does the diversity of human capital increase gdp?education/epc/img/symposiumseminar/...question 1:...
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Does the Diversity of Human Capital Increase GDP?:A Comparison of Education Systems
Katsuya Takii (Osaka U.)and
Ryuichi Tanaka (Titech)
Presentation at GRIPSFebruary 17, 2009
Question 1: What is the relationship between the variance of human capital and GDP?
Japan and Germany
Relatively homogenous labor force (low variance of human capital)
Advantage in production of automobile, consumer electronics, chemical products
Italy and U.S.
high variance of human capital, but many “talented” workers
Advantage in Software industries, financial services, movies
? GDP (h.k.) variance)h.k., averagegiven (For ↑↓⇒↑
Question 2: Which type of education policy is desirable?
Efficiency vs. Equity
• Efficiency: private education, ability tracking => variance (h.k.)
• Equity: public education, no tracking => variance (h.k.)
This paper
Dynamic model of human capital formation (OLG)
(1) Relationship between the variance of h.k. and GDP
(2) Comparison of education systems
The relationship between the variance of h.k. and GDP depends on production technology
Key parameters in production
Complementarity in final good production:
If the production of final good is complement in each individual’s production, the variance of h.k. decreases GDP
ex. Leontief production function
Increasing-returns-to-scale at individual level:
If the production of each individual’s good is IRS, the variance of h.k increases GDP (for given average h.k.)
ex. Superstar (Rosen 1981), Span of control (Lucas 1978)
Main result 1: variance of h.k. and GDP
Variance of h.k. and GDP
If the final good production is substitute in h.k. enough and the IRS at individual level is large enough, the variance of h.k. increases GDP
The determinants of the variance of h.k. and educational policy
Determinants of h.k. formation:education investment, individual ability and the quality of peers
Education Finance
• Private education (tuition-financed) => educational investment depends on household income
• Public education (tax-financed) => reduces variance of h.k.
Education program (ability tracking)
• Mixing: decreases the variance of h.k. (equalizing the quality of classmates)• Tracking: increases the variance
Main result 2: comparison of education systems
Program tracking mixing
Finance private A Dpublic C B
1. Under the same education program, public education generates higher GDP than private education (A<C, D<B)
2. Under the same education finance, the effects of tracking on GDP depend on the production parameters (A vs. D, B vs. C)
3. If tracking is available only in private school, private education can generate higher GDP than public education (A vs. B)
Related literatureComparison of Education Finance System
Public education generates higher steady-state GDP than private education if the production function of h.k. is concave and households are credit-constrained (ex. Loury 1981,, Fernandez and Rogerson 1998)
Private generates higher growth rate (Glomm and Ravikumar 1991)
Local vs. Global Complementarities
A stratified economy grows faster than an integrated economy if the local complementarity in education is stronger than the global complementarity in the final good production (ex. Benabou 1996)
Ability Tracking
Ability tracking affects H.K. formation through peer group effect (e.g., Epple and Romano 1998, Epple, Newlon and Romano 2002, Brunello, Giannini and Ariga 2007)
Model:final good production
[ ] )1,0(,)( :function Production/1
∈= ∫ ρρρ dixY i
tt
(GDP)output total:tY tdateat good teintermediath -i :itx
on substituti of elasticity :1)1/(1 >− ρ
[ ] dixpdix it
it
it ∫∫ −
ρρ /1)(max
:producer good final theof problemmax Profit
good teintermediath -i of price :itp
11 )( F.O.C. −−= ρρ itt
it xYp
Intermediate good production
The larger is the h.k., the larger is the use of physical capital (complementarity between h.k. and the size of the project she can manage)
)1,0(,)( :function Production ∈= λλit
it
it khx
i of capitalhuman :ith
good teintermediath -i of production for the used capital physical :itk
1978) (Lucas control ofspan theof degree :λ
itt
it
it krxp −max :problemmax profit si'
)1/(11 )( F.O.C.
λρρρλρ
−
−
= i
ttt
it hY
rk
rateinterest :tr
λρρ
λρρ
λρλρ
λρλρπ −−−
−
−= 11
11
)()1( :profit si' itt
t
it hYr
λ largefor strong is tendency This
GDP and total profit
ρλρ
λρρ
λλλ
λρ−
−−−
=
= ∫
1
1111
)(, GDP dihHHr
Y ittt
tt
GDP increases H.K. of variance the),1
1(11 If
λρ
λρρ
+>⇔>
−
λρρ
λρρλ
λρ
λρλρπ −−−−
−==Π ∫ 11
11)1( Profit Total tt
t
itt HY
rdi
tt Y)1( GDP andprofit talbetween to ipRelationsh λρ−=Π
Human capital formation
11 1
Pub Pri,1 1 1 1
Production function of h.k.
( ) ( ) ( ) , {0,1}, (0,1), 0
( ) , ( ) , : . . . with E( ) 1
i i it t t t
i i i i it t t t t t t t
h U R
U g R e i i d
αφ θ θ
υ υ
ξ θ α φ
ξ ξ ξ ξ
−+ +
+ + + +
= ∈ ∈ > = = =
it
itt 1
,Pri1
Pub1 ,1 :schools privatein only occurs ackingAbility tr 2.
0)(input effective for the negatively-non matters grouppeer ofquality The 1.
sAssumption
+++ ==
≥
ξξξ
υ
[ ]αθθαθυφξ )()()( :form Reduced 111
itt
it
it egh −+
++ =
Preferences3-period-lived OLG:
Continuum of households (normalized to 1) , Each household consists of child, young, and old
– Child receives education – Young earns profit by producing i-th intermediate good, and allocates the
after-tax income to consumption, child’s private education and saving– Old consumes her saving
, ,1 1
11 1
. . (1 ) , (budget constraint)( ) (( ) ( ) ) , (0,1), {0,1} (h.k. production function)
y i i i i o i it t t t t t t t
i i it t t t
s t c e s c r sh e gφ αθυ θ θ α
τ πξ α θ
+ ++ −
+ +
+ + = − == ∈ ∈
iot
it
iyt chc ,
11, lnlnlnmax :problemmax utility si' Young ++ ++ γβ
itt
it
itt
it se πτ
αβθγγπτ
αβθγαβθ )1(
1,)1(
1 F.O.C. −
++=−
++=
Physical capital market
Supply of physical capital at date t+1
Demand for physical capital at date t+1
Market clearing
tti
t dis Π++−
=∫ αβθγτγ
1)1(
∫ ++
+ = 11
1 tt
it Y
rdik λρ
111 )1(1 −
++ Π−++
= tt
tt Yrτγαβθγλρ
Tax choice (majority vote)
Voting
Young votes sincerely for the tax rate that maximizes her own utility for given future interest rate
Equilibrium policy
A policy supported by at lease 50 percent of voters in any pair-wise comparison
Equilibrium Tax rateγαβθαβτ++−
=1
)1(*t
Dynamics of income distribution
Dynamics of household income:
Income distribution),(~ln),2/(~ln),,(~ln 2
11122
12
++++ ∆⇒−∆ ttit
ittt
it mNNmN πσσξπ
BY ttit
it
it
λρθαλρ
λρλρ
λραθυφρ
λραθρ
πξπ −−+
−+−
+−+
++ Π=− 1
)]1([1
)1(1
11
)(
111
)()(
22
22
2
222
1
2
2
2
2
2
2
1
)1()(
)1()(
~2
1)1(
)()1(1)(
2)1(
)1()()1(1)(
tt
ttt
B
mm
∆−
+−+
=∆
+
−
−++−
++
∆
−++
−+−
++=
+
+
λραθρσ
λραθυφρ
σλραθυφρ
ρλραθυφ
θαλλρ
αθρρ
λρλα
Transition dynamics
GDP
GDP at date t+1
BYY ttt +
−
−+
++∆
−
−++=+ 2
11
)()(2
11
ln)(ln22
1σ
λρραθυφαθυφ
λραθραθλα
)1ln(2/ln 2 λρ−−∆+= ttt mY
.1 Assume :1n Propositio <+ λα
GDP speriod'-next the increasesability of variance the,)/(1 If (2) λαθυφρ ++>
GDP speriod'-next thedecreases income household of varianceThe (1)
Steady-state GDP
Implication: When the elasticity of substitution is larger than the threshold, the association between the steady-state variance of H.K. and steady-state GDP is positive
+
∆+−−
= ∞∞ BDY ~ln
2)(),,,,;(
11ln statesteady in GDP
2
2 αθυφρφυθλαρ
λα
.)1(1 and 1 Suppose :2n Propositio αθυφλλα −+−><+
.1))1(/(1 where)1,(iff0),,,,;( <+−+=∈> λυαθφρρρφυθλαρD
1)2( ))()((),,,,;( 2
−+++++−−−=
λαθυφρλαθυφλαθυφαθαθρφυθλαρD
Public ( ) vs. Private ( ) during transition
IntuitionThe negative effect of the variance of household income on GDP is absent
in public education system
If the variance of ability increases GDP, private education with ability tracking may generate higher GDP than public education
0=θ 1=θ
21
1)2(
21
1lnln
2
?
2Pri
1Pub
1σ
λρραυφαυ
λραρα
−
−+
−∆
−
−−=−
+
++t
tt YY
22Pri1
Pub1
Pri1
Pub1
)(1]1)2([ iff ),2/(1 When (2)
),2/(1 When (1)
3n Propositio
σλαρλαυφρυλαυφρ
λαυφρ
+−−++
>∆>++>
>++≤
++
++
ttt
tt
YY
YY
Public vs. Private in steady state
Implication: When the elasticity of peer group effects is large enough, the private system generates higher steady-state GDP than the public system
−
−+
−∆
−
−−
−−=− ∞
+
∞∞ 21
1)2(
21
111lnln
2
?
Pri,2PriPub σ
λρραυφαυ
λραρα
λα
YY
),,,( iff such that 0),,,( unique a exists there),,,,(each for (2), of case in the Moreover,
)(1]1)2([ iff ),2/(1 When (2)
),2/(1 When (1)
4n Propositio
*PriPub
*
2Pri,2PriPub
PriPub
φρλαυυ
φρλαυφρλα
σλαρλαυφρυλαυφρ
λαυφρ
<>
>
+−−++
>∆>++>
>++≤
∞∞
∞∞∞
∞∞
YY
YY
YY
Ability Tracking in Public School
Public vs. Private
If ability tracking is available both in public and private schools, the public system always generates higher GDP than the private system
Intuition:the negative effect of the variance of household income is absent in the public system
Tracking vs. Mixing in Public School
Whether tracking raises GDP depends on the production parameters
)2/(1 if GDPhigher generates Tracking λαυφρ ++>
ConclusionsVariance of h.k. and GDP
If the final good production is substitute in h.k. enough and the IRS at individual level is large enough, the variance of h.k. increasesGDP
Comparison of education systems
1. Under the same education program, public education generates higher GDP than private education
2. Under the same education finance, the effects of tracking on GDP depends on the production parameters
3. If tracking is available only in private education, private education can generates higher GDP than public education
DiscussionsDependence of child’s ability on parent’s ability
All results are qualitatively unchanged
Steady-state welfare comparisonNot only GDP but also income inequality matter for the comparisonIf tracking is given, public is better than private in terms of welfare
On the relevance of trade off between income inequality and diversity The elasticity of substitution of intermediate goods: ρ=5/6
(20 per cent markup as in Christiano, Eichenbaum and Evans, 2005)Labor share: 1−λρ = 2/3 => λ = 0.4 => ρ>1/(1+ λ)α <1−λ =0.6φ =1 (as in Benabou 1996)If αθν <φ=1, then Yt+1 is strictly increasing in σ2 .