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3.2 Human Capital
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Why Human Capital?
• It is intuituive: people with more human capital (due to more education) typically earn higher wages. Hence, on a
private level, human capital seems to generate more
income.
• It may help explain cross-country differences: education is
very different between high and low income countries.
Note: human capital is acquired by an individual throughlearning. Unlike ideas, it is rival (if you use your skills for
one task/employer, you cannot use it for another at thesame time) and excludable (you just have to say no).
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The Importance of Human Capital:
Mankiw, Romer, Weil (1992) A Contribution to the Empirics of Economic Growth, QJE 107, 407-437.
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Starting point
• Take Solow model seriously
• Account for the problem that the model cannot explain the
large quantitative differences between countries
• To do this, add human capital
• Result: this helps to explain the differences much better
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Model (1)
1( ) ( ) ( ) [ ( ) ( )]
( )( ) ( ) ( )
( ) ( )
Y t K t H t A t L t
Y ty t k t h t
A t L t
α β α β
α β
− −=
= =
( )
( )
( ) ( ) ( )
( ) ( ) ( )
k
h
k t s y t n g k t
h t s y t n g h t
δ
δ
= − + +
= − + +
�
�
Standard production function with human capital as input
Evolution of physical and human capital follows the same mechanism
where sk and sh are saving rates
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Model (2)
• Steady state requires dk/dt=dh/dt=0
• Set the respective equations to zero and solve
simultaneously (assume α+β<1)
• This yields
1/(1 )1
*
1/(1 )1
*
1/(1 )
* /(1 ) /(1 )1
k h
k h
k h
s sk
n g
s sh
n g
y s sn g
α ββ β
α βα α
α β
α α β β α β
δ
δ
δ
− −−
− −−
− −
− − − −
=
+ +
=
+ +
=
+ +
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Model (3)
• Similar to Solow model
• Now two produced factors
• per capita income growth on the steady state path fullydetermined by TP...
• ...but levels differ due to investment rates in physical and
human capital
*
*( )( )
( )
Y tA t y
L t
= ⋅
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Model (4)
• ...but levels differ due to investment rates in physical andhuman capital.
• Remember our example with Switzerland and India. Theyhave a tenfold difference in output per worker
• Without resorting to technology differences, the Solow
model (with α=1/3) required the capital intensity be of factor 1000
ˆˆ
ˆˆ
K
S S
I I
y k
y k
α
=
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Model (5)
• Now consider the augmented Solow model and assume
– α=β=1/3
– technology is the same across countries
– human capital per worker and thus hS/hI=3, which implies thatSwiss people attend school 3 times longer than Indians (empirically, this is a large number)
• This implies that a factor of 31/3=1.4 can be explaind by
human capital differences
• The remainder (10/31/3=6.9) still has to be explained by
physical capital. To achieve this, the relation needs to be
kS/kI=333 which is still too much but better than before:
1/3 1/3ˆ ˆˆ
10 333 3ˆ ˆˆ
S S S
I I I
y k h
y k h
α β
= = = ⋅
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Empirics (1)
• estimate one of the log-linear equations (g = rate of TP)
[ ] [ ]
*( )ln ln ( ) ln
( )
( )ln ln (0) ln( ) ln( ) ln( )
( ) 1 1 1
( )ln ln (0) ln( ) ln( ) ln( ) ln( )
( ) 1 1
k h
k h
Y tA t y
L t
Y tA g t n g s s
L t
Y tA g t s n g s n g
L t
α β α βδ
α β α β α β
α βδ δ
α β α β
= +
+= + ⋅ − + + + +
− − − − − −
= + ⋅ + − + + + − + +
− − − −
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Empirics (2)
Assumptions and measurement
• g: rate of TP – invariant across countries
• δ: depreciation rate – invariant across countries
• n: average population growth (age 15-64, 1960-1985)
• A(0): initial endowment with technology and anything else that is leftout – assume ln A(0) = const + e(i), where e(i) is a country-specificshock uncorrelated with the regressors
• Y/L: real GDP / working-age population in 1985
• sk: average investment share (1960-1985)
• sh: average fraction of working-age population in secondary school(1960-1985)
• Y/L: real GDP / working-age population in 1985
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Empirics (3)
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Empirics (4)
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Conclusions (1)
• pure Solow model: α=0.6, R²=0.59
• Solow model with HC: α=0.3, β=0.3, R²=0.78
• Hence, we do not need TP progress to account for most of the cross-country variation. The Solow-HC model does a
good job!
• Can we stop here?
• The literature did not because
– still 22 percent of the cross-country variation is unexplained
– the explanatory power of investment in physical and human capitalmight be exaggerated. Think about it ...
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Conclusions (2)
... and here ist the argument: the explanatory power of investment in physical and human capital might be
exaggerated in the MRW paper because
– physical and human capital are probably correlated with productivity/ TP/all other factors affecting income, unlike assumed in the paper
– reason: you invest more where the reward is higher
– econometrically, the estimated coefficients α and β are be biased if„technology and all the rest“ are left out of the equation
– the correct way to estimate the equation is to include a measure for„technology and all the rest“ or to use instruments
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3.3 Social Infrastructure
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Why Social Infrastructure?
• As we have seen, human capital is probably not enoughto explain cross-country difference
• It is intuituive: Social infrastructure – the institutions and
policies toward production of a society – may give more
or less incentives to do productive work.
• It may help explain cross-country differences: the social
infrastructure is very different between high and low
income countries. For example, the rule of law and the risk
of expropriation are quite different between, say, the US and Russia.
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The Importance of Social Infrastructure:
Robert E. Hall and Charles I. Jones (1999) Why Do SomeCountries Produce So Much More Output per Worker than
Others? Quarterly Journal of Economics 114: 83-116.
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Hall&Jones: Why Do Some Countries ProduceSo Much More Output per Worker than Others?
1. A Simple Model of Human Capital
2. Empirics 1: Can Human Capital Account for the
Observed Cross-Country Differences?
3. Empirics 2: Estimating the Effect of Social Infrastructure
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The Model Set Up (1)
αα −= 1)]t(H)t(A[)t(K)t(Y
Production function:
where Y, K and A are the same as in the Solow model.
,
)t(Ag)t(A A=•
)t(K)t(sY)t(K δ−=•
Dynamics of K and A:
(see Solow model)
,
.
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The Model Set Up (2)
( ) ( ) ( )H t L t G E= ⋅
Here: H – H is the total amount of productive service supplied by
workers. It therefore includes the contribution of both raw labour
and human capital.
where L is the number of workers and G(E) is a function giving
human capital per worker as function of education E per worker.
,
This implies that in this model the level of human capital
per worker depends on education, specifically on thenumber of years of schooling E per worker!
)E(G)t(L
)t(H= .Transformation yields
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Assumption: E is a shift variable. Easier than the produced factor view
of the preceding paper.
Microeconomic evidence: each additional year of education increases
an individual‘s wage by the same percentage amount. If wages reflect
the labour services that individuals supply, this implies that G‘(E) is
increasing. Specifically, it is assumed that G(E) takes the form
)t(nL)t(L =•
,e)E(GEΦ= 0>Φ
The Model Set Up (3)
Dynamics of L:
(see Solow model).
with ,
Specification of G(E):
where we have normalized G(0) to 1.
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Analyzing the Model (1)
• The dynamics of the model are exactly like those of the Solow
model.
[ ]L)E(AGK)t(k =Define
)t(k)gn())t(k(sf)t(k δ++−=•
)t(k)gn()t(sk δα ++−= .
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Analyzing the Model (2)
We also know that once k reaches k*, the economy is on a
balanced growth path with output per worker growing at
rate gA.
As in the Solow model, k converges to the point there ,0=•
k
From the equation above, this value k* is
[ ] )(*)gn(sk
αδ
−++=
11
.
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Analyzing the Model (3)
Effects of a change in the saving rate s: see Solow model.
To see this, note that since the equation of motion for k is
identical to that in the Solow model, the effects of changes
in s on the path of k are identical to those in the Solow
model.
Since output per unit of effective labour services, y, is
determined by k, it follows that the impact on the path of
y is identical.
)t(k)gn())t(k(sf)t(k δ++−=•
( )
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Analyzing the Model (4)
Finally, output per worker equals output per unit of
effective labor services, y, times effective labor services
per worker:
y)E(GAY ⋅⋅=L
The path of AG(E) is not affected by the change in the
saving rate: A grows at the exogenous rate gA, and G(E)
is constant.
Thus the impact of the change on the path of output per
worker is determined entirely by its impact on the path of y.
.
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Analyzing the Model (5)
Effects of a change in the number of years of schooling
per worker E:
Since E does not enter the equation for , the
balanced growth path value of k is unchanged, and so the
balanced growth path value of y is unchanged.
0=•
k
And since A/L equals AG(E)y, it follows that the rise in E
increases output per worker on the balanced growth path
by the same proportion than it increases G(E).
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Hall&Jones: Why Do Some Countries ProduceSo Much More Output per Worker than Others?
1. A Simple Model of Human Capital
2. Empirics 1: Can Human Capital Account for theObserved Cross-Country Differences?
3. Empirics 2: Estimating the Effect of Social Infrastructure
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Accounting for Cross-Country Income Differences (1)
• By using the model explained before, Hall and Jones
(1999) do growth accounting, but they do it across
countries rather than over time (what is often done).
• Measurement of the differences in physical and human
capital, and subsequently use of a framework like the model
of the previous section to estimate the quantitative
importance of those differences to income differences.
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Accounting for Cross-Country Income Differences (2)
• Procedure:
αα −= 1]HA[KY iiii
Use of the model of the preceding section, where output in a given
country is a Cobb-Douglas combination of physical capital and
effective labor services:
where i indexes the different countries.
,
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Dividing both sides of the production function by the number of
workers Li and taking logs yields
Accounting for Cross-Country Income Differences (3)
i
i
i
i
i
i
i Aln)(L
Hln)(
L
Kln
L
Yln ααα −+−+= 11
The basic idea is, as in growth accounting over time, to measure
directly all the ingredients of this equation other than Ai.
Thus the equation above can be used to decompose differences in
output per worker into the contributions of physical capital per
worker, human capital per worker, and a residual which is interpreted
as technology.
.
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Accounting for Cross-Country Income Differences (4)
However, this decomposition might not be the most interesting one.
The preceding decomposition therefore attributes – after an increase
in A – a fraction α of the long run increase in output per worker to
physical capital per worker.
It would be more useful to have a decomposition that attributes
all the increase to the residual, since the rise in A was the
underlying source of the increase in output per worker.
Suppose that the level of A rises with no change in the saving rate or
in education per worker. The resulting higher output increases the
amount of physical capital.
When the output reaches its new balanced growth path, physical
capital and output are both higher by the same proportion as the
increase in A.
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Accounting for Cross-Country Income Differences (5)
i
i
i
i
i
i
i
i
i Aln)(L
Hln)(
L
Yln
L
Kln
L
Yln)( ααααα −+−+
−=− 111
i
i
i
i
i Aln)(L
Hln)(
Y
Kln ααα −+−+= 11
Subtracting αln(Yi/Li) from both sides of our last equation yields
Dividing both sides by 1-α gives us
i
i
i
i
i
i
i AlnL
Hln
Y
Kln
L
Yln ++
−=
α
α
1
This equation expresses output per worker in terms of physical
capital intensity (that is, the capital-output ratio K/Y), human capital
per worker, and a residual.
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• Data on income shares suggest that α, the physical capital‘s share
in the production function, is around 1/3.
Data
• Hall and Jones construct estimates of the physical capital stock K
by using investment data from the Penn World tables and
assumptions about the initial capital stocks and depreciations.
• To calculate the stock of labour services H, Hall and Jones
consider only years of schooling. Specifically, they assume that Hi
takes the form , where Ei is the average number of years of
education in country i and is an increasing function. On the
basis of empirical evidence, they assume that is a piecewise
linear function with a slope of 0.134 for E below 4 years, 0.101 for E
between 4 and 8 years, and 0.068 for E above 8 years. These are
private return-to-schooling estimates (Psacharopoulos, 1994).
i
)E(Le iΦ
)E( iΦ
)E( iΦ
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Results (1)
• By using these data and assumptions, Hall and Jones use the
expression
to estimate the contributions of physical capital intensity, schooling
and the residual to output per worker in each country.
• They summarize their results by comparing the five richest and the
five poorest countries in their sample: average output per worker in
the rich countries exceeds the average in the poor countries by a
factor of 31.7.
i
i
i
i
i
i
i AlnL
Hln
Y
Kln
L
Yln ++
−=
α
α
1
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Results (2)
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Results (3)
• That is, only about a small fraction of the gap between the richest
and the poorest countries is due to differences in schooling. It leaves
a factor of 31.7/2.2=14.4 unexplained (this is roughly 1/6 of the log
difference).
• Given the relatively small variation in inputs across countries and
the small elasticities implied by neoclassical assumptions, it is hard
to escape the conclusion that differences in productivity – the
residual – play a key role in generating the wide variation in output
per worker across countries.
• The average differences between the two groups are
• in (K/Y)α/(1-α): factor 1.8
• in H/L: factor 2.2
• in A: factor 8.3.
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Results (4)
Further important finding:
The contributions of physical capital, schooling, and the residual are
not independent. Hall and Jones find
• a substantial correlation across countries between their estimates
of ln(Hi/Li) and ln(Ai): ρ = 0.52
• a modest correlation between their estimates of [α/(1-α)]ln(Ki/Yi)
and ln(Ai): ρ = 0.25).
This is an important criticism of the model of Mankiw, Romer and
Weil (1992), as they assume independency.
They also find a substantial correlation between physical and human
capital: ρ = 0.60. This suggests that there may be underlying forces
that affect all the determinants of output per worker.
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Problems
Two potential problems arise:
First, like in all growth accounting calculations, Hall and Jones
measure human capital’s contributions to output by its market
earnings. Remember: they use private return-to-schooling estimates
which may differ from social returns if there are externalities (and
externalities might be large, see Carstensen/Gundlach/Hartmann,
2009, GER). As result, the cross-country accounting procedure
might misestimate the contribution to cross-country income
differences.
Second, the calculations ignore all differences in human capital
other than differences in years of education. But there are many
other sources of variation in human capital. School quality, on the
job training, informal human capital acquisition, child caring, and
even prenatal care vary significantly across countries. The resulting
differences in human capital may be large.