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: rossana@decon.edu.uy. 2 Departamento de Economía, Facultad de Ciencias Sociales, email: marcel@decon.edu.uy.

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+ − −