ANNEX A. Technical approach to calculating tax and skills indicators
A.1. Introduction
This annex outlines an approach to estimating the financial incentives for individuals and governments to invest in individuals’ skills, and the effect of the tax system on these incentives. Specifically, the annex derives Marginal Effective Tax Rates (METRs) and Average Effective Tax Rates (AETRs) on Human Capital Investment, as well as Marginal Returns to Costs Ratios (MRCRs) and Average Returns to Costs Ratios (ARCRs) of Government Investment in Human Capital. The annex also explains how these measures relate to each other, and outlines the effect of government taxes and spending on incentives to invest in skills.
The annex revises and significantly extends the Effective Tax Rates on Skills Methodology outlined in Brys and Torres (2013). This annex clarifies some of their notation and incorporates more explicitly the tax treatment of scholarship income into their definition of the METR. The approach in this study relaxes their assumption that students finance their education using savings by allowing the student to borrow some fraction of the cost of their education. Their calculation of the METR and the Breakeven Earnings Increment (BEI) is extended by also calculating an AETR, for cases where an individual may earn economic rents on a skills investment. In addition to examining the financial returns to education from an individual’s perspective, the returns for governments are also examined. A Returns to Costs Ratio (RCR) of Government Investment in Education (a Tobin’s q of the government investment decision) is also calculated. This figure provides a summary statistic of the government’s financial incentives (in terms of tax revenue) to invest in the education of students. This is calculated for two cases; where an individual breaks even on a skills investment (the “marginal” case), and where an individual earns economic rents (the “average” case). Combining the RCR with the BEI for the individual highlights how the personal income tax (PIT) system apportions the returns and costs of skills investments between the government and the individual.
The formulae throughout this annex are designed to explicate the interaction with the OECD’s Taxing Wages models (OECD, 2014). Throughout, certain formulae are exactly the equations modelled in the calculations. These formulae are Equation 16 for the AETR, Equation 21 for breakeven income after education, Equation 22 for the BEI, Equation 28 for the METR, Equation 44 for the ARCR, and Equation 48 for the MRCR.
The approach for the individual taken is as follows.1 Section A.2 begins by outlining the key benefits of education – the extra earnings earned in the labour market – and how they are impacted by the tax system. The key individual costs of education are outlined: lost earnings, direct costs of education, and the offsetting benefits of scholarship income. The way in which the tax system can interact with these different factors is outlined. These costs and benefits, appropriately discounted, together form the NPV of the Educational Investment.2 This NPV is calculated with and without taxes; this is then used to calculate the AETR on Skills.
The focus then turns to calculating the BEI, which is the value of the earnings increment necessary to just breakeven on an educational investment (i.e. when the NPV of the investment is equal to zero). This is referred to as the BEI. This value is calculated with and without taxes; the difference forms the core of the METR. The relationship between the AETR and the METR is calculated, noting that as the amount of extra earnings from a skills investment converges towards the BEI, the AETR converges towards the METR. Conversely, as the earnings from a skills investment grow larger and move away from the BEI the AETR moves away from the METR and converges towards the marginal effective tax rate on the earnings increment. This result is analogous to the results in Devereux and Griffith (1998) for physical capital, where the AETR converges to the statutory corporate income tax rate.
Section A.3 moves to the perspective of the government.3 As discussed, two RCRs are calculated; a “Tobin’s q” of the government’s investment in skills. This is simply the NPV of the returns to education for the government over the replacement cost of the educational investment for the government. Two RCRs are calculated, an ARCR, where the earnings increment obtained by the individual is held fixed; and a MRCR, where the earnings increment is set to the BEI.
The consideration of the returns to the government is narrow; it is limited to returns in the form of tax revenue from higher wages. There is no accounting for the higher tax revenue that may result from the skills investment in the form of higher economic growth, increased productivity or increased employment. Nor is there any accounting for nontax benefits such as reduced spending on unemployment, greater wellbeing among the population and so on. This means that an ARCR value of less than one in the model does not mean that educational investments do not pay for themselves at all; it merely reflects that they do not pay for themselves solely in the form of recouped tax revenue from higher wages.
Section A.4 expands the analysis by relaxing the assumption that students finance their education with retained earnings. In this section, students are assumed to borrow some fraction of the cost of their investments in skills. It is assumed this borrowing occurs in the form of an interestonly bond. Interest is paid in each year of the duration of the loan, and the principal is repaid in the final year of the loan. The interest rate of the loan as well as its length is allowed to vary. The tax treatment of interest on the loan is considered, as well as the possibility that some fraction of the principal can be written off by the government.
In this section, the definitions of the key statistics, the BEI, METR, AETR, ARCR and MRCR are defined to account for the possibility of financing skills investments from debt as well as savings. It is assumed in the consideration of the returns to government that the government is the creditor of the student. Hence, reduced interest rates for the student constitute a cost for the government, as are parts of the loan principal that are written off. Finally, tax deductibility or exemptions of interest on the student loan are also a cost from the perspective of the government.
A.2. Private costs and benefits of education and their relationship to the tax system: deriving the AETR, BEI, and METR
The benefits of a skills investment
The first consideration is the returns to education. It is important to note that a key difference between human and physical capital is the fact that human capital cannot be sold by its owner at the end of use; when a worker retires or stops working; their skills cannot be readily sold to another worker for use. This key difference means that for a worker’s remaining years in the labour force, the skills investment will yield a return, but upon retirement these returns fall to zero.4 This also means that in order to break even on a skills investment, a worker must recoup the cost of the investment over the years of its use. In contrast, this cost need not be recouped for a physical asset which may be sold. This is discussed further in Box 1.
This per period return is defined as TR_{W} (Total Returns to the Worker). It can be written as:
(1)
Where g(⋅) are the returns dependent on the period, t is the time period, and 1/λ is the expected holding period of the investment, which in the model is the number of years to retirement. Equation 1 can be simplified as:
(2)
Where:
= and (3)
In this formulation, the following variables are defined:

EI = beforetax annual Earnings Increment during the holding period, which is the amount by which beforetax income after making the investment exceeds baseline earnings. This is written as perperiod income in the periods after education (I_{a}) less perperiod income before education (I_{b}). It is the extent to which education increases pretax wages.

T_{EI} = Marginal Effective Tax Rate on the Earnings Increment, where the ‘margin’ is the earnings increment.5 It is the rate at which the increase in earnings after education is taxed away. As with perperiod income, the perperiod tax rates faced before and after education are written as T_{Ib} and T_{Ia} respectively.
EI is the amount by which earnings after making the investment exceed baseline earnings. With progressive PIT systems, tax rates increase as incomes increase. Increasing amounts of tax will have to be paid when the earnings increment rise. This means that the tax rate T_{EI} is one of the key ways in which the tax system can affect the incentives to invest in skills.
The costs of a skills investment
The undiscounted cost for the individual of an investment in skills is written as:
(4)
Where:
Equation 4 can be simplified as:
(5)
Where: and . In this formulation:

FE is the beforetax annual Foregone Earnings during the period of skills acquisition. There is a distinction between earnings and income in each period to allow for the separate treatment of other forms of income such as scholarship and grant income. During education (in “d” periods) I_{d} = E_{d} + SG; income is equal to wage earnings and scholarship or grant income.

T_{FE} is the marginal effective tax rate on FE, also written as the extent to which the tax system increases or offsets the costs of education. It is also the change in the taxes paid, as a share of the change in earnings. As with T_{EI}, T_{FE} will be a core element of the AETR and METR.
Many countries provide tax relief for the costs of education. Many countries also subsidise education through reduced tax rates on scholarship or grant income. Like tax rates on labour earnings, these policy measures are characterised by a variety of deductions, exemptions, thresholds, rates and so on. The aggregate value of these skills tax expenditures (STEs) is encapsulated by the terms θ and ϕ.
The term θ refers to the fraction of the private direct costs of education (DC_{W}) that can be offset against a worker’s tax bill. For example, for countries with a refundable tax credit for educational expenses, the model defines θ = 1, the private costs of education are wholly offset by (nonwasteable) reductions in tax liability. By contrast, where no tax relief exists, the model defines θ = 0, and the full amount DC_{W} is borne by the worker.
The term ϕ refers to the tax paid on scholarship income. For countries where scholarship income is completely exempt from taxation, the model has ϕ = 0, and so the full amount of any scholarship or grant is deducted from the costs of education. In this case the amount of tax paid on total income during education will be T_{Id} I_{d} = T_{Ed} E_{d}. Where scholarship or grant income is taxed as ordinary earnings, the model taxes scholarship or grant income and earned income are taxed at the same rate, T_{Id} I_{d} = T_{Id} (E_{d} + SG).
The values of θ and ϕ may be a function of DC_{W}, SG, and E_{d}; for example, deductibility may decline as DC_{W} increases beyond a certain threshold. Furthermore, in the absence of transferability, refundability or full loss offset, θ begins to decline as DC_{W} exceeds E_{d} (or more precisely, taxable E_{d} before the deduction for education and training costs). It is assumed here that the deductibility of costs is nontransferable. However, if the deductibility of direct costs could be transferred to a higher income taxpayer (e.g. the student’s parent(s)), θ may also depend on the income of the transferee.
These interactions will vary across countries, indeed often education costs are deductible only net of scholarship income; conversely, in some countries scholarship income is taxable to the extent that it exceeds the direct costs of education. The term χ is used to define the overall extent to which these STEs alter the tax liability of the worker. Specifically, the term χ is defined as follows, noting by the brackets χ(DC_{W}, SG) that it is a function of DC_{W} and SG:
(6)
Where T^{*}_{Ed} is the average effective tax rate on taxable income corresponding to the earnings level E_{d} taking into account any tax provisions for the costs of education, as well as the taxation of scholarship income. The tax rate T_{Ed} does not take these provisions into account. Equation 6 therefore writes the effective tax gain χ as the total value of any STEs as a share of the direct costs of education less scholarship income.
When a tax allowance or tax credit for investment in skills is provided, Equations 4 and 5 can be rewritten as:
(7)
Where T_{FE}^{*} is the marginal effective tax rate on FE, where the tax on earnings during education E_{d} () takes into account any deduction for the costs of education. This means that . This formulation of the costs of education simplifies the exposition substantially.
is the tax rate on foregone earnings inclusive of STEs such as deductions for skills costs, and exemptions for scholarship income and the like. That is, – ; is equal to , the normal tax rate on earnings during education, less any allowances or credits based on direct costs, and including any tax paid on scholarship income
Defining the private net present value of a skills investment
Defining the private net present value with taxes
The NPV of an investment in skills, evaluated in period m, is given by:
(8)
The nomenclature used in this study differs in some respects to that used in Brys and Torres (2013) on which it is based. Here differences are outlined for readers wishing to link this annex with their study. There are several differences.
First, Brys and Torres explicitly break down income earned after education into the part that is the true “return” to the student, and the part that is required to repay the initial costs of the skills investment. In the physical capital literature this original outlay to pay for an asset does not need to be recouped in each period after the investment, as it is recouped when the asset is sold. As a skills asset cannot be sold (one cannot costlessly transfer skills to another person when one is finished with them), it must be recouped explicitly. This means that only the fraction of earnings after enough has been earned to repay the initial skills outlay really constitutes the “return” to the skills investment in the strict sense of the physical capital literature. This is the aftercost return that Brys and Torres define as income_{a}; the real “return” on a skills investment. Brys and Torres define the annual after tax net cash flow as a result of a skills investment as:
In this way, Brys and Torres separate the amount needed to recoup the initial pretax cost of the educational investment (income_{b} + income_{d} + DC_{W}) from the rest of the returns to the investment, income_{a}. In their formulation, λ is the reciprocal of the number of years left in the labour force after the student has completed the educational investment. In other words, λ is the fraction of the initial cost of the educational investment that must be repaid in each period, so that, in expectation, by the end of the remaining years in the labour force the entirety of the initial cost of the educational investment will have been repaid.
This formulation highlights a difference between the human capital and physical capital literature. In the physical capital literature, the initial cost of an investment can be recouped when the investment is sold at the end of the period of use. If this were possible for skills, the term λ (income_{b} + income_{d} + DC_{W}) would be equal to zero. In the physical capital literature the term income_{a} is the entire breakeven return to an investment. In the skills case, in order to break even, λ (income_{b} + income_{d} + DC_{W}) must be earned as well.
In the formulation in this study, the definition of the term income after education (I_{a}) as used by Brys and Torres differs to that term income after education. The formulation in this study is equivalent to BrysTorres in the following way:
This term for income after education encompasses the ‘return on investment’ as defined by the physical capital literature, and the amount necessary to recoup costs, as defined by Brys and Torres.
Second, Brys and Torres do not explicitly model interactions that may take place between the tax exemptions for scholarship income and tax deductions for the direct costs of upskilling. They explicitly model a tax allowance for skills costs, but do not account for other STEs. Specifically, Equation 5 in this study, a statement of the upfront costs reads:
The costs of education include:

Aftertax foregone earnings, (1 − T_{FE})FE

Plus the private costs of education DC_{W}

Which may be reduced by any STEs. The value of these STEs, expressed as a fraction of DC_{W}, is written θ

The costs of education are offset by scholarship income SG. This scholarship income is sometimes taxed at the same rate as earned income I_{d} but is often subject to a special rate ϕ.
The equivalent equation in Brys and Torres is equation 2, which reads:
It is important to note that there is no explicit accounting for scholarship income and the tax treatment it may receive. It is considered part of income_{d} and the effects of scholarship income are incorporated into T_{d}. This amendment to their approach allows the calculation of the parameter χ (defined in Equation 6 below), which is an overall summary parameter that captures the value of the taxpayer of STEs related to scholarship income, direct costs and so on. This allows the analysis of all these factors together.
Where TC_{W} and TR_{W} are as defined above, and

r is the real interest rate,

π is the inflation rate,

T_{c} is the average tax rate on capital income,

ρ = (1  Tc)(r + π), is the nominal aftertax return on an alternative capital investment,

m is the length of the period of education,

n is the expected number of years left until retirement,

λ is the inverse of the number of years left until retirement, or the fraction of the cost of the investment that must be paid back in each year to be paid off by retirement, 1/(nm).
It is assumed that r > 0, π > 0 and T_{c}<1, so that ρ>0.
As discussed, the first term of Equation 8 indicates the cost of the investment, TC_{W}, which is assumed to begin taking place in period 0 and takes m units of time. The second term (the double integral) is the expected present value6 of the return on the skills investment, where TR_{W} is the net annual cash flow generated by the investment. The term 0.e^{−(}^{ρ} ^{} ^{π}^{)(}^{n} ^{} ^{m}^{)} indicates that the human capital asset cannot be sold when the skills acquired are no longer used (e.g. when the worker retires), implying that its value eventually drops to zero.
The probability that the investment will yield a return in period n is assumed to decrease exponentially as n increases, as implied by the integral of the term λe^{−}^{λ}^{(}^{n} ^{} ^{m}^{)}. The inverse of λ, 1/λ, is the expected holding period of the investment, which is assumed to be positive. The holding period is assumed to be the period of time over which skills are used in the labour force; it decreases if the retirement age increases.
The net annual cash flow generated by the investment, TR_{W}, is discounted at the rate ρ and increases at the rate π for a net growth rate of −(ρ − π), where ρ is the nominal discount rate applicable to a human capital investment and π is the rate of inflation (assuming that wages increase with inflation).7 It is assumed that the investment can begin to yield a return in period m and that the cash flow stream begins to be discounted in period m. To adjust for the lag in discounting the stream of benefits, the costs incurred during the first period are grossedup by the real discount rate, ρ−π, which is assumed to be positive. These discounting assumptions imply that the present in the net present value calculation is assumed to be period m rather than period 0 as in traditional investment models.
Solving this equation yields, after some manipulations, Equations 9, 10, and 11:
(9)
(10)
(11)
Where , is the grossup of the current costs to the period in which the skills investment begins. Henceforth, a notational convenience is used, so that:

The grossup factor of the costs of investment in education as

The discount factor of the returns to investment in education as
The superscript T is used to denote that these discount factors incorporate the effects of taxes on capital income ( and will later be used to denote the discount factors without taxes on capital income). From Equation 11, the NPV of a skills investment becomes:
(12)
Defining the private net present value without taxes
To find the overall effect of the tax system on a skills investment, it is necessary to estimate the difference between the NPV of a skills investment with and without the tax system. Equation 12 above gives the NPV in the case with taxes, it remains to consider the NPV without taxes.
In the no tax case, each tax rate is simply set to zero. This includes the tax rate on income before education, during education, and after education, the tax rate on capital, as well as any deductions, or exemptions for direct costs and scholarship income. This means that:
(13)
There are several implications to this:

Given the previous formulae for the effective tax rates on foregone earnings and the effective tax rate on the earnings increment, and , it follows that .

Moreover, implies that the nominal aftertax return on an alternative capital investment, , now becomes simply .

This in turn implies that the discount factors, and , change also. and are defined as the discount factors in the notax case (that is, where ). This means that = , and .
Substituting in these various values into , a value for is obtained as follows:
(14)
This equation is analogous to Equation 12 above.
The average effective tax rate on skills investments
Following both Devereux and Griffith (2003) and Klemm (2008), the AETR is defined as simply the difference between the NPV of the skills investment with and without taxes, expressed as a share of the after tax cashflow from the investment.
(15)
Substituting in the values for and from Equations 12 and 14 above, a more complete definition can be obtained.
(16)
This is the expression that is used in the Taxing Wages modelling. In the absence of capital taxes, , , and , the expression simplifies considerably to become:
(17)
Which simplifies further to become:
(18)
This equation makes it clear that in the model the tax system interacts with the financial incentives to invest in skills in three key ways, expressed here as three key tax rates, , , and . First, the tax system taxes away the returns to skills in the form of the earnings increment (. Second, it reduces the cost of skills by reducing the cost of foregone earnings (. Third, the tax system offsets, or may offset, the cost of education through tax deductions or credits for the direct cost of education, or through tax exemptions for scholarship income ().
Breakeven earnings increment on skills investments
The AETR defines the effect of the tax system on a skills investment for any investment in skills, for any level of earnings that result from the investment. However, it can also be useful to define the effect of the tax system on a marginal skills investment; one where the worker is just indifferent between making the investment and not making it. In other words, the marginal worker faces the same net financial return to making the investment and not making it. In order to calculate the METR on skills the approach is to define what it means to be just indifferent between making a skills investment and not. Subsequently, the level of earnings at which a worker is indifferent between investing in skills and not investing is calculated. The METR is then simply a function of how this earnings level changes in the presence or absence of taxes.
The alternative to making a skills investment consists of earning the baseline earnings (I_{b}) and a return from investing the cost of education in an alternative capital investment (whose original cost can be fully recovered at the time of the asset’s disposition). An investment in skills is defined as marginal when a prospective student is indifferent between making this investment and the alternative. In other words, a marginal skills investment is one where the NPV of the skills investment is zero, where = 0, with the “hat” denoting the breakeven level of V. This is essentially saying that the costs of the skills investment are just covered by the returns.
(19)
All of the expressions follow from the AETR case. Substituting the expressions of and for , another expression for can be obtained.
(20)
Where, solving for , the result is:
(21)
Again, the pretax annual earnings increment defined such that V = 0 (the earnings increment required for a skills investment to break even) is the BEI. The BEI is referred to as the difference between the breakeven earnings level, and the previous income
(22)
Equation 20 can be reexpressed in a similar way by substituting the nowfamiliar formulation for .
(23)
Where, as before:
= and (24)
BEI (in the presence of taxes) is defined as:
(25)
To find the METR, it is necessary to find a similar breakeven income level and BEI in the absence of tax. As with the AETR case, the notax case involves setting each tax rate to zero. This means that , as mentioned before. To repeat, the implications of this are that:

Given that and , it follows that = 0.

implies that the nominal aftertax return on an alternative capital investment, , now becomes simply .

This in turn implies that the discount factors, and , change also. and are defined as the discount factors in the notax case, where This means that, = , and .
can be expressed as:
(26)
The notax BEI can similarly be expressed as:
(27)
The marginal effective tax rate on skills investments
The METR on skills is the difference between the Earnings Increment needed to make the investment in the presence and absence of taxes expressed as a share of the minimum earnings increment required to make the investment in the presence of taxes. Essentially it answers the question: for a marginal skills investor, what fraction of the required return is attributable to tax? This equation is expressed concisely below:
(28)
Substituting in the expressions for and from Equations 25 and 27 the definition of the METR in Equation 28 can be written as:
(29)
If it is assumed that , then this expression simplifies to:
(30)
Which, by adding and subtracting to the numerator, becomes:
(31)
Like Equation 18, this equation demonstrates that the METR is a function of differing tax effects: the tax rate on the earnings increment, as well as the tax rate on the costs of education, incorporating both the tax rate on foregone earnings, as well as the tax rate on the costs of education χ, both of which are incorporated into the rate .
Upper and lower bounds of the average effective tax rate
In a result similar to those found in the literature on the tax treatment of physical capital, it can be seen that the AETR is a weighted average of the METR and the statutory tax rate on human capital. The AETR on skills is equal to the METR when the earnings increment falls such that it is just equal to the amount needed to breakeven. Where the earnings increment from the skills investment is higher, the average tax rate is a mix between this marginal rate and the top tax rate on skills. In this case, the latter tax rate is the tax rate on the earnings increment. This can be expressed in Equation 32:
(32)
The intuition behind the BEI can further be explored by substituting the terms for and for , which yields the expression below:
This highlights the key components of the BEI:

The foregone return on the aftertax cost of investment borne by the student: ()

An amount to gradually recover the aftertax cost of the investment over the holding period

Enough to pay for any extra taxes incurred by these extra earnings,
This intuition is highlighted in the figure below, which shows the fall in earnings and the direct costs of education during an educational period, followed by the extra earnings needed to breakeven on the skills investment. The three key components of the BEI highlighted above: the tax wedge, the opportunity cost of capital and the recovery of the costs of a skills investment are highlighted. This graph is based on Norwegian data.
In other words, when a project just breaks even (when ); there are no economic rents earned. In this instance, the average tax rate on the project is just the marginal rate; the investment is on a knifeedge between taking place and not, there is no inframarginal return. This is shown in Equation 32; as , the ratio . The second term in the equation disappears, and .
Similarly, as the earnings increment grows large, the inframarginal return grows large as well. In this case the statutory tax rate becomes a larger share of the AETR. To see this simply note that as , the ratio . This will see the first term in the Equation 32 tend to 0, and .
To demonstrate the veracity of Equation 32, recall the definitions of METR, BEI and AETR. These are taken from Equations 31, 25, and 18, respectively.
(33)
(34)
(35)
Substituting Equation 33 into Equation 32 above, the following result is obtained:
(36)
Cancelling the first and last terms and adding in the definition of the BEI from Equation 34 above yields:
(37)
Simplifying, and noting that , yields Equation 35 above, demonstrating the result.
A.3. Public costs and benefits and the relationship to the tax system: deriving the ARCR and MRCR
The discussion so far has considered the costs and benefits, and the effect of the tax system on them, from the perspective of the worker or individual. Throughout, the other party to the educational investment has been the government.8 The second key set of indicators in this study examines the costs and benefits of skills investments from the perspective of the government. It is important to note that unlike in the individual case, no optimisation is assumed; there is no breakeven action on the part of the government. This section describes a ‘Tobin’s q’ for the investment in human capital from the perspective of the government; a ratio of the NPV of the earnings streams from education to the NPV of the costs of education. Two of these indicators are presented. The first is for a prespecified level of student earnings aftereducation. This is the ARCR for the government. The second RCR assumes that the earnings stream for the individual worker is such that the individual just breaks even. This indicator calculates the RCR for the government for this marginal student; this is the MRCR. These two indicators are analogous to the AETR and METR developed previously.
The key insight from these RCR indicators, as discussed in the main text, is that the tax system helps to divide the costs and benefits of the educational investment between the government and the individual worker. On the cost side, it does so partially; the tax rate on foregone earnings and the tax breaks given for scholarship income and direct costs perform this function. However, the ratio of individual direct spending to government direct spending matters too. On the benefits side, however, the tax system performs the entirety of the division of benefits between the government and the individual worker. This insight has been discussed in the main text, and will be outlined mathematically below.
The approach is as follows: first the overall costs of upskilling are modelled: lost earnings, direct spending by students, direct spending by the government. The benefits of the educational investment are then examined. The model then discusses how these benefits are apportioned, and calculates the costs and benefits for the government. Having done this, the model calculates the ARCR and the MRCR. The difference between these two indicators is simply that the ARCR calculates the government’s RCR for the average individual making an educational investment; the average student is assumed to receive a morethanbreakeven return to the investment. The MRCR calculates the RCR for the marginal individual, who just breaks even on the investment. In other words, the MRCR is simply the ARCR where the EI = BEI. Throughout this section, the assumption that is maintained for simplicity.
The total cost of upskilling
The total cost of upskilling is defined as follows:
(38)
Where , , and are as before, and is the direct costs of education borne by the government. Note that an assumption of this approach is that all scholarship and grant income is supplied by the government; this scholarship income is not a “cost” of education per se, but rather a transfer from the government to the individual worker; this means that it nets out of Equation 38.
The fraction of borne by the government ( is , where are the costs borne by the worker. Substituting the expression of from Equation 4, yields:
(39)
This can be divided into several parts:

The lost tax revenue from the worker’s reduced earnings, ,

The revenue losses result from the tax treatment of direct costs ,

The lost scholarship income (offset by any tax received on this scholarship income .

Plus the direct government spending on education,
Similarly, the total per period returns to upskilling are simply characterised by the earnings increment:
(40)
as defined in Equation 3. Recall that the fraction of borne by the government, is , where are the part of the return received by the worker. Substituting the expression for from Equation 2 yields Here, as before.
This means that the tax system apportions the returns between the student and the government; more progressive taxes will mean a higher , which will mean a higher share of returns from upskilling for the government.
The average returns to costs ratio of government spending
A Tobin’s q – a returns to costs ratio for a government investment in skills – can be defined. This is the ratio of the present value of the returns to the skills investment to the replacement cost of the skills investment. As with the second term in Equation 8, the NPV for the individual worker, the government’s present value of the returns to an educational investment can be defined as:
(41)
Whereas in Section A.2, the is a representation of the fact that the value of the investment falls to zero when a person retires. The ARCR can be defined as:
(42)
Solving the integrals gives:
(43)
It is important to note that in Equation 43 with respect to the government, “no tax” discount factors are used, and This is due to the assumption that capital taxation raises no wedge between the returns to skills and an alternative capital investment for the government; the government would not pay capital taxes to itself on an alternative capital investment. Note, however, that there is no accounting for lost capital taxes for the government when an individual invests in skills instead of investing in an alternative capital investment. The overall formula for will be:
(44)
Equation 44 is exactly the formula used in the Taxing Wages models. This formula is essentially a ratio of returns to costs; where it is greater than one; the discounted stream of benefits are larger than grossedup costs; in such situations, investing in skills is profitable for the government from a tax revenue perspective. Where it is less than one, the discounted stream of returns to the government is less than the grossedup costs. In such situations, the educational investment does not pay for itself in the form of tax revenue.
The marginal returns to costs ratio of government spending
The previous section takes the term as exogenous, however, under the assumption that the individual worker continues to invest in education up to the point where the returns fall to the breakeven level, the RCR for the government differs. In this case, the expected return from the investment is simply the breakeven earnings level. This is in contrast to the ARCR in Equation 44, where the earnings level is not specified.
Note the equation for the BEI derived previously (in Equation 25), stated below.
(45)
The MRCR is derived by substituting the BEI for the EI into Equation 44. This yields an expression for the MRCR as follows:
(46)
Multiplying by yields:
(47)
While this expression seems complicated, it can be simplified greatly by noting that
Simplifying, becomes:
(48)
To explain this, note that:

is the government’s discounted returns to the skills investment,

is the worker’s discounted returns to the skills investment,

is the government’s discounted costs for the skills investment,

is the worker’s discounted returns to the skills investment.
The ‘Tobin’s q’, the governments’ MRCR, is a ratio of ratios; it is a ratio of the ratio of discounted government and individual benefits to the ratio of discounted government and individual costs.
The ARCR (as defined in Equation 44) is a ratio of discounted benefits to discounted costs. Where this ratio is greater than one, the government is recouping the entirety of its benefits, in NPV terms, in later tax revenue. The MRCR functions in a similar way. The difference is rather that the breakeven response of the individual is built into the MRCR in a way that is not the case for the ARCR. The MRCR presumes that the amount invested by the individual worker is such that they break even. But this rate itself is a function of the division of costs and benefits for the government.
What matters for the government, in terms of the return on its investment, is not the ratio of its costs to its returns, but rather whether the ratio of its costs to the costs of the individual are higher or lower than the ratio of its returns to the individuals. Put another way, if the government receives a higher share of the returns than the share of the costs it bears, then the MRCR is greater than 1. If it receives a lower share of the returns than the share of the costs it bears, then the MRCR is less than one.
A.4. Including student debt and its effects on public and private costs and benefits of education
This section extends the analysis in Sections A.2 and A.3 to allow for the possibility that students finance part or all of the costs of their education by borrowing from the government. The section repeats the discussion and finds the same key equations for the AETRs, BEI, METR, ARCR and MRCR, but also incorporates several key terms that account for the impact of financing costs or subsidies that reduce the costs of skills financing, such as through reduced interest rates on student loans, loan writeoffs, and the tax treatment of student debt. The debt is constructed as a bond for ease of exposition, though the underlying principles remain the same as if the debt was constructed as a standard bank loan.
Several key areas of flexibility are including in the modelling.

The percentage of total costs borrowed is allowed to vary.

The length of the loan period is allowed to vary.

The rate at which the student borrows can also vary. It can be set equal to the riskfree real return on a capital investment r, or it can be a rate lower or higher than the risk free rate. In the case, for example, of a government subsidy in the form of a reducedinterest loan, the rate may be lower than the riskfree rate. Where the rate it is higher, it can be conceived to represent a premium for student debt, potentially due to higher riskiness of student debt.

The interest on student debt can be deductible from the student’s taxable income after they finish their schooling. The rate of this tax deduction can vary (it can be deducted at the marginal aftereducation tax rate, but limits can also be placed on the deductibility. Note that tax deductibility of repayments of the principal of the debt is not accounted for; only the interest is treated as being taxdeductible.

A fraction of the student debt can also be written off by the government, meaning that this part of the principal does not need to be repaid at the end of the loan period. The model does not account for the tax treatment of debtwrite offs.
The approach proceeds analogously to the approaches in Section A.2. Throughout, equations from these latter two sections are redefined incorporating loan provisions. The first section defines the adjusted costs of education incorporating student debt. The next section incorporates these revised costs and returns into the calculation of the AETR. The next section incorporates the same adjusted costs and returns into the calculation of the BEI. The next section incorporates this revised BEI into the calculation of the METR. The final two sections calculate revised versions of the ARCR and MRCR respectively.
Defining costs of education
The modelling proceeds as follows. The costs of education, involving lost posttax earnings , direct costs , scholarship income , and STEs ( and ), are all retained. It is then assumed that the student borrows % of the cost of upskilling.
(49)
It is important to note that remains to be paid by the student from their existing savings, as in Section A.2. Note that it could be assumed that the student only borrows the fixed costs of education that they bear . In this case the definition of would be .
However, the student could also borrow the full costs of the education, inclusive of any lost earnings. In this case the value of would be 1.
Interest on the loan is set at a nominal rate of , so the real rate of interest is . It is assumed, for ease of exposition, that interest starts to be paid when the student starts using the skills, so interest is not paid while the student continues in education. As mentioned, the real interest rate on the student loans can be set to equal to the riskfree real return on capital investments r, or can be set to be equal to some other value. The nominal amount of interest paid on the loan is assumed to be tax deductible at rate . Generally, it will be assumed that , though the modelling can also account for situations in which the size of the deductibility might be capped, and thus where .
The principal is paid back at the end of the expected holding period of the loan. This expected holding period is defined as years. In such cases as where the holding period of the loan is the student’s expected future years in the labour market, then . Further, it is important to note that the principal amount that is repaid is not indexed for inflation – only the nominal value of the loan taken out is repaid.
A portion of the nominal value of the loan is forgiven ; is repaid. It is assumed that this increase in wealth coming from loan forgiveness is not taxed.
By incorporating all these factors together, an expression for the NPV of the Costs of Education is obtained.
(50)
Rearranging yields the following equation:
(51)
This can be seen as analogous to the first term in Equation 8 in Section A.2 of this Annex, except that the expression for total costs is weighted by the extent to which borrowing these costs increases or decreases the costs in NPV terms. A term, can then be defined as the overall extent to which the treatment of student debt reduces or increases the costs of education to the student. This means that the true cost to the student is reduced or increased by (1 times the fraction of the total nominal costs of upskilling  as defined in Equation 4 of this Annex  borrowed by the student. This weight is defined as:
(52)
A series of rearrangements and simplifications yields a simplified expression for :
(53)
The definition of Total Costs of Education incorporating financing costs for the individual is as follows:
(54)
As mentioned, this is an expression analogous to the first term in Equation 8 above. Following from this, Equation 8, incorporating student debt, can be redefined as follows:
(55)
This can be rewritten as:
(56)
Defining the average effective tax rate on skills
Recall from Section A.2 that and that . Given this, V can also be expressed concisely as:
(57)
This equation is analogous to Equation 12 in Section A.2 of this Annex. In the notax case, as discussed, all taxes are set to zero. This means that . In accounting for student loans, this also means that T_{l} = 0. This in turn means that V_{NT} becomes:
(58)
Where , , and . Hence, in the notax case, the effects of the introduction of student loans on the overall NPV of upskilling are confined to two factors. The first is the difference in the real interest rate that the student pays and the real interest rate the student has to pay to the government on its student debt. The second is the value of any loan writeoff provided.
Building on this definition of the NPV of education, the AETR definition from Section A.2 is simply restated as the difference between the NPV of education with and without taxes, as a share of the discounted increase in earnings after education.
(59)
In this context, the AETR incorporating student debt is defined as:
(60)
Defining the breakeven earnings increment
Setting V= 0 in Equation 57 yields a definition for the BEI:
(61)
Which is analogous to Equation 25. , the necessary amount of pretax income needed to be earned to break even on a skills investment can be defined similarly as in Equation 21 as:
(62)
Disaggregating F^{T} yields:
(63)
Note that the first two terms in the above Equation 63 are the same as in Equation 21. They imply that breakeven income after education must be equal to income before education, plus enough to pay back the discounted costs of education, plus any additional taxes that may result due to higher earnings after education. The three last terms – new to this equation – are the three ways in which student debt impacts breakeven income. The first term accounts for potential gains from loan writeoffs ε. The size of this write off is multiplied by the posttax discounted value of borrowing.
The second term is the value of any special interest rate provisions provided to a student. This interest differential is defined as . Note that where capital taxes are zero it can be defined as , the difference between the riskfree interest rate and the rate at which the student borrows. Note further that if the student borrows at the rate r, this secondlast term falls to zero.
The third term is the value of any interest deductibility or any other student debt tax interest relief. Note that and . The marginal rate of interest deductibility T_{l} is the difference between tax paid in the presence of interest relief and tax paid without this relief . This is analogous to the definition of χ in Equation 6.
Defining the marginal effective tax rate on skills
Recalling the definition of the METR from Equation 28, Recalling also the definition of V_{NT} from Equation 58.
Whereas previously , and . This means that the definition of BEI_{NT} (analogous to Equation 27) is as follows:
(64)
Putting this equation together with the definition of the BEI in Equation 61 and the definition of the METR from Equation 28 yields the overall required equation. Assuming , this can expressed as follows:
(65)
Defining the average returns to costs ratio for governments
Recall from Equation 38 the perperiod total costs of education, including both the costs to the government and to the student, . Recalling the definition of incorporating student debt, . As in Equation 39, the difference in government costs are defined as the difference between total costs and the workers’ costs . This assumes no private loan provision to students. It follows that:
(66)
This can be rearranged as follows:
(67)
This is analogous to Equation 39 above, but where is defined as in Equation 53. Note that the addition of to the equation raises costs for the government; subsidised student debt provision costs the government, and does so more where the size of student debt is larger. The definition of the ARCR in this case is just as in Equation 43 in Section 3.2 above: , except where is defined as in Equation 67 above.
Defining the marginal returns to costs ratio
The approach to defining the MRCR in this case follows that in Section A.3; the BEI is substituted for the EI in the definition of the ARCR. Recalling the definition of the BEI:
(68)
Where, , , and . Noting from Section 3.3 that the definition of the MRCR is , substituting the BEI definition above yields the following expression:
(69)
Noting that as in Section A.3, = 0 throughout, can be defined as as follows:
(70)
Hence, the MRCR can be defined as follows:
(71)
Notes
← 1. Throughout, variables for the individual are denoted with a W subscript, for ‘worker’. Variables for the government are denoted with a G subscript.
← 2. For computational simplicity, the ‘Present’ of the ‘Net Present Value’ of an educational investment is taken to be the end of the educational investment, not the beginning. This means that costs of education, when they take place over several years, are ‘’grossed up’ or ‘reverse discounted’.
← 3. Throughout, the role of the firm is omitted from the analysis.
← 4. The discussion throughout abstracts from the returns to education that do not come in the form of increased expected earnings. For example, nonpecuniary benefits are omitted from the analysis, as are changes to employment probabilities.
← 5. is a marginal and not an average effective tax rate, although the margin EI can be quite large. Note that for each income level X > Y, the following relation holds (where and represent the average effective tax rates on income X and Y respectively and T_{XY} is the marginal tax rate on the earnings increment XY): ·X = ·Y + (XY). In cases where only PITs are considered and depending on the statutory PIT brackets, the TEI is a weighted average of the marginal PIT rates over the EI interval (i.e. a ‘weighted average’ marginal PIT rate).
← 6. The probability that the investment yields a return in period 1, after which no further return is earned is λ; the probability that the investment earns a return in periods 1 and 2, after which no further return is earned equals λe^{λ}; which is slightly lower than λ; the probability that the investment earns a return in periods 1, 2, and 3, after which no further return is earned is even smaller (λe^{2λ}). The probability of earning a return in every period until n but not thereafter decreases as n increases. Note that the periodic return is assumed to be constant over time but is discounted after period 1. This setup implies that the investment will earn a return in period 1 with full certainty while the probability that the investment yields a return in the following periods decreases over time until n reaches infinity, when the probability that the investment earns a return is 0.
← 7. It would be possible to assume that wages also increase with productivity by adding a productivity growth term to ρπ. It would also be possible to assume that wages increase by less than π, as if they were partly indexed, which could be incorporated as another extension of this work.
← 8. As is discussed in the main text, the role of firms or civil society in providing or financing education is ignored.