Dietmar Meyer1:
Human Capital and EU-Enlargement
Abstract
The enlargement of the European Union is an almost everywhere accepted necessity, but at
the same time of course also a compromise. Economies or regions of different economic,
social, institutional, etc. development become united in Europe with a territory from the
Atlantic to the Eastern borders of Poland, Slovakia and Hungary, from the Baltic Sea to the
Mediterranean Sea.
This integration process going along with the worldwide globalisation will imply a new
distribution, or a redistribution of the factors of production. First of all the human capital will
be touched by this development.2 One of the most important results found by social sciences
in the 20th century is the realisation of the immense role played by human factors in the
process of economic development. The extremely high efficiency of human capital and the
high mobility could diminish the regional differences in the economic development and
therefore in the social life. But even this is one reason for the mentioned re-allocation of the
human capital.
In the frame of a very simple static model (See e. g. Bishi – Kopel 2002) the flow of human
capital between different regions – called the European Union and the New Member States –
will be analysed. The introduction of search costs extends the field of policy-analysis.
1
Dietmar Meyer, Professor of Economics at Budapest University of Technology and Economic Sciences, Head
of Department of Economics. E-mail: [email protected]
Paper presented on the Conference “Economic Growth and Distribution: on the Nature and Causes of the
Wealth of Nations”, 16-18 June, 2004, Lucca, Italy. An earlier version of this paper was published in the
Andrássy Working Papers of the Andrássy University Budapest (Meyer 2003).
2
It should be remarked that human capital is not equivalent with the labour factor. Labour is – one – medium of
technical progress, and therefore closely connected to human capital. But the geographical flow of human capital
does not require the flow of labour, because the transfer of know-how can be realised by several technical
means, as e. g. Internet, etc.
1
A Simple Model
As a consequence of the enlargement of the European Union the old member states as well as
the new member states will have the opportunity to offer their human capital without any
serious difficulties in the other sphere of the unified Europe. Let us denote by H ij the human
capital produced in region i and used in region j, with i, j  E , M - E for the former European
Union and M for the new member states. Both markets, that of the old Union and that of the
enlarged Union, differ in several points, e. g. in the efficiency of production, in the demand
conditions, etc. Assume first of all different demand for human capital, i. e. (in the form of
inverse demand functions) for the European Union
pE  aE  bE H ME  H EE  ,
(1.1)
and in the case if the new member states
pM  aM  bM H MM  H EM  .
(1.2)
Therefore we have for the revenues: in the Union
RE  pM H EM  pE H EE ,
and in the new member states
RM  pM H MM  pE H ME .
Taking into account that H E  H EM  H EE , and H M  H MM  H ME , the profit functions of
the regions can be derived as
 E  aM  bM H MM  H EM H EM  aE  bE H ME  H EE H EE  CE H  ,
for the Union, and as
2
 M  aM  bM H MM  H EM H MM  aE  bE H ME  H EE H ME  CM H ,
for the new member states, where Ci H , i  E , M , denote the regions’ cost functions to
specified later.
The production of a certain quantity of human capital requires real capital and labor. The
technology applied in the production of human capital will be expressed by different
production functions: H i  Fi K i , Li , i  E , M . Let be both functions of Cobb-Douglastype, but different efficiency should be expressed by different parameters, i. e.
H i  Ai K i i Li i ,  i   i  1, i  E , M . Using these conditions the variable costs of the
production of human capital can be calculated as
  i wi
C H i   H i A 
 1   i ri
V
i
1
i



 i
1
wi , i  E , M ,
1i
where ri and wi , i  E , M , denote the regions’ interest rates and wage rates, respectively.
The model contains a special kind of transaction costs, the search costs, occurring after
human capital had been produced and before coming into use. Simplifying the problem, one
could consider a certain part of the search cost as expenditures when labour force – as the
human capital’s medium – is looking for a job. Search costs are assumed to be a non-linear,
convex function of human capital, expressing that the more human capital had been produced,
the lower are chances to find a job, and therefore the more expensive is it to be employed. But
is has to be remarked that search costs interpreted here in a much more extensive sense
include all financial burdens undertaken by the owners of human capital. From this point of
view these costs could be considered as a measure for the cooperation between the institutions
producing human capital and those using it – the better is the cooperation, the lower are the
search costs, etc. Without exaggeration it can be assumed that lower search cost are generally
related to higher developed economies, and vice versa, the lower developed is a society, the
higher will be the human capital’s search costs.
3
For the sake of simplicity let us assume that search costs are expressed by wiS H i2 , i  E , M .
Therefore, the total variable costs of human capital are
  i wi
C H i   H i A 
 1   i ri
1
i
V
i



 i
1
wi  wiS H i2 , i  E , M ,
1i
To these variable costs we have to add the fixed costs occurring in region i, and the costs of
purchasing the human capital produced in the other region, i. e., the cost function of human
capital is
  i wi
Ci H i   H i A 
 1   i ri
1
i



 i
1
wi  wiS H i2  FCi  w Hj H j , i  E , M , i  j ,
1i
where FCi , i  E , M denotes the fixed costs and wiH stands for the costs of 1 unit of human
capital produced in region i. The corresponding marginal cost functions are of the form
  i wi
MC i H   A 
 1   i ri
1
i



 i
1
wi  2wiS H i , i  E , M , i  j .
1i
The profit function of region i is now given by
 i  a j  b j H jj  H ij H ij  ai  bi H ji  H ii H ii 
  i wi
 H i A 
 1   i ri
1
i



 i
1
wi  wiS H i2  FCi  w Hj H j , i  E , M , i  j .
1i
Both regions, the European Union and the new member states try to maximize the profit due
to the quantity of human capital, i. e. they strive for that structure of human capital used in
their own regions maximizing the profit. The condition is well-known:
4
 i
 0 , i, j  E ,M .
H ij
This means for the old European Union
 Ei
  E wE
 E
 a M  bM H MM  2bM H EM  AE1 
H EM
 1   E rE



  E wE
 E
 a E  bE H ME  2bE H EE  AE1 
H EE
 1   E rE
 E
1
wE  2wES H E  0
(2.1)
1E
and



1
wE  2wES H E  0 .
1E
(2.2)
Equation (2.1) yields
H EM
  E wE
a
1
1
 M  H MM 
AE1 
2bM 2
2bM
 1   E rE



 E
wS
1
wE  E H E ,
1E
bM
and from Eq. (2.2) follows
H EE
a
1
1 1   E wE
 E  H ME 
AE 
2bE 2
2bE
 1   E rE



 E
wES
1
wE 
HE .
1E
bE
Summing up the last two expressions, one obtains the equations of the European Union’s
reaction curves due to the new member states:
a
bM bE
a
1
1  1
1

 M  E 
HE 

S
2 bM bE  wE bM  bE   bM bE A E  bM bE


  E wE

 1   E rE
bM bE
1
HM ,
2 bM bE  wES bM  bE 
Similarly we can derive for the new member states:
5



 E

1
wE  
1E

(3.1)
HM
a
bM bE
a
1
1  1
1   M wM

 M  E 

 
S
2 bM bE  wM bM  bE   bM bE A M  bM bE  1   M rM





 M

1
wM  
1M

bM bE
1
HE .
2 bM bE  wMS bM  bE 
(3.2)
From the assumed Cobb-Douglas-technology follows the linearity of the cost functions, and
therefore the linearity of the reaction curves. Their slopes depend on the search costs.
According to an earlier remark it should be assumed that the search costs if higher developed
economies will be lower, implying that the corresponding reaction curve is flatter than in the
case of lower developed regions. With this in mind the graphic of the reaction curves has the
following form:
HE
ˆ M
RM
E
P
H E
RE
H

M
HM
M
̂ E
Graphic No. 1
The reaction curves of the European Union and of the new member states
6
Here
a

bi b j
aj 1  1
1
i
  1   i wi

i 


2 bi b j  wiS bi  b j   bi b j A i  bi b j  1   i ri




 i

1
wi  ,
1 i 

and
a

aj 1  1
  1   i wi
ˆ i   i 


 bi b j A i  bi b j  1   i ri



 i

1
wi  i  E , M ,
1 i 

denote the point of intersection between the reaction curves and the axes; point P is the wellknown Cournot-equilibrium, H M and H E denote the equilibrium quantities of human capital
with
H

M


bE bM 2
bE bMˆ M
1


1 
S
2 
S
S
2 bE bM  wM bE  bM   4 bE bM  wE bE  bM  bE bM  wM bE  bM   bE bM  



4 bE bM


bE bM 2ˆ E
,
2
 wES bE  bM bE bM  wMS bE  bM   bE bM 
(4.1)
and
H E 


2 bE bM  wMS bE  bM 



bE bMˆ E 
4 bE bM  wES bE  bM  bE bM  wMS bE  bM   bE bM 


bE bM 2

2

4 bE bM  wES bE  bM  bE bM  wMS bE  bM   bE bM 
7
2
ˆ M
(4.2)
Analysis of the model
It can be seen that the search costs play an important role in the qualitative and quantitative
determination of the Cournot-equilibrium. If these costs would be equal in both regions, the
slopes of the reaction curves are 
1
. Therefore, the curves coincide, if  E  M , and they
2
have no joint point  E  M . In the earlier case, all combinations of the different region’s
human capitals satisfying one of the above conditions are optimal, while in the latter case
such an optimum does not exist. From this point of view, different search cost are conditions
for an economically meaningful solution of the model.
Increasing search costs will change the slope of the reaction curves. Assuming that the search
costs in the new member states grow up, the intersection of the new member states’ reaction
curve and the vertical axis will move upwards, and at the same moment  M will decrease
(Eq. (3.2)), i. e. the curve shifts to the origin. The first effect implies a lower production level
of human capital in the new member state, the second effect, however, acts in the direction of
an increasing production of human capital in the mentioned area. The final consequence, of
course, is depending on the relative strength of these effects.
After some calculations we would obtain for the change of the optimum quantities of human
capital as a function of varying search costs the following condition:

M
S
M
dH
dw





 0,
 0,
 0,






ˆ b b  wS  b b 
ˆ
8
M
E M
E
E M
E
S
ˆ
ˆ
8M bE bM  wE  bE bM E .
ˆ b b  wS  b b 
ˆ
8
M
E M
E
E M
E
Since ˆ i represents that amount of human capital produced in region j if the production of
human capital in region i is zero, the economically acceptable condition will be
dH M
 0,
dwMS
because any other relation would mean that in the new member states at least eight times so
much human capital has to be produced if the production of human capital in the European
8
Union would be 0, as human capital would have to be produced in the EU-region, if the new
member states would not produce any human capital.
As a final consequence it could be said, that increasing search costs in the region of the new
member states will imply a fall of the optimal quantity of human capital from H M to H M ,
while the human capital in the EU-region will increase – even under unchanged behavior in
this region - from H E to H E . (See Graphic No. 2)
HE
1
H E
H E
RE
H

M
H

M
HM
2
Graphic No. 2
The change of the optimal quantities of human capital when search costs increase in the lower
developed area
On the other hand, lower search costs could increase the production of human capital, and this
could imply a higher rate of economic growth. For economies to which this possibility is
given, an administrative intervention of this kind may be very useful.
9
Analysing the effect of a change in the technology parameter Ai , it can be seen that an
increase of this parameter in the region of new members will shift the reaction curve in the
NE-direction, meaning that the new member states’ more efficient technology let not only
grow up the level of human capital’s production, but the distribution between human capitals
produced in the different areas will also change for the benefit of the lower developed region.
(See Graphic No. 3)
HE
R M
H E
P
H E
RE
RM
H M
HM
H M
Graphic No. 3
The effect of the technology parameter on the quantities of human capital
Similarly we could investigate the effects of growing interest rates and of an increasing wage
rate. It can easily be seen that
10
dˆ i
1

dri
Ai
 1 1   i wi
  
 b b  1   r
j 
i i
 i



 i 1
 0.
(5)
Since equations (4.1) and (4.2) imply
dH i
dH i
 0 und
 0 , i, j  E ,M ,
dˆ i
dˆ j
(6)
The following result can be summarized:
An increase (decrease) of the interest rate in region i causes in this area a lower (higher)
production level of human capital; in region j the effect will be a higher (lower) production
level of human capital.
The effect of the wage level’s change depends of the relation between capital elasticity and
labour elasticity. To see this one has to derive ̂i with respect to wi obtaining
 1 1   i wi
dˆ i
1
  

dwi
Ai 1   i   bi b j  1   i ri
If  i 



 i
 i

 1 .

1   i

(7)
dH i
dˆ i
dH i
i
1
 0.
 0 . Using (6) we have
, then
 0 and
 1  0 and thus
2
dw j
dwi
dwi
1 i
For  i 
dH i
dH i
1
 0 .3 Therefore an increasing
it follows analogously that
 0 and
2
dw j
dwi
wage rate will imply a higher production level of human capital, if the capital elasticity in this
region is higher than the labor elasticity; otherwise the production level of human capital will
fall, if wages grow up.
3
The case  i 
ˆ
dH i
d
dH i
1
i
 0 , i. e. if the contributions

0
implies
and consequently
 0 and
dw j
2
dwi
dwi
of capital and labour to the production of human capital in a region are exactly the same, the change of wages
does not touch the output of human capital in this region.
11
As a résumé of the model’s analysis the effects of economic policy will be recapitulated in the
following table.
Steps of economic policy
Effects on the production of Effects on the production of
human capital in the region human capital in the other
where intervention has been region
done
Decreasing
(increasing) Increasing (decreasing)
Decreasing (increasing)
Increasing (decreasing) the Increasing (decreasing)
Decreasing (increasing)
search costs
efficiency of the production
of human capital
Decreasing
(increasing) Increasing (decreasing)
Decreasing (increasing)
interest rates
Increasing (decreasing) wage Increasing (decreasing), if Decreasing (increasing), if
rate
i 
1
2
i 
1
2
Decreasing (increasing), if Increasing (decreasing), if
i 
1
2
i 
1
2
Until this point it had been assumed that demand for human capital is given and unchanged.
Now it should be analysed how a change in demand will touch the production and the
distribution of human capital. Let us assume that a new member states will be faced by a
higher demand for human capital produced in their region, i. e. let a M increase.4 From Eq.
(4.1) one obtains
4
This would be not only a simple price-independent change in demand. Since demand for human capital in
region i has been described only as a function of the price p i , a probably change in the price for the human
capital in the other region, i. e. a change of p j , will also be reflected by parameter
12
i .
dH M
1 bE bM

da M
2 M
2
2
ˆ
ˆ

 d


bE bM 
bE bM 
d
M
E
1


,

2
2
da
da




4



b
b
4



b
b
M
E
M
E M
E
M
E M

 M
where  i  bE bM  wiS bE  bM  , i  E , M . Since
dH M
1 bE

da M
2 M
It can be shown that
ˆ
d
1
i

, i  E , M , we have
da M bM
 2bE bM 2  bE bM  M 
.
1 
2 
 2 4 E  M  bE bM  


dH M
 0 , therefore increasing demand in the new member states implies
da M
a higher production level of human capital in this region.
On the other hand, however, a higher demand for human capital in the lower developed region
will influence the production of human capital in the EU-region too. From Eq. (4.2) it can be
derived that
dH E
b 2 M  bE bM 
 E
 0.
2
da M
4 E  M  bE bM 
A generalisation of the model and the problem of stability
Present analysis can summarised with a very obvious formulation: In almost every situation
the enlargement of the European Union will be advantageous for the production of human
capital in the new member states, and therefore advantageous also for their economic
development and welfare. It could be shown that under economically acceptable conditions an
equilibrium solution does exist. But what will happen, if the initial distribution of human
capital is a disequilibrium one? Does there exist a mechanism ensuring the equilibrium, i. e.


will the pairs of human capitals H E (t ), H M (t ) tend to the equilibrium values H E , H M as
t ?
It has been proved that the solution of Cournot’s duopoly is for a linear demand function
stable, if the second derivative of the cost function is positive. (See e. g. Fisher 1961, Hahn
13
1962) In the present model, this condition is satisfied by using a convex function representing
the search costs. With other words: lower search cost are advantageous for the development of
the economies, but – on the other side – the higher are the search costs, the more probably is
the stability of the solution. These – from some points of view – inconsistent results could be
escaped by a more efficient technology, exactly spoken: by using production functions with a
degree of homogeneity higher than 1.
Using the more general form H i  Ai K i i Li i , i  E , M we will obtain for the second
derivative of the corresponding cost function the expression
1 2  i   i 
 i  i
d 2 Ci H i  1   i   i

H
dH i2
 i   i 2 i
 2wiS , i  E , M .
The condition for a stable Cournot equilibrium (Hahn 1962, p. 331) is now5

1   i  i
 i   i 
2
1 2  i   i 
 i  i
Hi
 2 wiS  0 , i  E , M .
(8)
For the trivial case H i  0 the stability condition is identical with that of the Cobb-Douglas
production function, for positive quantities of human capital condition (8) is fulfilled if
5
For general market demand functions
pi  i qi  , and at least twice differentiable cost functions Ci qi  ,
i  E , M , the following conditions have to be satisfied:
d i qi   d 2i qi  d 2Ci qi 
 qi

0
dqi
dqi2
dqi2
(i)
2
(ii)
2
and
(iii)
di qi   d 2i qi 
 qi
 0 , or as general condition
dqi
dqi2
d i qi 
d 2Ci qi 

 qi
, i  E , M , with q i , i  E , M , as equilibrium quantities. (Hahn
dqi
dqi2
1962)
Since in the present case a linear demand functions have been assumed, we obtain
d 2 i qi 
 0 , implying the earlier expression.
dqi2
14
di qi 
 bi and
dqi
 i   i  1 , i  E , M , i. e. if the degree of homogeneity of the production function is higher
than 1.
At the first moment expression (8) seems to imply that the equilibrium could be stabilized by
increasing search costs – the left hand side of the inequality will be more and more negative
the higher is wiS , i  E , M . The situation is more than paradox: On the one side high search
costs will retard the dynamics in the development of human capital, but on the other side they
seem to be advantageous for the stability of the distribution of human capital between the
regions – especially for increasing efficiency in the production of human capital..
More detailed analysis, however, throws light on the higher degree of complexity of the
relationship between the economics of scale and the government’s policy. It can be shown
(see Appendix) that in an early period high dynamics in the return to scale will really justify
increasing search costs, but they will (or at least could) switch into a decreasing development
if the degree of homogeneity of the production function has been passed a certain amount.
With other words: as long as a country produces its human capital with a more lower
efficiency increasing search cost would be advantageous for the stability of the Cournot
solutions, i. e., it will imply a mutually acceptable and stable distribution of human capital
between the regions. Technical progress and therefore an increasing efficiency in the
production of human capital will engender a qualitative change in the possibilities for
economic policy – government can the search costs keep on a lower level and in this way
stimulate the production of human capital while the stability is also ensured.
The implication for economic policy seems to be clear: the new member states have to
produce their human capital with efficiency as high as possible. To do that they have to use
the best qualified human capital wherever this had been produced. May be that a possible
strategy is to hinder the human capital of the new member states to leave them – e. g. by
introducing high search costs – and to stimulate human capital produced in the EU-region to
come to the lower developed region – by lower search cost, tax allowances, etc?
May be that there is nothing new under the sun? A similar concept had been suggested by
Friedrich List – 170 years ago – arguing that the integration of Germany into the European
Economy will be successful only after having developed the German industry, especially its
15
railway-system; before that it would be wise to protect the German manufactures. “If it would
become clear that the domestic factories remain behind those of other countries because they
are short of skilful, diligent and persistently working labour forces, then the wages for the best
domestic workers should be fixed on an appropriate level.” (List 1961, p. 155) To develop the
domestic manufactures the government should increase the tariffs step by step until a level
necessary that the whole supply on the domestic market will be produced by domestic firms,
because this will stimulate the competition among the nation’s factories. “This domestic
competition depresses the prices and develops welfare… then can the tariffs be decreased step
by step and the national industry can be exposed to the competition of the foreign factories.”
List 1961, p. 147-148)
16
Appendix
For the production function H i  Ai K i i Li i , i  E , M we obtain the cost function:
1
1
dCi
1

H ii  i i  2wiS H i .
dH i  i   i
The second derivative with respect to human capital has the form
1 2  i   i 
 i  i
d 2 Ci 1   i   i

Hi
dH i2  i   i 2
 i  2 wis .
From condition (iii) (s. footnote 5) follows  bi 
1 i  i
 i   i 2
H
1 2  i   i 
 i  i
i
 i  2wis . Since
H i  0 , wiS  0 and  i  0 we have to distinguish between the following situations:
a) If H i  0 , then the solutions are stable independent on the concrete conditions.
b) If H i  0 and  i   i  1 , then the expression on the right hand side of the inequality
is positive and therefore the conditions for stability are satisfied.
c) If we consider the case when the degree of homogeneity of the production function is
higher than 1, stability is not automatically given. The solution is stable, if
1 2  i   i 
 i  i
1
1 1  i  i
 bi 
Hi
2
2  i   i 2
 i  wis ,
1 2  i   i 
 i  i
1
1 1  i  i
w   bi 
Hi
2
2  i   i 2
s
i
where
 i  i  1.
Let’s
chose
 i   , with   0 . The condition for stability are
satisfied and the search costs could be considered as depending on the productions
function’s degree of homogeneity:
1 2 x
x
1
1 1 x
wiS   bi 
Hi
2
2 x2
i   ,
i  E ,M .
(A1)
where for the  i   i had been substituted by xi . From this function we can derive:
17
1 2 x
x
dwiS
1
  i H i
dx
2
1
x( x  2)  (1  x) ln H i  .
x2
The above expression can only be zero if x( x  2)  (1  x) ln H i   0 . From the present
point of view the quantity of human capital is constant, thus we have a quadratic
equation with the solutions
x1 
2  ln H i  
2  ln H i 2  4 ln H i 
2
,
and
x2 
2  ln H i  
2  ln H i 2  4 ln H i 
2
.
According to the assumption the degree of homogeneity should be higher than 1,
therefore xi  1, i  1,2 . This is true only for x1 , in the case of x2 this would imply the
contradiction 0  4 .
Substituting the second derivative of (A1) into x1 , one obtains
d 2 wiS
dx 2
 0.
x x1
Consequently x1 is a maximum of function A1. At the beginning the increasing degree
of homogeneity implies higher search costsm but passing the point x1 the amount of
1 2 x
x
1
1 1 x
 bi 
Hi
2
2 x2
 i   will decrease.
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References
Bischi, Gian-Italo - Kopf, Michael: The Role of Competition, Expectations and Harvesting
Costs in Commercial Fishing. In: Puu, T. – Sushko (2002), I., 85-110.
Fisher, F. M.: The Stability of the Cournot Oligopoly Situation: The Effects of Speed of
Adjustment and Increasing Marginal Costs. Review of Economic Studies, vol. XXVIIX
(1961), 125-135.
Hahn, F. H.: The Stability of the Cournot Oligopoly Solution. Review of Economic Studies,
vol. XXIX (1962), 329-331.
List, Friedrich: Das natürliche System der politischen Ökonomie. Akademie-Verlag, Berlin,
1961.
Meyer, D.: Humankapital und EU-Beitritt. Überlegungen anhand eines Duopolmodells.
Andrássy Working Papers No. VIII, 2003.
Puu, T. – Sushko, I. (Eds.): Oligopoly Dynamics. Springer-Verlag, Berlin – Heidelberg –
New York, 2002.
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