁛

Nj

󰀕󰁝

Vj (s j )d µ̄j (s j ) +g1󰁝

s̄(q)dµc1 + g2

jsubject to

S1c

󰁝

S2c

󰁝

Vjc (s cj )d µ̄j (s cj )󰀖s̄(q)dµc2 = Geand the government budget constraint where sj (q) = gj s̄(q).
The government recalculates the lifetime values with rational expectation.

30 / 33

Exercise 3: Year Dependent Subsidies That Maximize Welfare of Newborns
󰁛

Nj

󰀕󰁝

Vj (s j )d µ̄j (s j ) +g1󰁝

s̄(q)dµc1 + g2

jsubject to

S1c

󰁝

S2c

󰁝

Vjc (s cj )d µ̄j (s cj )󰀖s̄(q)dµc2 = Geand the government budget constraint where sj (q) = gj s̄(q).
The government recalculates the lifetime values with rational expectation.
Current state

Optimals1 (1)
s1 (2)
s1 (3)$13,599$11,447$10,922$10,721$9,025$8,611s2 (1)s2 (2)
s2 (3)$13,599$11,447$10,922$19,858$16,716$15,949

Optimal subsidies are back-loaded.
30 / 33

Aggregatesshare of college enrollees
share of college graduatesskill premium
welfare gain

Current state

Optimal

74.2%
32.9%
90.9%

73.8%
33.6%
87.3%
+0.15%

31 / 33

Aggregates

Current state

Optimalshare of college enrollees
share of college graduatesskill premium
welfare gain

74.2%
32.9%
90.9%

73.8%
33.6%
87.3%
+0.15%

Total

Level

Uncertainty

Inequality+0.07%+0.15%+0.04%−0.09%

Optimal

Back-loaded subsidies improve welfare.

31 / 33

Aggregates

Current state

Optimalshare of college enrollees
share of college graduatesskill premium
welfare gain

74.2%
32.9%
90.9%

73.8%
33.6%
87.3%
+0.15%

Total

Level

Uncertainty

Inequality+0.07%+0.15%+0.04%−0.09%

Optimal

Back-loaded subsidies improve welfare.
The level eﬀect is positive while inequality at the initial state increases.

31 / 33

Welfare
Y
K
C
wS
wU
std c
std a
std h
std wage

Current state

Optimal

0.318
0.413
0.211
0.355
0.405
0.129
0.478
0.0834
0.544

0.318
0.413
0.211
0.352
0.408
0.129
0.475
0.08330.540

32 / 33

Welfare
Current state

Optimal

0.318
0.413
0.211
0.355
0.405
0.129
0.478
0.0834
0.544

0.318
0.413
0.211
0.352
0.408
0.129
0.475
0.0833
0.540

Y
K
C
wS
wU
std c
std a
std h
std wage

θ=1
θ=2
θ=3
θ=4

q=1

q=2

q=3

+0.6%
+0.2%
−0.8%
−0.9%

+0.1%
−0.4%
−0.3%
−0.0%

+0.5%
+0.5%
+0.5%
+0.4%

High-ability poor-family enrollees lose welfare.
Consumption Response

Correcting Bias

No Optimism
32 / 33

Conclusion

Back-loaded subsidies maximize the number of college graduates and social welfare.

33 / 33

Conclusion

Back-loaded subsidies maximize the number of college graduates and social welfare.
The number of college graduates increases and the skill premium decreases as muchas the case with increasing the total budget by 50%.

33 / 33

Conclusion

Back-loaded subsidies maximize the number of college graduates and social welfare.
The number of college graduates increases and the skill premium decreases as muchas the case with increasing the total budget by 50%.
Enrollment decreases despite an increase in college graduates. Policies increasing
enrollment might be misguided.

33 / 33

Student Loan Transformation

The fixed payment to repay full debt for 20 years (10 periods) d is given by
a′ =

9
󰁛

1 − (1 + r s )−10dd 1 − (1 + r s )−10==d(1 + r s )t
1 + r s 1 − (1 + r s )−1
rst=0

To have the same payment schedule d with interest r − , the initial balance has to beã(a′ ) =9
󰁛

1 − (1 + r − )−10
1 − (1 + r − )−10
d
d
=
=
d
(1 + r − )t
1 + r − 1 − (1 + r − )−1r−
t=0

As a result,ã(a′ ) = a′ ×

1 − (1 + r − )−10
rs
×
s
−10
1 − (1 + r )
r−

Back

34 / 33

Working StageVj (a, e, θ, η) = maxu′c,h,a ,y󰀕c,1 − h1 + 1Jf ζ

subject to′

c + a − y + T (c, a, y ) =󰀫󰀖+ βEη′ |η Vj+1 (a′ , e, θ, η ′ )(1 + r )a(1 + r − )a

if a ≥ 0
if a < 0y = w e εej (θ, η)h, a′ ≥ −Ae c ≥ 0, 0 ≤ h ≤ 1, η ′ ∼ π e (·|η)
where 1Jf is an indicator function which is one when the individual lives with its children(j ∈ [jf , jb − 1]). Back

35 / 33

Transfer
Vj (a, e, θ, η) =maxc(θh′ ),h(θh′ ),a′ (θh′ ),y (θh′ )Eθh′ |e,θ {u(c(θh′ ), 1 − h(θh′ )) + Ṽjb +1 (a′ , θ, θh′ , e, η)}subject toc(θh′ ) + a′ (θh′ ) − y (θh′ ) + T (c(θh′ ), a(θh′ ), y (θh′ )) =

where

󰀫

(1 + r )a(1 + r − )a

if a ≥ 0
if a < 0y (θh′ ) = w e εej (θ, η)h(θh′ ), a′ ≥ −Ae c(θh′ ) ≥ 0, 0 ≤ h(θh′ ) ≤ 1, η ′ ∼ π e (·|η)

Ṽjb +1 (a, θ, θh′ , e, η) = max βEη′ |η Vjb +1 (a−b, e, θ, η ′ )+νEη′′ ,φ V0 (b, θh′ , η ′′ , q̃(w e εej (θ, η)), φ)b∈[0,a]for all θh′ .
Individuals can make parental transfers b to their children only at this age.
Before making any decisions, individuals observe only their children’s high school
ability θh′ from πθ (θh′ |e, θ).
Back36 / 33

Family income level

Family income level
󰀻
󰁁
󰀿1
e e
q̃(w εj (θ, η)) = 2󰁁󰀽3

if w e εej (θ, η) × 0.35 ∈ [0, q1 ]if w e εej (θ, η) × 0.35 ∈ [q1 , q2 ]else

where q1 and q2 correspond to $30,000 and $80,000.
Back

37 / 33

Retirement StageVj (a, e, θ) = maxu(c, 1) + βϕj+1 Vj+1 (a′ , e, θ)′
c,asubject to−1
c + a′ = (1 + r )ϕ−1
j a + p(e, θ) − T (c, ϕj a, 0)

a′ ≥ 0 c ≥ 0

The sources of income is asset earnings and retirement benefits p(e, θ).
The asset inflated by ϕ−1reflects that assets of expiring households are distributedjwithin cohorts (perfect annuity market).
BackSocial Security

38 / 33

Social Security

The average life time income isŷ (e, θ) =

󰁓jr −1j=ja +2 we eεj (θ, 1)h̄jr − 2The pension formula is given by󰀻
s1 ŷ (e, θ)󰁁󰁁󰁁󰀿s b + s (ŷ (e, θ) − b )1 1
21p(e, θ) =󰁁s1 b1 + s2 (b2 − b1 ) + s3 (ŷ (e, θ) − b2 )󰁁󰁁
󰀽
s1 b1 + s2 (b2 − b1 ) + s3 (b3 − b2 )for ŷ (e, θ) ∈ [0, b1 )for ŷ (e, θ) ∈ [b1 , b2 )for ŷ (e, θ) ∈ [b2 , b3 )for ŷ (e, θ) ∈ [b3 , ∞)where s1 = 0.9, s2 = 0.32, s3 = 0.15, b1 = 0.22ȳ , b2 = 1.33ȳ , b3 = 1.99ȳ ,ȳ = $28, 793 annually.
Back

39 / 33

Financial Market
There is no insurance market and individuals can self-insure using only risk-free
assets.
Borrowing wedge:◮

Overseeing cost ι for workers: r − = r + ι◮Overseeing cost ι + ιs for enrollees: r s = r − + ιs

Borrowing limit:◮

Ae for workers with education e◮

Acj for enrollees at age j

Back

40 / 33

Share of Skilled Labor from College Dropouts
Two separate data:◮

the fraction of jobs requiring each education level◮the fraction of workers acquiring each education level

Interpreting jobs for college dropouts and more as skilled labor.

Back

41 / 33

Government Budget

Government Budget Constraint
Gc + Ge +

J 󰁝
󰁛
j=jr

p(e, θ)dµj =
Sj

󰁛󰁝

j=1,2

Sjc

T (cjc (s cj ), ajc (s cj ), yjc (s cj ))dµcj+

󰁛󰁝j

T (cj (s j ), aj (s sj ), yj (s sj ))dµsj
Sj

where
Gc = gF (K , H)󰁛󰁝Ge =sj (q, θ)dµcj
j=1,2

Sjc

Back

42 / 33

Market clearing
Aggregate labor

H S + κE = H CG
H U = H HS + H CD

where
H CG =

jr −1 󰁝󰁛j=3

H CD =

jr −1 󰁝󰁛j=2

H HS =

jr −1 󰁝󰁛
j=1

Capital

SjCD

SjHS

K =

CG
󰂃CG
j (θ, η)hj (s j )dµj

CD
󰂃CD
+
j (θ, η)hj (s j )dµj

HS
󰂃HS
+
j (θ, η)hj (s j )dµj

jr −1 󰁝󰁛
j=1

Education

SjCG

aj′ (s j )dµj +
Sj

󰁝

󰁝

󰁛󰁝

j=1,2

E =

󰁛󰁝

j=1,2

Sjc

S2c

S1c

Sjc

c c
c
󰂃CD
2 (θ, η)h2 (s 2 )dµ2

c c
c
󰂃HS
1 (θ, η)h1 (s 1 )dµ1

aj′c (s cj )dµcj

dµcj

Back
43 / 33

Equilibrium
Definition
A stationary equilibrium is a list of value functions of workers and college enrollees{Vj (s j ), Vjc (s cj )}, decision rules of enrollment d0 (s 0 ) and graduation d1 (s c1 ), decision rulesof consumption, asset holdings, labor, output, parental transfers of workers{cj (s j ), aj′ (s j ), hj (s j ), yj (s j ), b(s j )}, decision rules of college enrollees{cjc (s cj ), aj′c (s cj ), hjc (s cj ), yjc (s cj )}, aggregate enrollees, capital, and labor inputs{E , K , H S , H U }, prices {r , w S , w U , pe }, policies τℓ , measures µ = {µcj (s cj ), µj (s j ), µej (s ej )}such that
1

2

Taking prices and policies as given, value functions {Vjc (s cj ), Vj (s j )} solve thehousehold Bellman equation*s and d0 (s 0 ), d1 (s c1 ),{cj (s j ), aj′ (s j ), hj (s j ), yj (s j ), b(s j )}, {cjc (s cj ), aj′c (s cj ), hjc (s cj ), yjc (s cj )} are associateddecision rules.
Taking prices and policies as given, K , H HS , H CG solve the optimization problem of
the good sector and E solves the optimization problem of the education sector.

3

The government budget is balanced.

4

Human capital, asset, and education markets clear.

5

Measures µ are reproduced for each period.
Market Clearing
44 / 33

Labor Productivity Process Estimation

PSID: SRC sample, only people with 8 or more individual-year observations
keep only positive hours of labor aged 25-63
eliminate extreme changes in earnings
quadratic ages are separately estimated by education group with year dummies
Back

Age
Age 2

HS

CD

CG

.0530181( .0030501)-.0005314(.0000356).0684129(.0040353)-.0006872(.0000474).0955783(.0036997)-.0009521(.0000429)

45 / 33

Labor Productivity

For high school graduates, θ = θh which is approximated by ln AFQT80.
For college dropouts and college graduates, I use high school ability (θc = θh + 󰂃c ).
ln 󰂃e + ln ψje + 󰂃eθ θc + ln η = ln 󰂃e + ln ψje + 󰂃eθ θh + (ln η + 󰂃eθ 󰂃c )because θh is uncorrelated with ln η + 󰂃eθ 󰂃c .

46 / 33

Markov Chain Approximation

Two state Markov chain with education-specific states for {−σe , σe } and transition
matrix󰀗󰀘πe
1 − πeΠ=
1 − πe
πe
where

Back

ρe2 = 2πe − 1
σe = 󰁳σηe1 − ρe247 / 33

Parameters Determined outside the Model
Parameters

Interpretation

Value

γ
h̄
ζ
α
δ
ρ

Coef of relative risk aversion
Study time
Adult equivalence scale
Capital share
Depreciation (annual)
Elasticity of substitution in production 1.41

4
0.25
0.3
33.3%
7.55%
0.2908

ιs
Ac1
Ac2
AHS
ACD
ACG
τc
τk
g

Staﬀord interest premium (annual)
Borrowing constraint for 1st half (Staﬀord loan)
Borrowing constraint for 2nd half (Staﬀord loan)
Borrowing constraint, HS (SCF)
Borrowing constraint, CD (SCF)
Borrowing constraint, CG (SCF)
Consumption tax rate
Capital income tax rate
Gov cons to GDP ratio

2.3%$6,125$23,000$17,000$20,000$34,000
7%
27%
17.1%

Back

48 / 33

Minimum Distance Estimator

The residual process is assumed to be
yia = αi + zia + uia
wherezia = ρzia−1 + 󰂃ηia , 󰂃ηia ∼ N(0, ση2 )Then
2cov (yia , yia−d ) = σα+ ρd1 − ρ2a 2ση +1 − ρ22d=0 σuBack

49 / 33

Responding to the consumption loss at the first period% of subsidy loss
Subsidies
Labor income(Price of an hour of working)(Leisure)
Transfer from parents
Reducing savings
Less tuition

-100%
+24%
+13%
(-0.061)+0.03%+65%+4%

Consumption-7%

Consumption at the first period does not decrease much because:◮

The wage of college enrollees increases due to a smaller skill premium.

50 / 33

Responding to the consumption loss at the first period% of subsidy loss
Subsidies
Labor income(Price of an hour of working)(Leisure)
Transfer from parents
Reducing savings
Less tuition

-100%
+24%
+13%
(-0.061)+0.03%+65%+4%

Consumption-7%

Consumption at the first period does not decrease much because:◮

The wage of college enrollees increases due to a smaller skill premium.

◮

They work for longer hours.

50 / 33

Responding to the consumption loss at the first period% of subsidy loss
Subsidies
Labor income(Price of an hour of working)(Leisure)
Transfer from parents
Reducing savings
Less tuition

-100%
+24%
+13%
(-0.061)+0.03%+65%+4%

Consumption-7%

Consumption at the first period does not decrease much because:◮

The wage of college enrollees increases due to a smaller skill premium.

◮

They work for longer hours.

◮

Parents increase transfer.

Back
50 / 33

Correcting bias

If we can correct bias, do we still need year-dependent subsidies?

51 / 33

Correcting bias

If we can correct bias, do we still need year-dependent subsidies?
Find the new optimal year-dependent subsidies given that government eliminates
bias costlessly.

51 / 33

Correcting bias

If we can correct bias, do we still need year-dependent subsidies?
Find the new optimal year-dependent subsidies given that government eliminates
bias costlessly.

Current state

Optimals1 (1)
s1 (2)
s1 (3)$20,344$17,124$16,339$21,750$18,308$17,469s2 (1)s2 (2)
s2 (3)$20,344$17,124$16,339$17,808$14,990$14,30251 / 33

Correcting bias

If we can correct bias, do we still need year-dependent subsidies?
Find the new optimal year-dependent subsidies given that government eliminates
bias costlessly.

Current state

Optimals1 (1)
s1 (2)
s1 (3)$20,344$17,124$16,339$21,750$18,308$17,469s2 (1)s2 (2)
s2 (3)$20,344$17,124$16,339$17,808$14,990$14,302

Front-loaded subsidies are optimal when correcting bias.

51 / 33

Correcting Bias

Correct bias
Correct bias (Optimal)

Total

Level

Uncertainty

Inequality+1.69%+2.05%−2.77%−2.31%+3.57%+3.51%−1.34%−1.37%

52 / 33

Correcting Bias

Correct bias
Correct bias (Optimal)

Total

Level

Uncertainty

Inequality+1.69%+2.05%−2.77%−2.31%+3.57%+3.51%−1.34%−1.37%

Current state

Correcting bias

Optimal

74.2%
32.9%
90.9%

45.5%
26.2%
124%
-9.28%

45.8%
26.0%
125%
-9.25%

share of college enrollees
share of college graduatesskill premium
welfare gain
Back52 / 33

Correcting Bias

Correct bias
Correct bias (Optimal)

Total

Level

Uncertainty

Inequality+1.69%+2.05%−2.77%−2.31%+3.57%+3.51%−1.34%−1.37%

Current state

Correcting bias

Optimal

74.2%
32.9%
90.9%

45.5%
26.2%
124%
-9.28%

45.8%
26.0%
125%
-9.25%

share of college enrollees
share of college graduatesskill premium
welfare gain
Back

Correcting bias reduces welfare significantly.

52 / 33

Correcting Bias

Correct bias
Correct bias (Optimal)

Total

Level

Uncertainty

Inequality+1.69%+2.05%−2.77%−2.31%+3.57%+3.51%−1.34%−1.37%

Current state

Correcting bias

Optimal

74.2%
32.9%
90.9%

45.5%
26.2%
124%
-9.28%

45.8%
26.0%
125%
-9.25%

share of college enrollees
share of college graduatesskill premium
welfare gain
Back

Correcting bias reduces welfare significantly.
Enrollment is excessively low due to no insurance on college ability.

52 / 33

No Optimism

In this paper, optimism is a key factor for college dropouts.
A diﬀerent approach to explain college dropouts: High option value due to highuncertainty of college ability.
I assume that the standard deviations of college ability can vary across high school
ability.
σc (θh ) = σc exp(σcθ θh )53 / 33

No Optimism: The Remaining Parameters
Parameter

Description

Value

λ
λθ
λφ1
λφ2
aS
󰂃CD
σc
σcθ
κ
µ
β
v
d
ι
m
σh

college utility intercept
college utility slope
first period college taste
second half college taste
productivity of skilled laborproductivity of CDs.d. of college ability intercept
s.d. of college ability slope
education costconsumption share of preferencetime discount rate
altruismlump-sum transfer ratio
borrowing wedge (r − = r + ι)intergenerational ability transmission intercept
intergenerational ability transmission s.d.

-16.6
287
68.8
40.0
0.435
0.985
0.721
0.158
0.422
0.422
0.931
0.06300.131
18.7%-0.0384
0.076454 / 33

No Optimism: Matched Moments

MomentModel

TargetEnrollment rate of ability quartile
Graduation rate of ability quartile
Enrollment rate of family income quartile
Graduation rate of family income quartile
Skill premium for CG
Skill premium for CD
Education cost/mean income at 48
Hours of work
K /YTransfer/mean income at 48
Log pre-tax/post-tax income
Borrowers
Mean of AFQT
Standard deviation of AFQT

(figure)(figure)(figure)(figure)
90.7%
20.1%
0.308
33.3%
1.241
67.2%
60.5%
6.07%
0.08800.204(figure)(figure)(figure)(figure)
90.2%
19.9%
0.33
33.3%
1.325
66%
61%
6.3%
0
0.213

55 / 33

No Optimism: Model Fit

Figure: Enrollment rates

Figure: Graduation rates

56 / 33

Mo Optimism: Optimal Policy

Current state

Optimals1 (1)
s1 (2)
s1 (3)$13,600$11,448$10,923$14,153$11,913$11,367s2 (1)s2 (2)
s2 (3)$13,600$11,448$10,923$12,478$10,503$10,021Back

57 / 33