OPTIMAL EDUCATIONAL CHOICE AND HIGHER EDUCATION FINANCING:
EFFICIENCY OF COSTS RECOVERY
RUSSAYANI ISMAIL
UNIVERSITY OF EXETER
R.Ismail @ exeter.ac.uk
School of Business and Economics
University of Exeter
Streatham Court, Rennes Drive
EX4 4PU
United Kingdom
1.0 Introduction
Education and earnings are positively linked and the study of the private returns to investment in
higher education around the world shows an average return of around 19 percent (Psacharopoulos &
Patrinos, 2002). Despite this overwhelming evidence of private benefits, higher education in general
is heavily subsidised by governments. On average OECD countries devote 12.7% of total public
expenditure to educational institutions and the expenditure has grown faster than total public
spending. Public subsidies for students are evident mainly at the tertiary level where on average
17% of public spending on tertiary education is devoted to supporting students, households or other
private entities (OECD Indicators 2004). Externalities, the existence of capital market imperfections
and the use of education as a redistributional policy instrument, justify the government's provision
of education. Among these three reasons, capital market imperfections appear to be the strongest
basis for the government's subsidization of higher education with many believing externalities to be
the least. As pointed out by many economists, such as Rivlin (1961) and Trostel (2002), these
externalities are rather vague and difficult to quantify. The study by Wyckoff (1984), which tried to
quantify the spillover benefits of education, found that the actual benefits are less than that usually
asserted by the proponents of public provision of education.
Over the last quarter century most governments have developed higher education systems which
combine grants from public funds, repayable loans and some private contributions as a scheme to
support those who intend to invest in higher education. It is a general belief that if education is
publicly provided, efficiency and equity are enhanced. However with constraints on the government
expanding higher education such a belief may no longer be true. Globally, over the last decades
pressure for reform in higher education financing has intensified. As the demand for higher
education expands, and with the government facing a very tight budget, heavy reliance on the
public purse to finance higher education no longer seems to be an ideal solution. As suggested by
Woodhall (1989):
1
`During the 1980s there have been significant changes in the level and mechanism of funding
higher education institutions in many countries …….., the institutions have been encouraged or
force by economic pressures to seek new sources of funds. These trends are likely to continue; the
funding of higher education institutions and the balance between public and private finance for
teaching and research will remain a subject of political debate in many countries.'
Though very controversial, the idea of cost recovery has become an alternative solution. According
to Albrecht and Ziderman (1992) cost recovery refers to the revenue generated from charging
tuition fees or delayed cost recovery which refers to the tuition deferment through the introduction
of loans or a graduate tax. Basically cost recovery indicates revenue generated from those that
directly benefit from education. Substantial private returns from investment in higher education
(Blundell et al. 2000), suggests that it is efficient that graduates bear some of the costs. Thus it is in
the intention of this paper to examine whether the move towards greater cost recovery will bring
about efficiency in higher education.
Among the previous studies which concentrate on higher education and take a public finance
perspective are Green and Sheshinki (1975), Creedy (1995) and Penalosa and Walde (2000). They
focused on the distributional effects of higher education subsidies. Penalosa and Walde (2000) took
further steps to focus on three main types of higher education funding i.e., traditional tax-subsidy, a
pure loan scheme and a graduate tax with the government objective of attaining efficiency, equality
of lifetime incomes and equality of opportunity. They found that the traditional system of taxsubsidies failed to achieve all these targets simultaneously and that loan schemes and the graduate
tax fare better. Green and Sheshinki (1975) and Creedy (1995) analysed the implications of the
government funding of higher education when there are externalities but abstract from capital
market imperfections and uncertainties where as Penalosa and Walde (2000) analysed the efficiency
and equity effects of subsidies to higher education under both imperfect capital market and
uncertainty. Under the assumption that the government maximises a utilitarian social welfare
function and with an exogenous quantity of funds available for higher education, Green and
Sheshinki (1975) found that increases in this given amount lead to a more regressive distribution of
government funds. They also found a positive relationship between the size of the externality
generated by higher education and the regressivity of the distribution. In contrast to the assumption
of the exogenous total amount available for higher education, Creedy (1995) analysed the general
equilibrium implications of the government budget constraint and how majority voting affects
education policy. His study reinforced the results obtained in Green and Shishenki (1975).
This study differs in several aspects with regard to the previous studies. Firstly, the analysis
abstracts from capital market imperfections and externalities. The only distortion in the market is
caused by the government method of financing. Secondly it focuses on the two main methods of
2
financing higher education i.e subsidised student loans and merit-based grants (scholarships). Under
the perfect capital market assumption, the educational choice of an individual will be modelled by
taking into consideration these two schemes simultaneously. However, similar to Green and
Shishenki (1975), the paper assumes that the total amount available for higher education funding is
exogenous and the government's objective is to maximise a utilitarian social welfare function. Apart
from that the analysis takes the same perspective as Creedy (1995) which considered the financing
of higher education as a `dual decision' aspect in public economics. First, individuals make
decisions to maximise their utility subject to certain constraints and, secondly, taking the
individuals' decisions to invest in higher education, the government then maximizes some social
welfare function. In addition there is no disutility of labour supply and higher education is
considered as a homogeneous product (to abstract from differences between academic subjects).
Furthermore, in this model government is the only provider of higher education. There is a
continuum of individuals who differ both in their ability and initial endowment (wealth). Higher
education is regarded purely as an investment good (so disregarding the consumption motive and
screening or credentialism).
In order to understand the issues at stake, the paper describes the first best solution. In the first best,
it is assumed that the government is able to achieve any level of desired redistribution through noncostly transfers, abstracting from redistributive issues. Efficiency means that the government choose
the threshold ability level of those who should take higher education so as to maximize the total
income in the economy after paying for the costs of education. Realising the fact that higher
education is heavily subsidized by the government and with the availability of scholarships and
subsidized government loans, the analysis focuses on positive effects on how these policies affect
the participation of population in higher education. Furthermore the paper attempts to assess the
relative efficiency of the policies by considering the effects of their variation upon welfare, given
the objective of the government is to maximise a utilitarian social welfare function.
The following section (Section 2) briefly discusses the model. Section 3 determines how population
choices are related to the policy instruments and characterises the welfare of society. Section 4
underlines the perspective of the paper. Section 5 introduces a specific functional form for the
utility function and then performs some policy experiments. The results are then presented. Section
6 draws a conclusion.
2.0 Educational Choice
In this section, the analysis begins by modelling an individual's choice of higher education with a
perfect capital market where the lending and borrowing rates are the same. Education is considered
purely as an investment (Becker, 1975), with no consumption motives integrated into the model and
3
the returns to human capital investment are known with certainty.1The intertemporal choice model
within the utility maximisation framework is similar to Kodde and Ritzen (1984). The departure
from the static model of utility maximising behaviour seems to be appropriate in the discussion of
educational choice since it involves investment in human capital which in turn will affect the
income streams of individuals. Thus any future changes, particularly in income, must be taken into
account in the present decision. The decision over time may involve many future time periods but
for the purpose of easy exposition this analysis is confined to a simplified two-period model as in
Eaton and Rosen(1980), Boadway & Bruce (1984) and Varian (1992).
Let the superscript 1 or 2 denote the period of life. The general form of utility function for an
individual can be written as
U  U 1 ( x1 )  U 2 ( x 2 ) ,
(1)
where U 1 ( x1 ) and U 2 ( x 2 ) are the utility derived from consumption of goods in periods 1 and 2
respectively and δ, which is the level or degree of impatience describes individual preferences over
time. Per-period utility functions are strictly concave and exhibits diminishing marginal utility. In
the absence of any consumption loan, the lower is the value of δ (the more impatient the individual
is), the more likely he is to choose not to invest in higher education. However with the presence of a
consumption loan, the value of the interest rate, r, will determine the individual's intertemporal
allocation. Low market rates of interest could make investment in higher education worthwhile. In
addition there is no disutility from the supply of labour i.e., we abstract from the demand for leisure,
so that utility maximization amounts to maximization of discounted wealth in a perfectly
functioning capital market.
The decision of an individual on whether or not to acquire higher education takes place in the
first period of life. Studying entails a direct financial cost, c, and also involves forgone earnings.
However an individual choosing education will benefit from future returns in terms of higher wages
in the second period compared to remaining uneducated. The choice of the individuals can only take
either one of the two values: no education (ne) and education (e). The choice between ne and e is
based on assessment on the net present value of life time earnings.
Table 1: Education-Wage Relationship
Level of Education
Period 1
Period 2
1) Without education (ne)
Work: Wage = wne
Work: wne ( a )
2) With Education (e)
Study: Cost = c
Work: we ( a )
1
1
2
2
The importance to consider the consumption motives apart from investment has been viewed upon by Blaug(1976)
and Kodde and Ritzen (1984).
4
Table 1 summarises the situation faced by the individual, where the subscripts denote the level of
education. If a person chooses not to acquire higher education, he will be able to work in period 1
and get the wage of w1ne . Ability denoted by a, is stochastically determined at birth. The wage in the
first period is not a function of ability since at this point in time the employer cannot determine the
ability of the worker. However as he works through time, his wages will change from w1ne to
2
wne
( a ) in period 2. The motivation for this assumption is based on the observation that, after a
certain period of working, the employer is able to identify the ability of the worker and pay a wage
according to his or her ability. On the other hand, if a person chooses to take higher education, he
will not only forgo the benefit in term of the first period wage but also have to face the educational
cost, c in period 1.2 The wage for an educated individual in period 2, which is denoted by we2 (a ) , is
2
( a ) representing that
assumed to be more than the wage of uneducated individual, we2 (a ) > wne
higher education has a positive effect on wages. Our assumption about the wage structure follows
Bowles (1972) where he posits that productivity and thus wages depend on education. 3 In addition
it is important to note that the wage function is increasing in ability i.e:
2
wne
w 2
 0 and e  0.
a
a
2.1 No Intervention
The educational choice decision is analysed first for the case of no intervention by the government.
This provides a baseline from which the later results can be judged. The decision to acquire higher
education or not is made by comparing utility with or without higher education. Let U ne and U e
refer to utility with no education and with education respectively. The utility function can then be
written as
U j  U i ( x ij )  U i ( x ij ) {i=1,2 and j=ne,e }
(2)
where x ij is consumption in period i given choice j. Let p¹ and p² be the two corresponding price
levels in the two periods and for the purpose of simplifying the analysis, both p¹ and p² are
normalized to one.
We will first begin the analysis by finding the maximum attainable utility without education.
Without education, an individual works in periods 1 and 2. If they begin period 1 with an initial
2
Let c denotes only the fees charged to those enter higher education even though a direct financial costs may also
includes other maintenance cost such as lodging, books and etc. This assumption will not in any way affect the result of
the analysis.
3
Following Arrow (1973), it is irrelevant for individual's investment point of view whether education enhances
productivity or functions as a market signal. What is important is the wage-schooling relationship.
5
endowment (wealth), m, and can save or borrow at interest rate, r, the inter-temporal budget
constraint is
x1 
w 2 (a)
x2
 m  w1ne  ne
1 r
1 r
(3)
An individual choosing no education then solves the following problem
2
MaxU ne  U 1 ( x1ne )  U 2 ( xne
)
(4)
2
{x1ne , xne
}
subject to (3). The solution can be written as the pair of demand functions:
i
i
2
x ne
 x ne
((1  r ), m, w1ne , wne
(a);  ) i=1,2
(5)
We can then construct an indirect utility function



2
2
2
Vne (.)  U 1 x1ne ((1  r ), m, w1ne , wne
(a);   U 2 x ne
((1  r ), m, w1ne , wne
(a); 

(6)
Vne (.) is the maximum utility attainable by an individual with ability a and income m if he chooses
not to obtain higher education.
The overall budget constraint for an individual who decide to take education is
we2 (a)
x2
x 
 mc
1 r
1 r
1
(7)
An individual who chooses education then solves
MaxU e  U 1 ( xe1 )  U 2 ( xe2 )
(8)
{xe1 , xe2 }
The solution can again be written as the pair of demand functions
xei  xei ((1  r ), m, we1 (a), we2 (a);  ) i=1,2
(9)
The indirect utility function can be constructed as



Ve (.)  U 1 xe1 ((1  r ), m, c, we2 (a);   U 2 xe2 ((1  r ), m, c, we2 (a); 

(10)
Ve (.) is the maximum utility attainable by an individual with ability a and income m if he chooses to
obtain higher education. With a perfect capital market utility maximization can be separated into
two stages (the separation theorem). In the first stage, an individual will allocate time between
education and work to maximize the present value of wealth over the life cycle and in the second
stage, given the optimal wealth, consumption is planned to maximize utility. In this model, the
separation theorem applies in the sense that an individual will choose in the first period either to
obtain education or not to maximize the present value of lifetime income and secondly, based on his
preferences he will maximize utility. It is important to note that in a perfect capital market where
the rate of borrowing is the same as the lending rate, the level of initial income will not affect the
6
choice of education. 4 Figure 1 shows the budget constraint facing by an individual in two different
circumstances, i.e., without education (line AB) and with education (line CD). In this example,
taking education will result in a higher lifetime income compared to without higher education. The
separation theorem says that when confronted with this situation, a rational individual will first
select the budget constraint which will give him the largest opportunity set (line CD). Then in the
second stage, faced with this budget constraint he will choose the most preferred allocation between
current and future consumption to maximize his utility. This is the point where the marginal rate of
substitution between future and current consumption is equal to the slope of the budget line. Figure
1 depicts three different equilibrium points of utility maximization. Point e₁
refers to an
equilibrium for a saver, point e₂ refers to someone who neither borrows nor saves and point e₃
refers to a borrower .
2
w e ( a )  ( m  c )(1  r )
C
e1
2
1
w ne ( a )  ( m  w ne )(1  r )
Future Consumption (x2)
2
we (a )
A
e3
Slope=-(1+r)
2
w ne ( a )
e2
B
mc
1
m  w ne
2
w ne ( a )
1
m  w ne 
1 r
D
mc
w2 ( a )
e
1 r
Present Consumption ( x1 )
Figure 1
The educational choice or investment decision by an individual can be illustrated using Figure 2. An
individual with ability, a and initial income, m will decide to acquire higher education or not by
4
Please refer appendix.
7
comparing the maximum utility attainable with higher education Ve (a, m) to that obtainable without
higher education Vne (a, m) . Whichever is higher for the given level of a and m will determine the
choice. Assuming it is not optimal for everybody to invest in higher education, there is a single
point at which the two curves cross, corresponding to an ability level, a. Following Creedy (1995)
this point is described as the educational choice margin. For those individuals with ability below a,
they will find that the investment in higher education is not worthwhile and those who are above it
will invest since the returns from education outweigh the costs.
Figure 2
V
e
V
ne
Utility
â
Ability
Indirect utility is a function of prices and income. The higher is income, other things being equal,
the more opportunities are available and higher utility is achieved. However initial income has no
effects on individual's choice for education. This is because both Ve (a, m) and Vne (a, m) rise to an
equal degree with income so their intersection remains at the same value a. This observation
permits the population to be partitioned on the basis of ability alone.
2.2 Financial Support
Assume that for exogenous reasons (possibly for political motives) the government intervenes in the
educational market to provide financial support for students. There are two types of financial
support available for those who decide to invest in education: a loan scheme, provided by the
government below the market rate of interest and a scholarship. For the loan support there is no
threshold ability required for entitlement. Once an individual chooses to enrol in higher education,
the loan will be made available. However, in order to be eligible for a scholarship, the student must
8
meet a certain ability level, a, which is determined by the policy maker. It is assumed that those
awarded scholarships cannot also take out a loan.
r the subsidised rate of interest.
Let g be the value of the scholarship, l be the value of the loan and ~
It is assumed that the value of government scholarship is larger than the discounted value of the
difference between the market rate of interest and the subsidized interest rate on the value of loan,
r
r~
so g  l 
 . This assumption ensures that at any ability level, a scholarship is preferred to a
 1 r 
r
r~
loan. It is also assumed that both g and l 
 are less than c.
 1 r 
With the introduction of financial support into the model, the educational choice of an individual
can be either: no education (ne); education with loan (eL); or education with scholarship (eS).
Letting the subscript denote the level of education chosen and the superscript the time period, the
utility levels from each choice are
U j  U i ( x ij )  U i ( x ij )
{i=1,2 and j=eL,eS }
(11)
Given the known structure of wages, individuals with different abilities and incomes will decide
whether to invest in higher education by comparing the utility from each choice. Each individual
will maximise utility based on the different budget constraints in three different circumstances. We
will now consider each situation in turn.
2.2.1 Education with Scholarship.
If an individual has ability a  a , which qualifies him for a scholarship, the budget constraint is
1
xeS

xeS2
w 2 (a)
 mc g  e
(1  r )
(1  r )
(12)
The demand choices will satisfy,
MaxU  U 1 ( xes1 )  U 2 ( xes2 )
(13)
{xes1 , xes2 }
subject to (12). The solution can be written as the demand functions
i
i
xeS
 xeS
((1  r ), m, c, g , we2 (a);  )
{i=1,2}
(14)
Using these demands we can construct an indirect utility function as
1
Ves (.)  U 1 ( xeS
((1  r ), m, c, g , we2 (a), r ;  )  U 2 ( xeS2 ((1  r ), m, c, g , we2 (a), r;  )
(15)
Where Ves (.) is the maximum attainable utility if an individual chooses to take higher education
and receives a scholarship.
2.2.2 Education with Loan
If an individual with ability a  a chooses to take higher education, and thus must finance it with a
loan, his inter-temporal budget constraint is
9
1
xeL

2
xeL
w 2 (a) (1  ~
r )l
 mcl  e

1 r
1 r
1 r
(16)
r  r implying a subsidized rate of interest.5 An individual will maximize utility
with ~
2
MaxU  U 1 ( xeL )  U 2 ( xeL
)
(17)
1
2
{xeL
, xeL
}
The solution will be the pair of demand functions
i
i
xeL
 xeL
((1  r ), m, c, l , ~
r , we2 (a);  ) {i=1,2}
(18 )
Thus we derive an indirect utility function in the form
1
2
VeL (.)  U 1 ( xeL
((1  r ), m, c, l , we2 (a), ~
r ;  )  U 2 ( xeL
((1  r ), m, c, l , we2 (a), ~
r ; )
(19)
where VeL (.) is the maximum attainable utility of an individual with ability a  a and income m
who chooses to take higher education. For a person with ability a, the educational choice results
from comparing the utility levels determined by Vne (.) , VeS (.) and VeL (.) . Given our assumption that
r
r~
g  l
 , then VeS  VeL for any value of a. Hence, if an individual is eligible for a scholarship
 1 r 
they would rather accept it than take a loan. Thus, as shown in Figure 3, there will be two crossing
points of interest. The first intersection occurs when V ne crosses Ves
at ability level a S . This is
referred to as the educational choice margin for an individual who receives a scholarship. An
individual with ability level a S is indifferent between not investing in higher education and
investing if they were to receive a scholarship. Below the ability level a S , it is not worthwhile to
invest since the cost of taking up higher education (the loss of first period wage plus the direct cost
of the fees) is larger than the returns even if a scholarship were granted by the government. Above
a S , an individual with a scholarship will find it is worthwhile to invest. The second intersection
occurs when the curve V ne crosses VeL at the ability level a L . This point is the educational choice
margin for an individual indifferent between investing and not investing in higher education when
he takes out a loan. An individual with a loan will only find it is worthwhile to invest if he has the
ability level above a L . The choice made by an individual of ability a will depend upon where the
scholarship entitlement level, a , is situated relative to a S and a L . This is analysed in the next
section.
5
With the purpose to encourage student to borrow, most of the loans programmes provide some interest subsidy. The
rate of interest charged varies, with certain countries are not charging interest at all like in West Germany or in Pakistan
where interest charge is considered as usury and prohibited by the Islamic law.
10
Figure 3
V
eS
V
eL
V
ne
Utility
VeS  VeL
as
aL
Ability
3.0 Population Welfare
Section 2 has described the individual choice of higher education in a perfect capital market with
government intervention in providing scholarships and loans affects the choice. This section
determines how population choices are related to the policy instruments and characterises the
welfare of society.
Assume that there is a continuum of individuals in the economy, who differ in their ability to
benefit from education and in their initial endowment. An individual is characterised by the pair
(a,m). Ability and income are distributed in the population with the density function f(a,m) where
f(a,m) >0 for a min  a  a max , mmin  m  mmax .
The threshold ability level for the scholarship entitlement, a , chosen by the government will affect
the educational choice of an individual depending on their value of (a,m). In deriving this
interdependence the first point to observe is that the value of income will not affect the choice. An
increase in income raises the values of V ne , VeS and VeL equally so the intersection points are
unaffected. The introduction of scholarships and loans does not therefore affect the observation that
the population can be partitioned into educational choices by level of ability.
The value of the scholarship entitlement can fall into one of three ranges.
i) 0  a  a S
11
In this case individuals with ability level below a S will choose not to take higher education. Hence
some of those entitled to scholarships will prefer to remain uneducated. Those with ability level
a S and above will choose to take education with a scholarship. No-one will take up a loan.
ii) a S  a  a L
Individuals with ability level below a will choose not to be educated. Those with ability level
a and above will choose education with a scholarship. No-one takes up a loan.
iii) a  a L
Individuals with ability level above a L but below a will choose education financed by a loan.
Individuals with ability level a and above choose education with scholarships.
These three cases can be summarised by noting that the population will therefore be partitioned
according to ability as follows:
i) No education: amin  a  max( a S , min( a , a L )) ;
ii) Education with a loan: min( a , aL )  a  a ;
iii) Education with a scholarship: min( amax , max( a , a S ))  a  amax ;
We assume that the government measures the economic welfare that results from a combination of
loans and scholarships using a utilitarian social welfare function. This welfare function aggregates
the individual welfare levels and subtracts the cost of the education policy to arrive at a measure of
net welfare. The government allocates its revenue between various government projects with part of
the expenditure going to higher education. By assuming that education is only one part of the
general government budget, we can abstract from the need to trace the source of government funds.
Let E be the government expenditure required to finance a place for one student. When a place in
higher education is financed through scholarships, the costs incurred by the government in
providing one place is the total expenditure, E, plus the government scholarship, g, minus the fee, c.
So, letting E S be the net cost of providing education through a scholarship, E S  E  g  c . When
higher education is financed through loans, the actual costs of providing one place is E minus c plus
(r  ~
r )l . Hence, defining E
as the net cost of providing education through
L
loans, EL  E  c  (r  ~
r )l .
We now turn to the analysis of the welfare effects of scholarships and loans. Total welfare is
calculated by summing the utilities of the population in three different circumstances, i.e without
education, education with scholarship and education with loans based on the ability level. To
determine the net welfare it is necessary to convert costs into the same units of measurement. This
is done by multiplying the cost by α, where α is the marginal social utility of income defined by
12

mmax max( a S , min( a , a L ))


mmin
amin
mmax
dm
amax

+
dVne( a ,m )
dVeS ( a ,m)

mmin min( amax , max( a , aS )
mmax
a

+
dVeL ( a ,m )

mmin min( a , a L )
f (a, m) da dm
dm
dm
f (a, m) da dm
(20)
f (a, m) da dm
The level of net welfare function can then be defined by
W  max 
mmax

mmin

mmax

mmax
mmin
mmin

mmax

mmax
mmin
mmin
max( a s , min( a , a L )
amin

amax

a
min( amax , max( a , a s )
min( a , a L )

a

amax
Vne (a, m) f (a, m) da dm
VeS (a, m) f (a, m) da dm
VeL (a, m) f (a, m) da dm
min( a , a L )
(21)
( E  c  (r  ~
r )l f (a, m) da dm
min( amax , max( a , a S )
( E  C  g ) da dm
4. Effects of Policy
The perspective of this paper is to accept the observed fact that governments intervene in the market
for education by providing subsidised loans and ability-dependent scholarships. The reasons why
these policies have been adopted is not addressed, they may be the outcome of the political process
or have been introduced for redistributive reasons. Instead, we are interested in the positive effects
of these policies in terms of how they affect the population that participates in higher education. We
also consider the effects that variations in the policies have upon welfare in order to assess their
relative efficiency.
5. Specification
In order to derive precise results, we adopt a specific functional form for the utility function, the
distribution of ability and income and the relation of wages to ability. It is unlikely that the choice
of functional forms has any significant effect upon the nature of the results. It is assumed that
preferences can be described by the logarithmic utility function
U i  log xi1   log xi2 i  ne, eS , eL
(22)
Given these preferences the indirect utility functions for the three cases are
2
 (1  r )m  (1  r ) w1ne  wne
 (1  r )m  (1  r )w1ne  wne2 (a) 
(a) 
Vne  log 


log



(1   )  (1  r )
(1   )




13
(23)
 (1  r )m  (1  r ) g  we2 (a)  (1  r )c 
 (1  r )m  (1  r ) g  we2 (a)  (1  r )c 
VeS  log 


log



(1   )(1  r )
(1   )




(24)
 (1  r )m  (r  ~
 (1  r )m  (r  ~
r ) L  we2 (a)  (1  r )c 
r ) L  we2 (a)  (1  r )c 
VeL  log 


log



(1   )(1  r )
(1   )




(25)
For those not choosing education the second-period wage is assumed to be
2
wne
(a )  w(1  a )
(26)
The second-period wage following participation in higher education is
we2 (a)  2w(1  a)
(27)
The fact that education increases productivity is reflected by the fact that we2 (a ) is higher than
2
wne
( a ) . The paper assumes a linear ability-earning relationship. This assumption supported by
Tobias (2003) which showed that the ability-earnings relationship was roughly linear for those
investing in education. Ability and income are assumed to be uniformly distributed across
population with
f ( a , m) 
a max
1
1
.
 a min mmax  mmin
(28)
The values of the other parameter are given in Table 2.
Table 2: Parameter Values
w

E
amin
a max
r
mmin
mmax
12000
2
20000
0
5
0.06
0
10000
The policy variables whose effects are studied in the following experiments are the value of the
government scholarship, g, the value of government loans, l, the interest rate on government loans,
~
r , and the fee for education, c.
5.1 The First Best
In order to evaluate the effect of government intervention in higher education, we construct the
first-best equilibrium in which the government faces no constraint in redistributing income. 6 The
first-best is obtained by maximizing total income in the economy net of the costs of higher
education by selecting the number of people who receive higher education. Equivalently, the level
of ability that forms the threshold between education and no education is chosen. Once the set of
consumers to be educated is chosen, income can be reallocated by lump-sum transfers
6
In this paper it is assumed that individuals have the same preferences (identical utility functions) and hence income
should be distributed equally as to maximise utilitarian social welfare. In addition, due to redistribution, initial income
does not matter.
14
Define a~ fb as the threshold level of ability that yields the maximum total income in the economy.
Given this value, there are two groups of individual defined by ability as follows:
i) amin  a  a~ fb : not be allocated to higher education;
ii) a~ fb  a  a max : allocated to higher education.
The total income in the economy is the present value of lifetime income of these two groups of
people. The present value of lifetime incomes for the first group is the unskilled wage they received
in the first period plus the present discounted value of the second period wages, w1ne 
2
wne
(a)
. For
(1  r )
the second group, the lifetime contribution to total income is the present discounted value of
second-period wages minus the cost of providing higher education,
we2 (a)
 E . Given that the total
(1  r )
income in society is to be maximized, the threshold ability, a~ fb is defined by
mmax
a~ fb  arg max 

mmin
1 mmax
1  r mmin
1 mmax

1  r mmin

 E
mmax
mmin

amin

a~ fb

amax
amin
amax
a~ fb
a~ fb
w1ne f (a, m) da dm
2
wne
(a, m) f (a, m) da dm
a~ fb
we2 (a, m) f (a, m) da dm
f (a, m) da dm
The first-order condition for this optimisation is
2 ~ fb
wne
(a ) we2 (a~ fb )
w 

E
1 r
1 r
1
ne
At the threshold level a~ fb the marginal benefit from investment in higher education is equal to
marginal cost, so total income in the economy is maximized. Using the (30) the threshold ability
level in the first-best situation is given by a~ fb =1.8267.
5.2 Policy Experiments
In the policy experiments the values of the scholarship, loan, subsidised rate of interest and the
student fee are varied from a baseline case. This provides an insight into their effect upon welfare.
Table 3: Initial Policy Variables
~
l
g
c
r
6000
6000
0.04
20000
15
The baseline values of the parameters are shown in Table 3. In the baseline case the student fee is
equivalent to the exact cost of providing education, meaning that the fee is not subsidised. At this
value of fee, the government is only intervening in the educational market through the scholarship
and the subsidised rate of interest on the student loan.
5.2.1 Value of Loan
The first experiment considers the effect of changing the value of the loan for various scholarship
entitlement levels. Since the minimum level of ability amin  0 , a value of a  0 implies that the
entire population qualify for a scholarship. Similarly, a  5 matches the maximum level of ability
and so is equivalent to there being no scholarships. The results for a loan of l=12000 and for a loan
of l=20000 are shown in Table 4.1 and 4.2. Wne , WeS ,WeL refer to welfare without education,
education with scholarships and education with loans respectively. TW is the total welfare and
NW is the net-welfare . E L , E S are cost of financing through loans and cost of scholarships
respectively and TC is the total cost.
Table 4.1: l  12000
a s  1.2967 , a L  1.8067
a

0
1.0
2.0
3.0
4.0
5.0
0.00005101 0.00005101 0.00005162 0.00005242 0.00005285 0.00005313
Wne
7.6816
7.6816
10.7825
10.7825
10.7825
10.7825
WeS
23.7912
23.7912
19.4718
13.1377
6.6346
0
0
0
1.1877
7.4698
13.9341
20.5379
TW
31.4728
31.4728
31.4420
31.3900
31.3512
31.3203
ES
0.2267
0.2267
0.1858
0.1258
0.0634
0
EL
0
0
0.0005
0.0030
0.0056
0.0081
TC
0.2267
0.2267
0.1863
0.1288
0.06910
0.0081
NW
31.2461
31.2461
31.2557
31.2612
31.2822
31.3122
4.0
5.0
WeL
Table 4.2: l  20000
a s  1.2967 , a L  1.7933
a

0
1.0
2.0
3.0
0.00005101 0.00005101 0.00005162 0.00005239 0.00005285 0.00005308
Wne
7.6816
7.6816
10.7009
10.7009
10.7009
10.7009
WeS
23.7912
23.7912
19.4718
13.1378
6.6346
0
16
0
0
1.2697
7.5531
14.0186
20.6231
TW
31.4728
31.4728
31.4424
31.3918
31.3541
31.3240
ES
0.2267
0.2267
0.1858
0.1257
0.0635
0
EL
0
0
0.0009
0.0051
0.0093
0.0136
TC
0.2267
0.2267
0.1867
0.1308
0.0728
0.0136
NW
31.2461
31.2461
31.2557
31.2610
31.2813
31.3104
WeL
The increase in the value of the loan reduces the educational choice margin, a L , from 1.8067 to
1.7933. In both cases this is lower than the first-best level. The margin between no education and
accepting a scholarship is a S =1.2967 in both cases. As a consequence of these values there will
always be more people choosing higher education than is efficient in the first-best even if no
scholarships are offered. This is due to the fact that in the second-best setting the subsidised rate of
interest increases the return from investment in higher education, prompting people with less ability
to invest. Reading across each table it can be seen that welfare is increased for a given value of loan
by increasing the level of ability for scholarship entitlement. Contrasting the tables, a reduction in
the loan also increases welfare. Comparing these results with the first-best it is clear that reduction
in the amount of loan takes the outcome closer to the first-best. These results demonstrate that in
order to raise welfare, the scholarship entitlement level a should be set equal to the maximum
ability level so that nobody receives a scholarship. It is preferable to provide only loan for those
who wish to invest in education rather than provide both loan and scholarship. As the threshold
ability qualifies for scholarship increases, the total benefits and the total costs both decrease but the
reduction in cost is higher than the reduction in benefit.
5.2.2 Value of Scholarship
Table 4.3 and 4.4 show the effects of changing the amount of scholarship. Increase in the amount of
scholarship will increase the returns for those who taking education with scholarship and thus will
reduce the educational choice margin with scholarship. Increase in scholarship from 12000 pounds
to 18000 pounds has reduced the educational choice margin with scholarship from 0.7667 to 0.2367
without affecting the educational choice margin with loan. Similar to the effects of changing the
value of a loan, reduction in the value of a scholarship increases net-welfare and takes the outcome
closer to the first best. Reading across Table 4.3, for a given value of a scholarship it shows that the
net-welfare increases as the threshold ability qualifies for scholarship increases. In contrast, Table
4.4 shows no direct relationship between the threshold ability qualifies for scholarships and the netwelfare. Increasing a from 0 to 1, has increased the net-welfare from 31.1153 to 31.1686, however
as a is increased to 2, the net-welfare decline to 31.1395 but starting to rise again as a increases.
17
r and c) determine the
This results signifies that the chosen value of the policy variables i.e ( g , l , ~
effects of variation in a has upon the net-welfare.
Table 4.3: g=12000
a s  0.7667 , a L  1.8167
a

0
1.0
2.0
3.0
4.0
5.0
0.00004823 0.00004835 0.00005028 0.00005178 0.00005262 0.00005317
Wne
4.5031
5.8965
10.7825
10.7825
10.7825
10.7825
WeS
27.1574
25.7593
19.5897
13.2059
6.6651
0
0
0
1.1875
7.4685
13.9320
20.5351
TW
31.6605
31.6558
31.5597
31.4569
31.3796
31.3176
ES
0.5274
0.4642
0.3621
0.2485
0.1263
0
EL
0
0
0.0002
0.0015
0.0027
0.0040
TC
0.5274
0.4642
0.3623
0.2500
0.1290
0.0040
NW
31.1331
31.1916
31.1974
31.2069
31.2506
31.3136
WeL
Table 4.4: g=18000
a s  0.2367, a L  1.8167
a

0
1.0
2.0
3.0
4.0
5.0
0.00004468 0.00004599 0.00004910 0.00005119 0.00005239 0.00005317
Wne
1.3767
5.8965
10.8437
10.8437
10.8437
10.8437
WeS
30.5048
25.9344
19.7000
13.2704
6.6941
0
0
0
1.1263
7.4073
13.8708
20.4740
TW
31.8815
31.8309
31.6700
31.5214
31.4086
31.3177
ES
0.7662
0.6623
0.5303
0.3685
0.1886
0
EL
0
0
0.0002
0.0015
0.0027
0.0041
TC
0.7662
0.6623
0.5305
0.3700
0.1913
0.0041
NW
31.1153
31.1686
31.1395
31.1514
31.2173
31.3136
WeL
5.2.3 Interest Subsidy
Changing the interest rate on government loans will not have any impact on the educational choice
margin with scholarship. In table 4.5 the government rate of interest is set to be equal to zero (full
subsidisation) where student just pay the principal amount of loan and in Table 4.6 where there is
18
no subsidise interest. By looking at each table, net-welfare increases as a is increased. Comparing
both tables, the result shows the reduction in interest subsidy increases net-welfare. Table 4.6 shows
when a is set to be equal to the maximum ability level, the educational choice margin with loan is
1.8267 (equal to the first best). This result suggests, in order to achieve the first best, in a perfectly
functioning capital market, is to remove all the distortions.
Table 4.5: ~
r 0
a s  1.2967, a L  1.7967
a

0
1.0
2.0
3.0
4.0
5.0
0.00005101 0.00005101 0.00005162 0.00005239 0.00005282 0.00005309
Wne
7.6818
7.6818
10.7213
10.7213
10.7213
10.7213
WeS
23.7910
23.7910
19.4718
13.1377
6.6346
0
0
0
1.2492
7.5323
13.9974
20.6018
TW
31.4728
31.4728
31.4423
31.3913
31.3534
31.3231
ES
0.2267
0.2267
0.1858
0.1257
0.0634
0
EL
0
0
0.0008
0.0045
0.0084
0.0122
TC
0.2267
0.2267
0.1866
0.1303
0.0718
0.0122
NW
31.2461
31.2461
31.2557
31.2610
31.2816
31.3109
4.0
5.0
WeL
Table 4.6: ~
r  0 .06
a s  1.2967, a L  1.8267
a

0
1.0
2.0
3.0
0.00005101 0.00005101 0.00005163 0.00005246 0.00005291 0.00005320
Wne
7.6816
7.6816
10.9049
10.9049
10.9049
10.9049
WeS
23.7910
23.7910
19.4718
13.1377
6.6346
0
0
0
1.0649
7.3448
13.8076
20.4100
TW
31.4728
31.4728
31.4416
31.3874
31.3471
31.3149
ES
0.2267
0.2267
0.1859
0.1259
0.0635
0
EL
0
0
0
0
0
0
TC
0.2267
0.2267
0.1859
0.1259
0.0635
0
NW
31.2461
31.2461
31.2557
31.2615
31.2836
31.3149
WeL
19
5.2.4 Educational Fee
Changes in educational fee has different effects from variation in the value of scholarship in the
sense that reduce fee will benefits those who taking education either they financed it through loans
or scholarships. Reduction in fee will reduce both the educational choice margin with scholarship
and loan, signifies people with less ability will now find it profitable to invest in education. By
comparing Table 4.7 and 4.8, it is clear that increment in educational fee will increase net-welfare
and hence brings the outcome closer to the first best.
Table 4.7: c=10000
a s  0.4133, a L  0.9333
a

0
1.0
2.0
3.0
4.0
5.0
0.00004597 0.00004672 0.00004797 0.00004858 0.00004894 0.00004918
Wne
2.4125
5.4974
5.4974
5.4974
5.4974
5.4974
WeS
29.3914
25.8776
19.6640
13.2494
6.6846
0
0
0.3995
6.5472
12.9157
19.4450
26.1007
TW
31.8039
31.7745
31.7086
31.6625
31.6270
31.5981
ES
0.6747
0.5981
0.4605
0.3109
0.3038
0
EL
0
0.0063
0.1036
0.2033
0.1566
0.4048
TC
0.6747
0.6044
0.5641
0.5142
0.4604
0.4048
NW
31.1292
31.1701
31.1445
31.1483
31.1666
WeL
31.1933
Table 4.8: c=15000
a s  0.8550, a L  1.3750
a

0
1.0
2.0
3.0
4.0
5.0
0.00004874 0.00004879 0.00005001 0.00005071 0.00005111 0.00005137
Wne
5.0294
5.8965
8.1553
8.1553
8.1553
8.1553
WeS
26.5976
25.7287
19.5706
13.1949
6.6601
0
0
0
3.8386
10.1650
16.6624
23.2923
TW
31.6270
31.6252
31.5645
31.5152
31.4778
31.4476
ES
0.4445
0.4294
0.2432
0.2231
0.1124
0
EL
0
0
0.0236
0.0844
0.1374
0.1907
TC
0.4445
0.4294
0.2668
0.3075
0.2498
0.1907
NW
31.1825
31.1958
31.2977
31.2077
31.2280
31.2569
WeL
20
6.0 Conclusion
In general this paper attempts to evaluate the current policy of financing higher education which
involves a mixture of merit-based grants and loans under the assumption of a perfect capital market.
The specific objective is to analyse the positive effects of these policies have in terms of how they
affect the population that participate in higher education and the effects that variations in the
policies have upon welfare. Using a standard approach of human capital theory and the
intertemporal utility maximizing behaviour of an individual, the paper describes how the
government policies regarding financing higher education affects individual’s choice for education.
The fact that in a perfect capital market initial income does not matter in determining individual's
choice for higher education allows the partition of population into educational choices by level of
ability. Given the government's objective is to maximize the utilitarian social welfare function,
utilities are then aggregated and the value of net-welfare is derived by subtracting the cost of
education policy. In order to assess the relative efficiency of various policies, the paper provides the
first best solution as the benchmark case. The policy variables include the threshold ability qualified
for a scholarship, a , the value of a loan, l, the value of a scholarship, g, the subsidise rate of interest,
~
r and educational fee, c. The results shows that for any given value of l , g and ~
r and c increasing
a will have ambiguous effect on the net-welfare, but the highest net-welfare always occur when a
is set to be at the maximum ability level (signifies no scholarships should be offered). With regards
r and c , for any value of a , increasing l , g and ~
r will reduce net-welfare
to variation in l , g and ~
and increase in c will increase net-welfare. This model provides a theoretical evidence that in a
perfectly functioning capital market the move towards costs recovery will bring about efficiency.
Thus the financial reform in higher education towards loan financing should be given more
consideration.
Appendix
i)
Utility Maximisation
U  U ( x1 )  U ( x 2 )
U ne  x1 
U ne
w2
x2
 m  w1ne  ne
(1  r )
(1  r )
2
wne
x2
 U (m  w 

)  U ( x 2 )
(1  r ) (1  r )
1
ne
dU ne  U ' ( x1 )

 U ' ( x 2 )  0
(1  r )
dx 2
21
ii)
Obtain an Indirect Utility Functions: Without Education ( V ne ) and Education with loan
( VeL )
  1
  1
w2 ( a )  
w2 ( a ) 
U ne  U ne  x1 
, m  w1ne  ne   U ne  x 2 
, m  w1ne  ne 
(1  r ) 
(1  r ) 
  (1  r )
  (1  r )
  1
  1


w2 ( a ) ( r  ~
r )L
w2 ( a ) ( r  ~
r )L
U eL  U eL  x1 
,m  e

 c   U eL  x 2 
,m  e

 c 
(1  r )
(1  r )
(1  r )
(1  r )


  (1  r )
  (1  r )
iii)Equate the Indirect Utility Functions
U ne  U eL
iv)Use Budget to Relate Demand Functions
 1

x1  x1 
,M 
1 r

x 1  x 2 (1  r )  M
 1

x2  x2 
,M 
1 r

1
x
x 12 
M
x 2
x 22 
M
v)Totally Differentiate Equality by Varying a and M

 

2
2
2
2



w (a )
w (a )
w (a )
w (a) 

da U ne ' x 12 ne
 U ne ' x 22 ne   U eL ' x 12 e
 U eL ' x 22 e
a
a  
a
a 


(1  r )
(1  r )  
(1  r )
(1  r ) 
dM (U ne ' x 12  U ne ' x 22 )  (U eL ' x 12  U eL ' x 22 )  0


22
vi)Simplify using First Order Condition


2
2
2

w (a )
w (a ) 
x
da U ne ' (1  2 ) ne
 U ne ' x 22 ne 
a
a 
1 r


(1  r )
(1  r ) 


2
2
2

w (a )
w (a ) 
x

 U eL ' (1  2 ) e
 U eL ' x 22 e
a
a 
1 r


(1  r )
(1  r ) 


x2
x1
 dM (U ne ' (1  2 )  U ne ' x 22 )  (U eL ' (1  2 )  U eL ' x 22 )  0
1 r
1 r




2
2

w (a )
w (a ) 
  dM U ne 'U eL '  0
da U ne ' ne
 U eL ' e
a
a 


(1  r )
(1  r ) 
 U ne 'U eL '
da

dM 

2
2

w (a )
w (a ) 
U ne ' ne

 U eL ' eL
a
a 


(1  r )
(1  r ) 
2
wne
(a)
we2 (a)
Where the numerator U ne 'U eL '  0 and and the denominator U ne ' a  U eL ' a  0
(1  r )
(1  r )
2
we2 (a)
da
wne
(a)
0
(since w (a )  w (a ) and
>0 ,
 0 so
dM
a
a
2
e
2
ne
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