Effective Tax Rates on Skill Formation: An
A Priori Approach
Glenn Michael Anderson1
Curtin University, Sydney Campus
25 September 2010
JEL Codes: H21, H24, H52, I22, I28, J24
Abstract
The few attempts that have been made to measure the effective tax rate on skill formation
are either limited by the fact that they inherit assumptions applicable to the theory of the
firm or have dubious theoretical foundations. I develop a forward-looking measure of the
effective tax rate that is grounded in human capital theory, allowing for features that
differentiate human capital formation from physical capital formation. These features
include concavity of the earnings-investment frontier, intertemporal productivity effects
and adjustments in capital utilization through leisure. In addition, the new measure allows
for three types of financing: personal equity, government sponsored bank loans (GSBL)
and income contingent loans (ICL). The measure is applied to Australian data.
1
Please address all correspondence to [email protected]. I gratefully acknowledge the
support of the Economics Department of the University of New South Wales, where much of the research
for this paper was undertaken as part of my PhD.
1
1. Introduction
Since the seminal contributions of Jacob Mincer, Theodore Schultz and Gary Becker,
post-compulsory education and training has been treated as an investment capable of
contributing to lifetime earnings (Becker, 1975; Schultz, 1961) and explaining personal
income differentials (Mincer, 1958). The focus of the empirical literature has been on
estimating the rate of return to education using techniques pioneered by Mincer (1974)
and addressing such problems as self-selection, the lack of counterfactual earnings
streams with which to compare actual earnings, and distinguishing between ex ante and
ex post returns2.
At the applied level there has been scant empirical work on the impact of taxes and
subsidies on human capital accumulation (Alstadsæter, 2005). Where work has been done
the focus has been on the impact on individual incentives, as measured by the effective
tax rates on human capital accumulation (Collins and Davies, 2004; Fuente and JimenoSerrano, 2005; Gordon and Tchilinguirian, 1998; Karzanova, 2005) or economy-wide
behaviour, using applied dynamic general equilibrium models (Heckman, Lochner and
Taber, 1998a,b; Trostel, 1993; Lee, 2005). This paper makes a contribution to the first of
these fields in developing a measure of the effective tax rate on skill formation that is
grounded in the theory of human capital.
The effective tax rate is a measure of the consolidated impact of taxes and subsidies on
the return from an investment. In the case of investments in tangible capital, it measures
the difference between the marginal return that a producer is able to achieve and the
return for those financing the marginal project once all taxes and subsidies have been
taken into account. The difference is known as the tax wedge. From the size of the tax
wedge on the marginal investment one infers the impact of policy on investment
incentives. The framework also allows for counterfactual experiments although this is
2
See Card (1999) for a review of the literature and Heckman, Lochner and Todd (2005) for a critical
review of Mincer earnings equation.
2
limited by only being able to measure the immediate impact on incentives. In terms of
methodology, King and Fullerton (1984), and more recently Devereau and Griffith
(1998a), have been key influences.
In the case of education, there have been surprisingly few attempts to measure the impact
of taxes and subsidies on incentives. However, of these there are two distinct approaches.
The first attempt to measure the effective tax rate appears to be an OECD research paper
by Gordon and Tchlinguarian (1996). In this paper the King-Fullerton methodology is
applied to derive an analytical expression for the social rate of return. However, apart
from arguing the case for ignoring discontinuities which may give rise to economic
profits, there is no attention to theory and in particular no attempt to examine the extent to
which their approach is compatible with human capital theory.
A second approach, adopted by Fuente and Jimeno-Serrano (2005), as well as Collins and
Davies (2004), calculates the internal rates of return for what is described as either an
“average student” or “representative agent”. The internal rates of return are calculated
before-tax and after-tax, and their difference is used as a measure of the tax wedge for the
average or representative student. However, as discussed below, how one can interpret
such measures for policy purposes is problematic.
In this paper I develop and apply an a priori measure for the effective tax rate which is
grounded in the theory of human capital, integrates rising marginal costs and falling
marginal revenues (due to aging), substitution between inputs and augmented for the
impact of leisure and dependence between investment decisions over time (due to an
intertemporal productivity effect). Three types of finance are modeled: personal equity,
government sponsored bank loans (GSBL) and repayments that are contingent on
graduate earnings or income contingent loans (ICL). In each case, the measure is used to
derive effective marginal tax rates over the career of an individual, during and after
formal education.
The paper is organized as follows. I begin with an overview of the principles underlying
the a priori approach for deriving effective tax rates based on the theory of the firm,
followed by a discussion of the important features distinguishing human capital theory
3
from the theory of the firm and finally a critical review of the few attempts to measure the
effective tax rate on skill formation (Section 1.1).
The derivation of the effective tax rate on human capital accumulation proceeds in two
steps. In Section 2 I derive a minimum cost function relating the costs of production to
investment, describing the technology of human capital production and deriving effective
tax rates on the component and overall costs of production. In Section 3, the cost function
is incorporated into a standard lifetime utility-maximising framework in order to derive a
set of first-order conditions.
Assuming human capital is equally productive in leisure as it is of work or education and
no intertemporal productivity effects, the first-order conditions are used to derive the
effective marginal tax rate is for a base-model, which assumes tuition is solely financed
from personal equity or normal loans (Section 4).
The model is extended in two ways to permit intertemporal productivity effects (Section
5) and allowing for the possibility that human capital is not productive in leisure (Section
6).
In Section 7 the effective marginal tax rate is derived assuming tuition is financed
through government sponsored bank loans (GSBL) and income contingent loans (ICL).
In Section 8 I present some concluding remarks drawing the implications for further
research at both the theoretical and applied level. An Appendix describes the data and
details some of the derivations.
The model formulated so far is of greater generality than the King-Fullerton model, but
does make several restrictive assumptions. These include assuming human capital is as
productive in home production as it is in work or education and what Ben-Porath (1967)
described as the “neutrality condition”. In the next two sections I discuss how these
assumptions may be relaxed and the reasons why one may want to do so.
1.1
CURRENT MEASURES OF THE EFFECTIVE TAX RATE ON EDUCATION
While the methods for measuring the effective tax rate on tangible and financial capital
have been well established since the 1980’s, there have been very few attempts to
4
measure the effective tax rate on investments in education3. An OECD research paper by
Gordon and Tchlinguarian (1998) uses a modified King-Fullerton model to compare
effective marginal tax rates on human and physical capital for the OECD, but with one
exception (Karzanova, 2005), there appears to have been no further interest in applying
this approach. More recently, an approach based on the difference between internal rates
of return before and after tax has been used to derive a measure of the tax wedge for an
“average” or “representative” student (Collins and Davies, 2004; Fuente and JimenoSerrano, 2005). In both cases however, there is a noticeable lack of attention to the
differences in the way investment is treated in the theory of the firm and human capital
theory. These differences can be considerable.
1.2
PHYSICAL AND HUMAN CAPITAL
From its inception, the theory of human capital has assumed the costs of education are
rising in the level of investment; an assumption which has received rigorous theoretical
support (Rosen, 1972) as well as empirical support (Heckman, 1976; Heckman, Lochner
and Taber, 1998b). Increasing marginal costs to education mean that the average costs
will be lower than marginal costs with implications for human capital accumulation under
conditions of certainty (Blinder and Weiss, 1976) and uncertainty (Levhari and Weiss,
1974).
In the case of physical capital, the cost of investment is simply the price paid on the
market for investment goods. The production of investment goods does not enter into
consideration. In the case of human capital, output is non-transferable so production
becomes an essential consideration in determining the costs of investment. The implied
cost function itself depends on the after-tax prices of inputs and the functional form of the
production function.
The stock of human capital may also influence marginal costs through an intertemporal
productivity effect. Intertemporal productivity effects, if positive, will arise whenever
3
Effective tax rates on wage-earnings, or the return to human capital, have been calculated by the OECD
on an annual basis since 2001 in its publication entitled Taxing Wages (e.g., OECD, 2006), but there have
been very few attempts to measure the effective tax rate on investment in human capital.
5
greater effort at school today enhances a person’s general ability to acquire knowledge in
the future. Because more of today’s investment in education will reduce the marginal
costs of future education, the marginal return to today’s investment will rise with future
investment or education plans. More generally, the intertemporal productivity effect
reflects the degree to which skills may be substitutes or complements over time (Cunha
and Heckman, 2007)4.
Intertemporal productivity effects are closely linked to the presence of increasing returns
to scale in reproductive factors: human capital and direct inputs. The first to model
human capital production, Ben-Porath (1967), assumed a zero intertemporal productivity
effect which, with concavity in the earnings-investment frontier, implies decreasing
returns to scale in reproductive factors. On the other hand, a number of the models arising
in the endogenous growth literature assume constant returns to scale5 which, given
concavity of the earnings-investment frontier, implies a positive intertemporal effect6.
While there is reasonable empirical support for Ben-Porath’s original model (Heckman,
1976; Heckman, Lochner and Taber, 1998b), recent empirical work by Trostel (2004)
suggests returns to scale which vary in a fashion typical of the textbook production
function: increasing in the early school years up to year twelve and decreasing thereafter.
Section 5 considers the implications of returns to scale and intertemporal effects and how
they can be integrated into the effective tax model.
The differences between measuring effective tax rates for human capital and physical
capital are not limited to the production/purchase of capital. A second set of distinctions
arises in relation to returns. The infinite time horizon that is generally employed for
physical capital reflects the transferability of ownership, something which is not possible
with human capital. The fact that human capital is embodied in the individual and not
4
Recent empirical evidence indicates that experience-earnings profiles become steeper with the level of
educational attainment (Brunello and Comi, 2004).
5
See Trostel (2004) and references therein.
6
See Section 5, below.
6
transferable means that the returns from education must to be realized within the
individual’s lifetime. As a result, marginal revenues will decline with age alone (Blinder
and Weiss, 1976; Rosen, 1972). Furthermore, the impact of the tax-transfer system on
after-tax wages may be compounded by utilisation effects: the impact of taxes and
transfers on leisure (Lucas, 1990). A higher tax on future earnings reduces after-tax
earnings not just directly, but indirectly by increasing the demand for leisure (assuming
the substitution effect dominates the income effect).
1.3
METHODOLOGY
In developing a measure of the effective tax rate on human capital, the natural starting
point is the established effective tax literature that is based on the theory of the firm. The
methods used to measure the effective tax rate on physical capital investment are of two
types. The first are a priori measures, which use the theory of the firm to derive a formula
for the hurdle rate of return that the firm needs to achieve on physical capital in order to
cover all taxes and yield a competitive after-tax rate of return for those financing the
project (Jorgenson and Yun, 1991; King and Fullerton, 1984). A second approach,
referred to as the ex post method, uses data on corporate tax revenues and some base
(such as reported operating profits), to derive a measure of the average ex post rate of
taxation. The measure I develop in subsequent sections is an a priori measure and for this
reason only the methodology underpinning the a priori approach is considered7.
7
See Sørensen, 2004, for a review. In practice the two methods yield significantly different results
(Devereux, 2004). The main advantage of the ex ante method is that it is forward looking, providing a
measure of the tax burden on a hypothetical project. The ex post method is backward-looking. Since tax
revenue data will reflect the impact of the tax system on capital of different generations, the accuracy of the
ex post approach in indicating the impact of taxes on current investment will depend on the extent to which
the economy’s trajectory approximates its balanced growth path (Gordon, Kalambokidis and Slemrod,
2003).
The main limitation of the a priori method is that it is impractical to incorporate all but the simplest of tax
regimes. On the other hand, since the ex post measure is based on outcomes, it automatically accounts for
the multiplicity of ways in which companies can alter their effective tax burden, legally and illegally.
However, unlike the ex post method, the forward-looking approach provides a convenient and useful
7
To begin, the effective tax rate, ETR, is defined by the following relationship,
(1-1)
   1  ETR  r  
where    defines the before-tax or social rate of return in real terms and r   is the
real after-tax, or private, rate of return8, after adjusting for the opportunity cost associated
with the prospective rise in the value of the physical asset,  . The effective tax rate is
therefore
(1-2)
ETR  1 
r 
 
While r is presumed to be known, the social rate of return,  , is unobservable. To derive
an expression for the effective tax rate an expression for the social rate of return,    ,
is required. This is achieved using theory to solve for the cost of capital upon which
   depends.
To illustrate, assume the asset will earn a before-tax rental of c j in each period, j, and
depreciates at a constant geometric rate,  . Let q0 be the acquisition price of the asset in
year 0, so c1 q0 denotes the cost of capital. The cost of capital, c1 q0 , is defined as the
rental rate that is just enough to cover the costs of acquiring the last unit of capital in
period 0. With r as the opportunity cost and t defined as the tax on rental income, the cost
of capital will satisfy the following equation,
(1-3)

 j 1
q0    
j 1  i 1 1  ri

1  t 1    j 1 c j

analytical expression for the effective tax rate in terms of statutory taxes and subsidies. This can be a
powerful tool for policy-makers in assessing the impact that changes in tax regime will have on the
financial incentives to invest.
8
This way of expressing returns is due to Jorgenson and Yun (1991) and is equivalent to the return after
depreciation used by, for example, Devereux (2004).
8
Equation (1-3) states that the acquisition price (which is also the marginal cost of
investment) will equal the present value of future after-tax rentals accruing to the
additional unit of capital (the marginal revenue from investment). It can be derived from
the first-order conditions for a firm maximizing the present discounted value of
outstanding shares9.
Assuming r is constant and c rises at a constant rate10,  , then the cost of capital can be
expressed as a function of known parameters,
c
1
r     1   

q0 1  t
(1-4)
It is clear from this expression that the social rate of return, the rate of return before tax,
will be given by
(1-5)
  
c
  1   
q0
Substituting (1-4) into (1-5), the expression for the social rate of return becomes
(1-6)
  
1
r     1      1   
1 t
To derive an analytical expression for the effective tax rate, (1-5) is substituted into (1-2)
to yield an expression for the effective tax rate which depends on known parameters,
(1-7)
ETR  1 
r 
1  t  r     1      1   
1
This simple example illustrates the basic methodology of what can be described as the
cost-of-capital approach11. More detailed specifications replace r with an expression
which allows for tax on interest-income and capital gains, while t would be expressed as
9
See Jorgenson and Yun (1991) or Devereux (2004) for a more detailed treatment
10
The rate of increase in c will be equal to the rate of increase in the acquisition price, q, as the life of the
asset approaches infinity.
11
See OECD (1991), Jorgenson and Yun (1991) or Devereux (2004) for more detailed treatments.
9
a function of the tax on dividends, capital gains tax and any tax offsets. Dividends
themselves will depend on the company tax rate, investment and depreciation allowances,
as well as the tax treatment of different modes of financing (retentions, new equity and
debt). The derivation of the cost of capital for more complex scenarios will generally
require explicit modelling of the firm’s wealth-maximising behaviour and derivation of
first-order conditions Devereux (2004). More recently, the approach has been extended to
measure the average effective tax rate on economic profits, with the aim of assessing the
impact of national tax regimes on the locational decisions of multinational enterprises
(Devereux and Griffith, 1998a; 1998b).
The approach of King and Fullerton (1984) has been adopted by the OECD for measuring
the impact of taxes on investment at both the national and international level (OECD,
1991). Gordon and Tchlinguarian (1998), use the King-Fullerton model to compare
effective marginal tax rates on human and physical capital for OECD-member countries
and, as far as I am aware, the only subsequent study along the same lines is Karzanova
(2005). Two features specific to human capital investment are incorporated into the KingFullerton model12:
i.
The “acquisition price” is expressed as a composite of direct and indirect inputs,
implicitly assuming that the underlying production technology is Leontief (fixed
coefficients);
ii.
An attempt is made to model the impact of a progressive tax regime on marginal
revenues.
The measure ignores the impact of leisure, intertemporal productivity effects and the
technology underpinning the costs of producing human capital. Since the measure is not
grounded in the theory of human capital it is difficult to draw on the literature in this area
to evaluate the assumptions being made and the extent to which they are appropriate. A
further point is that the ad hoc approach does not readily permit extensions for such
purposes as evaluating the impact of different modes of finance which are important
avenues through which governments intervene in the provision of education. In short, a
12
See the Appendix, Section 0, for the derivation of this measure.
10
measure of the effective tax rate on human capital that is grounded in the theory of
human capital is essential for further progress.
Of the a priori methods one can distinguish the cost-of-capital approach, as described
above, from a second approach which involves the derivation of internal rates of return
for the “average student”, before- and after- tax. Instead of deriving an analytical
expression for the “cost of capital”, or rental rate, and assuming the internal rate of return
is equal to the private rate of return on financial investments, under the internal-rate-ofreturn approach the rental rate for human capital is effectively treated as datum.
For example, to estimate the growth in earnings due to schooling, Collins and Davies
(2004) use the average wages at two levels of educational attainment. Fuente and JimenoSerrano (2005) combine estimated returns from Mincer regressions with OECD data on
the “average production wage” (allowing for the possibility of unemployment) to
estimate the alternative lifetime earnings profiles facing a “representative agent”. In each
case the effective tax rate is based on the difference between two internal rates of return:
the internal rates of return for an “average individual” or “representative agent”13 before
and after taxation (and transfers).
Several issues can be raised in relation to this approach. Consider first a labour market in
which every individual is identical in terms of their abilities14 and access to financial
markets15. If the internal rate of return is different from the opportunity cost, then in the
absence of barriers to competition this can only be a transitory state in which the flow of
entrants into the occupation with the higher discounted net present value would
eventually result in a compensating adjustment in efficiency wages between any two
occupations (Mincer, 1958; Willis, 1986). In this case, the internal rates of return will at
13
Collins and Davies (2004) use these measures to derive the effective tax rates on schooling in Canada,
while Fuente and Jimeno-Serrano (2005) apply their method to countries in the European Union.
14
This assumption is stronger than required. Differences in the skill levels are permitted as long as relative
endowments are the same (Willis, 1986).
15
This is normally interpreted as the cost of borrowing or opportunity cost being the same for everyone
(Willis, 1986).
11
best describe a transitory situation arising from past changes in the relative demand for
people differing in educational attainment.
However, a fundamentally more telling criticism is that the approach assumes an
individual who plans to spend a fixed amount of time at school or university. Yet, if the
internal rate of return on schooling is above the opportunity cost then why would the
individual be planning to leave school and therefore select an occupation with a lower
discounted net present value. Similarly, if the internal rate of return is below the
opportunity cost then individuals would have left school earlier, which is again contrary
to the maintained assumption. Therefore, the internal coherence of the approach requires
that the internal rate of return be equal to the opportunity cost, which of course is the
maintained assumption of cost-of-capital approach, as well as my own.
A further difficulty arises when one allows for heterogeneity among individuals16. In this
case average or median earnings will, almost by definition, not be of the individual who
is indifferent between two alternative occupations (characterized in terms of different
levels of schooling). Yet it is the marginal investor whose decisions will be influenced by
changes in taxes and transfers. Quite apart from the problematic nature of the empirical
exercise which relies on standard Mincer regressions to estimate the earnings profiles of
the marginal individual17, once again the necessity of doing so can be questioned. For the
16
Even if human capital is homogeneous in every other respect, differences in the skill-endowments will
translate into more or less of the homogeneous stock of human capital. Rates of return can still differ
between individuals as long as there are significant direct costs (Heckman, Lochner and Todd, 2003), with
rates of return being higher for those with higher ability (those who can utilize the tuition supplied more
effectively). This of course assumes that direct costs are not proportional to the ability of the student.
17
Heckman, Lochner and Todd (2005) find that some of the implications of the Mincer model are not
supported by U.S. Census data, 1940 to 1990, particularly the prediction that log-earnings to experience
profiles will be parallel across schooling levels for more recent periods (i.e. from the 1960s onwards,
compared to 1940s and 1950s which tend to support the prediction). Trostel (2004), using a comprehensive
international micro dataset, International Social Survey Program, finds a robust relationship between logearnings and a cubic function of schooling for the period of the survey, 1985 to 1995.
12
marginal individual the internal rate of return should equal the opportunity cost, which
again lends support to the cost-of-capital approach.
The internal-rate-of return approach is also limited by the fact that it assumes individuals
make a once in a lifetime decision about their schooling and occupation, and are not able
to revise their plans in the light of new information (Cunha, Heckman, Lochner,
Masterov, 2005). This may not be a serious limitation in times of reasonable certainty
about costs and revenues. However, in more uncertain times, individuals may place a
premium on being able to wait before enrolling in a year-long course, reflecting the
expectation of more reliable estimates of returns in the future (Jacobs, 2007). In other
words, the internal-rate-of-return method does not lend itself to accommodating
uncertainty. While I will not be introducing uncertainty, it will be seen that the cost-ofcapital approach does lend itself to a sequential decision-making approach which in turn
permits individuals to revise their decisions in light of new information18.
In the following sections I detail a methodology for calculating the effective tax rate on
skill formation before applying a measure characterized in terms of assumptions made
about the technology of human capital production, the nature of preferences and the form
of finance.
2. Costs of education
The cost of investing in physical capital is a market price and, trivially, marginal costs
equal average costs. The cost of investment in human capital involves the opportunity
cost of effort and direct costs which in turn will generally depend on the scale of
investment and the time available after work and leisure. For this reason, students can be
thought of as making two distinct decisions: the choice of inputs for a given level of
education output and the level of education itself.
18
Jacobs (2007) assumes that waiting to enrol has an advantage in revealing more accurate estimate of
wage-returns to educations, which in turn would raise the required return on education. In contrast Cunha,
Heckman, Lochner and Masterov (2005) assume high-school completion provides an option on potentially
higher returns from college education, which would reduce the return on completing high-school. See
Jacobs (2007) for further applications of real options to human capital.
13
I assume that as producers, individuals decide upon the cost-minimising combination of
inputs for meeting any given level of investment, leisure and work. These decisions are
embodied in a cost function for education. In this section, I specify the technology of
human capital production (Section 2), derive a marginal cost function (Section 2.2) and
examine the impact of taxes and transfers on costs, based on measures for the effective
tax rates on specific and overall costs (Section 2.3). The decision about the scale of
investment requires a solution within a broader framework involving the desired flows of
consumption, saving and leisure, and the desired portfolio of financial, physical and
human capital. This is modeled in Section 3.
2.1
TECHNOLOGY
I assume the production of human capital combines the individual’s existing human
capital stock, K, with a flow of education services, e, to produce human capital
investment,
(2-1)
I  K  e
Production can be thought of as generating a flow of education services, denoted by e,
allowing for the possibility that the ability and speed with which a student can absorb
education services will depend on their existing stock of human capital, K.
The coefficient,  , measures the relatedness of investments over time. If  is zero, then
past investments will have no impact on one’s current ability to acquire knowledge
through further education, e. If  is greater than zero then past and present investment
will be complements: past investment decisions will enhance the ability of the individual
to add to their human capital stock through education. A value for  that is less than zero
means that past and present investment will be substitutes. While the concepts of
intertemporal complementarity and substitutability relate to specific skills19, the basic
model will assume skills can be aggregated into a general skills index, K20.
19
For example, research tends to suggest that investment in cognitive skills during early childhood and
adolescence tend to be intertemporal complements: more time devoted to cognitive skills with children
before the age of ten will improve the ability to acquire cognitive skills in adolescence. On the other hand
14
A second parameter,  , defines the returns to education. These are generally assumed to
lie between zero and one reflecting concavity in the earning-investment frontier, which in
turn can be explained in terms of transactions costs associated with the movement
between jobs (Blinder and Weiss, 1976). If   1 , the frontier would be linear and
generally individuals will choose corner solutions: full-time schooling or no schooling at
all. However, a more realistic set of predictions arises from a model which assumes
0    1 (Rosen, 1972).
Human capital services will also enter as an input in the production of education services,
e, just as they do in the production of market goods and services. The flow of human
capital services will depend on how much time is spent on market activities, h (work and
education), and leisure, l = 1 – h. The flow of human capital services is equal to mhK,
where m is the share of market time that is spent on education. The production of e
combines the flow of human capital services, mhK, with two other inputs: “tuition”, x,
and “equipment”, u. The technology for the production of education takes the following
form21
non-cognitive skills (such as patience, perseverance, discipline) tend to be intertemporal substitutes: greater
investment in early childhood reduces the returns from further acquisition in the future. For a detailed
discussion of the literature in this area and its implications for the technology of human capital production,
see Cunha, Heckman, Lochner and Masterov (2005).
20
In Section 5 I return to this issue and, based on recent empirical research, consider a model in which 
itself varies with age.
21
A model consistent with the general formulation of Blinder and Weiss (1976) would channel time-
dependence through hK  , rather than K  , so that the production function would look like I  hK  e  .
There are compelling practical and theoretical reasons for adopting my own formulation. The theoretical
case is made by arguing that the intertemporal productivity effect reflects the impact of prior education on
the individual’s ability to absorb new information or adapt to new circumstance and there is no compelling
reason why this should depend on current decisions about leisure. At the practical level, my formulation
renders a tractable analytical solution for the social rate of return which incorporates the labour supply
elasticity. Under the Blinder-Weiss formulation there is no straight forward analytical solution when it
comes to deriving the social rate of return once leisure is included.
15
e  F mhK; x; u 
(2-2)
where F(.,.) is a linear homogeneous function in the three inputs. Substituting (2-2) into
(2-1), yields
I  K  F mhK ; x; u 

(2-3)
Specified in this way, the model highlights a distinction between the contribution of
human capital as a stock, through the term K  , and a flow, through the production
function, F()22.
2.2
MARGINAL COST FUNCTION
Let w be the after-tax efficiency wage; q the price of tuition after-subsidies; and p
denote the price of equipment, after subsidies and taxes23. The minimum cost function for
education is defined as follows
(2-4)
D  min wmhK  qx  pu
m, x ,u
subject to
(2-5)
I  K  F mhK ; x, u 

The objective is to derive the cost-minimizing combinations of mhK, x and u for
producing a given level of human capital, I, given the stock of human capital, K, the share
22
This distinction parallels two views concerning the contribution of human capital to market production
(Aghion and Howitt, 1998: Chapter 10). The earliest approach treats human capital as just another means of
production, augmenting the productivity of labour (Lucas, 1988). A second approach assumes the stock of
capital increases the ability to innovate and adopt new technologies (Nelson and Phelps, 1966). The
practical import of the distinction is reflected in the debate concerning the main drivers of productivity
growth. See Parham (2004) for a discussion of the impact of human capital in facilitating the adoption of
new technologies and methods versus its role in augmenting the effective supply of labour, in the context of
Australia’s high rate of productivity growth over the 1990s.
23
Before-tax prices are denoted with a superscript ‘b’. See Section 2.3.
16
of market activity, h, and input prices, w, q and p. The cost function for investment in
human capital will take the following form,
1
 I 
DI , K   Gw, q, p   
K 
(2-6)
where Gw, q, p  is linear homogenous in prices, w, q and p. For example, if F   , is a
Cobb-Douglas production function, then
Gw, q, p   aw m q x p1 m  x
(2-7)


where a   m m  x x 1   m   x 
1 m  x 1
and  m and  x are the elasticities of 'output', I,
with respect to m and x, respectively. Under the more general CES functional form,

Gw, q, p    m w1   x q1   u p1
(2-8)

1
1
where  is the elasticity of technical substitution and the δ’s are technical parameters that
depend on the underlying production technology24.
The cost function in (2-6) is increasing in investment whenever  lies within the open
interval 0,1 . Let  denote marginal costs which are given by

(2-9)
D 1
 K
I 


G w, q, p I
1

As can be seen marginal costs are also increasing in investment, at a rate which depends
on  25. For later reference, the impact of capital on total costs is given by the following
expression,
24
See for example Varian (1992). For reasons that will be explained below, it is assumed that 0    1 ,
ruling out Cobb- Douglas technology (see Section 6).
25
Since
 ln  1  

 0 for 0    1 .
 ln I

17
(2-10)
2.3
D
G w, q, p   I
 
K
K






1

I
K
1 
  
I
K
IMPACT OF TAX AND TRANSFERS ON COSTS
Let w b , q b and p b denote the before-tax efficiency wage, tuition price and equipment
price respectively. With these definitions, a set of effective taxes on the individual inputs,
as well as overall marginal costs, can be derived.
Wages and prices are measured in terms of consumer goods, so that the effective subsidy
on the real wage will depend not just on the marginal tax rate for labour but the taxes
imposed on consumer goods. Let t w denote the marginal tax rate on wage income and t c
the tax on consumer goods. Let wh denote the gross efficiency wage received by
households. The after-tax efficiency wage is
(2-11)
w
1 t w h
w
1 t c
Further, let t p denote the payroll tax on the gross wage and wb denote the before-tax
efficiency wage,
(2-12)


wb  1  t p w h
Using (2-11) and (2-12), the effective marginal subsidy on the use of human capital in
education, which is denoted by  h , is given by
(2-13)
 h  1
w wh
1 t w

1

w h wb
1 t c 1 t p



In addition, the effective subsidies on tuition and equipment may involve concessionary
treatment under a goods and services tax. Let sx and su denote the subsidy on tuition and
equipment, respectively. Furthermore assuming upfront payment of fees attracts a
discount of dx. If in addition tuition and equipment are partially exempt from a general
goods and services tax, by a fraction equal to  , then the effective subsidies for tuition
and equipment are, respectively,
18
(2-14)
 x  1
q
1   xt c

1

1 sx 1 sd
b
c
q
1 t
(2-15)
u  1
p
1   ut c

1

1  su
pb
1 tc





Drawing on the linear homogeneity of the cost function, the effective subsidy on the
marginal costs of producing education is
(2-16)
  1

Gw, q, p 
 1
 1  g 1   h ,1   x ,1   u 
b
b
b
b

Gw , q , p 
It is readily seen that  coincides with the effective subsidy on the average and total
costs as well as the marginal costs of investment. Under Cobb-Douglas technology, from
(2-7), the effective subsidy on the costs of investment is
(2-17)
 w
 1  a b 
w 
 CD

 1 a 1 h
m
 q 
 b 
q 
x
1 m  x
 p 
 b 
p 
 1    1   
m
x x
u 1 m  x
Under the CES specification, from (2-8),
(2-18)
 CES
  w 1
 q
 1   bm  b   bx  b
 w 
q



 1  bm 1  
where bm 
 m wb
 m wb
1
  xqb

h 1




1
 bx 1  
 p 
 1  bm  bx  b 
p 

x 1

1
 u pb
1
 xqb
 m wb
1
1
 1



 1  bm  bx  1  
1
and bx 
1

1
u 1 1

1
  xqb
1
 u pb
1
Consider the marginal costs of education over two consecutive periods,  0 and 1 . Let
 denote the proportional rise in  (after-tax marginal costs). Using (2-16),
(2-19)
 b   b       1  
1   b 

  b     1   



b


1    

19
The rise in before-tax marginal costs can only be equated to the rise in after-tax marginal
costs, and therefore the rise in marginal revenues, if  does not change over time. This
may not be the case at points when the marginal tax rates change or subsidies are
withdrawn.
3. The Investment decision
In the previous section, I derived a cost function corresponding to the production of
human capital. The minimum costs of producing human capital were shown to be a
function of the level of investment and the current stock of human capital, as well as
leisure. In this section the cost function enters through a series of budget conditions that
constrain the individual’s welfare-maximising choice of investment, consumption and
leisure over their lifetime.
In what follows I begin with the assumptions underpinning a general model allowing for
leisure and different forms of financing, including the concessional treatment of loans
financing tuition fees. The first-order conditions for utility-maximisation are then used
for the derivation of an expression for the cost of human capital and the corresponding
effective marginal tax rate.
3.1
PRELIMINARIES
The individual is assumed to mature in period 0 and live for a further J years before
passing away. They inherit a stock of human capital, K 0 , and real assets, A0 . Each
individual can choose how they will share their time between market activities, h, and
leisure, l. This will leave the individual with an earning capacity equal to whK , where w
is the real, post-tax, efficiency wage and hK is the stock of human capital utilized in
market production26. The total time available is normalized to one and the following
constraint is imposed,
(3-1)
26
h l 1
0  l 1
Recall that market production includes learning as well as work
20
Let A denote the real face value of accumulated savings at the beginning of the period.
Savings takes the form of one-period bonds with a real, after-tax, interest rate of r. By the
beginning of the next period, the real stock of bonds will increase by A  A and the real,
after-tax, interest earnings over the period will be equal to rA .
In addition, the individual may be eligible for a state-subsidized loan to finance expenses
arising from tuition. Let B denote the real accumulated value of the state-subsidised oneperiod loans at the beginning of the period and z be the real rate of interest that is charged
over the same period. These loans will depend on tuition expenses which in turn depend
on the level of investment27.
Further, let c j denote real consumption of period j and let D j be the total cost (inclusive
of indirect costs) of education for period j. All income and expenditure flows of period j
are due at the end of period j. The one-period budget constraint facing the individual is
(3-2)
c j  w j h j K j  D j  A j 1  A j 1  r j   B j 1  B j 1  z j 
0 j J
Finally, let  denote the rate of depreciation and I denote investment in human capital.
The equation of motion relating investment in human capital to changes in the stock is
(3-3)
I j  K j 1  1   K j
0 j J
In summary, there are three constraints. The first constrains market time and leisure time
to one unit of time, (3-1), the second is the one-period budget constraint, (3-2), and the
third constraint is the equation of motion relating investment in human capital to changes
in the stock over time, (3-3).
A no-default condition, AJ  0 , is imposed, which means that all debts are paid before
death. Since there are no bequests, non-satiation will imply AJ  0 . Both conditions will
of course imply zero savings at the end of the person’s life, or AJ 1  0 . Finally, I assume
27
In Section 4, several repayment schemes are specified in more detail.
21
the individual commences their working life with an initial stock of human capital of
K 0  K 0 and financial wealth of A0  A0 .
3.2
UTILITY MAXIMISATION
Lifetime utility is assumed to be a time separable function of consumption, c, and leisure,
with allowance being made for the possibility that human capital is productive of leisure,
 
 


V  V U c0 , l 0 K 0 ,U c1 , l1 K1 ,..., U c J , l J K J
(3-4)

where U c, lK 

 is strongly concave with positive first partials and
0    1 . In this
specification if   1 then human capital is as productive in leisure or work, while for
  0 human capital makes no contribution to leisure28.
The individual will select consumption and leisure for each period to maximise their
lifetime utility subject to the three constraints (3-1) to (3-3) and the end-point conditions
described in the previous section. Consider the total differential of lifetime utility,
J
J
j 0
j 0

dV  V jU jC dc j  V jU jL K j dl j  l j K j 1dK j
(3-5)

where, V j  V U j is the “subjective rate of discount”, and U jC  U c j , l j  c j and

 
U jL  U c j , l j K j  l j K j

are the marginal utilities of consumption and leisure,
respectively.
Totally differentiating the resource constraint, (3-1), the budget constraint, (3-2), and the
law of motion, (3-3), and substituting in for the initial conditions dK 0  dA0  dB0  0 ,
the following conditions are obtained
(3-6)
dl  dh
(3-7)
dc0  w0 K 0 dh0  dD0  dA1  dB1
28
In Heckman (1976) education increases productivity in consumption, effectively assuming   1 . King’s
(1989) model implies
  0.
22
dc j  w j h j dK j  w j K j dh j  dD j
 dA j 1  dA j 1  r j   dB j 1  dB j 1  z j 
1 j  J
dI 0  dK1
(3-8)
dI j  dK j 1  1   dK j
1 j  J
The total differential of the cost function, D, using (2-6), (2-9) and (2-10), is
dD j  dI j   j
(3-9)
Ij
Kj
dK j
Substituting (3-8) into (3-9),
dD0  dK1
(3-10)
dD j   j dK j 1   j 1   dK j   j
Ij
Kj
dK j
1 j  J
where it will be recalled that  is the marginal cost of human capital investment (see
(2-9)). I next substitute (3-6), (3-7), (3-8) and (3-10) into (3-5), to derive an expression
for the total differential of lifetime utility,
J

U jL  
dV   V jU jC w j K j 
K j dh j
U jC


j 0
(3-11)
J

U jL
I j 
l j K j 1   j 1      j
 V0U 0C dK1   V jU jC w j h j  
dK j
U jC
K j 

j 1
J
 V j 1U j 1C  j 1dK j  VJ U JC  J dK J 1
j 1
 V0U 0C dA1  V jU jC dA j 1  1  r j dA j 
J
j 1
 V0U 0C dB1  V jU jC dB j 1  1  z j dB j 
J
j 1
After substituting the initial conditions of the previous section into (3-11) and rearranging
23
 1

 U jL K j 

dV  V jU jC 1 
w j K j dh j

j 0
 U jC w j 

J
(3-12)
 w 


U jL K j 1    j 
I 

j 

 
1     j   j  1
 V j 1U j 1C  j 1 
1 l j 1
dK j


   j 1 
U
w
K
  j 1 
j 1
jC
j
j









J
 VJ U JC  J dK J 1
 V j 1U j 1C 1   j 1  r j dA j  VJ U JC dAJ 1
J
j 1
 V j 1U j 1C 1   j 1  z j dB j  VJ U JC dBJ 1
J
j 1
where  denotes the marginal rate of substitution between current and future
consumption,
(3-13)
 j 1 
V j 1U j 1C
V jU jC
As mentioned in the previous section, the no-default condition implies AJ 1  0 , while
non-satiation implies AJ 1  0 . Therefore,
(3-14)
dAJ 1  AJ 1  0
Similarly,
(3-15)
dBJ 1  BJ 1  0
Furthermore, maximization of lifetime utility requires dV  0 and a necessary condition
is
(3-16)
V J U J ,C  J dK J 1  0
Excluding corner solutions, or assuming non-satiation, so that U J ,C  0 , a necessary and
sufficient condition for this last equality is I J  0 , since this implies  J  0 ; namely
24
investment in the last period of the individual’s life is zero. Substituting (3-14), (3-15)
and (3-16) into (3-12),
 1
J

 U jL K j 

dV  V jU jC 1 
w j K j dh j
U
w


j 0
jC
j


(3-17)

 w
 V0U 0C  0  1


 0


U K  1   
1  l1 1   1L 1    1


U 1C w1    0




I 

1     1  1  1dK1
K1 



 w 


U jL K j 1    j 
I 

j 
 
1     j   j  1
 V0U 0C   m1  j 1 
1  l j 1  
dK j


   j 1 


U
w
K

j 2 m2
j

1
jC
j
j











J

U jL K j 1  
 w J 

   J  1
 V0U 0C  J 1   m 1 
1 l j 1 
dK J

 J 1 
U jC w j  
m2






J 1
j
 V0U 0C 1  1 1  r1 dA1  V0U 0C   m1 1   j 1  r j dA j
J
j
j 2 m2
 V0U 0C 1  1 1  z1 dB1  V0U 0C   m1 1   j 1  z j dB j
J
j
j 2 m2
where  j 1 
V j 1U j 1C
V jU jC
For a general model, the only first-order condition that will need to be imposed is the
intertemporal substitution condition for dA; namely
(3-18)
 j 1 1  r j 1   1
0  j  J 1
After substituting (3-18) into (3-17)
 1

 U jL K j 

dV  V jU jC 1 
w j K j dh j

j 0
 U jC w j 

J
(3-19)
25
 w 


U jL K j 1    j 
I 
1

j 
 
1     j   1  r j 
 j 1 
1  l j 1  
dK j


   j 1 


U
w
K

j 1 m 1 1  rm
j

1
jC
j
j











1
J


U jL K j  
1 
 wJ 
   1  rJ 
 V0U 0C  J 1 
1  l j 1  

dK J



U
w
m 1 1  rm   J 1
jC
j







J 1
j
 V0U 0C 
1
r j  z j dB j
j 1 m 1 1  rm
J
j
 V0U 0C 
Equation (3-19) will provide the first-order conditions for a series of models each
distinguished in terms of the assumptions made about financing and the parameters 
and  .
4. Basic Model
The first differential, (3-19), gives rise to a number of first-order conditions for
maximization of the individual's welfare. The exact form of these conditions will depend
on the specific assumptions made. A basic model will be characterised by three
assumptions
A1.
No concessional finance: dB j  0 for all j
A2.
Human capital useful in leisure:   1
A3.
No intertemporal productivity effect:   0
The basic model assumes tuition will be financed from personal savings or loans with no
concessional treatment. In Section 4, I allow for concessional tuition finance, examining
the impact of government sponsored bank loans (GSBL) and income-contingent loans
(ICL).
If   1 , then human capital is equally productive of leisure as it is in work or education,
an assumption adopted by Heckman (1976) in his augmented Ben-Porath model. When
  0 , then the wage-elasticity of labour plays an important role in determining the
returns on investment through the effect of leisure on capacity utilization (King, 1989).
The maintained assumption will be   1 and I will discuss the implications of relaxing
this assumption in Section 6.
26
A value other than zero for  implies optimal investment decisions of one period cannot
be solved independently of investment decisions of another period. The standard
assumption of the Ben-Porath model is that   0 which Ben-Porath referred to as the
“neutrality condition”. This condition appears to have strong empirical support
(Heckman, 1976; Heckman, Lochner and Taber, 1998b) and will be maintained for
throughout most of the analysis. In Section 5 I take a closer look at the empirical
evidence; explore the impact of “non-neutrality” on the rental rate and the effective tax
rate; and suggest an alternative modeling strategy.
4.1
PRIVATE COST OF CAPITAL AND RETURNS
The marginal rental rate is derived from the first-order conditions of the previous section
and represents the key element in deriving an a priori measure of the effective tax rate.
Taxes and transfers influence incentives only in so far as they influence the marginal
rental rate. By equating the coefficient of dh j 0  j  J  to zero, the work-leisure tradeoff implies,
(4-1)
U jL K j 1
U jC w j
1
0 j J
Further, using assumptions A1 to A3, and setting the coefficient for dK j , 1  j  J , in
(3-19) to zero, the first-order conditions for lifetime utility maximization are
(4-2)
(4-3)
wj
 j 1
wJ
 J 1
 1  r j  1   j 1   
1  j  J 1
 1  rJ
where w j  j 1 is the marginal rental rate and  j is defined above, (2-19), as the rise in
marginal costs. Corresponding to the private marginal rental rate is the private return on
an investment in human capital. Rearranging (4-2)
(4-4)
rj   j 
wj
 j 1
  1   j 
27
The right-hand side represents the opportunity cost of investing in education. The lefthand side of (4-4), r j   j , represents the opportunity cost of the last dollar invested in
education. This consists of the real rate of interest, r j , and the gain from a fall in the
marginal cost of education,   j .
An expression for  j is derived from the sequence of first-order conditions yielding the
following expression
j 
(4-5)
w j 1
1  r j 1

1   w j 2
1  r 1  r 
j 1
j 2
 ... 
1   J  j 1 w
J
 1  r 
1  j  J 1
J
m
m  j 1
The left-hand side denotes the marginal cost of education while the right-hand side of the
equality represents marginal revenue. I assume that the before-tax efficiency wage rises at
a constant rate of  29, and the real rate of interest is constant and equal to r. Let  j
denote the rise in the tax-factor on the efficiency wage as defined by
(4-6)
j 
1   hj
1   hj1
1
With these assumptions and definitions, the expression for marginal revenue can be
simplified as follows,
(4-7)
i  j 1 J
J

w j 1 


 1   1    


j 
1



1   

m 
1 r 
1 r


m j  2
 i j 2 

1  j  J 1
Hence, for 1  j  J  1 ,
29
This will depend on the growth in the marginal product of human capital in market production and
therefore factors such as the physical capital accumulation and technological change.
28
j 
(4-8)
j
 j 1
i  j 1 J
J


 1   1    
1   m 

1   

1 r

 i j 2 

m j  2
 1  1   j 1 1   
 1
i

j

1
J
J
 1   1    
 1
1   m  





1 r

i  j 1
m  j 1


While it is possible to simplify further by assuming  j is constant this would not
adequately reflect the discrete nature of changes arising from the progressive tax
schedules30. However, J will generally be sufficiently high to justify the following
approximation,
j 
(4-9)
j
 1  1   j 1 1     1
 j 1
An important aspect of this approximation is that this approximation would be
appropriate even with   0 31 .
4.2
SOCIAL COST OF CAPITAL AND EFFECTIVE TAX RATE
Let  j denote the internal rate of return that equates marginal costs before-tax with the
before-tax return on investment
(4-10)

b
j 1

wbj  1   bj 1   
1  j
An important feature of this definition is that it does not equate the before-tax marginal
cost with the before-tax marginal revenue at the private opportunity cost. There is no
reason to expect the discounted present value of marginal revenues and costs before-tax
to be equal. In this sense  j can be thought of as an internal rate of return.
30
Gordon and Tchilinguirian (1998) derive an average growth rate as an approximation, while conceding
the treatment is less than satisfactory.
31
When   0  needs to be replaced with   
Ij
Kj
in (4-9)
29
Rearranging (4-10), the following expression for the social rate of return is derived
 j  
b
j
(4-11)
w bj
 bj1

  1   bj

where  bj is the rise in before-tax marginal costs as defined by (2-19). To derive an
analytical expression for  j   bj , the first-order conditions are used to solve for the
marginal rental rate, before-tax, wbj  bj1 . Using (2-13) and (2-16), and substituting in for
(4-2),
w bj
(4-12)

b
j 1
11

 1  r j  1   j 1   
j 1
1  j  J 1
h
j
An expression for the social opportunity cost that provides an expression in terms of
known parameters and variables is obtained by substituting (4-12) and (2-19) into (4-11),
(4-13)  j   bj 
1   j 1
1
h
j
r
j

  j  1   j  
1   j 1
1 j
1   
j
Finally, recalling the definition of the effective marginal tax rate, the social rate of return
(4-13) is substituted into (1-2) to derive,
(4-14)
EMTR  1 
rj   j
1   j 1
1   hj
r
j

  j  1   j  
1   j 1
1 j
1   
j
5. Intertemporal productivity effects
From (3-19) the first-order condition can be written in the following form,
(5-1)
wj
 j 1

1  rj

Opportunity cost
of investment


I 
j 

1     K 1   j 

j



1  j  J 1
Savings from bringing
forward investment
In this equation there is what can be termed an intertemporal productivity effect,
 I j K j , which offsets the impact of depreciation. The effect reflects the impact of past
30
investment decisions on returns through the interaction with current investment decisions.
If  is positive then past and current investment will be complementary: higher current
investment will raise the return of past investment. On the other hand, if  is negative,
the past and current investments are substitutes: the returns on past investments will be
lower the higher is current investment.
The higher is  or the rate of investment planned for period j, I j K j , the higher are the
savings that are derived from bringing forward investment, which in turn reduces the
required rate of return. This is because of the impact that a rise in K j can have on the
future productivity of the student if they engage in further study. The effects are positive
if  is positive (investments are complementary over time) and negative if  is negative
(investments are substitutes over time).
While there is evidence to suggest the assumption of a zero  may not be too far from the
truth, more recent research suggests a need for caution. For example, Heckman (1976)
and Heckman, Taber, Lochner (1998) using two distinct sets of data, both pertaining to
the United States, have found that assuming a zero  works well in predicting observed
lifetime earnings profiles.
On the other hand, drawing on recent research into early childhood development, Cunha,
Heckman, Lochner and Masterov (2005) suggest that more complex technological
relations underpin returns. They report evidence that investments in cognitive skills
during early childhood and adolescence are complementary: higher early childhood
development in skills such as mathematics and writing will increase the optimal level of
investment in such skills later in life. On the other hand, early and later investments in
non-cognitive skills tend to be substitutes for one another. The early inculcation of skills
such as patience, discipline and perseverance reduces the need (and hence returns) for
later investment, while a deficiency in non-cognitive skills in early childhood can be
made-up through investment later in life.
In related research, Heckman, Lochner and Todd (2003) and Trostel (2004) provide
theoretical and empirical support for rejecting the Mincer earnings regressions as a
method of deriving the return on human capital investment. The typical Mincer
31
regressions assume a log-linear relationship between earnings and schooling and Trostel
(2004) shows that there is compelling empirical evidence that the relationship is cubic
with the return in terms of earnings initially rising, reaching a maximum and falling
thereafter.
Trostel’s findings are significant because they provide evidence that returns to scale vary
with schooling. Trostel shows that a cubic relationship between log-earnings and
schooling implies that returns to scale are initially increasing in the early years of
schooling but eventually decline, becoming constant at about year 12 or 13, and
exhibiting decreasing returns thereafter.
In terms of the single "general-skills modeled" used in this paper, Trostel's findings may
be reconciled with the evidence presented in Cunha, Heckman, Lochner and Masterov
(2005) by allowing  to change as the individual matures32. The early years of childhood
development are critical for the acquisition of cognitive skills, suggesting that  would
predominately reflect the intertemporal complementarity of cognitive skills. In later
years, non-cognitive skills make a more important contribution and this would be
reflected in a decline in  . This would give rise to returns to scale,    , which fall
with age.
One way to capture this effect would be to allow  to depend on the level of maturity of
the individual, as measured by age, A, with the expression for the marginal rental rate
taking the following form,
(5-2)
wj
 j 1


 1  r j  1   j  1  ˆ j  A
1  j  J 1
where
32
While Trostel’s empirical finding is that returns to scale fall with the level of schooling, the evidence
summarized in Cunha, Heckman, Lochner and Taber (2007) suggests that production technology will be
changing with the physical development of the child. I will therefore interpret Trostel's results as evidence
in support of returns to scale falling with age.
32
ˆ j  A      A
(5-3)
Ij
Kj
To find an explicit expression for   A , assume  is related to age, A, in the following
manner

(5-4)
a
b
A
Since lim   A  b , b represents the lower bound for  . Let A be the level at which
A
returns to scale are constant. For example, based on the estimates from Trostel (2004), A
is either 12 or 13 years. In order to identify a, substitute A for A and     1 into (5-4).
This yields a  A 1    and
(5-5)
  A 
A
1     1  A b
A
A

In order to identify b, let  denote the age at which  is zero. When (5-4) is subjected to
this constraint the expression for b is
(5-6)
 A Aˆ 
1   
b  
ˆ
1 A A 
Substituting (5-6) into (5-5) yields the following expression for   A ,
(5-7)

 A  A  A Aˆ 

1   
 1  

ˆ 
A
A
1

A
A








  A  
33
In order that   A be decreasing in A, it is necessary and sufficient that Aˆ  A 33, which
implies   A will be decreasing in A but at a falling rate34 and the lower bound, b, is
negative but rising towards zero in  35.
To measure the rate of investment, an augmented Mincer regression, along the lines of
the model used by Trostel (2004) to test for constant returns to scale, would need to be
estimated. Let  j S  denote the estimate of the marginal effect of schooling, S, on logearnings and assume S is a simple linear transformation of age, so that S can be replaced
by A. The rate of investment would be estimated using the following expression36,
(5-8)
Ij
Kj
1   j 1  A

1
1 
where  denotes the rate of growth in efficiency wages. Finally, substituting (5-8) and
(5-7) into (5-3) an expression for “net depreciation”, as a function of age is obtained,
(5-9)
1   j 1  A

 A  A  A Aˆ 

 1     1  
1   

ˆ
A  1  A A 
 1 

A 

ˆ j  A    

where A and  are derived empirically such that  A   1   and  Aˆ  0 .
In relation to  , the parameter measuring the degree of concavity in the earningsinvestment frontier, there are few reliable estimates to be found in the literature (Trostel,
33
Differentiating (5-7) with respect to A, yields   A  
1
A
A 1  A Aˆ
2
1   
which is negative if and only
if Aˆ  A .
34
35
The second derivative of (5-7) is   A 
2
A
A 1  A Aˆ
3
1    which is positive if and only if
From (5-6), lim b  0 .
Aˆ 
36
That is  j 1  A 
w j 1 K j 1
wjK j
 1  1   
K j 1
Kj
 1 , so that
K j 1
Kj

1   j 1  A
1 
.
Aˆ  A .
34
1993). Indeed, Lord (1989), Trostel (1993) and Hendricks (1999) all use values based on
the estimates of Heckman (1976) and Haley (1976). In particular, Trostel (1993) specifies
a human capital production function of the following form,
I  AmhK  x b
a
(5-10)
and derives a value of 0.45 for a and 0.15 for b. In terms of the parameters under the
Cobb-Douglas specification a   m and b  (1   m ) . These imply a value for  of
0.60 (   a  b ). More recently, Heckman, Lochner and Taber (1998b) derive an
estimate for  which is about 0.8. Therefore, based on current research a reasonable
range for  would be between 0.6 and 0.8.
6. A role for leisure
To introduce leisure the assumption that human capital is just as productive of leisure as
it is of work or education, namely   1 , needs to be relaxed. Despite its convenience in
terms of calculation, in certain cases the assumption may be unwarranted. For example,
mothers are often faced with a choice between remaining at home to care for their
children and work, and their vocational qualifications may have little to contribute to the
former. In this event, “leisure” may mean that human capital is less than fully utilized and
this will affect returns to education.
When   0 the first-order conditions derived from (3-19) imply
(6-1)
wj
 j 1
1  l   1  r  1   1   
j
j
j
A simple model that allows leisure to be incorporated without too much complication
assumes a lifetime utility of the following form37 ,
37
This utility function is adopted by Jacobs (2002) to examine the impact of taxes on human capital
investment.
35
1  l 1
U c, l   c 
1
(6-2)
1
1


In this case, the marginal rate of substitution is given by U l , j U c, j  1  l j

1
, where 
is the uncompensated wage elasticity of labour supply. This functional form effectively
rules out voluntary retirement, since the marginal rate of substitution will be zero when l
=138. However, the restriction does provide a convenient solution for h  1  l .
Substituting the marginal rate of substitution into the first-order condition for the workleisure choice, (4-1), yields
(6-3)
1  l j  w j K j 

Substituting (6-3) into (6-1),
(6-4)
w1j
 j 1
K j  1  rj  1   1   j 
Furthermore, the social rate of return in (4-11) is replaced by the following
(6-5)
 j  
b
j
w bj ,1
 bj1

K j   1   bj

This requires an expression for the cost of capital, before-tax. Using (2-13) and (2-16),
along with (6-4), this expression is
38
Retirement takes place when the marginal rate of substitution of consumption for leisure is greater than
the wage rate U l , j U c, j  w j K j . In the above model this condition will only hold when the wage rate is
zero. Since measuring the social rate of return close to retirement is not likely to be much of an issue this
seems a small price to pay for analytical simplicity. In this context, the retirement age is assumed to be
exogenous.
36
(6-6)
w bj,1

b
j 1
K j  1  r j  1   1   j 
1   
1   
j 1
h 1
j
Substituting (6-6) and (2-19) into (6-5),
(6-7)
 j   bj  1  r j  1   1   j 
1   
1   
j 1
h 1
j

1   j 1
1 j
1   
j
The wage-elasticity compounds the impact of a rise in the effective tax rate on future
returns. Not only will a rise in  hj reduce the after-tax efficiency wage, but it will also
reduce the utilization of human capital and raise the require pre-tax return further.
Allowing for leisure allows for heterogeneity in the population reflected in differences in
wage elasticities. There is strong empirical evidence to suggest that the uncompensated
wage elasticity of labour supply (in terms of hours of work) does vary significantly across
individuals. For example, research persistently finds that the uncompensated wage
elasticity is much higher for women than for men. For women the elasticity tends to be
within the range of 0.1 to 1.0, while for men the elasticity tends to fall within the range
0.0 to 0.1, although much higher estimates have been reported (Blundell and Macurdy,
1999)39. Birch (2005) reviews estimates of the wage elasticity for Australian women and
concludes that the best (hours to work) point-estimate for the uncompensated wage
elasticity is around 0.5, although she also recommends a sensitivity analysis over the
range 0.0 to 1.0, due to the number of estimates falling into this range.
7. Introducing concessional educational finance
In this section I extend the model to include two types of concessional loan schemes for
the financing the cost of tuition. I begin by modeling the impact of invest on tuition. I
then model the impact of a concessional loan with a duration of one period, before
extending the model to examine the impact of deferring repayment by more than one
period.
39
The elasticities from Blundell and Macurdy's survey are reported in the Appendix, Section 0.
37
The extended model permits a distinction between the impact of concessional interest
rates on the initial one-period loan and in the provision of roll-over finance at
concessional rates. For example, repayment of a loan to finance the first year of study
will generally be deferred for a longer period than a loan to finance the final year of
study.
The model allows for two types of repayments. The first, a Government sponsored bank
loan (GSBL), involves the government in the subsidization of the (real or nominal)
interest charged on an otherwise standard bank loan with repayments taking the form of a
fixed annuity.
The second form of repayment is an income-contingent loan (ICL) and involves a loan
from the government with size and timing of repayments contingent on the individual’s
income after completing their studies.
7.1
TUITION
Loans will be used to finance tuition. For this purpose we need an expression for tuition
in terms of the increment to human capital. An expression for tuition demand, x, is
derived by invoking Shephard's lemma using the optimal cost function derived in Section
2. Differentiating the cost function, (2-6), with respect to the private cost of tuition, q,
yields
(7-1)
where  jx 
qjxj  qj
 ln G j
 ln q j
D j
q j
  jx D j
represents the optimal share of tuition in total costs. This will take
different forms depending on the elasticity of substitution characterising the underlying
production technology. With Cobb-Douglas technology  jx is invariant to prices and
therefore taxes and subsidies, while under CES technology (including Leotief fixed
coefficients technology),  jx will depend on prices and any taxes and subsidies.
Using (2-7), Cobb-Douglas technology implies
(7-2)
 jx   x
38
where  x is a technical parameter. Using (2-8), CES technology implies
(7-3)
 q bj 
x
 j  x b 
G 
 j

1
 1   xj

1 j

1




0   1

where G b  G wb , q b , p b ,  is the elasticity of technical substitution and the  x is a
technical parameter. Under Leontief technology,   0 .
Recall that  x is the effective subsidy on tuition, defined by (2-14), and  is the
effective subsidy on marginal costs, defined by (2-16). Assuming 0    1 , when the
effective subsidy for tuition is one (i.e.,  x  1 ) then no loans are required and the
expression for the after-tax rental rate will be the same as the zero-finance case40.
After taking the first differential of (7-1) and substituting in for dD, using (3-10) and
dK 0  0 , the impact of investment on tuition is given by the following expressions
(7-4)
d q0 x0    0x  0 dK1



Ij 
dK j 
d q j x j    jx  j dK j 1  1    

K j 



7.2
1 j  S
ONE-PERIOD LOAN
To begin, consider the impact of a one-period concessional loan. This means that all
loans are repaid within one period. Recall that B j denotes a one-period concessional loan
at the beginning of period j41. If a concessional loan is taken out against tuition expenses
40
Under Cobb-Douglas technology, loans would still be required even when tuition is fully subsidised. This
is difficult to justify and in application I assume CES technology with 0    1 .
41
Recall that all flows occur at the end of the period while assets are reported for the beginning of each
period. Often universities require payment of fees at the beginning of the period. In the current model this is
treated as an advance payment, with the student effectively lending the university funds until the delivery
of services. Tuition services are delivered at the end of the period, so universities effectively discharge their
debt in-kind when fees are paid at the start of term.
39
of period j, and any previous loan cannot be refinanced under the concessional scheme
then42
(7-5)
dB j  d q j 1 x j 1 
Substituting (7-4) and (7-5) into (3-19), and imposing the first-order conditions (see
Section 4, above) the required rental rate, after-tax, takes the following expression
(7-6)
wj
 j 1

Ij 
 1  j
 1  1   j  jx1 1  r j   1  1   j 1  jx 1    


K
j 



 
 


where
(7-7)
j 
1 z j
1  rj
Implicit in the above expression is that the student is eligible for one-period loans for
both periods: the initial period j  1 and the subsequent period, j. For every dollar
invested in period j  1 , the rental rate needs to cover the opportunity cost of non-tuition
costs 1   jx1 1  rj  and tuition costs  j jx1 1  rj . The parameter  j is the discounted
value of a one dollar loan at the concessional interest rate of z j . Concessional finance
means that  j  1 , reducing the opportunity cost associated with tuition costs and
therefore the required rate of return.
On the other hand, investment in period j  1 reduces the need for investment in
subsequent periods (by construction, the impact on all future capital stocks is offset by
subsequent disinvestment). If the student is eligible for a concessional loan in period j ,
then  j 1  1, so that the savings from bringing investment forward would be less than if
 j 1  1 (no concessional finance in period j). This raises the required rate of return.
42
This is equivalent to a one-period concessional loan which is rolled over but charged at the market
interest rate.
40
Further, notice that the net effect on the required rental will depend not only on the share
of tuition and concessional interest rates, but also on the inter-temporal productivity
effect. The higher is  or the rate of investment planned for period j, I j K j , the higher
are the savings that are derived from bringing forward investment, which in turn reduces
the required rate of return. This is because of the impact that a rise in K j can have on the
future productivity of the student if they engage in further study. The effects are positive
if  is positive (investments are complementary over time) and negative if  is negative
(investments are substitutes over time).
7.3
DEBT ACCUMULATION PHASE
In this and the next section a more general model is introduced. The model allows for
several periods over which debts can be deferred, the debt accumulation phase, and then
repaid, or the repayment phase.
Let the maximum number of period’s over which debts can be accumulated and rolled
over before repayment be S and let N denote the number of periods in the repayment
phase. Education begins in year 0, is completed by the end of year S  1 , so that the
accumulation phase extends from year 0 to the beginning of year S. The first repayment is
made at the beginning of year S+1 (end of year S).
Recall that B j denotes the newly formed debt at the beginning of period j, while q j 1 x j 1
denotes tuition costs at the end of period j  1 (beginning of period j). The accumulated
debt carried over into period j, after j years of tuition financed by a concessional loan
(starting from period 0), is
(7-8)
B j  1  z j 1 B j 1  q j 1 x j 1
1 j  S
and
B1  q0 x0
The equations in (7-8) can be solved recursively in terms of tuition costs, yielding the
general expression,
j 1 j 1
(7-9)
B j   1  z m qi 1 xi 1  q j 1 x j 1 1  j  S
i 1 mi
and
B1  q0 x0
41
The present discounted value of the gain from concessional finance over the
accumulation phase is given by the following expression
1
 1  r r
S
(7-10)
j
j 1 m 1
j
 z j dB j  1, S d q0 x0 
m
S 1
  1  rm   j  1, S d q j x j 
j
1
j 1 m 1
where  j 
1 z j
1  rj
;
 j , S   1   j 
i
S
 1   
i  j 1
i
m1
; 1  j  S 1
mi
S , S   1   S
and
Next substitute (7-4) into (7-10), the expression for the net benefit derived from
concessional finance over the debt accumulation phase is given by
1
 1  r r
S
(7-11)
j
j 1 m 1
j
 z j dB j 
m
S 1
j
 1  r 
j 1 m 1
m
j 1

j 
I 

x
x
1     j 
dK j
 j , S  j 1 1  r j    j  1, S  j



K


j

1
j




 S 1
S
1  r
S , S  Sx1 1  rS dK S
m
m 1
7.4
ACCUMULATED DEBT
Using (7-8), the accumulated debt at the end of S years of government sponsored finance
can be expressed as follows
42
S 1 S 1
(7-12)
BS   1  z m q j 1 x j 1  q S 1 x S 1
j 1 m  j
The present discounted value of the total accumulated debt is
S 1
(7-13)
1
1 r
m 1
S 1
1
1
 jS11q j x j  
q S 1 x S 1
j 1 m 1 1  rm
m 1 1  rm
S 2
j
BS  1S 1 q0 x0   
m
where
k
 1  z 
m
(7-14)
 
k
j
m j
k
 1  r 
k
  m
1 j  k ;
m j
m
m j
The coefficient  jk represents the fraction of costs due at the end of period j that
effectively have to be paid after allowing for the benefit of deferring a payment until the
end of period k. The longer the duration before repayment, the greater the gain to be had
from the concessional interest rate.
Totally differentiating (7-13) and substituting in for (7-4),
S 1
(7-15)
1
1  r
m 1
dBS  1S 1 0x  0 dK1
m



I 
1


 jS11 jx  j dK j 1  1     j dK j 
Kj 
j 1 m 1 1  rm





S 1



I 
1

 Sx1  S 1 dK S  1     S 1 dK S 1 
K S 1 
m 1 1  rm



S 2
j
 
Rearranging (7-15), the following expression for the present discounted value of the
accumulated debt is obtained,
S 1
(7-16)
1
1  r
m 1
dBS 
m

j 
I 
1

1     j 
 j 1  jS 1 jx1 1  r j    jS11 jx
dK j



K
j 1 m 1 1  rm


j

1
j



S 2
j
 
43


1
 S 2  SS11 Sx2 1  rS 1    Sx1 S 1
 S 2
m 1 1  rm

S 1


I 
1     S 1 dK S 1
K S 1 

S
1
 S 1 Sx1 1  rS dK S
1

r
m 1
m

Under a concessional loans scheme, only a fraction of the present discounted value of the
student’s debt is effectively repaid. However, the form of repayment will differ between
the two financing schemes. In the next section I augment the basic model to allow for
government-sponsored bank loans, GSBL, and income-contingent loans, ICL.
7.5
REPAYMENTS UNDER GSBL AND ICL
Let A j denote the additional repayment at the beginning of period j and assume that prior
to undertaking education in period 0 there is no outstanding debt at concessional rates.
The sequence of loans that carry over the remaining debt, after repayments for a period of
N years, starting in year S is defined by the following expression
BS  j  1  z S  j 1 BS  j 1  AS  j ,
(7-17)
Within a given period, dK, will be driven by the level of investment that equates marginal
cost with expected marginal revenue. Arising from dK will be new debt, part of which is
financed at concessional terms.
Repayments are more usefully expressed in the following form
 S 1 1  1
1


r

z
dB

dBS 1



j
j
j
j  S 1 m 1 1  rm
 m1 1  rm  1  rS
S  N 1 j
(7-18)
 S 1 1 S  N 1 j 1
dB j 1  1  z j dB j 
 

 m1 1  rm  j  S 1 m S 1  rm
 S 1 1 S  N 1 1
1  z S  N 1 dBS  N 1
 

 m1 1  rm  mS 1  rm
44
Using dB j 1  1  z j dB j  A j 1 and dBS  N  1  z S  N 1 dBS  N 1  AS  N  0 , the following
expression is derived for the impact of concessional finance over the repayment phase
S  N j 1

 S 1 1 1  z S
1
1



dB

Aj 
r

z
dB






j
j
j
S
j  S 1 m 1 1  rm
j  S 1 m S 1  rm
 m1 1  rm  1  rS

S  N 1 j
(7-19)
Using (7-17) and dBS N  0 , dBS  N 1 can be solved recursively,
S  N 1
S  N 1S  N 1
m S
j  S 1 m j
1  z S  N 1 dBS  N 1   1  z m dBS    1  z m dA j  dAS  N
(7-20)
Further using (7-20) , the increment to indebtedness by the end of the debt accumulation
phase is expressed in terms of the discounted value of subsequent repayments, where the
discounting is at the concessional interest rate,
dBS 
(7-21)
SN
j 1
  1  z 
j  S 1 m  S
1
m
Aj
Substituting (7-21) into (7-19), the impact of concessional finance over the debt
repayment phase is given by the following expression
j 1
SN


1

dA

1  rm  dA j 


S

1

j
S

1
S  N 1
1

j  S  2 m  S 1
rj  z j dB j   1 1r 1 
(7-22)  
 S dBS
j
SN
1
j  S 1 m 1 1  rm
m 
 m 1
dAS 1    1  z m  dA j 


j  S  2 m  S 1
In the case of a government sponsored bank loan (GSBL) the duration of the loan is fixed
and the repayments for each period will rise by the same amount dA j  dA . Substituting
these two conditions into (7-22) yields
 S 1 1 
1


r

z
dB


1  GSBL  S dBS

j
j
j
j  S 1 m 1 1  rm
 m1 1  rm 
S  N 1 j
(7-23)
45
1
where
 GSBL 
1
j 1
SN
  1  r 
j  S  2 m  S 1
j
SN
1
m
  1  z 
j  S  2 m  S 1
1
m
In reference to income contingent loans, ICL, the annual payments and N, the length of
the repayment phase, will depend on the individual lifetime income profile. Let Y jb1 and
 Y jb1  be the level of gross income and educational surcharge, respectively.
The determination of N can be formally expressed as follows,
(7-24)



N  k   : 1  z S  k 1 BS  k 1   YSb k 1 YSb k 1

where 1  zS  k 1 dBS  k 1 is the debt outstanding in the last period. The repayments
themselves will take the form,
(7-25)
 
Aj
  Y jb1 Y jb1
AS  N
  YSb N 1  S  N 1YSb N 1

for
S 1  j  S  N

The term  S  N 1 is the fraction of maximum amount of income that would be repayable
under the ICL that needs to be earned to repay the debt remaining in the last year which,
by definition, is no greater than  YSb N 1 YSb N 1 ,
 S  N 1 
(7-26)
1  z S k 1 BS k 1
 YSb N 1 YSb N 1
A rise in K will raise BS k 1 and therefore  S  N 1 through (7-26) and possibly N, through
(7-24). The impact on repayments is represented as follows
(7-27)





dAS  N 0
  YSb N0 1 1   S  N0 1 YSb N0 1
dAS  N 0  k
  YSb N0 k 1 YSb N0 k 1
for
S  N 0  k  S  N 0  dN
46


  YSb N  dN 1  S  N  dN 1YSb N  dN 1
dAS  N  dN
In applying the model, I will assume that each successive period of tuition and debt
results in two periods of additional repayment. Further, I assume that the surcharge
remains unchanged. Let g j be the growth in gross income from period j  1 to period j.
As a result, repayments under a ICL scheme at the end of period j, A j 1 , will rise by g j
1  g S 1 2 k
1  rS 1 2 k
  S  2 k 1 S  2 k
1  g S 1 2 k
1
1  z S 1 2 k
1
 ICL ,k
(7-28)
1  g S 1
1  rS 1

1  g S 1
1
1  z S 1
1
 ICL , 0
k  1,2...S  1
1  g S 3
1  rS 3
1   S 1 S  2
1  g S 3
1
1  z S 3
1
and
where k relates to the change in the capital stock in period k, k  1,2...S  1.
7.6
DEBT REPAYMENT PHASE
Recall (7-16) which expresses dBS in terms of the increments to human capital. When
(7-16) is substituted into (7-22), one obtains
S  N 1 j
1
  1  r r
(7-29)
j  S 1 m 1
j
 z j dB j
m

j 
I 
1

1     j 
 j 1 1   j 1  jS 1 jx1 1  r j    jS11 jx
dK j



K
j 1 m 1 1  rm


j

1
j




S 2
j
  S 


1
 S 2 1   S 2  SS11 Sx2 1  rS 1    Sx1 S 1
 S 2
m 1 1  rm

S 1
S 

I 
1     S 1 dK S 1
K S 1 

47
S
 1   S 1  S 
1
 S 1 Sx1 1  rS dK S
m 1 1  rm
Combining (7-11) and (7-29) to derive the overall impact of concessional finance
(7-30)
S  N 1
j
j 1
m 1
1
  1  r r
j
 z j dB j 
m
S 1


1
1   j 1  S  jS 1   j, S   jx1 j 1 1  r j dK j
j 1 m 1 1  rm
j
 


1
1   j 1  S  jS11   j  1, S   jx  j 1     I j dK j
Kj 
j 1 m 1 1  rm

S 1

j
 

S
1
1   S 1  S  S , S  Sx1  S 1 1  rS dK S
m 1 1  rm

To derive the first-order differential equation, substitute (7-30) into (3-19). Assuming, as
with the basic model,   1 and   0 , the first-order conditions for the maximization of
lifetime utility now take the following form,
For 1  j  S  1
(7-31)
(7-32)
wj
 j 1
wS
 S 1







 1   1 j jx1 1  r j   1   2 j 1 jx 1    1   j
 1  1S  Sx1 1  rS   1   1   S 
where
1 j  1   j 1  S  jS 1   j, S 

48
 2 j  1   j 2  S  jS 1   j, S 
k
 jk   m .
m j
Under the GSBL scheme:
1
j 1
  1  r 
j  S  2 m  S 1
 k   GSBL 
(7-33)
SN
1
SN
1
m
j
  1  z 
j  S  2 m  S 1
1
m
Under an ICL scheme:
1  g S 1 2 k
1  rS 1 2 k
  S  2 k 1 S  2 k
1  g S 1 2 k
1
1  z S 1 2 k
1
 ICL ,k
(7-34)
k  1,2...S  1
Further, from (7-3)
 
x
j
(7-35)
b, x
j
 1   xj

1 j





1
0   1
1
Where 
b, x
j
 q bj 
 x b 
G 
 j
is the before-tax share of tuition.
8. Concluding remarks
In this paper a method was developed for deriving the effective tax rate on skill
formation. The method follows the tradition of the cost-of-capital approach used to
measure the consolidated impact of taxes and concessions on physical capital formation.
However, the use of effective tax rates derived from the theory of the firm may be
49
misleading if applied to examine the impact of taxes and transfers on the incentives for
skill formation. In particular, it was seen that human capital accumulation involves
production in an essential way.
An explicit modeling of production and the derivation of a cost function took into
account substitution in production and intertemporal relatedness of investment over time.
In addition it provides a platform for deriving marginal rental rates under different modes
of finance and for different stages of an individual's career. It was seen that the
intertemporal productivity effect may play an important role through its impact on returns
to scale and the risk premium under uncertainty.
An alternative approach, based on the derivation of the private internal rate of return, was
shown to have dubious theoretical foundations. The internal consistency of the approach
relies upon deriving a private internal rate of return equal to the opportunity cost. The
cost-of-capital approach derives the implicit marginal rental rate given the real
opportunity cost.
A further limitation of the internal-rate-of-return approach is that it assumes an individual
making a once-in-a-lifetime decision. On the other hand, the cost-of-capital approach can
accommodate a sequential decision-making process, and the potential benefits that may
accrue from real options arising under uncertainty. In other words, the new approach is in
a better position to accommodate extensions involving sequential decision-making and
uncertainty.
A possible limitation of the new methodology is the fact that it assumes all costs, and
rewards, will vary continuously with effort. It does not allow for fixed costs associated
with a decision to participate or withdraw. For instance, there are incentive schemes that
encourage a rapid completion of a qualification through a lump-sum payment, while other
schemes offer benefits to those who complete their qualifications, such as an
apprenticeship. However, in particular instances the assumption of continuity may not be
too far from the truth.
The scope for further theoretical exploration is wide open with more research required
into the technology of human capital at different levels of education and training, and the
implications of uncertainty. The recent work by Cunha and Heckman (2007) provides a
50
much needed framework for integrating advances in our understanding of the process of
skill formation across a range of disciplines. The framework allows for different
relationships between skills contemporaneously and over time, as well as changes in 'skill
technology' as the individual matures. The most important feature of this research is that
it provides a framework for integrating research from a wide range of disciplines and
assessing their implications for specific cases. A priority in this respect is to examine the
extent to which the “single-skill” framework of this paper is adequate for the task of
measuring the impact of taxes and transfers on incentives.
Uncertainty may play a role unique to skill formation through the intertemporal
productivity effect or more generally the relatedness of skills over time. If skills are
characterized by intertemporal complementarity, returns will be positively correlated with
future investment plans while the correlation will be negative in the event that skills are
predominately substitutes over time. Based on well-established principles of asset-pricing
(Cochrane, 2005), intertemporal complementarity or substitutability will give rise to a
risk premium that will depend on the correlation between consumption and the rate of
investment43. The integration of uncertainty would have an immediate application in
indicating the extent to which the lower effective subsidy for ICL, as against the GSBL,
adequately compensates for default risk associated with the GSBL.
The methodology developed in this paper provides a platform for a comprehensive
empirical analysis of the impact of the tax-transfer system on skill formation and a
framework for integrating advances in the theory of human capital for more reliable
measurement.
43
In those papers that do consider investment in an uncertain environment, it is either assumed that the
student is looking for a return from only one period’s investment (Eaton and Rosen, 1980, Judd, 2000;
Weiss, 1972; Levhari and Weiss, 1974), in which the intertemporal productivity effect is irrelevant, or the
model assumes linear homogeneous technology (Palacios-Huentia, 2003, Williams, 1978) and a zero
intertemporal productivity effect. Of course, with uncertainty taxes will also mean that the government
shares in the risk with the student and this will reduce the effective tax rate under the assumption of riskaversion (Devereux, 2004).
51
9. Appendix
MARGINAL REVENUE FUNCTION USED FOR OECD’S EMTR
In this section I show how the OECD expression for the marginal value of investment is
derived, although the no derivation is reported in the paper of Gordon and Tchlinguarian
(1996).
The source of the expression may be made more clearer by beginning with the underlying
earnings flows. For this purpose, consider the expression,

(0-1)
V 
Y
i
S

 Ti S   YiU  TiU  1   
1  r 
i
i 1
i 1
1   i  MRRi 1   

i
i 1
1  r 

i 1
where I have written the increase in after-tax earnings in the following form,
(0-2)
Y
i
S
 Ti S   YiU  TiU   1   i  MRRi
where MRRi  Yi  Yi
S
Ti S  TiU
and  i  S
. Assuming that the marginal rate of return
Yi  YiU
U
rises from a current level of MRR at a constant rate,  , equal to the rate of inflation, then
(0-1) can be written as
1   i 1   1   i 1
1  r i
i 1

(0-3)
V  MRR 1   
The final step is to deal with the non-linearity of the tax system. Notice that  i is a
“local” measure of the marginal tax rate. By this I mean that even if the personal income
tax regime is progressive, for most of the region relevant to the region for the current
investor, the marginal tax rate may be effectively constant. Nevertheless, Gordon and
Tchlinguarian assume that the tax factor rises at a constant rate of  . Letting  denote
the marginal rate of tax next period, (0-2) becomes
i 1


1   1   1   
V  MRR 1   1   
1  r i
i 1

(0-4)
52
Finally, I use the approximation, 1   1    1     , to derive the expression in
(0-1). The assumptions made to derive (0-1) are no different from those employed using
the original King-Fullerton procedure, which effectively uses average rates for inflation,
depreciation and interest. The one additional term is  , the term measuring the
progressivity of the tax system.
WAGE ELASTICITIES
In addition to the evidence cited in Section 8, a survey of empirical studies by Blundell
and Macurdy (1999), confirms that wage-elasticities can vary widely over the population
with the most well-established differences being between married men (see Table 1) and
married women (see Table 2).
Table 1 The elasticity of labour supply of married men
Authors
Sample
Uncompensated
Wage Elasticities
[0 ; 0.03]
Income Elasticity
[–0.95; –1.03]
Hausman (1981)
U.S.
Blomquist (1983)
Sweden
0.08
[–0.03; –0.04]
Blundell and Walker. (1986)
U.K.
0.024
–0.287
Triest (1990)
U.S.
0.05
0
Netherlands
0.12
–0.01
Bourguignon and Magnac (1990)
Source: Blundell and Macurdy (1999, Table 1, pp 1646-1648)
Table 2 The elasticity of labour supply of married women
Authors
Sample
Hausman (1981)
U.S.
Uncompensated
Wage Elasticities
0.995
Arrufata and Zabalza (1986)
U.K.
2.03
–0.2
Blundell, et al. (1988)
U.K.
0.09
–0.26
Arellano and Meghir (1992)
U.K.
0.29
–0.40
Triest (1990)
U.S.
0.97
–0.33
Source: Blundell and Macurdy (1999, Table 2, pp 1649-1651)
Income Elasticity
–0.121
53
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