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1
SKILL TO UNSKILLED LABOUR RATIO DRIVEN BY PUBLIC
EXPENDITURE IN EDUCATION
Rossana Patron1 Marcel Vaillant2
University of Uruguay
Preliminary draft
July, 2010
Abstract
The ratio of skill to unskilled labour stocks in the economy is widely acknowledged to have
an important role for growth. Can education policy do anything to affect the evolution of this
ratio? This paper shows that it can, and it also shows that the actual effect of education policy
depends on the allocation rule of the budget across educational levels. The consideration of a
stylised hierarchical education model allows us to develop analytical conditions under which
the allocation rule favours the accumulation of skills. The analysis has implication for policy
makers in developing countries, where skill formation is much needed, as it shows that
observed allocation rules usually violate the optimality condition by the assignment of higher
than optimal resources to higher education.
Key words: hierarchical education model, education budget allocation and development.
JEL: I21, I22, I28
1 Departamento de Economía, Facultad de Ciencias Sociales, email: [email protected]. 2 Departamento de Economía, Facultad de Ciencias Sociales, email: [email protected].
2
1 INTRODUCTION
Even when the links between skills and growth, and hence education and growth, is well
established on theoretical grounds (mainly in the endogenous growth theory, see for instance,
Lucas, 1988, Romer, 1986, etc.), the theoretical analyses about the effects of the structure of
human capital on growth is far less considered in the literature. Some exceptions are Birdsall
et al. (1998), Gemmel (2002), Papageorgiou (2003), who have stressed the importance of the
distinction between different stages of human capital creation for development, and therefore
the relevance of considering the internal allocation rules of the education budget.
As the composition of the human capital stock is relevant, therefore the structure of
the education system matters. The performance at earlier stages affects the output at higher
levels, so, the budget allocation rule should take this into account. Hierarchical education
models have been used to analyse this point by Driskill and Horowitz (2002, 2009) and Su
(2004). As noted by Su (2004) hierarchical structure in educational systems implies that levels
are not perfect substitutes, which means that different allocation of same size budget have
different effects on aggregate efficiency and distribution. Considering a desired target of skill
share, this paper discusses optimal allocation rules of educational budget in a stylised two-
level education system characterised by -poor quality caused -internal inefficiency. It is
shown that dealing with inefficiency optimally will allow to reach a maximum target or to
reach a set level in the shortest period. The simplicity of the model allows developing
analytical conditions under which the allocation rule favours the accumulation of skills, which
may assist as a guide to policy makers.
The point is not minor, as the skill to unskilled stock ratio in developing and
developed countries shows significant differences. According to UNESCO while the
proportion of population with below upper secondary as maximum educational attainment for
adult population in the OECD country members is 29% average (data for 2005) in most
developing countries the proportion of lower secondary as maximum attainment of
developing countries is much higher, for instance, in Brazil is 70.5%, in Kuwait 74.4, and in
many African countries is even higher as in Uganda 93.6% or Nigeria 98.9%. It is easy to
understand the underlying reasons of this gap. The creation of skills is the result of education
activities. The ratio skilled to unskilled ratio of the inflow of labour created depends on the
internal efficiency of the education sector: the higher the systemic performance the higher the
3
skill to unskilled ratio in the inflow of labour. In developing countries the accumulation of
skills is hindered by inefficient education systems, often aggravating funding troubles.
This article is organized as follows. Section 2 describes the current situation regarding
education budget allocation and accumulation of skills. Section describes the education model
and its properties. Section 4 the conditions to maximise the stock of skills are discussed.
Section 5 concludes. There is an Appendix with mathematic details.
2. DEVELOPMENT AND EDUCATION
Schooling affects individuals’ opportunities to access the labour market (and their
probabilities of being successful), as well as the rate at which a country experiences economic
growth. This fact is widely recognised in both academic and non-academic circles. For
instance, the International Labour Organisation report “Learning and training for work in the
knowledge society” (2003) states in its concluding remarks: “The knowledge and skills
endowment of a country’s labour force, rather than its physical capital, determines its
economic and social progress, and its ability to compete in the world economy. Promoting
innovation, productivity and competitiveness of individuals, enterprises and countries is
therefore the first pillar that underlies contemporary learning and training policies and
provision. Similarly, individuals’ possession of knowledge and skills increasingly determines
their employment outcomes and lifetime.”
However, the distribution of skills across countries varies considerably, especially
between developed and developing countries. UNESCO data illustrates the dimension of this
disparity, as shown in Table 1. The reality is that while in developing countries the majority of
the population (above a half) has primary education or less as maximum educational
attainment, in developed countries the proportion of the population with tertiary education is
above a third of total population, and it is as high as 46% in the case of Canada or 40% in the
case of Japan.
The observed gap is indeed a consequence of the difference performance of
educational systems. Many developing countries often show several weaknesses, especially in
quality and coverage, causing spread illiteracy in the population or early exit. In many cases
the expenditure on public education per student is far behind that in the developed world, but
4
as argued by Birdsall et al. 1998, Gemmel 2002, Papageorgiou, 2003, it is both the size and
the efficiency of the allocation of the public funds for education are relevant for the overall
systemic performance. In particular, inefficient allocation of the education budget is singled
out as the reason of the poor performance of Latin American countries by Birdsall et al.
(1998) rather than the size of the budget.
Table 1 Educational attainment of the adult population. Distribution of the population aged 25
and over, by highest level of education attained (in percentages). Selected countries.
COUNTRY YEAR
NO SCHOOLING
AND PRIMARY
COMPLETE OR
INCOMPLETE
LOWER SECONDARY
UPPER SECONDARY
TERTIARY EDUCATION
Argentina 2004 43.8 14.2 28.4 13.6 Bangladesh 2001 73.3 9.6 12.9 4.2 Botswana 2000 75.3 15.7 5.9 3.1 Brazil 2004 57.5 13.0 21.2 8.1 Chile 2004 24.0 26.0 36.9 13.2 Costa Rica 2007 50.9 13.8 18.5 15.0 India 2000 77.7 12.4 6.5 3.3 Kuwait* 2006 55.2 19.2 17.2 8.3 Mauritius* 2000 60.5 18.6 17.6 2.6 Mozambique 2000 96.9 2.3 0.8 0.1 Nigeria 2000 97.1 1.8 0.7 0.4 Uganda 2002 88.5 5.1 1.6 4.8 Uruguay 2006 52.8 22.4 15.1 9.6 Australia 2005 9.1 25.8 33.3 31.5 Canada 2004/2005 4.9 9.9 39.2 46.1 Finland* 2006 22.0 8.9 38.8 30.3 Ireland* 2006 23.7 16.3 31.2 26.4 Japan 2004/2005 60.1 40.0 New Zealand 2005 21.3 51.6 27.1 Republic of Korea 2005/2006 11.9 12.6 43.9 31.6 United Kingdom 2004/2005 14.4 55.9 29.6 United States 2005 6.3 8.5 49.0 36.2
Notes: Last data available. Totals may differ from 100% due to missing information or rounding.* Upper secondary includes post-secondary non-tertiary
Source: Own elaboration with data from UNESCO/UIS WEI (www.uis.unesco.org/publications/wei2007); UNESCO, Global Education Digest 2009; Barro and Lee data set.
On observed budget allocation rules, also UNESCO data can shed some light. Table 2
shows the public expenditure per pupil as a % of GDP per capita by education level, and the
public expenditure on education as % of GDP for a selection of developing and developed
5
countries. This table shows that, in general, while the government preferences do not differ
much across developing and developed countries in terms of the size of the budget (measured
by the public expenditure on education as % of GDP, in the last column of the table) there are
significant differences in the preference over the internal distribution of the budget (measured
by the public expenditure per pupil as a % of GDP per capita by education level, first, second,
and third data columns of the table). While countries like US and Japan have almost perfectly
flat allocation rules (levels “equally preferred”), in general developed countries have a quite
even distribution (around average), with the exception of Korea, that allocates half than
average to higher education. The situation among developing countries is more
heterogeneous. For instance, while in countries like Chile and Argentina the distribution is
quite flat, there are many countries that display strong preferences for higher education, some
of them extraordinary high as Mozambique and Botswana.
Table 2: Public expenditure per pupil as a % of GDP per capita by education level, and Public expenditure on education as % of GDP. Selected countries.
PUBLIC EXPENDITURE PER PUPIL AS A % OF GDP PER CAPITA
COUNTRY Primary Secondary Tertiary All levels
PUBLIC EXPENDITURE
ON EDUCATION AS % OF GDP
Argentina 12 19 13 14 4
Bangladesh 9 15 46 13 2 Botswana 16 41 450 34 10
Brazil 14 12 34 15 4 Chile 12 13 13 13 3
Costa Rica 17 17 36 19 5 India 9 17 58 15 3
Kuwait 12 16 102 22 5 Mauritius 12 19 37 17 4
Mozambique 16 69 570 23 5 Nigeria 31 46 366 34 3 Uganda 11 32 179 14 5 Uruguay 8 10 18 11 3
Australia 17 16 24 18 5 Finland 18 32 35 28 6 Ireland 15 22 25 19 5 Japan 22 22 20 22 4
New Zealand 19 22 28 22 6 R of Korea 18 23 9 17 4
United Kingdom 20 25 30 24 5 United States 22 24 24 23 6
Note: Averages of available years 2004-2006. Source: Own elaboration with data from Unesco data base. http://stats.uis.unesco.org/unesco/ReportFolders/ReportFolders.aspx
6
The observed differences in educational budget allocation rules and systemic
performance (measured by the percentage of population with high educational attainment)
lead to the obvious question of the role of the allocation of the budget in skills formation. As
noted by Gemmel there is a key skill level for each development stage: that human capital
effects on growth are most evident at the primary level in low income countries, for higher income
developing countries the key is the secondary level, while the tertiary level is the relevant in developed
countries. Thus, according to this, many African countries are using allocation rules contrary to
their development needs.
Noteworthy, in the long term, the links between education and growth are
bidirectional, as richer countries are allowed to invest more money expanding and improving
their educational services, generating virtuous or vicious circles. Considering a desired target
of skill share, policy makers could allocate resources accordingly. The conditions to do this
efficiently will be discussed in the rest of the paper.
3 THE EDUCATION MODEL
Consider a two-level education system, consisting of basic and higher education ( HBj ,= ).
The output of education activities is given by ( , )j j j jQ F K E= , where jK is the resources, jE
is enrolment, and the subindex j represents the level. The function F is subject to constant
returns to scale, so the output per student can be written as ( )j j j j jq Q E q k= = , where jk
measures the resource intensity per student, and 0>∂∂ jj kq , 022 <∂∂ jj kq . The variable
jq is the amount of knowledge embodied in successful students, and it is assumed to measure
“school quality.” Students leave the system early when the quality of education they receive is
poor, output per student is taken as the determinant of early dropouts, where 0<∂∂ Bqθ and
022 <∂∂ Bqθ .
As the learning process is cumulative, the indicator jf is defined as ∑=j
jj qf ,
which is the knowledge accumulated per student who has completed up to level j . The
7
indicator jf measures the human capital accumulated during the schooling process; the jq
accumulated at different levels.
The composition, of the inflow of labour produced, depends on time of exit and on
school quality. Thus, the accumulation process is driven by
BBU fEdL θ= (1)
HBHHS fEfEdL )1( θ−== (2)
where θ is the early exit rate, and UdL and SdL are the inflow of units of unskilled and
skilled labour, respectively.
The marginal ratio of skilled to unskilled labour can be defined as:
B
H
U
SHB f
fdLdL
kkθ
θξ −== 1),( (3)
The ratio of skilled to unskilled labour in the economy is modified by ξ . When
US LL>ξ , where SL and UL are the stocks of skilled and unskilled labour, respectively, the
ratio of skilled to unskilled labour in the economy rises, it declines when US LL<ξ , and
remains unchanged when growth is balanced.
As the marginal ratio ξ is dependent in the capital intensity of basic and high
education, totally differentiating and after some manipulation results:
^
ˆ(1 )
H
B
ff
θξθ
⎛ ⎞= −⎜ ⎟ −⎝ ⎠
(4)
where a hut (^) placed over the variables denote rate of growth.
Changes in ξ depends on the effects of allocation on the survival to exit rate and on
the relative human capital accumulation across levels. This is presented in figure 1,
considering 01 22 <∂⎟⎠⎞
⎜⎝⎛ −∂ Bk
θθ , ( ) 0<∂∂ BBH kff and ( ) 022 >∂∂ BBH kff , which
implies the variability of ξ over Bk taking Hk as given.
8
Figure 1
θθ−1
B
H
ff
Bk
Considering equation (4) then the condition to ˆ 0ξ > is the following:
( )( )1
H B
H B
d f ff f
θθ
>−
(5)
Proposition 1: The capital intensity in basic education will have a positive effect on the
marginal ratio of skilled to unskilled labour if the elasticity of the drop out variable to the
quality of basic education (Bqθε ) is greater than the proportion of the population that survive
(1 )θ− to the high level of education times the share of human capital accumulation in the
high level as a proportion of overall human capital accumulation ( HH
H
qsf
= ).
Taking the total differentiation of expression ξ (see equation (3)) and making some
manipulations the condition under which 0>∂∂ Bkξ is found:
( )1 0Bq H Bs kθε θ ξ> − ⇔ ∂ ∂ > (6)
9
where HHH fqs = and θθεθ BBq qqB
∂∂−= . (See Appendix for demonstration).
For a given Hs , if the survival parameter is too low is more possible that the increase
capital intensity in basic education will have a positive effect on the marginal ratio ( , )B Hk kξ .
Proposition 2: The partial effect of capital intensity in higher education over the marginal
ratio is always positive ( 0Hkξ∂ ∂ > ) and equal to the elasticity of education quality in high
education to the capital intensity times the share of capital accumulation in total capital
accumulation:
H H Hk H q ksξε ε= (7)
Considering the educational budget constraint B HK K K= + , it can be shown that
HB kkk )1( θ−+= , then (1 )
BH
k kkθ
−=−
. We analyses the effect of a marginal change in capital
intensity in basic education over capital intensity in high education.
2
(1 ) ( ) 1( )(1 ) ( ) (1 )
B BB k kHH
B B
k kdk kdk k k
θ θ θθ θ
− − + − −= = − +
− − − (8)
After some manipulation it can also be shown that:
1 1H B B B B B
B Bk k k q q k
H H
K KK Kθ θ
θ θε ε ε εθ θ
= + = +− −
(9)
where θθεθ BBk kkB
∂∂−= and HBBHkh kkdkdkBH
−=ε .
10
Figure 2
Budget restriction set
Bk
Hk
The determinants of the elasticity of ξ respect to the resource intensity in basic
education, can be shown using the expression:
( )1B B B
B B B H H H B
q k qk H q k q k k ksθ
ξ
ε εε ε ε ε
θ= − +
− (10)
Proposition 3: The allocation or more resources to basic education (with fixed budget and
enrolment) will increase the marginal ratio of skilled to unskilled labour, i.e. 0>Bdkdξ , if
( )1 ( )B B B
B B H H H B
q k qH
q k q k k k
sθε εθ ε ε ε
>− +
3 (10)
Or:
( )11 ( )
B B B
B B H H h B
q q kB
q k q k k k
s θε εθ ε ε ε
> −− +
(11)
3 See the Appendix for demonstration.
11
Substituting (8) in expression (9), and considering the “quasi-neutral” assumption on
education technology thatB B H Hq k q k qkε ε ε= = , it results:
(1 ) (1 )(1 )(1 ) (1 )
B
B
B
q H qk Hk B
B q H qk
s sEE s
θξ
θ
ε θε θεε θε
− −= − −− −
(12)
where B BE K K= is the participation of higher education in total budget.
Using the assumption about education technology –including the drop out function-
then it is possible to obtain an explicit expression of the previous equation:
(1 ) (1 )(1 )1(1 ) ( ))B
B Hk B
BB
H
q l s EE q l
s
ξα θα θε
α θα
− −= − −− −
(13)
Where: drop out function is BlqBeθ −= , B- is the scale parameter of the drop out variable and ;
l - measures the responsiveness to quality in basic education, that is specified as BB Bq A kα=
with 0 1Bα< < .
Proposition 4: The elasticity of the marginal ratio of skilled to unskilled labour relative to the
resource intensity in basic education is higher ( 0BkddB
ξε> ) when the scale parameter of the
early exit rate (B) is higher for every value of the elasticity of the marginal ratio of skilled to
unskilled labour relative to the resource intensity (see Appendix for demonstration and figure
3).
12
Figure 3 Effect in the elasticity of the marginal ratio of skilled to unskilled labour to the resource
intensity in basic education- drop out parameter
-0,1
-0,08
-0,06
-0,04
-0,02
0
0,02
0,04
0,06
0,08
0,000000 0,010000 0,020000
B=1,1 B=1 B=0,95
We use equation (13) to make simulations in the parameters space to show how
Bkξε .changes with variations in the parameters. In some cases it is not possible to obtain
general conditions as we did with the scale level of the early drop out variable. According
with simulations (see figure 4) for every value of Bk where Bkξε is positive the smaller is α
the higher is the elasticity of the marginal ratio (Bkξε ). See figure 4.
Bkξε
Bk
13
Figure 4 Effect in the elasticity of the marginal ratio of skilled to unskilled labour relative to the
resource intensity in basic education. Production function parameter
-0,1
-0,08
-0,06
-0,04
-0,02
0
0,02
0,04
0,06
0,08
0,000000 0,010000 0,020000
alfa=0,15 alfa=0,21 alfa=0,09
In many developing countries with bad outcomes (high drop out and low efficient
values and elasticity values) of education at low and the intermediate level of the system
(basic education) and too low allocation of resources to education (low k), the increase in the
share of resources to basic education ( B BE K K= ) could be more effective in terms of
increasing the amount of skill labour in relation to unskilled.
Bkξε
Bk
14
Figure 5
Relationship between ( )BEξ and B BE K K= Developed and developing countries pattern
B BE K K=
( )BEξ Developed
, , , ,k B l Aα
Developing
, , , ,k B l Aα
Given a certain amount of income devoted to the public education function it is
possible to specify and objective function to maximize in the structure of education
expenditure in each level of the system.
The objective is to maximize the marginal ratio ( ( , ) SB H
U
dLk kdL
ξ = ).
,max ( , ) : max ( , )
(1 ) (1 )B H B
B BB H H Bk k k
k k k kk k k kξ ξθ θ
− −= ⇒− −
(12)
According with the first order condition then:
( ) 0 ( ) 0(1 )
B B B
H H H B B B
q q kBH q k k k q k
B
d k sdk
θε εξ ε ε εθ
= ⇔ − + =−
(13)
Using (9) and the quasi neutral assumption about education technology and the
explicit function specify in the Appendix, then follows that:
15
(1 (1 ) ) (1 )(1 ) (1 )(1 (1 ))(1 (1 ) ) (1 (1 ) )
B B B
B B
q B qk B q B q qkB
q B qk q B qk
s s sk k k
s sθ θ θ
θ θ
ε θε θ ε θ ε εε θε ε θε
− − − − − − − − −= =
− − − − (14)
A second objective could be that the central authority has an objective over the amount
of human capital accumulation then the objective function could be specifying in the
following way:
( (1 )( )) (1 )U S B B B H UU dL dL E q q q dLθ θ γ ξ= + = + − + = + (15)
Taking the first order condition it is possible to show that:
(1 )(1 ) (1 )B B B
B B B
B B B H H H
q k qq k q H
q k k k q k
sθθ
ε εξε ε θξ ε ε ε+− + = −
+ (16)
The third possibility considered is achieve a certain objective level of skill to unskilled
labor (ξ ) in a minimum period of time considered a constant amount of income devoted to
education. Assuming a constant variation in each magnitude it is possible to show that the
number of period is:
0
( ) ( ) ( )
(1 )( )
( 1) ( 1)
S SS S U U S S U U
U U
U
S U U S
U U
L TdL L TdL L TdL L TdL L TdLL TdL
LL L T dL dL T
dL dL
ξ ξ ξ ξ
ξπξξ ξ ξ ξ
ξ ξ
+ = ⇒ + = + ⇒ + = ++
−− = − ⇒ = =
− −
(17)
Minimize de number of period is similar to solve the following problem:
max ( 1)B
UkdL ξ
ξ− (18)
Taking the first order condition it is possible to show that:
( )(1 ) ( (1 )B B B
B B B
B B B H H H
q k qq k q H
q k k k q k
sθθ
ε εξ ξε ε θξ ε ε ε−− + = −
+ (19)
16
We summarize the three public policy problem in table 3.
Objective Function a) Solution
Marginal
ratio
(U&SL)
( )Bkξ (1 )( )
B B B
H H H B B B
q q kH
q k k k q k
sθε εθ
ε ε ε= −
+
Human
capital
accumution
(U&SL)
(1 ( , ))U B SdL k kξ+ (1 )(1 ) (1 )B B B
B B B
B B B H H H
q k qq k q H
q k k k q k
sθθ
ε εξε ε θξ ε ε ε+− + = −
+
Time in
achievement
a certain
level of skill
to unskill
labour
(T&S/U)
( )( ( ) 1)U B BdL k kξ ξ − ( )(1 ) ( (1 )B B B
B B B
B B B H H H
q k qq k q H
q k k k q k
sθθ
ε εξ ξε ε θξ ε ε ε−− + = −
+
17
5 CONCLUSIONS
The ratio of skill to unskilled labour stocks in the economy is widely acknowledged to have
an important role for development. Can education policy do anything to affect the evolution
of this ratio? This paper shows that it can, and it also shows that the actual effect of education
policy depends on the allocation rule of the budget across educational levels.
The cumulative nature of the education process lead to asymmetries between educational
stages, basic and higher education. This is so as inefficient allocation leads to high dropout
rates at basic education, which leaves the few entrants to higher education with high output
per student: few an highly qualified graduates.The consideration of a stylised hierarchical
education model allows us to develop analytical conditions under which the allocation rule
favours the accumulation of skills.
The analysis has implication for policy makers in developing countries, where skill formation
is much needed, as it shows that observed allocation rules usually violate the optimality
condition by the assignment of higher than optimal resources to higher education. A further
implication is that, as the marginal ratio regulates the wage gap, a less than optimal value will
worsen the wage distribution.
18
REFERENCES
Birdsall, N., J. Londono and L. O’Connell 1998. Education in Latin America: demand and
distribution are factors that matter. Cepal Review, vol. 66, 39-52.
Driskill, R. and Horowitz, A. 2009. Hierarchical human capital and economic growth theory
and evidence. Journal of Institutional and Theoretical Economics 165, 723-743.
Driskill, R. and Horowitz, A. 2002. Investment in hierarchical human capital. Review of
Development Economics, vol. 6, no.1, 48-58.
Gemmel, N. 1996. Evaluating the impacts of human capital stocks and accumulation on
economic growth. Some new evidence. Oxford Bulletin of Economics and Statistics,
vol. 58, 9-28.
Lucas, R. (1988): “On the mechanics of economic development”. Journal of Monetary
Economics, vol. 22.
Papageorgiou, C. 2003 Distinguishing Between the Effects of Primary and Post-primary
Education on Economic Growth. Review of Development Economics, 7(4), 622–635.
Romer, P. (1986): “Increasing returns and long run growth”. Journal of Political Economy
vol. 94.
Su, X. 2004. The allocation of public funds in a hierarchical education system. Journal of
Economic Dynamics and Control 28, 2485-2510.
19
APPENDIX
CALCULATIONS AND PROOFS
The marginal ratio of labour ξ is:
( ) (1 )( ) (1 )S H H H B H B B H H B
U B B B B B
dL E f E q q E q q Q QdL E f E q Q
γ θ γ θ γθ θ θ
= = + = − + = + −= = =
(A.1)
So,
(1 )( , )
(1 )( ) (1 ) ( ) ( )
S HB H
U B
B H H H H
B B B B
dL fk kdL f
q q q dLQ qq q dLQ q
θξθ
θ γ θ γ γθ θ
−= = =
− + −= = + = + (A.2)
Differentiating the above expression:
^
(1 ) (1 )( ) ( )
(1 )( )ˆ
(1 ) (1 )
H H
B B
H
B H
H B
B
f fd d df f
fd df fdf ff
θ θξθ θ
θξ θθ ξθξ θ
θ
− −= + =
−⎛ ⎞
= + ⇔ = −⎜ ⎟− −⎝ ⎠
(A.3)
See that: 2
1 (1 ) (1 )1 1 (1 ) (1 ) (1 )
d d d dd θ θ θ θ θ θ θ θ θ θθ θ θ θ θ θ θ θ θ− − − − − −⎛ ⎞ = = = − =⎜ ⎟ − − − − −⎝ ⎠
.
We solve each of both terms of A.3:
2 2
2
( ) ( )´ ´
( ) ( ) ( ) ( ) ( )
(1 )´(1 )
(1 ) (1 ) (1 )
H H
B B H B H B H H H B B
H H H H HB B B
B B B B B
B
q qd dq q dq q q dq q dk q q dk
q q q q qq q qq q q q q
d dkd d
γ
γ γ γ γ γ
θθθ θθ θ θθ
θ θ θ θ θθ
+−= = = −
+ + + + +
−−− − −= =− − −
(A.4)
Putting the A.4 in A.3 then follows:
20
2
´ ´ ´( )(1 )( ) ( )
H H H B B B
H HB B
B B
q dk q q dk dkdq qq qq q
θξξ θ θγ γ
= − −−+ +
(A.5)
The partial elasticity of ξ respect to resource intensity in basic education is given by:
2
´ ´( )(1 )( )
( ) ( ) 0(1 ) (1 )
B B
B B B B B
B H B B B
HBB
B
k qPk H q k q k H
k q q k kqk qq
s s orθ θξ
θξξ θ θγ
ε εε ε ε
θ θ
∂ = − + =∂ −+
= = − + = − > <− −
(A.6)
Where: B B B B
B B Bk q q k
B B B
q q kq q kθ θ
θε ε εθ
∂−∂= =∂ ∂
and HH
H
qsf
= .
The partial elasticity of ξ respect to resource intensity in high education is given by:
´( ) H H
H H H HH q k
H B H H
k q q k sk q q q
ξ εξ γ∂ = =
∂ + (A.7)
To find the global effect of capital intensity in basic education, it is necessary to
consider the budget restriction: Considered now the budget constraint associated with the total
amount of education expenditure:
(1 ) (1 )
(1 )
EE
B B H H B B B H B HB
BH
KK E k E k E k E k k k kE
k kk
θ θ
θ
= + = + − ⇔ = = + −
−=−
(A.8)
The total amount of education expenditure EK must be divided between basic and
high education and taking into account the drop out function. This implies a relationship
between capital intensity in high and basic education. So equation 11 and figure 2 show the
set of points in the education capital intensity space such that the budget constraint is fulfil. It
is easy to show that is a decreasing function:
2
(1 ) ( ) ´ 1 ´( ) 0(1 ) ( ) (1 )
( )( ) (1 ) (1 ) (1 )B B
H BH
B B
H B B Bk k
B H B H
dk k k kdk k k
dk k k kdk k k k kθ θ
θ θ θθ θ
θ θε εθ θ θ
− − −= − = − − <− − −
− = + = +− − − −
(A.9)
21
It can be shown that 0Bd dkξ > when
22
2
2
´ (1 ) ( ) ´ ´1 ´( )(1 ) (1 )( ) ( )
(́(1 ) ( ) )́ ´1 1 ´( )(1 ) (1 )( )
(́(1 ) ( ) )́ ´( )( ) (1 )
H B H B
H HBB B
B B
H B H B
HB BB
B
B H B H H B B B
B B H H H B
q k k q qdq qdk q qq q
q k k q qdqdk qqq
k q k k q k k k qddk q q k q q
θ θξ θξ θ θ θγ γ
θ θξ θξ θ θ θγ
θ θξξ γ θ
− − −= − − + =− −+ +
− − − −= − + +− −+
− − −= − + ++ −
2 2
´(1 )
((1 ) ( ) ´ ((1 ) ( ) ´ ´( )(1 ) (1 ) (1 ) (1 )
( ) ( )(1 ) (1 )
(1( )
H H H H H H
B B B B
B B H H H B H H H B B B
B B B
H H H B B B
B
B B B B B Bq k q k q k
H H H
k q q kH q k q k k k H q k k k q k
q k qH
q k k k q k
k
k k k k k k k kk k k
s s
s
θ θ
θ
θθ θ
θ θ θ θ θε ε εθ θ θ θ
ε ε εε ε ε ε ε ε
θ θε ε
ε ε ε
− =−
− − − − − −= = −− − − −
= − + = − + ⇒− −
>+
)θ−
(A.10)
If B B H Hq k q k qkε ε ε= = then:
( )1
1
( )1 1
( 1 ( )1 1
(1 ) (1 )( (1 )) (
(1 ) (
B
B H B
H B B
B
B B
B
B
B B
qk qk H qk qk k k
Bk k qk q
H
qk q Bk H qk qk qk q
H
q Bqk H qk q
H
q H qk q H qkBqk H qk
H
s
KK
KsK
KsK
s sKsK
θξ
θ
θξ θ
θθ
θ θ
ε εε ε ε ε
θθε ε ε
θε ε θε ε ε ε ε
θ θε θε ε ε
θ θε θε ε θε
ε εθ
= − +−
= +−
⎛ ⎞= − + + =⎜ ⎟− −⎝ ⎠
⎛ ⎞= − + + =⎜ ⎟− −⎝ ⎠
− −= − + =
−( ))
1 )(1 ) (1 ) (1 )( ) (1 )(1 ) 1 (1 ) (1 )
B B
B
HB
q H qk q H qkH Hqk B
B B q H qk
KsK K
s ss sBB B s
θ θ
θ
θε θε ε θε θε
θ ε θε
− =− −
− − −− = − −− − − −
(A.11)
We can obtain B
ddk
ξ using the same assumptions and particular functions for the early
drop out variable BlqBeθ −= and the education production functions B Bq Ak α= and H Hq Ak α= .
We assume a neutral assumption about efficiency and the quality elasticity in education to
capital intensity ( B HA A A= = and B Hα α α= = ).
22
( )( (1 ) ( )
(1 )
1 ( ((1 )( ) )
( ((1 )(1 ( )) )1
(1 )( ((
B
B B H H H B
B
B B H H H B B
B
B B
B
k B HB B HB q k q k k k
B H B B B
k Hq k q k k k k
B B B
qqk H Bqk q q
B B H
qqk H
B B
q qk q qd kdk f k q q
qddk k q
q Kk q K
qk q
θ
θθ
θθ θ
θ
ε γξ θ θ ε ε εξ θ θ θ
ε γξ θ ε ε ε εθ θ
ε γε θθ ε ε εθ θ θ
ε γε θθ θ
+= − − +
−
= − − + − =
− − + + − =−
−− +
2
2
2
(1 )( )) )
(1 )(1 )( (( ( )) )
(1 )( ( ) ( )(1 )
(1 ) 1( (1 ) )(1 )
B
B
qBqk q
B
B H B BB
B B B
HB H B
B B B
H B
B B H B B
KK K
lq q K lqlqk q K K
ql q q lqk q E
q l qddk k q q l E
θθ
εθ ε εθ θ
α γ θ αα θ αθ θ θ θ
α θα γ α θθ
αξ θγ αθθ α
− + − =−
−−− + + − =−
−+ − + =−
−= + − −−
(A.12)
The effect of the exogenous parameter (B) of the early drop variable in the elasticity:
2
(1 )( ))
1 1
(1 )0
(1 )
B
B
B
q H qk Hk qk
B
H Bk
s sB
s lqB
B
θξ
ξ
ε θεε ε
θθα αε
αθ
−= − ⇒
− −
−∂= >
∂ −
(A.13)
The first order condition of the objective functions over human capital
accumulation:
23
( (1 )) ( ) ( )( )
( ) (1 )
(1 )(( ) ) 0
(1 )( ) 0
(1 )(1 ) 0
B B B B
B B B B
B
B B B BB
B B B
B B B B B B B
B B B B
B B B B B
B B B B B
q k k k
q k q k
k
d E q d q d qEdk dk dk
q d dq d q q d q d dq q ddk dk dk dk
q k d k dq k dk dk q dk dk
θ ξ
θ ξ
ξ
θ ξ θ θ ξ
θ θ ξ θ θ ξ θ θ θ ξξ
θ ξ θ ξξθ ξ ξ
ξε ε εξ
ξε ε εξ
ε
+ = + =
+ + ++ = + +
++ + =
+− + =
+− + =
= (1 )
(1 )(1 ) (1 )
B B B
B B B H H H
B B B
B B B
B B B H H H
q k qH
q k k k q k
q k qq k q H
q k k k q k
s
s
θ
θθ
ε εθ
ε ε εε εξε ε θ
ξ ε ε ε
− −+
+− + = −+
(A13)
If the objective function is minimize time to arrive to certain level of the ratio then the
first order condition is the following:
( ( 1))( ) ( )1( )
( ) ( )
( ( 1)) ( )1 1( (1 ) )
( )(1 ) 1 1( (1 ) (1 )) (1 )B B B
B B B
B B B H H H
B BB B
BB B B
B B
B B B B
B B BB
B B B B
q k q B Bq k q H
q k k k q k B
d E qd q d qE
dk dk dkd q d qdHC
dk E dk dk
d E q d qdHCEdk E dk dk
q d qsk dk
θθ
ξθθ ξ θξ
ξθ ξ θ
θ ξξ θξ ξ
ε ε θ ξ θξε ε θξ ε ε ε ξ ξ
−= − =
= −
− = − +
+− + − − − ++
(1 )
( )1( (1 ) (1 ))
(1 )( 1) ( (1 )) 0
( )(1 ) (
B B B
B B B
B B B
B B B H H H
B B B
B B B
B B B H H H
B B B
B B B
B B B H
B
B B Bq k q
B B
q k q B Bq k q H
q k k k q k B B
q k qq k q H
q k k k q k
q k qq k q
q k k k
q d dq qdk k
q d qsk dk
s
θ
θθ
θθ
θθ
θ θ θε ε
ε ε θ ξ θε ε θε ε ε ξ
ε εξ ξε ε θε ε εξ ξ
ε εξ ξε εξ ε ε ε
+ = −
− + − − −+
− − + − − =+
−− ++
(1 )H H
Hq k
s θ= −(A.14)
24
1
(1 ) (1 )(1 ) (1 )(1 ) (1 )
(1 )(1 )( (1 ))(1 )(1 )
( (1 ))
1(1 )( 1)1
( (1 ) ( ) (1 ))(1 )(1 )
( (1 )
B
H HB B
q H qk B H
HB
B B H
BB
BH
BB
B B
B BB
s sB Bs lq s
qBlq q q
B qlqq
BBlA k k k
B BlA k
θ
α
α α α
α αα α
θ θε θε α θα
θα γ θα
θγα θα
θ
α γ θ θαθ
α γ θ−
− −− − = − − =− −
−= − − =+ −
−− − =+ −
− −− =
− + − −−− =
− ( ) (1 ))Bk k α θα+ − −