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Capital income taxation and welfare in a small open economy
O T T E IR V¢ 0 R T H
Journal oflnternational Money and Finance, Vol. 14, No. 6, pp. 785-800, 1995
I~! E i N E M A N N
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Capital income taxation and welfare in a small open economy CEM
KARAYALCIN*
Florida International University, Department of Economics Miami, FL 33199, USA The paper studies the effects of various capital income tax policies in a small open economy. Households are infinitely-lived and possess endogenous time preferences. Investment by firms is subject to adjustment costs. It is shown that welfare paradoxes may exist, in that increases in tax instruments may improve welfare. (JEL H20).
The continuing process of global economic integration, as exemplified by the unification of the European market as well as by the recent liberalization of capital markets in several developing countries has given rise, inter alia, to a growing literature focusing on the international effects of taxation. Rapid progress seems to have been made since 1988, when Slemrod (1988) made the observation that the study of tax incidence in the context of international capital mobility was still in its infancy. Recent analyses of capital income taxation by Sen and Turnovsky (1990), Lucas (1990), Razin and Sadka (1991), Frenkel et al. (1991), Nielsen and Sorenson (1991), Frenkel and Razin (1992), Turnovsky and Bianconi (1992) attest to the dynamism of the field. 1 This paper focuses on the welfare effects of various capital tax instruments in a small open economy, intertemporal utility maximization framework, an issue hitherto unexplored in this context. The economy is populated by infinitely-lived households which can borrow or lend freely in the world capital market. For simplicity it is assumed that the economy produces a single good used for consumption and investment which is subject to adjustment costs. 2 The paper differs from its predecessors by modeling the households as possessing endogenous time preferences, rather than the commonly-used time-additive constant rate of time preference. It is well-known that to ensure the existence of long-run equilibrium the time-additive preference structure, which implies a constant rate of time preference, requires that this rate be arbitrarily set equal to the after-tax world rate of interest. Taxes that change this after-tax interest rate disturb this equality and create instability (Epstein and Hynes, 1983). To avoid such instability severe restrictions have to be imposed on international capital mobility. Even in the absence of tax instruments that alter the after-tax world rate of interest, the time-additive preferences lead to a *I would like to thank an anonymous referee for helpful comments.
Capital income taxation: C Karayalcin
hysteretic adjustment towards the steady state, rendering the analysis dependent on initial conditions.3. Overlapping-generations models overcome these problems at the expense either of the ability to study short-run adjustment (Buiter, 1981) or the adoption of the rather unrealistic assumption of an age-independent probability of death (Nielsen and Sorensen, 1991). The endogenous time preference framework, developed by Epstein and Hynes (1983), and adopted here avoids these problems.4 It also provides a rather parsimonious way of measuring the effects of changes in capital tax instruments on the lifetime welfare of households. Given this setup, I show that increases in the capital income tax rates (such as the tax rate on capital gains, on personal interest income, on corporate income) as well as a rise in the investment tax credit may increase lifetime welfare. The intuition for this result is easy to grasp. Such taxes and subsidies drive a distortionary wedge between the marginal productivity of capital at home and the world real rate of interest. If, for example, initially the former exceeds the latter, policies that increase the home capital stock will, by decreasing its marginal productivity, reduce the distortion and raise lifetime welfare. The significance of this result for the analysis of tax policies of such OECD countries as Canada and Britain, which have recently been in the process of dismantling their investment tax credit programs, is quite obvious. The paper also traces the time path of the economy in response to various policy changes and shows that the current account may adjust non-monotonically. 5 The rest of the paper is arranged as follows. The next section sets up the model. Section II studies the effects of changes in tax policy instruments. Section III concludes the paper. I. The model
Consider a small open economy with a constant number of identical, infinitely-lived households that have perfect foresight. Normalize, without loss of generality, the number of households to one. Suppose, also, that perfectly competitive firms in this economy produce a single good that can be used for consumption and investment. I now turn to a detailed analysis of household and firm behavior. 1.4. H o u s e h o l d s
Households inelasticaUy supply one unit of labor services, for which they receive the wage, wt, per unit of time. They also receive interest income r(1 - rr)a t (where r and ~'r denote the constant world real rate of interest and the personal tax on interest income) on their non-human wealth, a t . If households have Epstein-Hynes variable rates of time preference, they maximize lifetime welfare U over consumption path C, that is, they maximize oo
(1)
U(C) = - f
e x p ( - z t ) e x p [ - (1 - z r ) r t ] d t ,
J0
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subject to (2)
2 t = u ( c t) - (1
-
'rr)r ,
(3)
ti = (1 - rr)ra t + w t - c t + 'rt ,
<4>
z 0 = 0,
where u ( c ) > 0 is the felicity function with u ' > 0, u " < 0 and r t is the lump-sum government transfer. Since it is not the purpose of this paper to study the effects of government debt policy, I shall assume that the domestic government follows a policy of balanced budgets, with the implication that the lump-sum transfer received by the representative household comprises the net tax revenue. The lifetime welfare functional U differs from the conventional time-additive utility functionals used in the literature by its recursivity, which implies that the marginal rate of substitution between times t and s (s > t) is independent of consumption before t but not after s. The structure gives rise to a variable rate of time preference 12 at time s: (5)
12s={fs°~exp[-fstU(c)dr]dt}
-1 .
12 at s is the following function of the utility functional U ( C ) (6)
12(6~)
=
--6
-1 ,
~bs =
U(sC),
where sC stands for that part of the consumption path C beyond time s and ~bs denotes lifetime welfare at time s. When the consumption path is globally constant, as in long-run equilibrium, U ( s C ) = - 1 / u ( c ) , and the rate of time preference is given by (7)
12"(6") = u ( c * ) ,
where asterisks denote long-run equilibrium. Assuming u' > 0 implies that 12 is increasing in consumption path C. This assumption is labeled 'increasing marginal impatience' by Lucas and Stokey (1984). Though several arguments have been advanced in support of it, for our purposes it is sufficient to point out that local stability will fail to obtain in the absence of such an assumption. 6 Henceforth, I specialize the felicity function t o 7 <8)
u ( c , ) = to + lnc,,
where to is a parameter measuring generalized time preference. Appendix 1 shows that with (8) the solution of the lifetime welfare maximization problem yields s (9)
d = [(1 - zr)r - 12(~b)1c.
The dynamics of lifetime welfare ¢ are obtained by differentiating (1) with respect to time: (10)
d= 1 +u(c)eh.
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LB. Firms At a point in time perfectly competitive firms employ capital, k, and labor to produce the single good under constant returns to scale. New investment undertaken by these firms is financed both by the issue of corporate bonds, b c, and by retained earnings. Remaining profits net of corporate income tax are paid out as dividends to holders of equity. Firms are assumed to deduct interest payments on outstanding debt and adjustment costs T associated with investment i in determining taxable corporate profits. Consequently, total dividends before personal tax, ~, are given by 9 (11)
z r =- [ f ( k ) - w - r b
c - T](1 - zc) +b c - (1 - zrt)i ,
where zc and ~'t denote the corporate income tax rate and the rate of investment credit; f ( k ) represents a constant-returns-to-scale production function with the conventional Inada properties. It is well-known that in the absence of uncertainty one cannot adequately account for the differences in the forms of financing. Thus, I shall assume that the representative firm finances a fraction 6 of its new investment through retained earnings, and (1 - 6) of it through the issue of corporate bonds, i.e.
b c = (1 - 6)k, b c = (1 - e)/~.
(12)
Since installing investment goods is costly it takes i[1 + T(i/k)] units of output to increase the capital stock by i units. The installation-cost function T is assumed to have the following properties: (13)
T(0) = 0,
T'(.) > 0,
2T' + ( i / k ) T " > O.
Corporate bonds, b c, foreign bonds, b f, and equities are perfect substitutes in the portfolios of households. Let E denote the market value of outstanding equity. Then the arbitrage condition
7r
(14)
r(1 -
Tr) = "-~ 7!-
(1 -- ~'g)E g
(where rg stands for the tax rate on accrued capital gains) must hold at all times. The term on the left-hand side of (14) represents the after-tax of return on foreign bonds, while the expression on the right-hand side denotes the after-tax rate of return on equity. This expression is the after-tax sum of current yield and capital gains. Using (11) and (12) in (14) and integrating, one obtains the market value of equity as of time 0: oo
(15)
E = f0 0gl "ff exp(--rOrOg lt)dt'
Oi =- 1 - zj, j = c , g , r .
The representative firm chooses the time path of investment by maximizing the market value E subject to the constraint i = k. This yields (16)
d1 =rOrO~lq - O¢o~l[f'(k) - r ( 1 - 6) + ( i / k ) 2 T ' ( i / k ) ] ,
(17)
q = 0g-1 { ( 6 - ~'t) + Oc[T+ ( i / k ) T ' ( i / k ) ] } ,
(18)
w =f(k) -f'(k)k,
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where q denotes the shadow value of capital and can be easily shown, as in Hayashi (1982), to be equal to the stock market price of a unit of equity relative to the replacement cost of capital (that is q of Tobin's Q). Substituting <17) into (16), the latter is seen to be an arbitrage equation setting the after-tax rate of return on foreign bonds to the after-tax rate of return on equity. (18) is the familiar condition that requires the marginal productivity of labor to be equal to the real wage rate. Equation <17) implies that the rate of investment, i / k , is the following function of q (19)
~ = k- = ~o(q),
~o'(q*) > 0,
which states that the rate of investment is an increasing function of 'Tobin's Q'. The most salient feature of (16) and (19) is that neither q nor investment depends on the consumption and savings decisions of households. Thus, (16) and (19) constitute a system of two autonomous differential equations in q and k. The system is depicted in Figure 1. I.C. The current account
To obtain the dynamics of the foreign asset holdings of the representative household (or the dynamics of the current account as these two coincide here) use (3), (11)-(19), a = b f + qk + b c, and recall the assumptions of balanced government budgets to obtain (20)
bf=rbf+f(k)
- i ( 1 + T) - c,
q
k--O
k FIOURE1. The adjustment of q and k. Journal of InternationalMoneyand Finance1995 Volume14 Number6
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that is the current account is equal to the sum of output and interest earnings from foreign assets less the sum of consumption and investment spending.
LD. Equilibrium The model has five differential equations: (9),(10),(16), (19) and (20). To obtain the steady-state value of x = x(q, k, c, 4~, b) set these equations to zero. This yields TM (21) (22)
q* = 0 g l ( S - ~-l),
q*=q('rg,~'l),ql>O,
f'(k*)=r[(1-e)+OrO~lOgl(6-rt)], kt < 0 (i = 1,2),
q2
k*=k(zg,rc,rr,%),
kj > 0 ( j = 3,4),
(23)
rb f* + f ( k * ) = c*, b f* = b(rg, ~'c,zr, ~-,), bi > 0 (i = 1,2), bj < 0 ( j = 3,4), (24)
u ( c * ) = rOr,
(25)
~b* = - ( r O t ) - 1 ,
c* = C(Tr) , q~* = ~('rr) ,
~t < 0, ~ ' < 0,
where I assume in (21) that e > ~'l so that the equity price q is positive in the steady state) 1 Equation (22) shows that capital income taxes and the investment tax credit drive a distortionary wedge between the marginal productivity of capital f ' ( k ) and the world real rate of interest r. Whether this wedge causes the former to exceed the latter or not depends on the configuration of the tax rates Zg, ~-~,Zr, and the investment tax credit zt:
f ' ( k * ) <>r ,~
(26)
1-
0c0])
+ ---~-r
<>0.
If income taxes are uniform (so that rg = Zr), the corporate tax rate is fully integrated (so that ~'~= 0), and there is no investment tax credit (~'1= 0), it follows from (22) that f ' ( k * ) = r; that is, one obtains the well-known result that a Schanz-Haig-Simons income tax is neutral as regards investment. Further, (17) indicates that in the absence of the investment credit and the tax on capital gains q = e. Thus, in this case, the market value of equity E -- qk = e k equals the equity-financed portion of the accumulated investment of firms. However, actual tax systems do not generally have fully integrated corporation taxes, nor are the rates of investment tax credit set equal to zero. It is, therefore, useful to look at some representative tax rates for such countries as Canada, Sweden, and the Netherlands, which are generally considered to be small open economies, to have an understanding of the difference between f ' ( k * ) and r as implied by the rates in (26). The effective average tax rates are roughly: Tr 0.45, r~ = 0.25, r t = 0.10, z~ = 0.27; in addition e = 0.75.12 These indicate that f ' ( k * ) < r for the economies in question. Now, note from ( 2 3 ) - ( 2 5 ) that the variable rate of time preference implies well-defined long-run target utility, consumption and wealth levels) 3 =
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At a point in time, given the parameters and the steady-state level of x = (q, k, c, ~b,b) the five differential equations solve for x. (15) determines the wage rate. As Appendix 2 shows the dynamic system ((9), (10), (16), (19), (20)) possesses two negative (denoted by A1 and A2) and three positive eigenvalues, which (given the two predetermined variables k and b f) renders it saddle-path stable. TM Its motion along the convergent saddle-path is characterized by (27) (28) (29)
k, - k* = ( k o -
q, - q* =
k*)exp(Alt),
[ A1/k*q~' ( gl ) l( k t - k * ),
ct-c*=(r-Az)[(bfo-bf*)+l~(ko-k*)]exp(Azt)
(30)
,
~), -- 6 * = /3( Ct - C* ),
(31) b y - b f* = - ~ ( k o - k *
)exp(All) + [ ( b 0/ -
b f* ) t- t16(k 0 - k * )]
exp( A2t),
where Ix=[f'(k*)-A1][r-Aa]-l>o,
fl=[rc*Or(rOr-A2)]
-1
>0.
In the perfect-foresight, intertemporal-equilibrium framework adopted here the dynamics of the variables are dictated by the long-run changes they undergo. Thus, (27) indicates that along the convergent path the dynamics of investment are uniquely determined by its speed of adjustment and the long-run change in the domestic capital stock, while (31) shows that the dynamics of the current account depends on long-run changes in both foreign assets and the domestic capital stock, as well as the adjustment speeds of both. Straightforward manipulation of (27) and (31) yields (32) (33)
k = Xl(k t - k*), bY = t~(A2 - '~a)(kt
- k* ) +
A2(b/ - b I* ).
It follows from (32) and (33) that if /~1 < }~2 the capital stock, k, will adjust faster than the foreign asset holdings, b y, or, to put it differently, the current account will be more persistent than domestic investment. 15 This, however, contradicts the findings of recent studies (for instance Backus and Kehoe, 1992; Backus et al., 1992). Consequently, I shall concentrate on the case /~1 > /~2 "16 Figure 2 depicts the adjustment of k and b f. Further, since /3 > 0, (29) and (30) indicate that consumption c and lifetime welfare ~b always rise and fall together along the convergent path shown in Figure 3. II. The effects of taxation
I now put the model to work by considering the effects of changes in the tax rates. 17 I start by focusing on the effects of alterations in the investment tax credit ~t, in the tax rates on capital gains ~-g,and on corporate income ~c. They Journal of International Money and Finance 1995 Volume 14 Number 6
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b =0
~o
t+ f .b
Fxoum~ 2. The adjustment of the current account.
~l
~
F
6'=0
FIGURE3. The adjustment of consumption and lifetime welfare.
are treated as a group, for changes in these rates do not impinge upon the long-run target utility of households (equation (24)), thus do not entail any effects on the long-run level of consumption. Consider now the effects of an unanticipated permanent increase in the corporation tax rate. TM On impact this reduces dividends and the rate of return on equity. The consequent incipient excess stock demand for foreign assets leads to an immediate drop in the price of equity q (and to expectations of 792
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capital gains), which starts a process of capital decumulation. The process continues until the decline in the capital stock pushes the marginal productivity of capital and the dividends up sufficiently enough so that, in the absence of expectations of capital gains on it, the rate of return on equity equals the rate of return on foreign bonds. Similarly, an unanticipated permanent increase in the tax rate ~-~ on capital gains reduces the long-run capital stock. Since this has the effect o]~ raising the steady-state marginal productivity of capital and dividends, the long-run equity price q must also rise to ensure the equality of asset yields. As the short-run adjustment of the foreward-looking equity-price q is driven by the long-run changes in k and q--which in the instance put opposing pressures on q - - t h e impact effect of the increase in ~-g on q is ambiguous. However, regardless of whether it drops or jumps on impact, along the convergent path, q will rise towards its higher steady-state level (equation (28)). By contrast, an unanticipated permanent rise in the investment tax credit r t increases the long-run capital stock, by reducing the effect replacement cost of capital. The resulting decrease in the marginal productivity of capital and dividends requires a long-run fall in q to raise the rate of return on equity and to ensure the equality of asset yields. As in the case of the rise in 7g, the opposing pressures these long-run changes yield on q may result in a drop or jump of the equity price on impact. Yet, q unambiguously falls in the medium-run along the adjustment path. To see the consequences of these policies on consumption and lifetime welfare I use (22) and (23) in (29) and (30) and set t = 0 to obtain (34) (35)
c o - c* = ( - A1)(r - A2)(r - A 1 ) - l k j [ f ' ( k * ) - r],
j = 1,2,4,
th0 - th* = / 3 ( c 0 - c*),
which (recall that the long-run levels of c and 4, remain unaltered) show the changes in consumption and lifetime welfare on impact. Note that since ~b0 is the present discounted value of the future felicity stream as of time t = 0, and since ~b* is unaffected by them, the sign of ~b0 - 4 , * directly measures the welfare effects of the policies under consideration. Further, given /3 > 0, the sign of ~b0 - ~b* is the same as the sign of c o - c*. To understand the changes in consumption c on impact recall that households strive to attain the target utility level u(c*), which is not affected by the policies we are considering. This implies that the long-run level of consumption and the level of wealth (and, thus, of income) required to support it also remain constant. Thus, in response to these policies foreward-looking households choose transition paths that allow them to attain the original utility level. Consider, now, the effects of the increase in the tax rate on corporate income. This, as we saw, decreases the economy's capital stock ((22)) and GDP. To attain the original utility level households must increase their long-run holdings of foreign assets ((23)). On impact, the long-run decrease in the domestic capital stock makes a short-run consumption binge possible. On the other hand, the required long-run increase in the foreign asset holdings call for an immediate increase in savings and a drop in consumption on impact. If Journal of International Money and Finance 1995 Volume 14 Number 6
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the long-run effect of the decrease in the capital stock on income, that is f ' ( k * ) , exceeds the interest earned on foreign bonds, r, consumption must drop on impact (thus c o c* < 0 in (34)) to point A in Figure 3. Otherwise c will jump on impact (to point B in Figure 3). Since ¢0 follows c o if consumption jumps on impact so will lifetime welfare. 19 The following alternative way of looking at this result proves rather useful. As we saw, whether the rise in r c increases lifetime welfare or not depends on the sign of f ' ( k * ) - r . Recall that the difference between the marginal productivity of capital and the world real rate of interest is caused by the presence of distortionary taxes. If initially f ' ( k * ) > r, the fall in the capital stock caused by the rise in ~c will accentuate the distortion and reduce lifetime welfare by increasing the marginal productivity of domestic capital. On the other hand, if f ' ( k * ) < r initially as the tax rates discussed above indicate, the same policy will reduce the distortion and increase lifetime welfare. 2° Since the increase in zg has similar long-run effects on the domestic capital stock and the foreign asset holdings ((22), (23)) as the increase in r c, it follows that it will have similar consequences for consumption and lifetime welfare for the same reasons. The increase in the investment tax credit, on the other hand, has the opposite consequences for k* and b£*: it increases the long-run domestic capital stock and reduces the long-run foreign asset holdings ((22),(23)). Thus, if f ' ( k * ) > r initially, the rise in ~'t will decrease the marginal productivity of domestic capital and reduce the distortion, thereby raising lifetime welfare, and vice versa. However, since the inequality is reversed with the tax rates discussed, the opposite case, i.e. a reduction in lifetime welfare, seems empirically more plausible. I now turn to the effects of an increase in the tax rate on personal interest income, which, unlike the former policies does change the long-run target level of utility. First, note that an anticipated permanent rise in ~'r reduces the rate of return on foreign bonds. On impact, this creates an incipient excess stock demand for equity, which is eliminated by an immediate jump in the price q of equity that lowers the yield on it. The consequent rise in investment in the medium run will increase the domestic capital stock, and decrease its marginal productivity until the equity price q and the rate of investment return to their initial levels in the long-run. As their long-run utility target falls with the rise in rg, households will reduce their long-run consumption. To see the consequences of the policy for household lifetime welfare and consumption, rewrite (34) and (35) by taking into account the long-run changes in c and ~b: -
(36)
cff
-- C O =
( -- Al)(r
-
A2)(r
-
A1)-l~:3[f'(k
* ) -
r]
-
A2c*
,
(37) ~b~- - ~bo = f l ( ( - A , ) ( r - A 2 ) ( r - A1)-ak3[f'(k * ) - r ] + ( r - A2)c* } -- ( OrnOr)--l(,.i-r n -- Tr) ,
where superscripts - and + refer to the values of the variables before and after the discrete change on impact and the superscript n denotes the new 794
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levels of ~-g and Og. Equation (36) indicates that if initially the marginal productivity of the capital stock exceeds the world real rate of interest, consumption will jump on impact as before. However, unlike the previous cases, here we cannot infer from the jump that lifetime welfare increases. This is because, though an initial jump in c increases welfare, consumption decreases across steady states; consequently the new long-run lifetime welfare is lower. Thus, in Figure 3, c may jump to either point D (with an increase in 4') or to point E (with a decrease in Oh).As equation (37) indicates f ' ( k * ) > r is a necessary but not sufficient condition for welfare to improve. Yet, for small changes in the tax rate, the reduction in the distortion, implied by an initial f ' ( k * ) > r and the rise in the capital stock, will raise lifetime welfare. Otherwise, as in the empirically more plausible case, consumption and lifetime welfare may drop on impact (F and G in Figure 3, representing two such possible points). Note also that across steady-states, given the increase in the domestic capital stock and the decrease in their long-run target utility level, the rise in ~-g will lead forward-looking households to decrease their foreign asset holdings ((23)). Next I discuss briefly the motion of the current account along the convergent path. Since the underlying logic is, mutatis mutandis, the same in all the exercises carried out above, I shall focus on the effects of an increase in the tax rate on capital gains. Consider now the path originating from point A in Figure 2, which is drawn under the assumption that A1 > A2 for the reasons mentioned above. This path indicates that though the foreign asset holdings of households rise in the long run, initially the economy runs a current account deficit. To see why consider the following. On impact, the drop in the equity price q leads to an immediate decrease in investment, which by itself would give rise to a current account surplus. Yet, we also saw that consumption c may jump or drop on impact. If c jumps sufficiently enough to outweigh the drop in investment, domestic absorption may in fact initially rise, causing a current account deficit depicted by the path starting at A. As the economy increases its long-run holdings of foreign assets, it must, however, run a current account surplus later. This implies a non-monotonic adjustment of the current account. On the other hand, if c drops on impact, or if the jump in c is outweighed by initial domestic capital decumulation, the economy will run a current account surplus on impact, and will adjust monotonically, as shown by the path originating from point B in Figure 2. IIl. Conclusion
The purpose of this paper has been to study the effects of various capital income tax policies in the framework of a small open economy which is populated by infinitely-lived households, possessing endogenous time preference rates, and in which investment is subject to adjustment costs. I have traced the time path of investment, equity prices, consumption, current account, and lifetime welfare in response to changes in tax instruments. The paper has shown the possibility of welfare paradoxes, in that increases in tax rates may improve welfare. This is because the tax instruments drive a Journal of International Money and Finance 1995 Volume 14 Number 6
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distortionarywedge between the marginal productivityof domestic capital and the world interest rate. Policies that reduce the distortion have, therefore, proven to be welfare-improving. The paper has also demonstrated that the current account may adjust non-monotonically.
Appendix I The current value Hamiltonian for the lifetime welfare maximization problem of Section 2 can be written as (A1) H=
-exp(-z)
+ p ' [ ( 1 - zr)ra + w - c + z] - d/[u(c) - (1 - Zr)r],
where p' and ~b' are costate variables associated with a and z, respectively. First order conditions for the maximum are then
chu'(c) + p = O,
(A2)
p = p [ u ( c ) - (1
(A4)
-
~'r)r],
6 = 1 + ¢u(c),
where I have used the normalizations p =p'exp[z(t)] and 4) = ~b'exp[z(t)]. Equation (9) of the text is obtained from ( A 2 ) - ( A 4 ) . N o t e that - e x p ( - z ) is strictly concave in z, while (3) is linear in a. Also u(c) is strictly concave in c. Therefore the maximized Hamiltonian H is strictly concave in a and z. Impose the conditions
(A5) limt_~®p'(t)exp[- (1 - ~-~)rt] _> 0, lim t -~oo- ~b'(t)exp[- (1 - ~-~)rt] >_ 0, limt_.®p'(t)a(t)exp[-(1 - 7~)rt ] = limt_, ~ - d / ( t ) z ( t ) e x p [ - ( 1 - ~-~)rt] = 0. That the convergent path is optimal then follows from the sufficiency theorem for optimal controls (Arrow and Kurz, 1970; Obstfeld, 1990).
Appendix II Linearization of the dynamic system ((9), (10), (13), (14), (19)) yields k = M ( x - ~),
(A6) where
M =
796
rO, O i l
-O¢Og- 1 f ,, (k , )
0
0
O-
kg'(q)
0
0
0
0
0
0
0
--c(rOr )2
0
0
0
--(rcOr) -1
r er
0
-kq¢(q)
f'(k)
-1
0
r
,
Journal of InternationalMoney and Finance 1995 Volume 14 Number 6
Capitalincometaxation:C Karayalcin and where x = (q, k, c, th, b). Standard methods yield three real positive and two real negative eigenvalues. The latter, denoted by A1 and A2, are (a7)
Al=(1)(rOrOg 1 - ¢(rOrOgl)2-4f"(k)k~o'(q)OcOg I }, A2---- (1){rOr - ¢(rOr )2 + 4rOr }"
Note, from the assumed properties of the adjustment cost function in (13), that the higher is the cost of adjustment, the lower is ~p'(q) and the higher is h 1. AS adjustment costs approach zero, ¢'(q) goes to infinity and hi approaches minus infinity. To see whether the assumption that A1 > A2 holds under the tax rates given in the text and empirically relevant adjustment costs (as estimated by Craine (1975) and used by Mendoza (1991) in a small open economy model based on Canadian data), I set up the following numerical example. The functions used are:
f ( k , l) = A [ p k - ~ + (1 - p ) l -~ ]-~/~, bi T = ~--~. The first of these is the conventional CES production function, whereas the second is the T function used by Lipton and Sachs (1983). The parameters are calibrated to fit stylized facts and to deliver a meaningful steady state. Their values are: a = 0.5, p = 0.4, r = 0.06, b = 0.02, l = 1, b = 0.02. The value for the adjustment cost parameter b is approximately equal to the estimate by Craine (1975) of 0.025 and the lower limit 0.023 of the range adopted by Mendoza (1991). The parameters yield -0.081 = h I > h 2 = -0.166, supporting the empirical plausibility of the assumption in the text.
Appendix III The derivatives in (21)-(25) are as follows, (A8)
ql = 0; 2 ( ~ ' - 'TI) ~>0,
rOr( I~ TI) -
(A9)
q2 = - - 0 ; 1 ' ( 0 ,
-
kl = O~02f,,(k) < 0 ,
~:2 = oclOgkl < 0 ,
r( 6 - r l)
rOr
k3 = _f,,(k)OcOg > O, k4 -- _f,,(k)OcOg > O, (A10)
bl = -f'(k)r-1]¢l > O, b2 = - f ' ( k ) r - l k 2
b3 = - [c +f'(k)r-l~:3] < 0,
b4 :
(All)
?'= -rc
(A12)
dp'= --(rOr2) -1 < 0 .
> O,
-f'(k)r-1]¢4 < 0,
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Notes 1. Chamley (1986), which characterizes the Ramsey-efficient tax structure, and Lucas (1990), which reviews the research on capital income taxation, are two prime examples of the same line of inquiry in dosed economy models. 2. Adjustment costs are modeled after Blanchard (1983) and Blanchard and Fischer (1989). 3. See, for instance, Sen and Turnovsky (1990). 4. Shi (1994) adopts a similar time preference structure to study the effects of a different set of tax instruments. The analysis there is restricted, however, to long-run welfare effects. The present paper measures the effects of changes in tax rates on lifetime welfare. 5. Karayalcin (1994) demonstrates that such non-monotonicity is helpful in explaining the observed correlation between domestic savings and investment. 6. See Epstein (1987) and Obstfeld (1990) for the arguments and empirical studies in support of u' > 0. 7. Given the constancy of the real rate of interest, it can easily be shown that the use, for instance, of a felicity function of the C R R A family yields qualitatively similar results for the disturbances studied below. 8. Henceforth, I drop the time subscripts except when there is risk of confusion. 9. I follow Nielsen and Sorensen (1991) in the modeling of tax instruments. 10. See Appendix III for the derivatives. 11. This assumption implies that the replacement cost of capital 1 - ~'l (inclusive of the investment tax credit) exceeds the debt issue per unit of capital 1 - e; that is, the firm does not 'overfinance' its investment by excessive debt issue (see Nielsen and Sorensen, 1991). 12. These numbers are calculated from King and FuUerton (1984), Boadway et al. (1987), and Knoester (1993). 13. The empirical plausibility and analytical advantage of these target levels have been emphasized by Obstfeld (1990). 14. It is a straightforward exercise to show that the regularity conditions necessary for stability (Epstein, 1987) are satisfied here. 15. Karayalcin (1994) shows that, absent taxes, the relative size of the two eigenvalues depends on adjustment costs, such that if adjustment costs exceed a critical level A1 > A2. It can also be easily demonstrated that here a similar critical level exists that depends on the tax rates and the level of adjustment costs. For reasons of space and at the expense of completeness, I shall not discuss the case A1 = A2 which may hold by fluke. Standard methods can be used to obtain the solution for this case as well. 16. Appendix II provides a numerical example which shows that this configuration of eigenvalues holds with the tax rates given above and with the adjustment costs as estimated by Craine (1975) and recently used by Mendoza (1991) in a small open economy model that uses Canadian data. 17. The discussion in the section follows the analysis of, for example, Frenkel and Razin (1992), Nielsen and Sorensen (1991), and Turnovsky and Bianconi (1992), in treating the existing tax structure as given and studying the effects of changing the tax rates. It does not, therefore, attempt to answer the questions as to whether it may be optimal to tax capital or what would an optimal tax mix be. Such questions are taken up, for instance, by Imrohoroglu (1994), who points out that capital income taxes may have an 'insurance' role, and by Gordon (1992), which shows that if there is a dominant capital exporter that acts as a Stackelberg leader when setting its tax policy, there may be game-theoretic reasons for taxation of capital income. 18. Unanticipated temporary, as well as anticipated permanent, changes in the tax instruments can easily be shown to give rise to short-run dynamics similar in direction to unanticipated permanent changes. The exception is the investment tax credit, an anticipated increase in which may be contractionary in the short run. Further, note that the results obtained in this section are valid for small changes in the tax rates. 798
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19. Here and in what follows it is a straightforward exercise to show that if changes in policy instruments are temporary, the discrete changes in c and ~b will be equal to the present discounted values of changes associated with the permanent ones. 20. As is common with the familiar static analyses of the effects of changes in tax rates, if f ' ( k * ) = r initially, the increase in tax rates here and in what follows leaves lifetime welfare unchanged.
References
ARROW,KENNETHAND MORDECAIKURZ, Public Investment, the Rate of Return, and Optimal Fiscal Policy, Baltimore: Johns Hopkins Press for Resources for the Future, 1970. BACKUS,DAVIDAND PATRICKKEHOE,'International evidence on the historical properties of business cycles', American Economic Review, September 1992, 82: 864-888. BACKUS, DAVID, PATRICKKEHOE AND FINN KYDLAND,'International real business cycles', Journal of Political Economy, August 1992, 100: 745-775. BLANCHARD,OLIVIER, 'Debt and the current account deficit in Brazil', in Pedro Aspe Armella, Rudiger Dornbusch and Maurice Obstfeld, eds, Financial Policies and the World Capital Market The Problem of Latin American Countries, Chicago: University of Chicago Press, 1983. BLANCHARD,OLIVIERAND STANLEYFISCHER, Lectures on Macroeconomics, Cambridge, MA: The MIT Press, 1989. BOADWAY,ROBIN,N., NEIL BRUCE, AND JACK M. MINTZ, Taxes on Capital Income in Canada: Analysis and Policy, Toronto: Canadian Tax Foundation, 1987. BUITER WILLEM, 'Time preference and international lending and borrowing in an overlapping-generations model', Journal of Political Economy, August 1981, 89: 769-797. CHAMLE¥, CHRISTOPHE,'Optimal taxation of capital income in general equilibrium with infinite lives,' Econometrica, May 1986, 54: 607-622. CRAINE, ROGER, 'Investment, adjustment costs, and uncertainty,' International Economic Review, August 1975, 16: 648-661. EPSTEIN, LARRY, 'A simple dynamic general equilibrium model,' Journal of Economic Theory, February 1987, 41: 68-95. EPSTEIN, LARRY AND ALLAN HYNES, 'The rate of time preference and dynamic economic analysis', Journal of Political Economy, August 1983, 91: 611-635. FRENKEL, JACOBAND ASSAF RAZIN, Fiscal Policies and the World Economy, Cambridge, MA: The MIT Press, 1992. FRENKEL, JACOB, ASSAF RAZIN AND EFRAIM SADKA, International Taxation in an Integrated World, Cambridge, MA: The MIT Press, 1991. GORDON, ROGER, 'Can capital income taxes survive in open economies?', Journal of Finance, July 1992, 47: 1159-1180. HAVASHI, FUMIO 'Tobin's marginal and average q: a neoclassical interpretation', Econometrica, January 1982, 50: 213-224. IMROHOROGLU,SELAHATrlN,'A quantitative analysis of the optimal tax structure under incomplete markets,' mimeD, Department of Finance and Business, University of Southern California, August 1994. KARAVALCIN, CEM, 'Adjustment costs in investment, time preferences, and the current account', Journal ofInternationalEconomics, August 1994, 37: 81-95. KING, MERVYN AND DON FULLERTON, The Taxation of Income from Capital, Chicago: University of Chicago Press, 1984. KNOESTER,ANTONIO,Taxation in the United States and Europe, New York: St. Martin's Press, 1993. Journal of International Money and Finance 1995 Volume 14 Number 6
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Journal oflnternational Money and Finance, Vol. 14, No. 6, pp. 785-800, 1995
I~! E i N E M A N N
Copynght©1995EIsevierScienceLtd Printed in Great Britain. All fights reserved 0261-5606/95 $10.00 + 0.00
0261-5606(95)00032-1
Capital income taxation and welfare in a small open economy CEM
KARAYALCIN*
Florida International University, Department of Economics Miami, FL 33199, USA The paper studies the effects of various capital income tax policies in a small open economy. Households are infinitely-lived and possess endogenous time preferences. Investment by firms is subject to adjustment costs. It is shown that welfare paradoxes may exist, in that increases in tax instruments may improve welfare. (JEL H20).
The continuing process of global economic integration, as exemplified by the unification of the European market as well as by the recent liberalization of capital markets in several developing countries has given rise, inter alia, to a growing literature focusing on the international effects of taxation. Rapid progress seems to have been made since 1988, when Slemrod (1988) made the observation that the study of tax incidence in the context of international capital mobility was still in its infancy. Recent analyses of capital income taxation by Sen and Turnovsky (1990), Lucas (1990), Razin and Sadka (1991), Frenkel et al. (1991), Nielsen and Sorenson (1991), Frenkel and Razin (1992), Turnovsky and Bianconi (1992) attest to the dynamism of the field. 1 This paper focuses on the welfare effects of various capital tax instruments in a small open economy, intertemporal utility maximization framework, an issue hitherto unexplored in this context. The economy is populated by infinitely-lived households which can borrow or lend freely in the world capital market. For simplicity it is assumed that the economy produces a single good used for consumption and investment which is subject to adjustment costs. 2 The paper differs from its predecessors by modeling the households as possessing endogenous time preferences, rather than the commonly-used time-additive constant rate of time preference. It is well-known that to ensure the existence of long-run equilibrium the time-additive preference structure, which implies a constant rate of time preference, requires that this rate be arbitrarily set equal to the after-tax world rate of interest. Taxes that change this after-tax interest rate disturb this equality and create instability (Epstein and Hynes, 1983). To avoid such instability severe restrictions have to be imposed on international capital mobility. Even in the absence of tax instruments that alter the after-tax world rate of interest, the time-additive preferences lead to a *I would like to thank an anonymous referee for helpful comments.
Capital income taxation: C Karayalcin
hysteretic adjustment towards the steady state, rendering the analysis dependent on initial conditions.3. Overlapping-generations models overcome these problems at the expense either of the ability to study short-run adjustment (Buiter, 1981) or the adoption of the rather unrealistic assumption of an age-independent probability of death (Nielsen and Sorensen, 1991). The endogenous time preference framework, developed by Epstein and Hynes (1983), and adopted here avoids these problems.4 It also provides a rather parsimonious way of measuring the effects of changes in capital tax instruments on the lifetime welfare of households. Given this setup, I show that increases in the capital income tax rates (such as the tax rate on capital gains, on personal interest income, on corporate income) as well as a rise in the investment tax credit may increase lifetime welfare. The intuition for this result is easy to grasp. Such taxes and subsidies drive a distortionary wedge between the marginal productivity of capital at home and the world real rate of interest. If, for example, initially the former exceeds the latter, policies that increase the home capital stock will, by decreasing its marginal productivity, reduce the distortion and raise lifetime welfare. The significance of this result for the analysis of tax policies of such OECD countries as Canada and Britain, which have recently been in the process of dismantling their investment tax credit programs, is quite obvious. The paper also traces the time path of the economy in response to various policy changes and shows that the current account may adjust non-monotonically. 5 The rest of the paper is arranged as follows. The next section sets up the model. Section II studies the effects of changes in tax policy instruments. Section III concludes the paper. I. The model
Consider a small open economy with a constant number of identical, infinitely-lived households that have perfect foresight. Normalize, without loss of generality, the number of households to one. Suppose, also, that perfectly competitive firms in this economy produce a single good that can be used for consumption and investment. I now turn to a detailed analysis of household and firm behavior. 1.4. H o u s e h o l d s
Households inelasticaUy supply one unit of labor services, for which they receive the wage, wt, per unit of time. They also receive interest income r(1 - rr)a t (where r and ~'r denote the constant world real rate of interest and the personal tax on interest income) on their non-human wealth, a t . If households have Epstein-Hynes variable rates of time preference, they maximize lifetime welfare U over consumption path C, that is, they maximize oo
(1)
U(C) = - f
e x p ( - z t ) e x p [ - (1 - z r ) r t ] d t ,
J0
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subject to (2)
2 t = u ( c t) - (1
-
'rr)r ,
(3)
ti = (1 - rr)ra t + w t - c t + 'rt ,
<4>
z 0 = 0,
where u ( c ) > 0 is the felicity function with u ' > 0, u " < 0 and r t is the lump-sum government transfer. Since it is not the purpose of this paper to study the effects of government debt policy, I shall assume that the domestic government follows a policy of balanced budgets, with the implication that the lump-sum transfer received by the representative household comprises the net tax revenue. The lifetime welfare functional U differs from the conventional time-additive utility functionals used in the literature by its recursivity, which implies that the marginal rate of substitution between times t and s (s > t) is independent of consumption before t but not after s. The structure gives rise to a variable rate of time preference 12 at time s: (5)
12s={fs°~exp[-fstU(c)dr]dt}
-1 .
12 at s is the following function of the utility functional U ( C ) (6)
12(6~)
=
--6
-1 ,
~bs =
U(sC),
where sC stands for that part of the consumption path C beyond time s and ~bs denotes lifetime welfare at time s. When the consumption path is globally constant, as in long-run equilibrium, U ( s C ) = - 1 / u ( c ) , and the rate of time preference is given by (7)
12"(6") = u ( c * ) ,
where asterisks denote long-run equilibrium. Assuming u' > 0 implies that 12 is increasing in consumption path C. This assumption is labeled 'increasing marginal impatience' by Lucas and Stokey (1984). Though several arguments have been advanced in support of it, for our purposes it is sufficient to point out that local stability will fail to obtain in the absence of such an assumption. 6 Henceforth, I specialize the felicity function t o 7 <8)
u ( c , ) = to + lnc,,
where to is a parameter measuring generalized time preference. Appendix 1 shows that with (8) the solution of the lifetime welfare maximization problem yields s (9)
d = [(1 - zr)r - 12(~b)1c.
The dynamics of lifetime welfare ¢ are obtained by differentiating (1) with respect to time: (10)
d= 1 +u(c)eh.
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LB. Firms At a point in time perfectly competitive firms employ capital, k, and labor to produce the single good under constant returns to scale. New investment undertaken by these firms is financed both by the issue of corporate bonds, b c, and by retained earnings. Remaining profits net of corporate income tax are paid out as dividends to holders of equity. Firms are assumed to deduct interest payments on outstanding debt and adjustment costs T associated with investment i in determining taxable corporate profits. Consequently, total dividends before personal tax, ~, are given by 9 (11)
z r =- [ f ( k ) - w - r b
c - T](1 - zc) +b c - (1 - zrt)i ,
where zc and ~'t denote the corporate income tax rate and the rate of investment credit; f ( k ) represents a constant-returns-to-scale production function with the conventional Inada properties. It is well-known that in the absence of uncertainty one cannot adequately account for the differences in the forms of financing. Thus, I shall assume that the representative firm finances a fraction 6 of its new investment through retained earnings, and (1 - 6) of it through the issue of corporate bonds, i.e.
b c = (1 - 6)k, b c = (1 - e)/~.
(12)
Since installing investment goods is costly it takes i[1 + T(i/k)] units of output to increase the capital stock by i units. The installation-cost function T is assumed to have the following properties: (13)
T(0) = 0,
T'(.) > 0,
2T' + ( i / k ) T " > O.
Corporate bonds, b c, foreign bonds, b f, and equities are perfect substitutes in the portfolios of households. Let E denote the market value of outstanding equity. Then the arbitrage condition
7r
(14)
r(1 -
Tr) = "-~ 7!-
(1 -- ~'g)E g
(where rg stands for the tax rate on accrued capital gains) must hold at all times. The term on the left-hand side of (14) represents the after-tax of return on foreign bonds, while the expression on the right-hand side denotes the after-tax rate of return on equity. This expression is the after-tax sum of current yield and capital gains. Using (11) and (12) in (14) and integrating, one obtains the market value of equity as of time 0: oo
(15)
E = f0 0gl "ff exp(--rOrOg lt)dt'
Oi =- 1 - zj, j = c , g , r .
The representative firm chooses the time path of investment by maximizing the market value E subject to the constraint i = k. This yields (16)
d1 =rOrO~lq - O¢o~l[f'(k) - r ( 1 - 6) + ( i / k ) 2 T ' ( i / k ) ] ,
(17)
q = 0g-1 { ( 6 - ~'t) + Oc[T+ ( i / k ) T ' ( i / k ) ] } ,
(18)
w =f(k) -f'(k)k,
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where q denotes the shadow value of capital and can be easily shown, as in Hayashi (1982), to be equal to the stock market price of a unit of equity relative to the replacement cost of capital (that is q of Tobin's Q). Substituting <17) into (16), the latter is seen to be an arbitrage equation setting the after-tax rate of return on foreign bonds to the after-tax rate of return on equity. (18) is the familiar condition that requires the marginal productivity of labor to be equal to the real wage rate. Equation <17) implies that the rate of investment, i / k , is the following function of q (19)
~ = k- = ~o(q),
~o'(q*) > 0,
which states that the rate of investment is an increasing function of 'Tobin's Q'. The most salient feature of (16) and (19) is that neither q nor investment depends on the consumption and savings decisions of households. Thus, (16) and (19) constitute a system of two autonomous differential equations in q and k. The system is depicted in Figure 1. I.C. The current account
To obtain the dynamics of the foreign asset holdings of the representative household (or the dynamics of the current account as these two coincide here) use (3), (11)-(19), a = b f + qk + b c, and recall the assumptions of balanced government budgets to obtain (20)
bf=rbf+f(k)
- i ( 1 + T) - c,
q
k--O
k FIOURE1. The adjustment of q and k. Journal of InternationalMoneyand Finance1995 Volume14 Number6
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Capital income taxation: C Karayalcin
that is the current account is equal to the sum of output and interest earnings from foreign assets less the sum of consumption and investment spending.
LD. Equilibrium The model has five differential equations: (9),(10),(16), (19) and (20). To obtain the steady-state value of x = x(q, k, c, 4~, b) set these equations to zero. This yields TM (21) (22)
q* = 0 g l ( S - ~-l),
q*=q('rg,~'l),ql>O,
f'(k*)=r[(1-e)+OrO~lOgl(6-rt)], kt < 0 (i = 1,2),
q2
k*=k(zg,rc,rr,%),
kj > 0 ( j = 3,4),
(23)
rb f* + f ( k * ) = c*, b f* = b(rg, ~'c,zr, ~-,), bi > 0 (i = 1,2), bj < 0 ( j = 3,4), (24)
u ( c * ) = rOr,
(25)
~b* = - ( r O t ) - 1 ,
c* = C(Tr) , q~* = ~('rr) ,
~t < 0, ~ ' < 0,
where I assume in (21) that e > ~'l so that the equity price q is positive in the steady state) 1 Equation (22) shows that capital income taxes and the investment tax credit drive a distortionary wedge between the marginal productivity of capital f ' ( k ) and the world real rate of interest r. Whether this wedge causes the former to exceed the latter or not depends on the configuration of the tax rates Zg, ~-~,Zr, and the investment tax credit zt:
f ' ( k * ) <>r ,~
(26)
1-
0c0])
+ ---~-r
<>0.
If income taxes are uniform (so that rg = Zr), the corporate tax rate is fully integrated (so that ~'~= 0), and there is no investment tax credit (~'1= 0), it follows from (22) that f ' ( k * ) = r; that is, one obtains the well-known result that a Schanz-Haig-Simons income tax is neutral as regards investment. Further, (17) indicates that in the absence of the investment credit and the tax on capital gains q = e. Thus, in this case, the market value of equity E -- qk = e k equals the equity-financed portion of the accumulated investment of firms. However, actual tax systems do not generally have fully integrated corporation taxes, nor are the rates of investment tax credit set equal to zero. It is, therefore, useful to look at some representative tax rates for such countries as Canada, Sweden, and the Netherlands, which are generally considered to be small open economies, to have an understanding of the difference between f ' ( k * ) and r as implied by the rates in (26). The effective average tax rates are roughly: Tr 0.45, r~ = 0.25, r t = 0.10, z~ = 0.27; in addition e = 0.75.12 These indicate that f ' ( k * ) < r for the economies in question. Now, note from ( 2 3 ) - ( 2 5 ) that the variable rate of time preference implies well-defined long-run target utility, consumption and wealth levels) 3 =
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At a point in time, given the parameters and the steady-state level of x = (q, k, c, ~b,b) the five differential equations solve for x. (15) determines the wage rate. As Appendix 2 shows the dynamic system ((9), (10), (16), (19), (20)) possesses two negative (denoted by A1 and A2) and three positive eigenvalues, which (given the two predetermined variables k and b f) renders it saddle-path stable. TM Its motion along the convergent saddle-path is characterized by (27) (28) (29)
k, - k* = ( k o -
q, - q* =
k*)exp(Alt),
[ A1/k*q~' ( gl ) l( k t - k * ),
ct-c*=(r-Az)[(bfo-bf*)+l~(ko-k*)]exp(Azt)
(30)
,
~), -- 6 * = /3( Ct - C* ),
(31) b y - b f* = - ~ ( k o - k *
)exp(All) + [ ( b 0/ -
b f* ) t- t16(k 0 - k * )]
exp( A2t),
where Ix=[f'(k*)-A1][r-Aa]-l>o,
fl=[rc*Or(rOr-A2)]
-1
>0.
In the perfect-foresight, intertemporal-equilibrium framework adopted here the dynamics of the variables are dictated by the long-run changes they undergo. Thus, (27) indicates that along the convergent path the dynamics of investment are uniquely determined by its speed of adjustment and the long-run change in the domestic capital stock, while (31) shows that the dynamics of the current account depends on long-run changes in both foreign assets and the domestic capital stock, as well as the adjustment speeds of both. Straightforward manipulation of (27) and (31) yields (32) (33)
k = Xl(k t - k*), bY = t~(A2 - '~a)(kt
- k* ) +
A2(b/ - b I* ).
It follows from (32) and (33) that if /~1 < }~2 the capital stock, k, will adjust faster than the foreign asset holdings, b y, or, to put it differently, the current account will be more persistent than domestic investment. 15 This, however, contradicts the findings of recent studies (for instance Backus and Kehoe, 1992; Backus et al., 1992). Consequently, I shall concentrate on the case /~1 > /~2 "16 Figure 2 depicts the adjustment of k and b f. Further, since /3 > 0, (29) and (30) indicate that consumption c and lifetime welfare ~b always rise and fall together along the convergent path shown in Figure 3. II. The effects of taxation
I now put the model to work by considering the effects of changes in the tax rates. 17 I start by focusing on the effects of alterations in the investment tax credit ~t, in the tax rates on capital gains ~-g,and on corporate income ~c. They Journal of International Money and Finance 1995 Volume 14 Number 6
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Capital income taxation: C Karayalcin k .f
b =0
~o
t+ f .b
Fxoum~ 2. The adjustment of the current account.
~l
~
F
6'=0
FIGURE3. The adjustment of consumption and lifetime welfare.
are treated as a group, for changes in these rates do not impinge upon the long-run target utility of households (equation (24)), thus do not entail any effects on the long-run level of consumption. Consider now the effects of an unanticipated permanent increase in the corporation tax rate. TM On impact this reduces dividends and the rate of return on equity. The consequent incipient excess stock demand for foreign assets leads to an immediate drop in the price of equity q (and to expectations of 792
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capital gains), which starts a process of capital decumulation. The process continues until the decline in the capital stock pushes the marginal productivity of capital and the dividends up sufficiently enough so that, in the absence of expectations of capital gains on it, the rate of return on equity equals the rate of return on foreign bonds. Similarly, an unanticipated permanent increase in the tax rate ~-~ on capital gains reduces the long-run capital stock. Since this has the effect o]~ raising the steady-state marginal productivity of capital and dividends, the long-run equity price q must also rise to ensure the equality of asset yields. As the short-run adjustment of the foreward-looking equity-price q is driven by the long-run changes in k and q--which in the instance put opposing pressures on q - - t h e impact effect of the increase in ~-g on q is ambiguous. However, regardless of whether it drops or jumps on impact, along the convergent path, q will rise towards its higher steady-state level (equation (28)). By contrast, an unanticipated permanent rise in the investment tax credit r t increases the long-run capital stock, by reducing the effect replacement cost of capital. The resulting decrease in the marginal productivity of capital and dividends requires a long-run fall in q to raise the rate of return on equity and to ensure the equality of asset yields. As in the case of the rise in 7g, the opposing pressures these long-run changes yield on q may result in a drop or jump of the equity price on impact. Yet, q unambiguously falls in the medium-run along the adjustment path. To see the consequences of these policies on consumption and lifetime welfare I use (22) and (23) in (29) and (30) and set t = 0 to obtain (34) (35)
c o - c* = ( - A1)(r - A2)(r - A 1 ) - l k j [ f ' ( k * ) - r],
j = 1,2,4,
th0 - th* = / 3 ( c 0 - c*),
which (recall that the long-run levels of c and 4, remain unaltered) show the changes in consumption and lifetime welfare on impact. Note that since ~b0 is the present discounted value of the future felicity stream as of time t = 0, and since ~b* is unaffected by them, the sign of ~b0 - 4 , * directly measures the welfare effects of the policies under consideration. Further, given /3 > 0, the sign of ~b0 - ~b* is the same as the sign of c o - c*. To understand the changes in consumption c on impact recall that households strive to attain the target utility level u(c*), which is not affected by the policies we are considering. This implies that the long-run level of consumption and the level of wealth (and, thus, of income) required to support it also remain constant. Thus, in response to these policies foreward-looking households choose transition paths that allow them to attain the original utility level. Consider, now, the effects of the increase in the tax rate on corporate income. This, as we saw, decreases the economy's capital stock ((22)) and GDP. To attain the original utility level households must increase their long-run holdings of foreign assets ((23)). On impact, the long-run decrease in the domestic capital stock makes a short-run consumption binge possible. On the other hand, the required long-run increase in the foreign asset holdings call for an immediate increase in savings and a drop in consumption on impact. If Journal of International Money and Finance 1995 Volume 14 Number 6
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the long-run effect of the decrease in the capital stock on income, that is f ' ( k * ) , exceeds the interest earned on foreign bonds, r, consumption must drop on impact (thus c o c* < 0 in (34)) to point A in Figure 3. Otherwise c will jump on impact (to point B in Figure 3). Since ¢0 follows c o if consumption jumps on impact so will lifetime welfare. 19 The following alternative way of looking at this result proves rather useful. As we saw, whether the rise in r c increases lifetime welfare or not depends on the sign of f ' ( k * ) - r . Recall that the difference between the marginal productivity of capital and the world real rate of interest is caused by the presence of distortionary taxes. If initially f ' ( k * ) > r, the fall in the capital stock caused by the rise in ~c will accentuate the distortion and reduce lifetime welfare by increasing the marginal productivity of domestic capital. On the other hand, if f ' ( k * ) < r initially as the tax rates discussed above indicate, the same policy will reduce the distortion and increase lifetime welfare. 2° Since the increase in zg has similar long-run effects on the domestic capital stock and the foreign asset holdings ((22), (23)) as the increase in r c, it follows that it will have similar consequences for consumption and lifetime welfare for the same reasons. The increase in the investment tax credit, on the other hand, has the opposite consequences for k* and b£*: it increases the long-run domestic capital stock and reduces the long-run foreign asset holdings ((22),(23)). Thus, if f ' ( k * ) > r initially, the rise in ~'t will decrease the marginal productivity of domestic capital and reduce the distortion, thereby raising lifetime welfare, and vice versa. However, since the inequality is reversed with the tax rates discussed, the opposite case, i.e. a reduction in lifetime welfare, seems empirically more plausible. I now turn to the effects of an increase in the tax rate on personal interest income, which, unlike the former policies does change the long-run target level of utility. First, note that an anticipated permanent rise in ~'r reduces the rate of return on foreign bonds. On impact, this creates an incipient excess stock demand for equity, which is eliminated by an immediate jump in the price q of equity that lowers the yield on it. The consequent rise in investment in the medium run will increase the domestic capital stock, and decrease its marginal productivity until the equity price q and the rate of investment return to their initial levels in the long-run. As their long-run utility target falls with the rise in rg, households will reduce their long-run consumption. To see the consequences of the policy for household lifetime welfare and consumption, rewrite (34) and (35) by taking into account the long-run changes in c and ~b: -
(36)
cff
-- C O =
( -- Al)(r
-
A2)(r
-
A1)-l~:3[f'(k
* ) -
r]
-
A2c*
,
(37) ~b~- - ~bo = f l ( ( - A , ) ( r - A 2 ) ( r - A1)-ak3[f'(k * ) - r ] + ( r - A2)c* } -- ( OrnOr)--l(,.i-r n -- Tr) ,
where superscripts - and + refer to the values of the variables before and after the discrete change on impact and the superscript n denotes the new 794
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levels of ~-g and Og. Equation (36) indicates that if initially the marginal productivity of the capital stock exceeds the world real rate of interest, consumption will jump on impact as before. However, unlike the previous cases, here we cannot infer from the jump that lifetime welfare increases. This is because, though an initial jump in c increases welfare, consumption decreases across steady states; consequently the new long-run lifetime welfare is lower. Thus, in Figure 3, c may jump to either point D (with an increase in 4') or to point E (with a decrease in Oh).As equation (37) indicates f ' ( k * ) > r is a necessary but not sufficient condition for welfare to improve. Yet, for small changes in the tax rate, the reduction in the distortion, implied by an initial f ' ( k * ) > r and the rise in the capital stock, will raise lifetime welfare. Otherwise, as in the empirically more plausible case, consumption and lifetime welfare may drop on impact (F and G in Figure 3, representing two such possible points). Note also that across steady-states, given the increase in the domestic capital stock and the decrease in their long-run target utility level, the rise in ~-g will lead forward-looking households to decrease their foreign asset holdings ((23)). Next I discuss briefly the motion of the current account along the convergent path. Since the underlying logic is, mutatis mutandis, the same in all the exercises carried out above, I shall focus on the effects of an increase in the tax rate on capital gains. Consider now the path originating from point A in Figure 2, which is drawn under the assumption that A1 > A2 for the reasons mentioned above. This path indicates that though the foreign asset holdings of households rise in the long run, initially the economy runs a current account deficit. To see why consider the following. On impact, the drop in the equity price q leads to an immediate decrease in investment, which by itself would give rise to a current account surplus. Yet, we also saw that consumption c may jump or drop on impact. If c jumps sufficiently enough to outweigh the drop in investment, domestic absorption may in fact initially rise, causing a current account deficit depicted by the path starting at A. As the economy increases its long-run holdings of foreign assets, it must, however, run a current account surplus later. This implies a non-monotonic adjustment of the current account. On the other hand, if c drops on impact, or if the jump in c is outweighed by initial domestic capital decumulation, the economy will run a current account surplus on impact, and will adjust monotonically, as shown by the path originating from point B in Figure 2. IIl. Conclusion
The purpose of this paper has been to study the effects of various capital income tax policies in the framework of a small open economy which is populated by infinitely-lived households, possessing endogenous time preference rates, and in which investment is subject to adjustment costs. I have traced the time path of investment, equity prices, consumption, current account, and lifetime welfare in response to changes in tax instruments. The paper has shown the possibility of welfare paradoxes, in that increases in tax rates may improve welfare. This is because the tax instruments drive a Journal of International Money and Finance 1995 Volume 14 Number 6
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distortionarywedge between the marginal productivityof domestic capital and the world interest rate. Policies that reduce the distortion have, therefore, proven to be welfare-improving. The paper has also demonstrated that the current account may adjust non-monotonically.
Appendix I The current value Hamiltonian for the lifetime welfare maximization problem of Section 2 can be written as (A1) H=
-exp(-z)
+ p ' [ ( 1 - zr)ra + w - c + z] - d/[u(c) - (1 - Zr)r],
where p' and ~b' are costate variables associated with a and z, respectively. First order conditions for the maximum are then
chu'(c) + p = O,
(A2)
p = p [ u ( c ) - (1
-
~'r)r],
6 = 1 + ¢u(c),
where I have used the normalizations p =p'exp[z(t)] and 4) = ~b'exp[z(t)]. Equation (9) of the text is obtained from ( A 2 ) - ( A 4 ) . N o t e that - e x p ( - z ) is strictly concave in z, while (3) is linear in a. Also u(c) is strictly concave in c. Therefore the maximized Hamiltonian H is strictly concave in a and z. Impose the conditions
(A5) limt_~®p'(t)exp[- (1 - ~-~)rt] _> 0, lim t -~oo- ~b'(t)exp[- (1 - ~-~)rt] >_ 0, limt_.®p'(t)a(t)exp[-(1 - 7~)rt ] = limt_, ~ - d / ( t ) z ( t ) e x p [ - ( 1 - ~-~)rt] = 0. That the convergent path is optimal then follows from the sufficiency theorem for optimal controls (Arrow and Kurz, 1970; Obstfeld, 1990).
Appendix II Linearization of the dynamic system ((9), (10), (13), (14), (19)) yields k = M ( x - ~),
(A6) where
M =
796
rO, O i l
-O¢Og- 1 f ,, (k , )
0
0
O-
kg'(q)
0
0
0
0
0
0
0
--c(rOr )2
0
0
0
--(rcOr) -1
r er
0
-kq¢(q)
f'(k)
-1
0
r
,
Journal of InternationalMoney and Finance 1995 Volume 14 Number 6
Capitalincometaxation:C Karayalcin and where x = (q, k, c, th, b). Standard methods yield three real positive and two real negative eigenvalues. The latter, denoted by A1 and A2, are (a7)
Al=(1)(rOrOg 1 - ¢(rOrOgl)2-4f"(k)k~o'(q)OcOg I }, A2---- (1){rOr - ¢(rOr )2 + 4rOr }"
Note, from the assumed properties of the adjustment cost function in (13), that the higher is the cost of adjustment, the lower is ~p'(q) and the higher is h 1. AS adjustment costs approach zero, ¢'(q) goes to infinity and hi approaches minus infinity. To see whether the assumption that A1 > A2 holds under the tax rates given in the text and empirically relevant adjustment costs (as estimated by Craine (1975) and used by Mendoza (1991) in a small open economy model based on Canadian data), I set up the following numerical example. The functions used are:
f ( k , l) = A [ p k - ~ + (1 - p ) l -~ ]-~/~, bi T = ~--~. The first of these is the conventional CES production function, whereas the second is the T function used by Lipton and Sachs (1983). The parameters are calibrated to fit stylized facts and to deliver a meaningful steady state. Their values are: a = 0.5, p = 0.4, r = 0.06, b = 0.02, l = 1, b = 0.02. The value for the adjustment cost parameter b is approximately equal to the estimate by Craine (1975) of 0.025 and the lower limit 0.023 of the range adopted by Mendoza (1991). The parameters yield -0.081 = h I > h 2 = -0.166, supporting the empirical plausibility of the assumption in the text.
Appendix III The derivatives in (21)-(25) are as follows, (A8)
ql = 0; 2 ( ~ ' - 'TI) ~>0,
rOr( I~ TI) -
(A9)
q2 = - - 0 ; 1 ' ( 0 ,
-
kl = O~02f,,(k) < 0 ,
~:2 = oclOgkl < 0 ,
r( 6 - r l)
rOr
k3 = _f,,(k)OcOg > O, k4 -- _f,,(k)OcOg > O, (A10)
bl = -f'(k)r-1]¢l > O, b2 = - f ' ( k ) r - l k 2
b3 = - [c +f'(k)r-l~:3] < 0,
b4 :
(All)
?'= -rc
(A12)
dp'= --(rOr2) -1 < 0 .
> O,
-f'(k)r-1]¢4 < 0,
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Notes 1. Chamley (1986), which characterizes the Ramsey-efficient tax structure, and Lucas (1990), which reviews the research on capital income taxation, are two prime examples of the same line of inquiry in dosed economy models. 2. Adjustment costs are modeled after Blanchard (1983) and Blanchard and Fischer (1989). 3. See, for instance, Sen and Turnovsky (1990). 4. Shi (1994) adopts a similar time preference structure to study the effects of a different set of tax instruments. The analysis there is restricted, however, to long-run welfare effects. The present paper measures the effects of changes in tax rates on lifetime welfare. 5. Karayalcin (1994) demonstrates that such non-monotonicity is helpful in explaining the observed correlation between domestic savings and investment. 6. See Epstein (1987) and Obstfeld (1990) for the arguments and empirical studies in support of u' > 0. 7. Given the constancy of the real rate of interest, it can easily be shown that the use, for instance, of a felicity function of the C R R A family yields qualitatively similar results for the disturbances studied below. 8. Henceforth, I drop the time subscripts except when there is risk of confusion. 9. I follow Nielsen and Sorensen (1991) in the modeling of tax instruments. 10. See Appendix III for the derivatives. 11. This assumption implies that the replacement cost of capital 1 - ~'l (inclusive of the investment tax credit) exceeds the debt issue per unit of capital 1 - e; that is, the firm does not 'overfinance' its investment by excessive debt issue (see Nielsen and Sorensen, 1991). 12. These numbers are calculated from King and FuUerton (1984), Boadway et al. (1987), and Knoester (1993). 13. The empirical plausibility and analytical advantage of these target levels have been emphasized by Obstfeld (1990). 14. It is a straightforward exercise to show that the regularity conditions necessary for stability (Epstein, 1987) are satisfied here. 15. Karayalcin (1994) shows that, absent taxes, the relative size of the two eigenvalues depends on adjustment costs, such that if adjustment costs exceed a critical level A1 > A2. It can also be easily demonstrated that here a similar critical level exists that depends on the tax rates and the level of adjustment costs. For reasons of space and at the expense of completeness, I shall not discuss the case A1 = A2 which may hold by fluke. Standard methods can be used to obtain the solution for this case as well. 16. Appendix II provides a numerical example which shows that this configuration of eigenvalues holds with the tax rates given above and with the adjustment costs as estimated by Craine (1975) and recently used by Mendoza (1991) in a small open economy model that uses Canadian data. 17. The discussion in the section follows the analysis of, for example, Frenkel and Razin (1992), Nielsen and Sorensen (1991), and Turnovsky and Bianconi (1992), in treating the existing tax structure as given and studying the effects of changing the tax rates. It does not, therefore, attempt to answer the questions as to whether it may be optimal to tax capital or what would an optimal tax mix be. Such questions are taken up, for instance, by Imrohoroglu (1994), who points out that capital income taxes may have an 'insurance' role, and by Gordon (1992), which shows that if there is a dominant capital exporter that acts as a Stackelberg leader when setting its tax policy, there may be game-theoretic reasons for taxation of capital income. 18. Unanticipated temporary, as well as anticipated permanent, changes in the tax instruments can easily be shown to give rise to short-run dynamics similar in direction to unanticipated permanent changes. The exception is the investment tax credit, an anticipated increase in which may be contractionary in the short run. Further, note that the results obtained in this section are valid for small changes in the tax rates. 798
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19. Here and in what follows it is a straightforward exercise to show that if changes in policy instruments are temporary, the discrete changes in c and ~b will be equal to the present discounted values of changes associated with the permanent ones. 20. As is common with the familiar static analyses of the effects of changes in tax rates, if f ' ( k * ) = r initially, the increase in tax rates here and in what follows leaves lifetime welfare unchanged.
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