Taxation and the User Cost of Capital: An Introduction

Creedy, John; Gemmell, Norman Working Paper Taxation and the User Cost of Capital: An Introduction New Zealand Treasury Working Paper, No. 15/02 Provided in Cooperation with: The Treasury, New Zealand Government Suggested Citation: Creedy, John; Gemmell, Norman (2015) : Taxation and the User Cost of Capital: An Introduction, New Zealand Treasury Working Paper, No. 15/02, ISBN 978-0-478-43623-5, New Zealand Government, The Treasury, Wellington This Version is available at: https://hdl.handle.net/10419/205680 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Taxation and the User Cost of Capital: An Introduction John Creedy and Norman Gemmell New Zealand Treasury Working Paper 15/02 March 2015 DISCLAIMER The views, opinions, findings, and conclusions or recommendations expressed in this Working Paper are strictly those of the author(s). They do not necessarily reflect the views of the New Zealand Treasury or the New Zealand Government . The New Zealand Treasury and the New Zealand Government take no responsibility for any errors or omissions in, or for the correctness of, the information contained in these working papers. The paper is presented not as policy, but with a view to inform and stimulate wider debate. NZ TREASURY Taxation and the User Cost of Capital: An Introduction WORKING PAPER 15/02 MONTH/YEAR March 2015 AUTHORS John Creedy Professor of Public Economics and Taxation/Principal Advisor Victoria University of Wellington/New Zealand Treasury No. 1 The Terrace Wellington New Zealand Email: [email protected] Telephone: ++64 +4 917 6893 Norman Gemmell Professor of Public Finance Victoria University of Wellington Wellington New Zealand Email: [email protected] Telephone: ++64 +4 463 5843 ISBN (ONLINE) 978-0-478-43623-5 URL Treasury website at March 2015: http://www.treasury.govt.nz/publications/research-policy/wp/ 2015/15-02/twp15-02.pdf Persistent URL: http://purl.oclc.org/nzt/p-1724 ACKNOWLEDGEMENTS We should like to thank Martin Keene, Helen Miller and Florian Misch for their comments on an earlier draft. NZ TREASURY New Zealand Treasury PO Box 3724 Wellington 6008 NEW ZEALAND Email: [email protected] Telephone: +64 4 472 2733 Website: www.treasury.govt.nz Abstract The aim of this paper is to provide an introduction to the concept of user cost and its determinants. Particular attention is given to the influence of taxation. The concept of user cost relates to the rental, the rate of return to capital, that arises in a profit maximising situation in which further investment in capital produces no additional profit. This paper sets out in some detail the range of assumptions involved in obtaining alternative expressions for the user cost. The user cost refers to a before-tax capital rental, the rate of return that ensures that the (after-tax) cost of capital is equal to the post-tax returns over its life. Hence, associated with the user cost measure is an effective marginal tax rate. This can differ substantially from the statutory marginal rate applicable to the investor. A related effective average tax rate is also defined. WP15/02 Taxation and the User Cost of Capital: An Introduction i Executive Summary The concept of user cost relates to the rate of return to capital, referred to as the rental, that arises in a profit maximising situation. This is one in which further investment in capital produces no additional profit. Despite this apparently simple statement, the concept gives rise to a complex range of cases which need to be distinguished. The importance of taxation and the link with optimising behaviour by firms means that the user cost concept has a central role in investment and location decisions. Differences in tax regimes among countries can influence the decision regarding where to locate production and the head office of multinational firms. The relevant features of tax regimes relate not only to the treatment of companies but to the individuals who are the ultimate owners. Any change in a tax rate or tax structure which implies an increase in the user cost of capital implies that firms need to obtain a higher pre-tax rate of return or rental for an investment to be worthwhile. The aim of this paper is to provide an introduction to the concept of user cost and its determinants, paying particular attention to the influence of taxation. This paper sets out in some detail, using a consistent terminology, the range of assumptions involved in obtaining alternative expressions for the user cost. The user cost refers to a before-tax capital rental, the rate of return that ensures that the (after-tax) cost of capital is equal to the post-tax returns. Hence, associated with the user cost measure is an effective marginal tax rate. This can differ substantially from the statutory marginal rate applicable to the investor. Particular attention is given in this paper to the properties of the effective marginal tax rate in different circumstances, drawing attention to the difference between tax-inclusive and exclusive rates. It is shown that the relationship between the statutory tax rate and the effective tax rate can vary substantially, depending on the rate of interest. A related effective average tax rate is also defined for the context in which the firm obtains economic rents (that is, earnings above those needed for it to remain in its present position). This may be important in the context of multinational investment where the firm is operating below its profit maximising output. The link between the user cost, effective tax rates and investment is also briefly discussed. WP15/02 Taxation and the User Cost of Capital: An Introduction ii Contents Abstract i Executive Summary ii 1 Introduction 1 2 User Cost: The Simplest Case 3 3 Allowing for Taxation 6 3.1 Taxation of Corporations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Depreciation and Tax Credits . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 User Cost in Terms of the After-Tax Nominal Interest Rate . . . . . . . . . . 9 3.4 User Cost in Terms of Before-Tax Real Interest Rate . . . . . . . . . . . . . 9 3.5 User Cost in Terms of Before-Tax Nominal Interest Rate . . . . . . . . . . . 10 4 The After-Tax Real Interest Rate 10 4.1 ForeignInvestors................................. 10 4.2 Domestic Residents and Imputation . . . . . . . . . . . . . . . . . . . . . . 11 5 The Effective Marginal Tax Rate 11 5.1 The User Cost and the EMTR . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.2 AFormalStatement ............................... 14 5.3 Variation in EMTRIwith Interest and Inflation Rates . . . . . . . . . . . . . 15 5.4 Variation in EMTRIwith the Statutory Tax Rate . . . . . . . . . . . . . . . . 16 5.4.1 The Role of the Nominal Interest Rate . . . . . . . . . . . . . . . . . 16 5.4.2 The Role of Fiscal Depreciation and Tax Credit Rates . . . . . . . . 19 6 An Effective Average Tax Rate 22 7 Investment and the User Cost 27 8 Conclusions 28 List of Figures Figure 1 – The User Cost: No Taxation . . . . . . . . . . . . . . . . . . . . . . . . . 5 Figure 2 – User Cost and The Effective Marginal Tax Rate . . . . . . . . . . . . . 13 Figure 3 – Variation in Effective Marginal Tax Rate with Nominal Interest Rate . . . 16 Figure 4 – Variation in EMTR and User Cost with Statutory Tax Rate: Nominal Interest Rates of 0.03 and 0.05 . . . . . . . . . . . . . . . . . . . . . . 17 Figure 5 – Variation in EMTR and Net User Cost with Statutory Rate: Alternative Fiscal Depreciation Rates . . . . . . . . . . . . . . . . . . . . . . . . . 18 Figure 6 – Variation in EMTR and Net User Cost with Statutory Tax Rate for Alternative Tax Credit Rates: Nominal Interest Rate of 0.03 . . . . . . . . . 20 Figure 7 – Variation in EMTR and Net User Cost with Statutory Tax Rate for Alternative Tax Credit Rates: Nominal Interest Rate of 0,05 . . . . . . . . . 21 Figure 8 – Effective Average Tax Rate . . . . . . . . . . . . . . . . . . . . . . . . . 25 WP15/02 Taxation and the User Cost of Capital: An Introduction iii List of Tables Table 1 – List of Variables Used . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 WP15/02 Taxation and the User Cost of Capital: An Introduction iv Taxation and the User Cost of Capital: An Introduction 1 Introduction The aim of this paper is to provide an introduction to the concept of user cost and its determinants, paying particular attention to the influence of taxation. In the standard neoclassical model of production, the term ‘cost’ generally refers to an opportunity cost rather than simply a direct pecuniary cost of a good or service. In the present context, user cost relates to the rental, the rate of return to capital, that arises in a profit maximising situation in which further investment in capital produces no additional profit. This paper hopes to make this rather loose statement more precise and clear by setting out in some detail the range of assumptions involved.1 To provide a little more context at this preliminary stage, consider a firm’s decision to increase its investment in a capital asset. This decision depends on a complex range of factors, including the cost of financing the investment and their tax treatment. Suppose, for simplicity, that a single type of capital good is used in production. Assuming that capital can be varied continuously, a basic implication of profit maximisation is that the firm increases its investment until its total cost is equal to the present value of the after-tax and depreciation returns from the flow of capital services, discounted at a suitable rate over the life of the project. With an assumption of decreasing marginal returns, capital is increased until the condition is satisfied. It is not necessary to suppose that the firm actually owns the capital good, the firm may be considered to rent the corresponding capital services and for this reason the return is referred to as a rental, for comparison with a wage rate. Once this profit maximising position is achieved, a further increase in the use of capital services produces zero profit. The marginal revenue product at that point determines the capital rental. It is the before-tax rate of return, the capital rental at the profit maximising level of investment, that defines the user cost. This concept refers to the rate of return that ensures that the (after-tax) cost of capital is equal to the after-tax return. Hence, associated with the user cost measure is an effective marginal tax rate. As shown below, the effective marginal tax rate is typically not 1 This paper is not a literature review, so only selected references are made. The seminal paper is Hall and Jorgensen (1967). See also Auerbach (1983, 2002), King and Fullerton (1984), Benge (1997, 1998) and Fabling et al. (2013). WP15/02 Taxation and the User Cost of Capital: An Introduction 1 equal to the statutory marginal rate applicable to the investor: they are equal only under special conditions. Additional important distinctions, other than between beforeand after-tax values, must also be made. The existence of inflation means that in practice a distinction must be made between real and nominal values. The existence of depreciation means that a distinction must be made between gross and net values. The simple statement above must therefore be modified. The gross user cost is defined as the before-tax and before-depreciation real capital rental, obtained from a marginal investment which must be earned if the after-tax real rate of return (the rental adjusted for taxation, depreciation and capital gains or losses) is equated to the cost of capital. The latter cost is given by the after-tax real rate of interest. The net user cost is the gross user cost adjusted for depreciation. As shown below, care is needed to disentangle the various components in view of the complexity of tax structures. The importance of taxation and the link with optimising behaviour by firms means that the user cost concept has a central role in investment and location decisions. Differences in tax regimes among countries can influence the decision regarding where to locate production and the head office of multinational firms. The relevant features of tax regimes relate not only to the treatment of companies but to the individuals who are the ultimate owners. 2 Any change in a tax rate or tax structure which implies an increase in the user cost of capital implies that firms need to obtain a higher pre-tax rate of return or rental for an investment to be worthwhile. An understanding of precisely how taxation can affect the user cost in different circumstances is needed in order to appreciate the incentives facing firms. Although the concept is central in the neoclassical analysis of firms, and has important policy implications, its treatment is typically given very little attention in economics texts, despite the extensive and often technical literature in which it features. Therefore, the present introductory review seems warranted. Discussions of user cost are often not easy for the newcomer to follow. There appears to be no settled terminology, let alone notation, and even the concept itself is often described merely ‘in passing’ and is given various definitions when put into words rather than equations, which may appear confusing to the uninitiated. The assumptions behind its use are often not clear. Some authors even use the term ‘cost of capital’ when referring to user cost, while others use the term ‘cost of capital’ (as above) to refer to an appropriate rate of interest. Other authors 2 The situation is considerably complicated if the firm is not taxed as a separate entity, and is owned by a number of individuals who may themselves face different marginal tax rates, as well as different borrowing rates of interest. With such a diversity, there may not necessarily be unanimity regarding the desired level of investment. WP15/02 Taxation and the User Cost of Capital: An Introduction 2 3.3 User Cost in Terms of the After-Tax Nominal Interest Rate The real after-tax rate of interest, r∗ , can be expressed in terms of the nominal after-tax interest rate, i∗ . As defined above, the inflation rate is π , and Fisher’s equation gives the relationship between nominal and real interest rates as:14 (1 + r∗) (1 + π) = 1 + i∗(8) so that: r∗=i∗−π 1 + π(9) Substitution into (7) gives: cg=i∗−π 1 + π+δ{1−τ(k+Z)}1 1−τ(10) 3.4 User Cost in Terms of Before-Tax Real Interest Rate The user cost can also be related to the before-tax real interest rate. First, the relationship between nominal and real before-tax rates is: (1 + r) (1 + π) = 1 + i(11) so that: i=r(1 + π) + π(12) Using i∗ = i(1 −τ) , and substituting into (9), the real after-tax rate of interest is given in terms of i,πand τby: r∗=i(1 −τ)−π 1 + π(13) Finally, substitute (12) into (13) to get: r∗=r(1 −τ)−τπ 1 + π(14) Substituting this expression for r∗into (7) gives: cg=r(1 −τ)−τπ 1 + π+δ{1−τ(k+Z)}1 1−τ(15) 14 As mentioned in the introduction, some authors use the approximation r∗+π=i∗. WP15/02 Taxation and the User Cost of Capital: An Introduction 9 3.5 User Cost in Terms of Before-Tax Nominal Interest Rate The user cost of capital in terms of the nominal before-tax interest rate, i , is obtained by substituting for r∗using (13), and for Zusing (5), into (7):15 cg=i(1 −τ)−π 1 + π+δ1−τk+δ0 i+δ0 1 1−τ(16) Writing ηa,b to indicate the elasticity of a with respect to changes in b , the following results can be obtained. ηcg,π =−π 1 + π1 + i(1 −τ) δ(1 + π)−π+i(1 −τ)(17) ηcg,τ =−τk+Z 1−τ(k+Z)+i δ(1 + π)−π+i(1 −τ)−1 1−τ(18) ηcg,i =i"i+δ(1 + π)−π 1−τ−1 +1 1 + δ0(1 + δ0) (1 −τ) δ0−τ−1#(19) 4 The After-Tax Real Interest Rate In the previous section it was mentioned that distinctions can be drawn between debt and equity financing and the location and identity of the marginal investor. These distinctions can be viewed in terms of the determination of the appropriate value of r∗, the cost of funds. Alternatives are briefly examined in this section. 4.1 Foreign Investors For foreign-sourced equity funds, the investor is not usually subject to domestic taxation on the real rate of return on equity, rE.16 Hence: r∗=rE(20) Alternatively, suppose foreign-source debt finance is available at a world real interest rate of rW. Then: r=rW(21) 15 This is equivalent to the result in Benge (1997, p. 11), with the rate facing an individual investor, m, replacing the corporate tax rate, τ. 16 This assumes the existence of double-tax agreements. WP15/02 Taxation and the User Cost of Capital: An Introduction 10 Nominal interest rate expenses are tax deductible, so the required real after-tax rate of return is lower than rW . Substituting for r = rW in equation (14) gives the required after-tax rate of return with foreign debt financing. 4.2 Domestic Residents and Imputation In an imputation system, the appropriate tax rate depends on the personal tax status of the individual investor. Imputation is now less common than formerly, but applies for example in New Zealand and Australia. If all profits are distributed as dividends, with associated imputation credits, the corporate tax acts as a withholding tax only. Whether the investment is debt or equity financed, the appropriate rate is thus the effective rate on the investor’s investment income, say m . Hence, m simply replaces τ in the above expressions. 17 Even without imputation, it may also be argued that, to the extent that investment decisions of firms depend on their marginal investors, the relevant rate is the (marginal) investor’s effective rate. Nevertheless analyses of non-imputation regimes often ignore personal-level taxation and concentrate on the corporate rate. 5 The Effective Marginal Tax Rate In general, the effective marginal tax rate can be defined in terms of the proportional difference between relevant beforeand after-tax rates of interest. Define ep as the required equilibrium pre-tax rate of return that is necessary to produce a post-tax rate of return of r∗ . Denote the tax-inclusive effective rate (the rate applied to the return that includes the tax paid) by EMTRI and the equal-revenue tax-exclusive rate (the rate applied to the return that excludes the tax paid) by EMTRE . These are marginal rates since the context is of a marginal investment. Thus: r∗=ep−ep(EMTRI)(22) so that: EMTRI= 1 −r∗ ep(23) Furthermore: r∗=ep−r∗(EMTRE)(24) giving: EMTRE=ep r∗−1(25) 17 However, in practice it is not clear whether the top marginal personal income tax rate or the trust rate, or some other rate is appropriate for the investor. WP15/02 Taxation and the User Cost of Capital: An Introduction 11 The inclusive and exclusive rates are therefore related by EMTRE = EMTRI/(1 −EMTRI) and EMTRI = EMTRE/(1 + EMTRE) . 18 The definitions given here make no reference to the user cost. The following subsection shows how the effective marginal rate is related to the user cost concept. In what follows, any reference to the effective marginal rate is to the tax-inclusive rate, as in (23); this is the rate that compares most closely with statutory rates such as the corporate rate, τ , and the personal income tax rate, m. 5.1 The User Cost and the EMTR The direct link between the EMTR and the concept of user cost is provided by considering the equilibrium condition that defines the user cost. 19 As defined earlier, the user cost net of depreciation, cn , is the before-tax rental which ensures that the after-tax and depreciation return from the marginal investment is equal to the after-tax real rate of return, r∗ . Hence cn can be interpreted as being equivalent to the real before-tax rate of return, ep. Hence from (23): EMTRI= 1 −r∗ cn (26) The connection between the user cost and the effective marginal tax rate is illustrated in Figure 2. As in Figure 1, the profit maximising position in the absence of taxation is at point A: in that case the user cost is equal to the real rate of interest. But in the presence of taxation, which is assumed to be fully shifted, the firm now needs a before-tax real rate of interest, ep , that generates an after-tax real rate equal to r∗ , which in turn is equal to the cost of capital in the absence of taxation. The ‘tax wedge’ between r∗ and ep , represented by the height CD = ep−r∗ , determines – along with the shape of the MRPK curve – the desired amount of capital, OE, in the presence of taxation. Given the downward sloping nature of MRPK , a larger tax wedge reduces the desired capital further below the profit maximising position that would arise in the absence of taxation.20 The net user cost was defined earlier as the before-tax and after-depreciation capital rental, cn , which ensures that the after-tax and depreciation return from the 18 In the case of a goods and services tax imposed at the tax-exclusive rate of τ , the preand post-tax goods prices, p0 and p1 , are related by p1 = p0(1 + τ) . In this case, the tax-inclusive tax rate is τ/ (1 + τ). 19 On effective marginal tax rates, see King and Fullerton (1984), McKenzie and Mintz (1992), McKenzie et al. (1997), Egger et al. (2009) and Fabling et al. (2013). 20 The marginal revenue product curve can, in its earlier stages, slope upwards but only the downward sloping range is relevant. WP15/02 Taxation and the User Cost of Capital: An Introduction 12 Figure 2: User Cost and The Effective Marginal Tax Rate marginal investment is equal to the after-tax real rate of return, r∗ . Hence the net user cost is represented in Figure 2 by the height EC = cn = ep . For a marginal investment, starting from point E, the tax paid is the height CD, which is cn−r∗ . The effective marginal rate is thus CD divided by the tax base, where the latter is either the length DE or CE depending on whether the tax-exclusive or inclusive rate is required. If the tax structure is proportional, as assumed here, then the effective average tax-inclusive rate, given by the area epCDr∗ divided by the area epCEO , is clearly equal to the effective marginal tax-inclusive rate, CD/CE . However, in a different situation in which the use of capital does not correspond to the profit-maximising equilibrium, and where economic rents are thereby obtained, it has been suggested that a slightly different definition of effective tax rates is useful, for which the effective average tax rate does not equal the effective marginal rate. 21 As is often the case in this literature, care is needed using terms: there is an important distinction between ‘economic rent’ and ‘capital rental’. The analysis is extended in section 6 to deal with the average effective tax rate in cases where rents are obtained, but first it is useful to consider the properties of the marginal rate in more detail. 21 However, a desirable property is that as the economic rent tends to zero, the average rate tends towards the marginal rate. WP15/02 Taxation and the User Cost of Capital: An Introduction 13 5.2 A Formal Statement To express the effective marginal rate in terms of the various components used above, first use (7) and cn=cg−δto get the user cost net of depreciation: cn=(r∗+δ) (1 −ξ)−δ(1 −τ) 1−τ(27) Second, substituting for ep = cn into (23) gives the tax-inclusive effective marginal tax rate as: EMTRI= 1 −r∗(1 −τ) (r∗+δ) (1 −ξ)−δ(1 −τ)(28) In view of the fact that r∗ can be negative, the effective marginal tax rate can exceed 1, just as it can exceed 100%for individuals who are subject to the means-testing of benefits in addition to income taxation. Defining the tax component, T= (1 −ξ)/(1 −τ), (28) can be expressed as:22 EMTRI= 1 −r∗ (r∗+δ)T−δ(29) or: EMTRI=T−1 T−δ r∗+δ (30) Another way to write (29) is: EMTRI=τ+δ r∗+δτ−ξ 1−ξ 1−τ−δ r∗+δ(31) Which indicates how the effective rate differs from the statutory rate. When δ = 0 and k = 0, so that ξ = 0, the effective marginal rate is equal to the statutory rate, τ . For domestic shareholder-level taxation, the corporate marginal rate is replaced by the appropriate shareholder rate, m. The effective marginal rate can be zero under a number of circumstances. From (29), EMTRI = 0 when r∗ = −δ , which requires the real after-tax rate of interest to be negative. Substituting for r∗ from (13) shows that this requires the nominal interest rate to equal {π−(1 + π)δ}/(1 −τ) . The lower bound for the nominal rate is zero, so that an EMTRI of zero also requires δ < π/ (1 + π) . Alternatively (29) shows that the marginal tax rate can be zero if T = 1, which arises when k + Z = 1: this is the case when i = δ0k/ (1 −k) . A further possibility is where k= 0 and Z= 1, which requires i= 0 and hence r∗=−π/ (1 + π).23 It is useful to consider the variation in EMTRI further. For example, the net user cost, cn , is a linear increasing function of the nominal interest rate, i , and a 22 Using this definition of T, it can be seen that cn= (r∗+δ)T−δ. 23 The extreme case where k= 1 and z= 0 is too unrealistic to be of interest. WP15/02 Taxation and the User Cost of Capital: An Introduction 14 decreasing function of the inflation rate, π . The tax component, T , is a nonlinear increasing function of i , and is independent of π . The variation in the effective marginal tax rate is complicated in the present context by the fact that cn can become negative for some values of i and π . Hence the expression for EMTRI in (29) can have a singularity when cn = 0: it is subject to positive and negative asymptotes. Numerical examples are given below: all cases examined are for the tax-inclusive effective marginal rate. 5.3 Variation in EMTRIwith Interest and Inflation Rates An example of the variation in EMTRI with the nominal interest rate, i , is shown in Figure 3 for two values of the inflation rate, π = 0 . 02 and π = 0 . 04. The values are obtained for τ = 0 . 3, k = 0 . 2and δ = 0 . 15 = δ0 . For low values of i , and the low inflation rate, the EMTRI is increasing and above the statutory rate of τ = 0 . 3, while at higher nominal interest rates the EMTRI is increasing but below the statutory tax rate. This relationship is highly sensitive to the inflation rate, as can be seen by a comparison with the profile for π = 0 . 04, where the nature of the variation is reversed. 24 The EMTRI declines as i increases: at low values of i the effective tax rate moves below the statutory rate, but for higher values of i , the EMTRI moves towards τ . In both the cases, the lower ranges of i are associated with negative real rates of interest, r∗ , and negative user costs, cn . The higher ranges of iare associated with positive r∗and cn.25 24 For examples of profiles with similar characteristics, see King and Fullerton (1984, p. 288). 25 In both cases the value of zfalls from 0.94 to 0.52 as irises from 0.01 to 0.14. WP15/02 Taxation and the User Cost of Capital: An Introduction 15 Figure 3: Variation in Effective Marginal Tax Rate with Nominal Interest Rate 5.4 Variation in EMTRIwith the Statutory Tax Rate It is perhaps tempting to think that the EMTRI increases systematically as the statutory marginal tax rate increases. However, the variation is again complicated by the existence of the singularity in the expression for the effective rate, combined with the fact that this can arise for relevant ranges of the statutory rate (in combination with other variables). 5.4.1 The Role of the Nominal Interest Rate Figure 4 shows the variations in the effective marginal tax rate and the net user cost of capital as the statutory rate, τ , varies, when the nominal interest rate, i , is held constant at 0.03 (the dashed line) and at 0.05 (the solid line). These results are obtained for depreciation rates of δ = δ0 = 0 . 15 and an inflation rate of π = 0 . 02. For the lower interest rate, the net user cost is positive at low statutory rates and EMTRI is negative and declines as τ increases. But at higher values of τ the value of cn turns negative and EMTRI , now positive, decreases with increasing τ . The EMTRI relationship is substantially modified for a higher nominal interest rate of 0.05. 26 However, the net user cost decreases steadily, as before, although it remains positive. 26 The singularity arises for a much higher and unrealistic statutory rate. WP15/02 Taxation and the User Cost of Capital: An Introduction 16 Figure 4: Variation in EMTR and User Cost with Statutory Tax Rate: Nominal Interest Rates of 0.03 and 0.05 WP15/02 Taxation and the User Cost of Capital: An Introduction 17 Figure 5: Variation in EMTR and Net User Cost with Statutory Rate: Alternative Fiscal Depreciation Rates WP15/02 Taxation and the User Cost of Capital: An Introduction 18 Figure 8: Effective Average Tax Rate OF with a marginal revenue product of p per unit, and an after-tax real interest rate, r∗ . As in Figure 2, for a marginal project the before-tax return, required to yield an after-tax return of r∗ , is ep . As argued above, for a non-marginal investment to earn some positive economic rent it is required that OF is less than OE, such that p > ep . Measured in per unit of investment terms, and before discounting, the before-tax rent is therefore given by p−ep , or the distance GH in Figure 8. It follows that the after-tax rent is given by ( p−ep )(1 −τ )as shown; namely in (36), and in Figure 8 is a proportion (1 −τ )of the distance GH; namely GG 0 . However, total tax on the income stream includes tax on the non-rent component; that is: ep−r∗ (= HJ). Hence total tax liability is equal to the sum of those two components, HJ + GG 0 , or ( ep−r∗ ) + (p−ep)τ . This can be seen to be equivalent to the numerator of (37), except that r∗replaces r, as discussed above. Figure 8 also illustrates the alternative methods of defining the tax rate, which should be measured using the corresponding tax base. If the tax base is defined as the total return, then the tax rate could be defined as a fraction of the tax base, p , such as EATRDG in (37). However the tax could be thought of as applying to a base measured as the return in excess of a measure of the cost of capital, r or r∗ in Figure 8, where this cost is regarded as deductible from the total tax base in determining tax liability. 36 In this case the tax base would be, for example, p−r , which, suitably discounted, is shown in (35) to be equal to R . Devereux and 36 Of course, the extent to which this cost is deductible, typically depends on a number of conditions such as whether the investment is equity or debt financed. WP15/02 Taxation and the User Cost of Capital: An Introduction 25 Griffith (2003) argue in favour of the total tax base, p , in part because it facilitates comparison with backward-looking average tax rates based on actual tax and capital income (profit) data, rather than economic rents, and because a tax rate based on R , such as EATRI-DG in (33), is undefined for a marginal investment where R= 0. Average and marginal tax rates therefore differ for an investment involving economic rents. The relationship between the two rates can be see by rearranging (37). Using the definition of EMTRDG above, the two effective rates can be shown to be related as follows: EATRDG =EMTRDG cn p+τ1−cn p(38) This is equivalent to the decomposition given by Devereux and Griffith (2003, p. 112). Their effective average rate is thus a weighted average of their effective marginal tax rate and the statutory tax rate, with weights depending on the ratio of the user cost to the capital rental associated with the non-marginal investment. For small projects, well below the profit maximising scale, p is likely to be much larger than cn , so that cn/p is small (except where investment returns decline only slowly with the scale of an investment; that is, if the MRPK curve in Figure 8 is relatively flat). For this ‘small cn/p ’case the effective average rate is relatively close to the statutory tax rate. For larger projects, as p approaches cn the ratio moves closer to 1 and the average effective tax rate is closer to the marginal rate. For p = cn , average and marginal rates are equal, and this clearly corresponds to the case discussed at the end of subsection 5.1. Devereux and Griffith further show that, where it is desired to allow for differences between personal and corporate-level taxation, the statutory rate, τ , is replaced by an ‘adjusted statutory tax rate’, τ0 . In particular, this adjustment takes account of any differences in the personal tax treatment of new equity and distributions, and discounting uses the shareholders’ nominal discount rate, ρ , rather than the nominal interest rate, i , faced by the firm. Devereux and Griffith (2003, p. 113) show that the appropriate adjusted statutory rate is: τ0= 1 −γ(1 −τ)(1 + r) (1 + π) (1 + ρ)(39) where γ reflects the differential personal treatment of new equity and dividend distributions. The term (1 + r) (1 + π) is, by Fisher’s equation, equal to (1 + i) , so that if ρ=iand γ= 1,τ0=τ.37 37 Devereux and Griffith differ from most other literature in using the shareholders’ discount rate to derive present values of returns. WP15/02 Taxation and the User Cost of Capital: An Introduction 26 7 Investment and the User Cost It has been seen that the user cost concept is intimately related to optimal investment by a profit maximising firm. The firm invests, that is adjusts its capital stock, to the point where the returns match the cost of capital. This gives rise to an equimarginal condition which can be used to express the user cost in terms of the interest rate, the inflation rate, depreciation, taxation and so on. Only values at the time of investment are relevant because of the assumption that investment is reversible and there are no adjustment costs. In this case there are clear implications for the optimal capital stock in terms of the user cost. The gross user cost, as a pre-tax rental, is the capital rental associated with profit maximisation. Expressing the production function as a function of only capital, K , output is Y = F(K) and the capital rental is the marginal revenue product, equal to the product of marginal revenue and the marginal physical product of capital. In a competitive market, marginal revenue and price are equal, and the latter can be normalised to unity. Hence cg = ∂F (K)/∂K = F0 K . Consider the Cobb-Douglas production function, with an exponent of α on capital services. Then F0 K = αY/K . If K∗ represents the desired capital stock, it is simply given by rearranging αF (K∗)/K∗=cg, so that: K∗=αY cg (40) Hence the logarithm of the desired capital stock is a linear function of the logarithm of output and the logarithm of the user cost. This can easily be extended to deal with imperfect output markets and, say, the constant elasticity of substitution production function. For inputs of labour and capital of L and K , and normalising the efficiency term to unity, the Constant Elasticity of Substitution (CES) function is (with a re-definition of α): Y= (αKρ+βLρ)1/ρ (41) where ρ = 1 −1 σ and σ is the elasticity of substitution between labour and capital. The marginal physical product of capital is: ∂Y ∂K =αKρ−1Y1−ρ(42) If the price of the good per unit is p and the elasticity of demand is η , then using the well-known property that marginal revenue, MR = p1−1 η , the capital rental WP15/02 Taxation and the User Cost of Capital: An Introduction 27 is given by: cg= (MR)∂Y ∂K (43) =αpKρ−1Y1−ρ1−1 η(44) and using 1−ρ= 1/σ, desired capital stock is: K∗=Y α 1−1 ρ!σcg p−σ (45) As with the simple Cobb-Douglas case, the desired capital stock is a loglinear function of output and user cost, but the coefficient on the logarithm of user cost is −σrather than −1. Sometimes the expression for desired capital stock is used along with a specified adjustment process in order to produce an investment function. However, this necessarily involves a serious conflict, since the fundamental user cost derivation discussed earlier explicitly assumes there are no adjustment costs (the cost of capital is fixed independent of the amount of investment). However, from the basic relationship relating capital at time, t, and t−1, and investment, It: Kt=Kt−1+It−δKt−1(46) Rearrangement gives the following expression for the growth rate of capital: Kt−Kt−1 Kt−1 =It Kt−1 −δ(47) This proportional change can be approximated by the change in logarithms, ∆ kt . For example, a simple partial adjustment specification has ∆ kt = θ(k∗ t−kt−1) . Alternatively, error-correction or distributed lag models can be applied. 38 The effect on investment of changes in tax regulations or rates can therefore be traced via the effect on user cost. 8 Conclusions The aim of this paper has been to provide an introductory review of the concept of user cost and its determinants. The concept of user cost was seen to relate to the rental, the rate of return to capital, that arises in a profit maximising situation in which further investment in capital produces no additional profit. Despite this apparently simple statement it has been seen that the concept gives rise to a 38 On alternative specifications, see Bond and Van Reenen (2003). WP15/02 Taxation and the User Cost of Capital: An Introduction 28 complex taxonomy or range of cases which need to be distinguished. This paper sets out in some detail, using a consistent terminology, the range of assumptions involved in obtaining alternative expressions for the user cost. The user cost refers to a before-tax capital rental, the rate of return that ensures that the (after-tax) cost of capital is equal to the post-tax returns. Hence, associated with the user cost measure is an effective marginal tax rate. This can differ substantially from the statutory marginal rate applicable to the investor. Particular attention was given to the properties of the effective marginal tax rate in different circumstances, drawing attention to the difference between tax-inclusive and exclusive rates. A related effective average tax rate was also defined for the context in which the firm obtains economic rents. This may be important in the context of multinational investment where the firm is operating below its profit maximising output. The link between the user cost, effective tax rates and investment was only briefly discussed as this warrants separate extensive treatment. WP15/02 Taxation and the User Cost of Capital: An Introduction 29 Appendix A: Derivation of the Basic Hall and Jorgensen Result In their seminal paper, Hall and Jorgenson (1967) used a continuous-time approach. This appendix explains how their first result is derived. They began by taking the simplest case of no taxation, no depreciation and no inflation. A firm makes a marginal increase in its input of capital by obtaining a new capital good at time, t , with a supply price of capital of q(t) . Capital services at time, s≥t , are valued at c ( s ). Hence c(s) measures the marginal revenue product, and this clearly depends on the price of the good produced by the firm as well as the productivity of the equipment. Strictly, this rental depends on the marginal revenue, but on the assumption that the good is sold in a competitive market, price and marginal revenue are equal. Investment continues up to the point where the supply price, q(t) is equal to the present value of additional returns. Hence, at the profit maximising position, and with continuous discounting at the rate, r , Hall and Jorgenson (1967) write: q(t) = Z∞ t c(s)e−r(s−t)ds (A.1) An implication is that a further marginal increase in capital, made in only period t , involving an increase in q(t) , is exactly matched by the change in the present value of returns, measured by the right-hand side of ( A.1 ). Writing ˙q(t) = ∂q(t) ∂t , a further marginal increase in capital, made in only period t , involving an increase in q(t) , is exactly matched by the change in the present value of returns, measured by the right-hand side of (A.1). Hence: ∂q (t) ∂t =∂ ∂t Z∞ t c(s)e−r(s−t)ds (A.2) This equation therefore expresses the kind of investment that was discussed above. There is a marginal investment in one period which is reversed in the subsequent period. The right-hand side can be obtained by using the Leibniz Integral Rule. This states, for the general function f(s, t) and limits of integration given by a(t) and b(t), that: ∂ ∂t Zb(t) a(t) f(s, t)ds =Zb(t) a(t) ∂f (s, t) ∂t ds +f(b(t), t)∂b (t) ∂t −f(a(t), t)∂a (t) ∂t (A.3) Consider the term in (A.2) corresponding to the first term in (A.3). Then: ∂ ∂t c(s)e−r(s−t)=c(s)re−r(s−t)(A.4) so that: Z∞ t ∂ ∂t c(s)e−r(s−t)ds =rZ∞ t c(s)e−r(s−t)ds (A.5) WP15/02 Taxation and the User Cost of Capital: An Introduction 30 which is equal to rq (t). Furthermore, it can be seen that the second term in ( A.5 ) is zero and the third term is simply c(t) : this is because the term f(a(t), t) is equal to c(t)e−0=c(t)and ∂a(t) ∂t = 1. Hence, writing ˙q(t) = ∂q(t) ∂t , (A.2) becomes: ˙q(t) = rq (t)−c(t)(A.6) This is the Hall and Jorgensen result. The fact that only periodt values are relevant in ( A.6 ) arises from the strong assumption that the project is reversible. Furthermore, assume that the price of the capital good does not depend on the amount already invested, so that there are no adjustment costs: the supply curve is essentially horizontal. This means that ˙q(t) is assumed to be zero. With constant consumer prices, and dropping the time subscript, t, (A.6) becomes: c q=r(A.7) The rental per unit of capital is thus equal to the rate of interest. It is conventional to normalise the price of a unit of the capital good, so that setting q = 1 gives the simple result that c = r . The rental associated with profit maximisation is, by definition, the user cost of capital. Hence the user cost is equal to the interest rate. WP15/02 Taxation and the User Cost of Capital: An Introduction 31 Appendix B: Allowing for Uncertainty Modification of the above results to allow for uncertainty and risk aversion rapidly becomes very complicated. However, some insight can be obtained by first considering the simplest possible case where there is no taxation, no depreciation and no capital gain. In the deterministic case, the user cost of a dollar invested in capital is simply cg = r . Suppose now that r is uncertain, although the nature of the distribution is known. Risk aversion is modelled by supposing that there is a concave utility function, U(cg) associated with the user cost. If the investor is assumed to have constant relative risk aversion of ε6 = 1, then utility takes the form U(cg) = c1−ε g/(1 −ε). The certainty-equivalent user cost is that rental which, if received with certainty, gives the same utility as the expected utility from the distribution. Hence if r has the distribution function, F(r), the user cost is given by: c1−ε g 1−ε=Zr1−ε 1−εdF (r)(B.1) so that: cg=Zr1−εdF (r)1/(1−ε) (B.2) Thus cg is the power mean of order ε of the distribution of r . Some insight may be obtained by assuming that r is lognormally distributed, so that F(r) = Λ (r|µ, σ2) , with mean and variance of logarithms of µ and σ2 respectively. From the moment generating function of the lognormal distribution, it is known that: Zr1−εdΛ (r) = exp (1 −ε)µ+1 2(1 −ε)2σ2(B.3) and therefore (supposing that the range of ris 0< r ⩽∞): cg= exp µ+(1 −ε) 2σ2(B.4) Consider the effect of an increase in uncertainty. If this is assumed to result simply from an increase in σ2 , the arithmetic mean as well as the variance of r changes, since the arithmetic mean is given by E(r) = exp µ+1 2σ2 . However, increasing risk can be modelled as a mean-preserving spread of the distribution. Hence, when σ2 increases, the value of µ must fall to maintain a constant arithmetic mean. From the total differential: dE (r) = E(r)dµ +1 2E(r)dσ2(B.5) WP15/02 Taxation and the User Cost of Capital: An Introduction 32 It can be seen that: dµ dσ2E(r) =−1 2(B.6) Thus the effect on the user cost of a mean-preserving increase in uncertainty is given by totally differentiating (B.4): dcg dσ2=cg dµ dσ2E(r) +cg(1 −ε) 2=−cg 2+cg(1 −ε) 2=−cgε 2(B.7) and the elasticity of cgwith respect to a mean-preserving increase in σ2is: ηc,σ2=σ2 cg dcg dσ2=−εσ2 2(B.8) The elasticity of user cost with respect to a mean-preserving spread in r is thus (minus) half the product of the degree of relative risk aversion and the variance of logarithms of r . In the risk-neutral case, increased uncertainty (as a meanpreserving spread) has no effect since only expected values matter. WP15/02 Taxation and the User Cost of Capital: An Introduction 33 References Auerbach, A.J. (1979) Wealth maximization and the cost of capital. Quarterly Journal of Economics, 93, 433–446. Auerbach, A.J. 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