Introducing
Advanced
Macroeconomics:
Chapter 6 – first
lecture
Growth and business
cycles
EDUCATION AND
GROWTH: THE
SOLOW MODEL WITH
HUMAN CAPITAL
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Introduction
1. The steady state prediction of the Solow model,

*
lnyt  lnAt 
lns  ln  n  g    ng   ,
1
suggests the following regression:
i
lny00
  0   1 lnsi  ln  ni  0.075 ,
where  1   / 1    should be around ½. OLS estimation
across 86 countries gives:
i
lny00
 8.812  1.47 lns i  ln  ni  0.075  , adj. R 2  0.55,
 se  0.14
This equation has many nice properties, but the
estimated  1 is too big compared to the model-predicted
value (taking uncertainty into account, also).
Seemingly, s and n have a much stronger impact
on y in real life than they should have according
to the steady state prediction of the Solow
model.
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Companies, 2005
2. The convergence prediction of the Solow model,
lnyT  lny0
g
T
1  1   
T

T
1  1   
T
T
1  1   
T
lnA0 
T
lny0

lns  ln  n  g    ng   ,   1    n  g    ,
1
suggests the following regression:
gTi ,0   0  1lny0i   2 lns i  ln  ni  0.075   ,
1  1   
T

1 
and  2 
1 .
T
1
OLS estimation across 90 countries gives:
i
lny00
 8.812  1.47 lns i  ln  ni 0.075  , adj. R 2  0.55.
 se  0.14
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We have the following two estimates of the rate of
convergence:
1. From theory:   1    n  g       4  5% at least!
T
1/ T
2. From empirics: 1  1  1    / T    1  1  T 1  .
With an estimated 1 of 0.006 and T  40 this gives   0.7%.


The empirical results do not contradict
convergence, but in reality convergence is much
slower than predicted by the Solow model.
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Human capital
• Is there a way to modify the Solow model that can work to
rectify both empirical problems?
• Yes: Mankiw, Romer, Weil: ”A Contribution to the Empirics
of Economic Growth”, QJE, 1992 (MRW).
• Remember that the rate of convergence in the Solow model
is:   1    n  g    . A larger  gives a lower  : Capital
accumulation takes time. If only we could increase  …
But   1 / 3, because labour’s share is around 2 / 3 .
• If there were another type of capital with its own output
elasticity,  , accumulating the same way as physical
capital, but with the income share accruing to the
workers, then:
• Convergence would be slower (because      ) and
”labour’s share” would still be 1   (because only the share,
©The McGraw-Hill Companies, 2005
 , would not go to the workers).
• Human capital: the accumulated stock of what is
invested in training of the labour force: education, lost
production and income etc.
• Assume that human capital is accumulated in the same
way as physical capital: every year a constant share of
GDP is used for investment in human capital. Then,
accumulation of human capital should affect the rate of
convergence in the same way as accumulation ofphysical
capital.
• Human capital cannot be separated from the workers.
One can seperate a man from his computer, but not from
his education. Hence, the return on human capital
accrues to the workers. This keeps the ”labour’s share”
unchanged even though a new kind of capital is being
accumulated.
• It looks as if human capital could help solving the
problem concerning the rate of convergence in the Solow
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model.
• What about the problem concerning the steady state prediction
of the Solow model?
• Consider an increase in the rate of investment in physical
capital (called s until now). This increases capital and income
per worker in steady state through the usual well-known
channels.
• With a constant rate of investment also in human capital, larger
income per worker means more investment in human capital
per worker and in the end more human capital per worker in
steady state. Since human capital is productive income per
worker will increase more than without human capital.
• Seemingly, human capital could also contribute to solving the
problem concerning the impact of the investment rate on GDP
per capita.
• Hence, incorporating human capital might solve both of the
empirical problems of the Solow model.
• Furthermore it is an aim in itself to improve the model by
adding human capital, since, by intuition, the skills of the
labour force must be an important input factor.
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The Solow model with human capital
• More or less the same micro world as in the Solow model.
• The same types of agents: one representative firm and
one representative consumer (and possibly a government
sector).
• But the firm now uses human capital in its
production and
• the consumer also accumulates human capital.
• Output is used for consumption, investment in physical
capital or investment in human capital. There is still just
one production sector.
• The same markets: output and (services of) physical
capital and labour.
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• No separate market for human capital services since human
capital is linked to labour and its services are sold
inseparably together with the labour.
• The services traded in the labour market are no longer manyears of ”raw labour”, but man-years endowed with human
capital. The real wage is a mix of compensation for raw
labour and return to human capital.
• The number of workers, the labour force, is Lt . Every
worker is endowed with the same amount of human capital,
ht . The total level of human capital in the economy is Ht  ht Lt .
• Raw labour input, Ldt , cannot be varied independently of
human capital input. The input of human capital, ht Ldt , is
proportional to Ldt , because each worker brings his ht . Here
is an important distinction from physical capital.
• We can now describe the new elements of the economy.
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The production function with human
capital
Yt  Kt H t  At Lt 


1 
 K t  ht Lt 


 At Lt 
1 
 K t ht At1  L1t  ,
where 0    1, 0    1,     1, At  1  g  A0
t
• Constant returns to Kt ,Ht ,Lt : the replication argument.
• The per capita production function is:
yt  kt ht At1   gty   gtk   gth  1      gtA .
• When calculating the marginal product of labour, one has
to take into account that one additional worker comes
with a given amount of human capital, ht . Hence, ht , not
H t , should be taken as given when we differentiate wrt. Lt .
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  1  1
Lt the returns to
• Remembering that Yt  K t ht At
capital and labour, respectively, are computed as:
 1
 1 
1  1
t
t
rt   Kt ht A
L
 Kt 
 

A
L
 t t

 Ht 


A
L
 t t

wt  1    Kt ht A


1 
t

t
L

 Kt   H t 
 1    
 
 At .
 At Lt   At Lt 
• Observe that rt Kt / Yt   and wt Kt / Yt  1   : the workers
get the return to human capital.
• As usual, empirical observations suggest   1 / 3 , but
also (a little less firm perhaps)   1 / 3 , or a bit larger:
average and minimum wage rates in US manufacturing.
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The consumers’ accumulation of capital
• As before, each of the Lt consumers supplies one unit of
labour inelastically and
• the number of consumers grows at a constant rate, n  1 ,
• and all physical capital, K t , is supplied as long as rt  0 .
• As in the general Solow model, the representative
consumer has to decide Ct given Yt , which in turn
determines St  Yt  Ct .
• But now the consumer also has to decide how to divide
gross savings, St , into gross investment in physical capital,
I tK, and gross investment in human capital, I tH.
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• Given I tK and I tH :
Kt 1  Kt  I tK   Kt
H t 1  H t  I tH   H t
where we have assumed the same rate of depreciation,
0    1 , for physical and human capital.
K
H
• The restriction that I t and I t have to fulfill is:
I tK  I tH  Yt  Ct  St
• We assume that the consumer’s considerations result in
the standard Solow assumptions:
I tK  sK Yt and I tH  sH Yt
0  sK  1, 0  sH  1, sK  sH  1
• Hence St   sK  sH  Yt 
Ct  1  sK  sH  Yt .
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The complete Solow model with human capital
Yt  K t H t  At Lt 
1 
 1
 Kt 
rt   

A
L
 t t

 Ht 


A
L
 t t

 Kt   H t
wt  1    
 
A
L
 t t   At Lt
Kt 1  Kt  I tK   Kt , K 0
H t 1  H t  I tH   H t ,


 At

given
H 0 given
Lt 1  1  n  Lt ,
L0 given
At 1  1  g  At ,
A0 given
• Parameters:  , ,sK ,sH , ,n,g.
• Given K0 ,H 0 ,L0 , A0 the model determines the sequences
 Kt  , Ht  , Lt  , At  ,Yt  , rt  , wt  , St  .
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The law of motion
1. Define:
kt
Kt
kt 

,
At At Lt
ht
Ht
ht 

,
At At Lt
2. From Yt  K t H t  At Lt 


1 
yt
Yt
yt  
.
At At Lt
we get that
yt  kt ht .
3. Restatement of the capital accumulation equations:
Kt 1  Kt  I tK   Kt
H t 1  H t  I tH   H t
4. Dividing on both sides by At 1Lt 1 gives:
1
kt 1 
sK yt  1    kt
1  n 1  g 
1
ht 1 
sH yt  1    ht .
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1  n 1  g 




5. Inserting yt  kt ht gives the transition equations:
1
kt 1 
sK kt ht  1    kt
1  n 1  g 
1
ht 1 
sH kt ht  1    ht .
1  n 1  g 
• The law of motion: two coupled, first-order difference
equations in kt and ht . Given k0 and h0 they determine kt
and ht .
• There is no easy diagrammatic way to show convergence
towards a steady state. Today and in the next lecture we
will show:
 




 
1. There is a well-defined steady state.
2. Numerical simulations suggest convergence towards a steady state
for reasonable parameter values.
3. Convergence holds for a linear approximation around a steady
state.
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6. Subtracting kt and ht , respectively, on both sides of the transition
equations gives the Solow equations:
kt 1  kt 
ht 1  ht 
s k

1  n 1  g 
1

K t
1
1  n 1  g 

ht   n  g    ng  kt

sH kt ht   n  g    ng  ht .
•
These are just another way of stating the law of motion.
•
Each has a ususal Solow equation interpretation:
–
–

Left hand side is the addition to capital per effective worker.
Right hand side contains contributions from investment per effective
worker and replacement requirement from population growth,
technological growth, and depreciation.
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Steady state
• In steady state, kt 1  kt  ht 1  ht  0 :
sK kt ht   n  g    ng  kt  0
sH kt ht   n  g    ng  ht  0.
• Solving for kt  k * and ht  h* gives the steady state values:


s1K sH
k 

n

g



ng


*
1
1 
1
1 
a 1


s
*
K sH
, h 

n

g



ng


.
• Inserting these into yt  kt ht gives:


sK
y*  

n

g



ng



1 


sH


n

g



ng



1 
©The McGraw-Hill Companies, 2005
and then, from yt  yt At , the steady state growth path:

1 

1 




sK
sH
y  At 
.



 n  g    ng 
 n  g    ng 
• Taking logs gives:

*
lnyt  lnAt 
lnsK   n  g    ng  

1 


lnsH   n  g    ng   .

1 
*
t
*
• The elasticity of yt with respect to:
– s K is now  / 1      , where in the Solow model it was  / 1   . It
is now around 1, where before it was around ½.
– sH is  / 1      , that is, around one.
–  n  g    is now      / 1      , where in the Solow model it was
 / 1    , that is, now around -2, where before it was around -½.
These features are exactly as we conjectured and wished.
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Empirics for steady state
• The above equation suggests the following regression:
i
lny00
  0   1 lnsKi  ln  ni  0.075   2 lnsHi  ln  ni  0.075


1 
and  2 
.
1 
1 
• We use cross country data for y00i , s Ki and n i as usual.
i
How about sH?
forgone GDP due to education people of working age under education

s 
all people of working age
GDP if everybody worked
i
H

number of 12-17 year olds in secondary school

total number of 12-17 year olds
total number of 15-19 year olds
.
people of working age
©The McGraw-Hill Companies, 2005
•
OLS estimation across 77 countries:
i
lny00
 9.70  0.59 lnsKi  ln  ni  0.075   0.97 lnsHi  ln  ni  0.075  ,
se 0.11
 se 0.14
adj. R 2  0.79.
A very large R 2 , but more importantly,  1   / 1     
and  2   / 1      mean that:
1. With     1 / 3 , we get that  1   2  1 . This is almost
within the range of the 95% confidence interval.
2. If (e.g.)     0.3 we would have  1   2  0.75. This is
actually within two standard deviations of both estimates!
1
2
• Or:

and  
 0.38.
1  1   2
1  1   2
•
•
A remarkably good estimation! Even though we have
assumed that A00 is the same in all countries (but
remember: no D-countries in the sample).
©The McGraw-Hill Companies, 2005
Structural policies for steady state

1 

1 




sK
sH
y  At 
.



 n  g    ng 
 n  g    ng 


1  
1 
sK
sH
*
ct  At 1  sK  sH  
.



 n  g    ng 
 n  g    ng 
*
t
1. The effects of s K and n  g   on yt* are qualitatively
as in the Solow model, but the quantitative effects
are now much stronger and much more plausible. And
n  g   has a stronger effect than s K!
2. The effect of sH is of the same size as the effect of s K .
Golden rule:
sK   and sH   .
An interesting new question is:
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3. Should the government promote investment in
education (up to golden rule) by subsidizing education?
•
•
•
•
•
Note: this happens to a large extent in many Western
countries.
Also note: the empirics suggest that investment in
human capital is (at least) as important as investment in
physical capital.
Does this make investment subsidies in physical and
human capital equally-well or equally-badly motivated?
A general view of welfare economics: intervene if (and
only if) private decisions cannot be expected to lead to
socially optimal outcomes.
Government subsidies have to be motivated by specific
market imperfections.
©The McGraw-Hill Companies, 2005
• Some imperfections that specifically point to education
subsidies:
–
–
–
–
Imperfect credit markets (adverse selection problem)
Imperfect insurance markets (moral hazard problem)
Externalities.
Imperfect private knowledge.
• Another motive for education subsidies lies in distributive
concerns, e.g. to make up for negative social
inheritance.
©The McGraw-Hill Companies, 2005
Summing up
•
•
•
•
•
In this first lecture on the Solow model with human
capital we have:
Explained intuitively why incorporating human capital into
the Solow model should work to overcome both of its
empirical problems.
Set up explicitely a Solow model with accumulation of
both physical and human capital.
Derived the model’s law of motion.
Found its steady state and realized that indeed it
conforms with our intuitions.
Tested the model’s steady state prediction empirically and
found it to perform remarkably well.
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• Discussed the model’s implications for structural policy:
– Traditional policies for saving and investment in physical capital
and for population growth are qualitatively supported as in the
Solow model, but with stronger and more plausible qualitative
effects.
– Policies for investment in human capital are supported as
strongly as policies for investment in physical capital. Given the
imperfections related to educational decisions, public subsidies
to education seem to be well-motivated.
©The McGraw-Hill Companies, 2005