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Neutral taxation under uncertainty
Journal
of Public
Economics
NEUTRAL
33 (1987) 95-105.
TAXATION
North-Holland
UNDER
UNCERTAINTY
Received July 1986. revised version received
December
1986
The analogs under uncertainty of two well-known certainty results are derived: first, if there are timing differences between tax payments and accruals. neutrality is preserved if the resulting tax credits or liabilities are carried forward at the risk-free interest rate. provided th.rt tax credits and liabilities are sure to be redeemed eventually. Second, the invariance of asset valuations with respect to the rate of income tax. at a given pre-tax interest rate, proved by Johansson and Samuelson under certainty. can he extended to cover the case of uncertainty, given analogous ceteris paribus conditions.
1. Introduction This paper analyzes the neutrality of cash-flow and income taxes under uncertainty. The cash-flow tax under uncertainty is analyzed in section 2. It has been widely assumed that, if cash-flow tax payments and accruals are not equal period by period, the rate at which the resulting tax credits and liabilities must be carried forward, in order to preserve neutrality, depends on the risk characteristics of the projects being undertaken and/or on the financial structure of the tirm. In contrast to this assumption, section 2 shows that if tax credits and liabilities are sure to be redeemed eventually, the rate at which they must be carried forward, if neutrality is to be preserved, is simply the risk-free nominal interest rate. If tax credits and liabilities are not certain to be redeemed eventually, the appropriate carry-forward interest rate depends on the risk-characteristics of the project and the financial structure of the firm only to the extent that these factors affect the probability that tax credits or liabilities will never be redeemed. Section 3 is concerned with the income tax. Johansson (1961,1969) and Samuelson (1964) established that, in the absence of uncertainty, a tax on true economic income, defined as net cash-flow plus capital gains, is neutral in the sense that, for given relative prices and for a given pre-income tax nominal interest rate, the conditions which determine optimal investment decisions are independent of the rate of the income tax. This ‘neutrality’ result is quite different from the neutrality result for a cash-flow tax: at given relative prices and at a given pre-tax nominal interest 004772727,‘87/‘83.50
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G. Fane, Neutral
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under uncertainty
rate, a cash-flow tax at rate u reduces the value of the project being taxed to a fraction (1 -u) of its former value, and therefore leaves unchanged the decisions which maximize the value of the project. In contrast, given the same ceteris paribus assumptions, the introduction of a tax on true income, applied to all assets, leaves each firm’s privately optimal investment decisions unaffected because all asset values are left unchanged. But whereas the introduction of a compensated cash-flow tax really will leave all relative prices and interest rates unchanged, the introduction of an income tax will not do so, even if the revenue is returned in a lump-sum fashion to the taxpayers: at the original pre-tax interest rate the incentive to save is reduced by the introduction of an income tax. If the tax revenue is returned in a lump-sum fashion, the pre-tax interest rate will rise and the post-tax interest rate will fall, thereby causing households to reduce savings and firms to reduce investment. Samuelson dces not describe a tax on true economic income as being ‘neutral’. Rather, he claims, correctly, that only if income is defined as net cash-flow plus capital gains, will it be the case that individuals facing different marginal rates of income tax are never able to reduce their aggregate tax burdens by exchanging assets. If cash-flows and capital gains are taxed at different rates there will be distorting incentives for individuals in different tax-brackets to hold assets which yield predominantly cash-flows, or predominantly capital gains. This property of a tax on true economic income can therefore be described as neutrality with respect to portfolio decisions. Domar and Musgrave (1944), Sandmo (1969), Stiglitz (1976) and Mintz (1981) have analyzed the non-neutral effects of an income tax in models in which investors are risk averse and trading in state-contingent securities is not possible. Richter (1986) shows that the Johansson--Samuelson result carries over to the case of uncertainty, even if trading in state contingent securities is not possible, provided that investors are risk-neutral. Bulow and Summers (1983) and Auerbach (1983) analyze the problem in the context of the capital asset pricing model (CAPM). The main focus of these two papers is on the actual properties of the U.S. corporate income tax, rather than on neutrality results per se. Bulow and Summers analyze the effects of applying an income tax just to a small sector of a large economy, on the assumption that the untaxed interest rates on the ‘safe asset’ and on the ‘market portfolio’ in the rest of the large economy remain unchanged. They show that neutrality is not preserved, unless allowable deductions in the small sector include the interest rate which could have been earned on the untaxed safe asset, as well as the ex post value of economic depreciation in the small sector subjected to the income tax. Equivalently, given that the forgone interest on the safe asset in the large economy is not allowed as a deduction for investors in the small sector, neutrality would be preserved if
G. Fanr,
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taxation
under uncertainty
97
the income tax were applied uniformly both to the exogenous large economy and to the small sector, provided that the pre-tax interest rate in the large economy remains unchanged. Throughout this paper it is assumed that the value of the asset whose income is being taxed is equal to the sum of its future state-contingent payouts, valued at the corresponding state-contingent prices. The markets in state-contingent securities need not necessarily be complete, but it is assumed that a risk-free security exists. The analysis is conducted for given values of the risk-free nominal interest rate and for given values of the relative prices of state-contingent securities. Under these ceteris paribus conditions, section 3 shows that the Johansson-Samuelson result carries over to the case of uncertainty. In general, the introduction of an income tax will not leave these variables unchanged for the reasons given above. A full general equilibrium analysis of the income tax under uncertainty would therefore entail an analysis of the changes in these variables. Since a similar caveat applies to the certainty result of JohanssonSamuelson, it is not surprising that it also applies to the result under uncertainty discussed in this paper. Concluding comments are given in section 4.
2. The cash-flow tax under uncertainty Since a cash-flow tax is equivalent to a tax on rent it is obvious that such a tax is neutral even under uncertainty: the unexpected introduction of such a tax at rate u will reduce the value of the asset being taxed to a fracti,on ( 1 -u) of its previous value. This result was noted by Brown (1948, p. 313). It does depend on the assumption that the tax being analyzed is a compensated tax: the introduction of an uncompensated tax will, of course, give rise to income effects which may change relative prices. If tax payments and accruals are not equal period by period the question arises of determining the appropriate interest rate at which the resulting tax credits or liabilities should be carried forward if neutrality is to be preserved. This section proves that, provided that all tax credits and liabilities are certain to be redeemed eventually, the appropriate rate at which they must be carried forward, in order to preserve neutrality, is the risk-free nominal interest rate. If the interest on tax credits is assessed for income tax in subsequent periods, then, to preserve neutrality, the tax credits must be carried forward at the nominal interest rate gross of income tax; whereas, if the interest on tax credits is not itself assessable for income tax, then the credits must be carried forward at the nominal interest rate net of income tax. In this section, the rate of income tax is set at zero so that this distinction disappears. The assumption that tax liabilities are always redeemed does imply that firms with outstanding tax liabilities never go bankrupt. However, this
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consideration does not entirely destroy the applicability of the analysis to real world situations: some tax arrangements which approximate to cash-flow taxes allow firms to build up tax credits when accruals are negative, but do not allow them to build up tax liabilities when accruals are positive. An example of such a tax is the ‘resource rent tax’, which the Australian government has announced that it will apply to new off-shore petroleum projects. This tax will allow firms to build up tax credits in the early years of a project, when cash-flows are likely to be negative, but never allow them to build up positive tax liabilities. Tax credits will be carried forward at the long-term government bond rate, plus 15 percentage points per annum, but firms with unused tax credits at the end of unsuccessful projects will not be allowed to sell them. The analysis of this section proves that, in such cases, neutrality would be preserved if the tax system allowed credits to be carried forward at the risk-free nominal rate and guaranteed ‘full loss of offsets’; i.e., guaranteed that firms were always allowed, eventually, to make use of their tax credits. One way of guaranteeing full loss offsets is that suggested by Boadway and Bruce (1984, p 234): firms with unused tax credits could be allowed to sell them to firms with positive tax liabilities. The proposed Australian resource rent tax therefore deviates from neutrality in two respects, both of which will produce systematic distortions. First, it does not allow unused tax credits to be sold; second, it allows firms to accumulate credits at a rate of 15 percentage points per annum above the government bond rate. In contrast to the results of this section. many authors have simply assumed that the interest rate at which tax credits, or liabilities, should be carried forward, in order to preserve neutrality, must reflect the riskiness of the investments being undertaken and the firm’s financial structure. For example, in their discussion of the practical problems involved in implementing their proposal, Boadway and Bruce assert that the ‘nominal financial cost’ is not readily observable: ‘ . . . In have a of debt willing
principle, each firm will have a different value of r(t) since it will different value of debt to equity and a different maturity structure finance . nominal interest rates alone could be used if one were to ignore differences between the costs of debt and equity finance.’
(P. 236) Dowell (1979, pp. 115 and 129), Ball and Bowers (1984, pp. 2, l&l 1 and 16), Garnaut and Clunies Ross ( 1975, p. 280 and 1979, p. 196) and Mayo (1979, pp. 211 and 213) also assume that the rate at which firms should be allowed to carry forward resource rent tax credits should depend on the risk characteristics of the project. In discussing the neutrality of the resource rent tax proposals of Garnaut and Clunies Ross, Heaps and Helliwell (1985, p. 460) state that
G. Fnne, Neutral taxation under uncertainty
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’ . this result [i.e. the neutrality of a resource rent tax] depends on the rate at which the firm’s profits are discounted for tax purposes being the same as the discount rate used by the firm. If the industry uses a discount rate containing a risk premium reflecting the particular economic, geological, and political risks of a project, it would be possible in principle, but very difficult in practice, to use an estimate of this discount rate for tax purposes.’ In contrast to the assertions of these various authors, the analysis of this section shows that if cash-flows could be easily measured and if rents were substantially positive, it would not be difficult to design a revenue-raising neutral business tax. This could be achieved by using the risk-free nominal interest rate for accumulating unused tax credits and by guaranteeing full loss offsets. Since government bonds provide an (almost) certain nominal return, it is not difficult to estimate the risk-free nominal interest rate.’ The present value of the representative firm in period t, in the absence of all taxation, may be expressed as the sum of an infinite stream of future, state-conditional, net cash-flows: V’(t)=~I7(a)~[A’(a,t+ (I
+c WC,b,u).
1)+~17(b,a)+V(b,a,t+2) b
[NC, b,u, t+3)+...]]],
c
(1)
where a, b and c are the representative states of nature in periods t + 1, t + 2 and t +3, respectively; n(u) is the value in t of a state-contingent security which pays $1 in state a of t + 1; I7(b,u) is the value in t+ 1 of a statecontingent security which pays $1 in state b of t +2, given that state a occurred in t + 1; U(c, b,u) is the value in t+2 of a state-contingent security which pays $1 in state c of t + 3, given that states b and a occurred in t +2 and t+ 1, respectively; V(t) is the value of the firm in t; N(u, t+ l), N(b,u, t + 2), N(c, b, a, t + 3) are the net cash-flows in t + 1, t + 2 and t + 3, respectively, conditional on the occurrence of the indicated states. Eq. (1) may be re-written to correspond to Samuelson’s ‘fundamental equation of yield’: V(t) where
=I n(u) [N(u, t + 1) + V(u, t + l)], a
V(u, t + 1) is the value of the firm in state a in t + 1. When
(2) the firm is
‘If one wished to allow for the possibility that firms with positive tax liabilities may go bankrupt, the best administratively feasible rule would be to carry forward the firms’s tax liabilities at the market interest rate which the firm has to pay on its own debentures, not at the ‘nominal financial cost’.
G. Fane, Neutral taxation under uncrrtaint)
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subjected to a pure cash-flow tax at rate 11, with payments always equal to accruals, its value in period t, V,(t), will be a fraction (1 -u) of its value in the absence of such a tax. This follows trivially from the fact that the formula for c(t) is obtained by multiplying every term on the RHS of eq. (1) by (1 -u). Now suppose that tax payments and accruals do not exactly balance period by period. Let C(a, t + 1) denote the excess of payments over accruals in state u in period t + 1: C(u,t+l)=T(u,t+l)-u.N(u,t+l),
(3)
where T(u, t + 1) is the tax payment in state II in period t + 1. Initially, it is assumed that all tax credits, or liabilities, must t + 2, so that the tax payment in state h in t + 2 is given by:
be cleared
7J~,u,r+2)=u~N(h,u,t+2)-[1+p(u,t+1)]~C(c~,r+1),
in
(4)
where p(u,t+ 1) denotes the rate at which the tax rules allow tax credits to be carried forward to period t + 2 from state a in t + 1. Let V,*(t), V,*(u,t+ 1) and V:(h,u, t+2) denote the values of the firm in the indicated states and periods given these arrangements. Since all tax credits and liabilities are assumed to be cleared at the beginning of t +2 it is clear that V,*(h, CI,t + 2) equals V,(h, u, t + 2) and that:
V,*(u,1+1)=V,(a,t+l)+
.C(u,t+l).
~n(h,u).[l+p(u,r+l)]
(5)
h
Applying
the fundamental
equation
of yield in period
t now gives: (6)
v,*(t)=Cn(u).(N(u,t+l)-T(u,t+l)+I/,*(u,f+l)). a Under the obviously neutral pure cash-flow tax system, in which and accruals are always equal, period by period, the corresponding would be:
payments equation
(7)
v,(t)=Cn(u).CN(u.t+l).(l-U)+I/,(u,t+l)]. ll Subtracting
(7) from (6), and using (3) to eliminate
T(u, t + l), now gives:
v:(t)-v,(t)=~n(a)~[V,*(u,t+1)-I/,(u,t+1)-c(u,t+1)]. c1 Given
(5), it is clear
that
a suhicient
condition
(8) for neutrality,
i.e. for both
G. Fan@, Neutral
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101
sides of (8) to be zero, is:
C(a,t+
1). -l+~~(h,a)~[l+p(a,t+l)]
(9)
h
Let r*(a, t + 1) denote the risk-free nominal rate of interest between state u of period t + 1 and period t+2. Therefore, by definition, the reciprocal of [l +r*(u, t+ I)] is the value in state a of period t+ I of a security which guarantees to pay $1 in every state in period I + 2: l/[l +r*(U,t+
l)]=CH(h,u).
(10)
By comparing eqs. (9) and (10) it is clear that setting the rate at which tax credits, or liabilities, are carried forward to period t +2 from state a in t+ 1 equal to the risk-free nominal interest rate, denoted by r*(u, t + l), is sufficient to ensure neutrality. If the tax system is to be neutral for arbitrarily chosen values of C(u,t+ l), then setting the carry forward interest rate on tax credits and liabilities equal to the risk-free nominal rate becomes necessary as well as suflicient. Re-applying the above arguments proves that if tax credits, or liabilities, can be carried forward beyond period t+2, neutrality will be preserved if, between any two periods, the interest rate at which outstanding tax credits or liabilities are carried forward is the risk-free nominal rate. The intuition behind this result is straightforward: tax credits are equivalent to bonds, and the building-up of tax credits by a firm is therefore equivalent to its using equity finance to pay-off debt. That such equity-debt swaps do not alter the value of the firm is an implication of the Modigliani-m Miller theorem. The above proof established that the risk-free nominal interest rate is the appropriate rate at which cash-flow tax credits should be carried forward in order to preserve neutrality, but the same logic would c!early also show that the neutrality properties of other taxes would be preserved if all credits and liabilities were accumulated at the risk-free nominal interest rate.2
3. The income tax under uncertainty This section extends the result of Johansson to the case of uncertainty: for a given pre-tax
(1969) and Samuelson (1964) risk-free nominal interest rate,
‘Whether credits and liabilities should be carried forward at the pretax. or post-tax, risk-free nominal interest rate depends, in the way indicated at the start of section 2, on whether the interest on credits (or liabilities) is included in the base of the income tax (or allowed as a deduction).
102
G. F‘ane. Neulral taxation
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uncertainty
and for given relative prices of the state contingent nominal securities, asset values are independent of the rate of income tax, provided that the tax base is defined as true economic income, i.e. net cash-flow plus capital gain. In the absence of all taxation the value of the firm is given by eq. (l), from which eq. (2), the funddmental equation of yield, may be derived. It is helpful to re-write eq. (2) in the form: ~(t)=~(v(u)/[1+r*(r)l]‘(N(u,,t+l)+I/(u,t+1)}, ” where
l/[l +r*(t)]
=x0
n(a)
and
(11)
v(a) = n(u). I and t+ 1; i.e.
l/[l
+r*(t)]
is the value
in t
V,( of in particular state and period, when an income tax is applied at rate 0 to every income-generating asset in the economy. In these circumstances, eq. (11) must be replaced by:
v,(t)=C(v(a)/[l+(l-O)r*(t)]j.(N(a,t+l)-T(a,r+l) a +
Ve(a,t+I)),
(12)
where T(a, t + 1) denotes the tax payments in state a of period t + 1. Eq. (13) can now be obtained by multiplying both sides of eq. (12) by [l +( 1 -@r*(t)] and subtracting QV,(t) from both sides of the resulting equation: (1-U)~V~(t)~[1+r*(r)]=~v(u)~{iV(u,t+1)-T(u,t+1) a + &(a, t + l)} - 0. b(t). Note that the term in 0. b(t) may be enclosed within the curly the RHS of eq. (13) since, by definition, c v(u) = 1. Therefore: (1 -0).
V,(t).[l
+r*(t)]=Cv(u).
(1
(Z(u,t+
l)},
(13) brackets
on
(14)
where
au,t+ l)= The tax payments
N(c4r+ 1) + V&u, t + l)in state a of period
T(u,t+l)=U[N(u,t+l)-D(u,t+l)],
T(u, t + l)m-6,. Q(t).
(15)
t+ 1 are given by
(16)
G. Fane, Neutral
taxation
where D(u,t+ 1) denotes the amount of depreciation allowed in state u of period t+ 1. Eq. (15) may now be re-written as:
Z(a,t + 1) =(1-O). where of the Eq. ( 1 - fl)
103
under uncertainty
(N(a,t + 1) + Vo(u, t + l)} +O.X(a,
by the tax rules
t + I),
(17)
X(a,t+ l)=D(a,t+ 1)-[V&t)V,(u,t+ l)]. i.e. X(u,t+ 1) is the excess depreciation allowed by the tax rules over true economic depreciation. (17) may now be substituted into eq. (14); dividing both sides by . [ 1 + r(t)] gives: ~(t)=~~(U)‘[~(u,t+1)+~(a,t+1)+H’X(u,t+l,)/(1-e)]. a
(18)
A comparison between eqs. (2) and (18) shows that a sufficient condition for valuations to be invariant with respect to the rate of income tax, for given values of n(u), H(h,u), etc., is that the tax base be true economic income, as defined by Samuelson (1964); i.e., that allowable depreciation be defined as loss of economic value, so that X(u, t + 1) and the corresponding variables in all subsequent periods are identically zero: D(u,t+
l)=
b(t)-
v@(u,t+ 1).
If the rule for determining allowable depreciation must be specified independently of the realized value of I/e(u, t c l), I/,(b, a, t + 2), etc. then this condition is necessary as well as sufficient.
4. Conclusions This paper analyzed the neutrality of income and cash-flow taxes under uncertainty. The result established in section 2 is the analog under uncertainty of the certainty result established by Broadway and Bruce (1984). They showed that, even if there are timing differences between tax payments and accruals, cash-flow taxes are neutral, in the absence of uncertainty, provided that the tax credits or liabilities, which result from the timing differences, are carried forward at the going rate of interest. The main result of section 2 is that, to preserve neutrality under uncertainty, nominal tax credits and liabilities must be carried forward at the risk-free nominal interest rate, provided that these credits or liabilities are sure to be redeemed eventually. The intuition behind this result is a straightforward application of the Modigliani-Miller theorem. If it is not certain that tax credits and liabilities will eventually be redeemed - perhaps because of the possibility of bankruptcy, or because the
tax laws do not guarantee full loss offsets ~ the appropriate interest rate only depends on the risk characteristics of the asset being taxed to the extent that these characteristics determine the probability that the tax credits or liabilities will never be redeemed. Since capital gains and losses often result from the resolution of stochastic events, rather than from the mere unfolding of a perfectly anticipated future, it is important to know how the analyses of income taxation under certainty, due to Johansson (1969) and Samuelson (1964), are modified by allowing for uncertainty. This problem was studied in section 3, where it was shown that the Johansson-Samuelson result can be extended to the case of uncertainty even if investors are risk averse and even if the assumptions of the CAPM do not hold, provided that (a) each asset’s price is equal to the sum of its payouts in each state. valued at the prices of the corresponding statecontingent securities; and (b) the risk-free pre-tax interest rate and the relative prices of the state contingent securities are not affected by the introduction of the tax. These qualifications are the analog under uncertainty of the qualification needed to establish the Johansson-Samuelson result rate is not affected by the under certainty ~ that the pre-tax interest introduction of the tax.
References Auerbach, A.J., 1983. Corporate taxation in the United States, Brookings Papers on Economic Activity, 451-505. Ball, R. and J. Bowers, 1984. The resource rent tax: A penalty on risk-taking, Policy Monograph 5 (Centre for Independent Studies, Sydney). Roadway, R. and N. Bruce. 1984, A general proposition on the design of a neutral business tax, Journal of Public Economics 24. 231 239. Brown. E.C.. 1948. Business-income taxation and Investment incentives, in: Income. employment and public policy. essays in honor of Alvin H. Hansen (Norton, New York). Bulow, J.I. and L.H. Summers. 19X4, The taxation of risky assets. Journal of Political Economy 92. X&39. Domar. E. and R. Musgrave. 1944. Proportional income taxation and risk taking. Quarterly Journal of Economics 58. 3X2-422. Dowell. R.. 1979. Profits based royalties and productive efficiency. Resources and Energy 2. 103 130. Garnaut. R. and A.C. Clunies Ross. 1975. Uncertainty. risk aversion and the taxing of natural resource projects, Economic Journal 85, 272--2X7. Garnaut. R. and A.C. Clunies Ross, 1979. The neutrality of the resource rent tax. Economic Record 55. 193-201. Heaps, T. and J.F. Helliwell, 1985, The taxation of natural resources, in: M.S. Feldstein and A. Auerbach. eds.. Handbook of public economics (North-Holland, Amsterdam) ch. X. Johansson, S-E., 1969, Income taxes and investment decisions, Swedish Journal of Economics 71, 104110. Based on Johansson. S.-E.. 1901. Skatt investering vardering (Taxes investments valuations) Stockholm. Mayo, W., 1979, Rent royalties, Economic Record 55. 202-213. Mintz, J.M., 1981, Some additional results on investment, risk taking, and full loss offset corporate taxation with interest deductibility, Quarterly Journal of Economics 96. 631-642.
G. Fane, Neutral taxation under uncertainty
105
O’Driscoll, G.P., Jr, 1979, The Ricardian nonequivalence theorem, Journal of Political Economy 85, 207-210. Richter, W.F., 1986, Comprehensive versus neutral income taxation, Mimeo, Universitat Dortmund. Samuelson, P.A., 1964, Tax deductibility of economic depreciation to insure invariant valuations, Journal of Political Economy 72, 604-606. Sandmo, A., 1969, Capital risk, consumption and portfolio choice, Econometrica 37, 586599. Stiglitz, J.E., 1976, The corporation tax, Journal of Public Economics 5, 303-311.
J.P.E.
F
of Public
Economics
NEUTRAL
33 (1987) 95-105.
TAXATION
North-Holland
UNDER
UNCERTAINTY
Received July 1986. revised version received
December
1986
The analogs under uncertainty of two well-known certainty results are derived: first, if there are timing differences between tax payments and accruals. neutrality is preserved if the resulting tax credits or liabilities are carried forward at the risk-free interest rate. provided th.rt tax credits and liabilities are sure to be redeemed eventually. Second, the invariance of asset valuations with respect to the rate of income tax. at a given pre-tax interest rate, proved by Johansson and Samuelson under certainty. can he extended to cover the case of uncertainty, given analogous ceteris paribus conditions.
1. Introduction This paper analyzes the neutrality of cash-flow and income taxes under uncertainty. The cash-flow tax under uncertainty is analyzed in section 2. It has been widely assumed that, if cash-flow tax payments and accruals are not equal period by period, the rate at which the resulting tax credits and liabilities must be carried forward, in order to preserve neutrality, depends on the risk characteristics of the projects being undertaken and/or on the financial structure of the tirm. In contrast to this assumption, section 2 shows that if tax credits and liabilities are sure to be redeemed eventually, the rate at which they must be carried forward, if neutrality is to be preserved, is simply the risk-free nominal interest rate. If tax credits and liabilities are not certain to be redeemed eventually, the appropriate carry-forward interest rate depends on the risk-characteristics of the project and the financial structure of the firm only to the extent that these factors affect the probability that tax credits or liabilities will never be redeemed. Section 3 is concerned with the income tax. Johansson (1961,1969) and Samuelson (1964) established that, in the absence of uncertainty, a tax on true economic income, defined as net cash-flow plus capital gains, is neutral in the sense that, for given relative prices and for a given pre-income tax nominal interest rate, the conditions which determine optimal investment decisions are independent of the rate of the income tax. This ‘neutrality’ result is quite different from the neutrality result for a cash-flow tax: at given relative prices and at a given pre-tax nominal interest 004772727,‘87/‘83.50
,(‘ 1987, Elsevier Science Publishers
B.V. (North-Holland)
96
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taxation
under uncertainty
rate, a cash-flow tax at rate u reduces the value of the project being taxed to a fraction (1 -u) of its former value, and therefore leaves unchanged the decisions which maximize the value of the project. In contrast, given the same ceteris paribus assumptions, the introduction of a tax on true income, applied to all assets, leaves each firm’s privately optimal investment decisions unaffected because all asset values are left unchanged. But whereas the introduction of a compensated cash-flow tax really will leave all relative prices and interest rates unchanged, the introduction of an income tax will not do so, even if the revenue is returned in a lump-sum fashion to the taxpayers: at the original pre-tax interest rate the incentive to save is reduced by the introduction of an income tax. If the tax revenue is returned in a lump-sum fashion, the pre-tax interest rate will rise and the post-tax interest rate will fall, thereby causing households to reduce savings and firms to reduce investment. Samuelson dces not describe a tax on true economic income as being ‘neutral’. Rather, he claims, correctly, that only if income is defined as net cash-flow plus capital gains, will it be the case that individuals facing different marginal rates of income tax are never able to reduce their aggregate tax burdens by exchanging assets. If cash-flows and capital gains are taxed at different rates there will be distorting incentives for individuals in different tax-brackets to hold assets which yield predominantly cash-flows, or predominantly capital gains. This property of a tax on true economic income can therefore be described as neutrality with respect to portfolio decisions. Domar and Musgrave (1944), Sandmo (1969), Stiglitz (1976) and Mintz (1981) have analyzed the non-neutral effects of an income tax in models in which investors are risk averse and trading in state-contingent securities is not possible. Richter (1986) shows that the Johansson--Samuelson result carries over to the case of uncertainty, even if trading in state contingent securities is not possible, provided that investors are risk-neutral. Bulow and Summers (1983) and Auerbach (1983) analyze the problem in the context of the capital asset pricing model (CAPM). The main focus of these two papers is on the actual properties of the U.S. corporate income tax, rather than on neutrality results per se. Bulow and Summers analyze the effects of applying an income tax just to a small sector of a large economy, on the assumption that the untaxed interest rates on the ‘safe asset’ and on the ‘market portfolio’ in the rest of the large economy remain unchanged. They show that neutrality is not preserved, unless allowable deductions in the small sector include the interest rate which could have been earned on the untaxed safe asset, as well as the ex post value of economic depreciation in the small sector subjected to the income tax. Equivalently, given that the forgone interest on the safe asset in the large economy is not allowed as a deduction for investors in the small sector, neutrality would be preserved if
G. Fanr,
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taxation
under uncertainty
97
the income tax were applied uniformly both to the exogenous large economy and to the small sector, provided that the pre-tax interest rate in the large economy remains unchanged. Throughout this paper it is assumed that the value of the asset whose income is being taxed is equal to the sum of its future state-contingent payouts, valued at the corresponding state-contingent prices. The markets in state-contingent securities need not necessarily be complete, but it is assumed that a risk-free security exists. The analysis is conducted for given values of the risk-free nominal interest rate and for given values of the relative prices of state-contingent securities. Under these ceteris paribus conditions, section 3 shows that the Johansson-Samuelson result carries over to the case of uncertainty. In general, the introduction of an income tax will not leave these variables unchanged for the reasons given above. A full general equilibrium analysis of the income tax under uncertainty would therefore entail an analysis of the changes in these variables. Since a similar caveat applies to the certainty result of JohanssonSamuelson, it is not surprising that it also applies to the result under uncertainty discussed in this paper. Concluding comments are given in section 4.
2. The cash-flow tax under uncertainty Since a cash-flow tax is equivalent to a tax on rent it is obvious that such a tax is neutral even under uncertainty: the unexpected introduction of such a tax at rate u will reduce the value of the asset being taxed to a fracti,on ( 1 -u) of its previous value. This result was noted by Brown (1948, p. 313). It does depend on the assumption that the tax being analyzed is a compensated tax: the introduction of an uncompensated tax will, of course, give rise to income effects which may change relative prices. If tax payments and accruals are not equal period by period the question arises of determining the appropriate interest rate at which the resulting tax credits or liabilities should be carried forward if neutrality is to be preserved. This section proves that, provided that all tax credits and liabilities are certain to be redeemed eventually, the appropriate rate at which they must be carried forward, in order to preserve neutrality, is the risk-free nominal interest rate. If the interest on tax credits is assessed for income tax in subsequent periods, then, to preserve neutrality, the tax credits must be carried forward at the nominal interest rate gross of income tax; whereas, if the interest on tax credits is not itself assessable for income tax, then the credits must be carried forward at the nominal interest rate net of income tax. In this section, the rate of income tax is set at zero so that this distinction disappears. The assumption that tax liabilities are always redeemed does imply that firms with outstanding tax liabilities never go bankrupt. However, this
J.P.E
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98
G. Fane, Neutral taxation
under uncertainty
consideration does not entirely destroy the applicability of the analysis to real world situations: some tax arrangements which approximate to cash-flow taxes allow firms to build up tax credits when accruals are negative, but do not allow them to build up tax liabilities when accruals are positive. An example of such a tax is the ‘resource rent tax’, which the Australian government has announced that it will apply to new off-shore petroleum projects. This tax will allow firms to build up tax credits in the early years of a project, when cash-flows are likely to be negative, but never allow them to build up positive tax liabilities. Tax credits will be carried forward at the long-term government bond rate, plus 15 percentage points per annum, but firms with unused tax credits at the end of unsuccessful projects will not be allowed to sell them. The analysis of this section proves that, in such cases, neutrality would be preserved if the tax system allowed credits to be carried forward at the risk-free nominal rate and guaranteed ‘full loss of offsets’; i.e., guaranteed that firms were always allowed, eventually, to make use of their tax credits. One way of guaranteeing full loss offsets is that suggested by Boadway and Bruce (1984, p 234): firms with unused tax credits could be allowed to sell them to firms with positive tax liabilities. The proposed Australian resource rent tax therefore deviates from neutrality in two respects, both of which will produce systematic distortions. First, it does not allow unused tax credits to be sold; second, it allows firms to accumulate credits at a rate of 15 percentage points per annum above the government bond rate. In contrast to the results of this section. many authors have simply assumed that the interest rate at which tax credits, or liabilities, should be carried forward, in order to preserve neutrality, must reflect the riskiness of the investments being undertaken and the firm’s financial structure. For example, in their discussion of the practical problems involved in implementing their proposal, Boadway and Bruce assert that the ‘nominal financial cost’ is not readily observable: ‘ . . . In have a of debt willing
principle, each firm will have a different value of r(t) since it will different value of debt to equity and a different maturity structure finance . nominal interest rates alone could be used if one were to ignore differences between the costs of debt and equity finance.’
(P. 236) Dowell (1979, pp. 115 and 129), Ball and Bowers (1984, pp. 2, l&l 1 and 16), Garnaut and Clunies Ross ( 1975, p. 280 and 1979, p. 196) and Mayo (1979, pp. 211 and 213) also assume that the rate at which firms should be allowed to carry forward resource rent tax credits should depend on the risk characteristics of the project. In discussing the neutrality of the resource rent tax proposals of Garnaut and Clunies Ross, Heaps and Helliwell (1985, p. 460) state that
G. Fnne, Neutral taxation under uncertainty
99
’ . this result [i.e. the neutrality of a resource rent tax] depends on the rate at which the firm’s profits are discounted for tax purposes being the same as the discount rate used by the firm. If the industry uses a discount rate containing a risk premium reflecting the particular economic, geological, and political risks of a project, it would be possible in principle, but very difficult in practice, to use an estimate of this discount rate for tax purposes.’ In contrast to the assertions of these various authors, the analysis of this section shows that if cash-flows could be easily measured and if rents were substantially positive, it would not be difficult to design a revenue-raising neutral business tax. This could be achieved by using the risk-free nominal interest rate for accumulating unused tax credits and by guaranteeing full loss offsets. Since government bonds provide an (almost) certain nominal return, it is not difficult to estimate the risk-free nominal interest rate.’ The present value of the representative firm in period t, in the absence of all taxation, may be expressed as the sum of an infinite stream of future, state-conditional, net cash-flows: V’(t)=~I7(a)~[A’(a,t+ (I
+c WC,b,u).
1)+~17(b,a)+V(b,a,t+2) b
[NC, b,u, t+3)+...]]],
c
(1)
where a, b and c are the representative states of nature in periods t + 1, t + 2 and t +3, respectively; n(u) is the value in t of a state-contingent security which pays $1 in state a of t + 1; I7(b,u) is the value in t+ 1 of a statecontingent security which pays $1 in state b of t +2, given that state a occurred in t + 1; U(c, b,u) is the value in t+2 of a state-contingent security which pays $1 in state c of t + 3, given that states b and a occurred in t +2 and t+ 1, respectively; V(t) is the value of the firm in t; N(u, t+ l), N(b,u, t + 2), N(c, b, a, t + 3) are the net cash-flows in t + 1, t + 2 and t + 3, respectively, conditional on the occurrence of the indicated states. Eq. (1) may be re-written to correspond to Samuelson’s ‘fundamental equation of yield’: V(t) where
=I n(u) [N(u, t + 1) + V(u, t + l)], a
V(u, t + 1) is the value of the firm in state a in t + 1. When
(2) the firm is
‘If one wished to allow for the possibility that firms with positive tax liabilities may go bankrupt, the best administratively feasible rule would be to carry forward the firms’s tax liabilities at the market interest rate which the firm has to pay on its own debentures, not at the ‘nominal financial cost’.
G. Fane, Neutral taxation under uncrrtaint)
100
subjected to a pure cash-flow tax at rate 11, with payments always equal to accruals, its value in period t, V,(t), will be a fraction (1 -u) of its value in the absence of such a tax. This follows trivially from the fact that the formula for c(t) is obtained by multiplying every term on the RHS of eq. (1) by (1 -u). Now suppose that tax payments and accruals do not exactly balance period by period. Let C(a, t + 1) denote the excess of payments over accruals in state u in period t + 1: C(u,t+l)=T(u,t+l)-u.N(u,t+l),
(3)
where T(u, t + 1) is the tax payment in state II in period t + 1. Initially, it is assumed that all tax credits, or liabilities, must t + 2, so that the tax payment in state h in t + 2 is given by:
be cleared
7J~,u,r+2)=u~N(h,u,t+2)-[1+p(u,t+1)]~C(c~,r+1),
in
(4)
where p(u,t+ 1) denotes the rate at which the tax rules allow tax credits to be carried forward to period t + 2 from state a in t + 1. Let V,*(t), V,*(u,t+ 1) and V:(h,u, t+2) denote the values of the firm in the indicated states and periods given these arrangements. Since all tax credits and liabilities are assumed to be cleared at the beginning of t +2 it is clear that V,*(h, CI,t + 2) equals V,(h, u, t + 2) and that:
V,*(u,1+1)=V,(a,t+l)+
.C(u,t+l).
~n(h,u).[l+p(u,r+l)]
(5)
h
Applying
the fundamental
equation
of yield in period
t now gives: (6)
v,*(t)=Cn(u).(N(u,t+l)-T(u,t+l)+I/,*(u,f+l)). a Under the obviously neutral pure cash-flow tax system, in which and accruals are always equal, period by period, the corresponding would be:
payments equation
(7)
v,(t)=Cn(u).CN(u.t+l).(l-U)+I/,(u,t+l)]. ll Subtracting
(7) from (6), and using (3) to eliminate
T(u, t + l), now gives:
v:(t)-v,(t)=~n(a)~[V,*(u,t+1)-I/,(u,t+1)-c(u,t+1)]. c1 Given
(5), it is clear
that
a suhicient
condition
(8) for neutrality,
i.e. for both
G. Fan@, Neutral
tcrxation
under uncertainty
101
sides of (8) to be zero, is:
C(a,t+
1). -l+~~(h,a)~[l+p(a,t+l)]
(9)
h
Let r*(a, t + 1) denote the risk-free nominal rate of interest between state u of period t + 1 and period t+2. Therefore, by definition, the reciprocal of [l +r*(u, t+ I)] is the value in state a of period t+ I of a security which guarantees to pay $1 in every state in period I + 2: l/[l +r*(U,t+
l)]=CH(h,u).
(10)
By comparing eqs. (9) and (10) it is clear that setting the rate at which tax credits, or liabilities, are carried forward to period t +2 from state a in t+ 1 equal to the risk-free nominal interest rate, denoted by r*(u, t + l), is sufficient to ensure neutrality. If the tax system is to be neutral for arbitrarily chosen values of C(u,t+ l), then setting the carry forward interest rate on tax credits and liabilities equal to the risk-free nominal rate becomes necessary as well as suflicient. Re-applying the above arguments proves that if tax credits, or liabilities, can be carried forward beyond period t+2, neutrality will be preserved if, between any two periods, the interest rate at which outstanding tax credits or liabilities are carried forward is the risk-free nominal rate. The intuition behind this result is straightforward: tax credits are equivalent to bonds, and the building-up of tax credits by a firm is therefore equivalent to its using equity finance to pay-off debt. That such equity-debt swaps do not alter the value of the firm is an implication of the Modigliani-m Miller theorem. The above proof established that the risk-free nominal interest rate is the appropriate rate at which cash-flow tax credits should be carried forward in order to preserve neutrality, but the same logic would c!early also show that the neutrality properties of other taxes would be preserved if all credits and liabilities were accumulated at the risk-free nominal interest rate.2
3. The income tax under uncertainty This section extends the result of Johansson to the case of uncertainty: for a given pre-tax
(1969) and Samuelson (1964) risk-free nominal interest rate,
‘Whether credits and liabilities should be carried forward at the pretax. or post-tax, risk-free nominal interest rate depends, in the way indicated at the start of section 2, on whether the interest on credits (or liabilities) is included in the base of the income tax (or allowed as a deduction).
102
G. F‘ane. Neulral taxation
under
uncertainty
and for given relative prices of the state contingent nominal securities, asset values are independent of the rate of income tax, provided that the tax base is defined as true economic income, i.e. net cash-flow plus capital gain. In the absence of all taxation the value of the firm is given by eq. (l), from which eq. (2), the funddmental equation of yield, may be derived. It is helpful to re-write eq. (2) in the form: ~(t)=~(v(u)/[1+r*(r)l]‘(N(u,,t+l)+I/(u,t+1)}, ” where
l/[l +r*(t)]
=x0
n(a)
and
(11)
v(a) = n(u). I and t+ 1; i.e.
l/[l
+r*(t)]
is the value
in t
V,( of in particular state and period, when an income tax is applied at rate 0 to every income-generating asset in the economy. In these circumstances, eq. (11) must be replaced by:
v,(t)=C(v(a)/[l+(l-O)r*(t)]j.(N(a,t+l)-T(a,r+l) a +
Ve(a,t+I)),
(12)
where T(a, t + 1) denotes the tax payments in state a of period t + 1. Eq. (13) can now be obtained by multiplying both sides of eq. (12) by [l +( 1 -@r*(t)] and subtracting QV,(t) from both sides of the resulting equation: (1-U)~V~(t)~[1+r*(r)]=~v(u)~{iV(u,t+1)-T(u,t+1) a + &(a, t + l)} - 0. b(t). Note that the term in 0. b(t) may be enclosed within the curly the RHS of eq. (13) since, by definition, c v(u) = 1. Therefore: (1 -0).
V,(t).[l
+r*(t)]=Cv(u).
(1
(Z(u,t+
l)},
(13) brackets
on
(14)
where
au,t+ l)= The tax payments
N(c4r+ 1) + V&u, t + l)in state a of period
T(u,t+l)=U[N(u,t+l)-D(u,t+l)],
T(u, t + l)m-6,. Q(t).
(15)
t+ 1 are given by
(16)
G. Fane, Neutral
taxation
where D(u,t+ 1) denotes the amount of depreciation allowed in state u of period t+ 1. Eq. (15) may now be re-written as:
Z(a,t + 1) =(1-O). where of the Eq. ( 1 - fl)
103
under uncertainty
(N(a,t + 1) + Vo(u, t + l)} +O.X(a,
by the tax rules
t + I),
(17)
X(a,t+ l)=D(a,t+ 1)-[V&t)V,(u,t+ l)]. i.e. X(u,t+ 1) is the excess depreciation allowed by the tax rules over true economic depreciation. (17) may now be substituted into eq. (14); dividing both sides by . [ 1 + r(t)] gives: ~(t)=~~(U)‘[~(u,t+1)+~(a,t+1)+H’X(u,t+l,)/(1-e)]. a
(18)
A comparison between eqs. (2) and (18) shows that a sufficient condition for valuations to be invariant with respect to the rate of income tax, for given values of n(u), H(h,u), etc., is that the tax base be true economic income, as defined by Samuelson (1964); i.e., that allowable depreciation be defined as loss of economic value, so that X(u, t + 1) and the corresponding variables in all subsequent periods are identically zero: D(u,t+
l)=
b(t)-
v@(u,t+ 1).
If the rule for determining allowable depreciation must be specified independently of the realized value of I/e(u, t c l), I/,(b, a, t + 2), etc. then this condition is necessary as well as sufficient.
4. Conclusions This paper analyzed the neutrality of income and cash-flow taxes under uncertainty. The result established in section 2 is the analog under uncertainty of the certainty result established by Broadway and Bruce (1984). They showed that, even if there are timing differences between tax payments and accruals, cash-flow taxes are neutral, in the absence of uncertainty, provided that the tax credits or liabilities, which result from the timing differences, are carried forward at the going rate of interest. The main result of section 2 is that, to preserve neutrality under uncertainty, nominal tax credits and liabilities must be carried forward at the risk-free nominal interest rate, provided that these credits or liabilities are sure to be redeemed eventually. The intuition behind this result is a straightforward application of the Modigliani-Miller theorem. If it is not certain that tax credits and liabilities will eventually be redeemed - perhaps because of the possibility of bankruptcy, or because the
tax laws do not guarantee full loss offsets ~ the appropriate interest rate only depends on the risk characteristics of the asset being taxed to the extent that these characteristics determine the probability that the tax credits or liabilities will never be redeemed. Since capital gains and losses often result from the resolution of stochastic events, rather than from the mere unfolding of a perfectly anticipated future, it is important to know how the analyses of income taxation under certainty, due to Johansson (1969) and Samuelson (1964), are modified by allowing for uncertainty. This problem was studied in section 3, where it was shown that the Johansson-Samuelson result can be extended to the case of uncertainty even if investors are risk averse and even if the assumptions of the CAPM do not hold, provided that (a) each asset’s price is equal to the sum of its payouts in each state. valued at the prices of the corresponding statecontingent securities; and (b) the risk-free pre-tax interest rate and the relative prices of the state contingent securities are not affected by the introduction of the tax. These qualifications are the analog under uncertainty of the qualification needed to establish the Johansson-Samuelson result rate is not affected by the under certainty ~ that the pre-tax interest introduction of the tax.
References Auerbach, A.J., 1983. Corporate taxation in the United States, Brookings Papers on Economic Activity, 451-505. Ball, R. and J. Bowers, 1984. The resource rent tax: A penalty on risk-taking, Policy Monograph 5 (Centre for Independent Studies, Sydney). Roadway, R. and N. Bruce. 1984, A general proposition on the design of a neutral business tax, Journal of Public Economics 24. 231 239. Brown. E.C.. 1948. Business-income taxation and Investment incentives, in: Income. employment and public policy. essays in honor of Alvin H. Hansen (Norton, New York). Bulow, J.I. and L.H. Summers. 19X4, The taxation of risky assets. Journal of Political Economy 92. X&39. Domar. E. and R. Musgrave. 1944. Proportional income taxation and risk taking. Quarterly Journal of Economics 58. 3X2-422. Dowell. R.. 1979. Profits based royalties and productive efficiency. Resources and Energy 2. 103 130. Garnaut. R. and A.C. Clunies Ross. 1975. Uncertainty. risk aversion and the taxing of natural resource projects, Economic Journal 85, 272--2X7. Garnaut. R. and A.C. Clunies Ross, 1979. The neutrality of the resource rent tax. Economic Record 55. 193-201. Heaps, T. and J.F. Helliwell, 1985, The taxation of natural resources, in: M.S. Feldstein and A. Auerbach. eds.. Handbook of public economics (North-Holland, Amsterdam) ch. X. Johansson, S-E., 1969, Income taxes and investment decisions, Swedish Journal of Economics 71, 104110. Based on Johansson. S.-E.. 1901. Skatt investering vardering (Taxes investments valuations) Stockholm. Mayo, W., 1979, Rent royalties, Economic Record 55. 202-213. Mintz, J.M., 1981, Some additional results on investment, risk taking, and full loss offset corporate taxation with interest deductibility, Quarterly Journal of Economics 96. 631-642.
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O’Driscoll, G.P., Jr, 1979, The Ricardian nonequivalence theorem, Journal of Political Economy 85, 207-210. Richter, W.F., 1986, Comprehensive versus neutral income taxation, Mimeo, Universitat Dortmund. Samuelson, P.A., 1964, Tax deductibility of economic depreciation to insure invariant valuations, Journal of Political Economy 72, 604-606. Sandmo, A., 1969, Capital risk, consumption and portfolio choice, Econometrica 37, 586599. Stiglitz, J.E., 1976, The corporation tax, Journal of Public Economics 5, 303-311.
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