3.2 Human Capital - LMU

Ifo Institute for Economic Research at the University of Munich

3.2 Human Capital

2


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).

3


The Importance of Human Capital:

Mankiw, Romer, Weil (1992) A Contribution to the Empirics of Economic Growth, QJE 107, 407-437.

4


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

5


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

6


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

− −−

− −−

− −

− − − −

=

+ +

=

+ +

=

+ +

7


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

= ⋅

8


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

=

9


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

= = = ⋅

10


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

= +

+= + ⋅ − + + + +

− − − − − −

= + ⋅ + − + + + − + +

− − − −

11


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

12


Empirics (3)

13


Empirics (4)

14


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 ...

15


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


3.3 Social Infrastructure

17


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.

18


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.

19


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

20


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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( ) ( ) ( )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

22


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


Dynamics of L:

(see Solow model).

with ,

Specification of G(E):

where we have normalized G(0) to 1.

23


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 ++−= .

24



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

.

25



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++−=•

( )

26



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.

.

27



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).

28


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

29


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.

30



• 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


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.

.

32



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.

33



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.

34


• 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

35


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

36


Results (2)

37


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.

38


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.

3.2 Human Capital - LMU

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