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Uncertain lifetime, the annuity market, and estate taxation
Journal
of Public
Economics
UNCERTAIN
40 (1989) 217-235.
North-Holland
LIFETIME, THE ANNUITY AND ESTATE TAXATION Sheng Cheng
MARKET,
HU*
Krannert Graduate School oJ Management, Purdue University, West LaJayette, IN 47907, USA Received June 1988, revised version
received July 1989
This paper extends traditional studies of estate taxation to allow for the possibility that taxpayers may avoid the burden of the estate tax by strategically timing bequests to their children. In this model, the lifespan of the individual is uncertain and bequests can be left either in the middle or at the end of the lifespan. Bequests are subject to the estate tax only when they are left upon death. It is shown that the effects of the estate tax depend on the ability of the individual to make intra vivos gifts, which in turn is affected by the existence of a competitive annuity market.
1. Introduction The estate tax has been one of the measures taken by many industrial countries, Britain and the United States included, to reduce wealth concentration caused by intergenerational transfers. However, there are ways in which individuals can minimize or entirely eliminate estate-tax liabilities, the best known of which has been to make intra vivos gifts.’ Kay and King (1986) reported that avoidance of estate duty was so easy in Britain that it was sometimes described as a ‘voluntary tax’. Bernheim (1987) estimated that the U.S. federal estate tax represented less than 1 percent of all federal revenues each year in the first half of the 1980s despite the imposition of high statutory marginal tax rates. Although the overhaul of the British estate tax system in 1975 transformed estate duty to capital transfer tax and added gifts to the base of the transfer tax, the tax rates on gifts were considerably lower than the rates applicable to transfers on death. Likewise, although the U.S. Tax Reform Act of 1976 made an attempt to unify gift and estate taxes, there *The author has benefited from discussions with J.E. Foster and H. Yu. Thanks are due to the two anonymous referees for numerous helpful comments and suggestions. ‘See Bernheim (1987) for a discussion of other tax-avoiding methods. Feldstein (1976) has suggested that for most families the voluntary transfers are not bequests at death, but the bequests that their heirs enjoyed as children. Some of this consumption may be the purchase of education. 0047~2727/89/$3.50
c
1989, Elsevier Science Publishers
B.V. (North-Holland)
218
XC. Hu, Estate taxation
is still substantial room for individuals to completely avoid the gift tax.’ Traditional studies of estate taxation, such as Atkinson (1971), Sheshinski (1974) and Stiglitz (1978), have assumed that bequests are left upon death regardless of the differential tax treatments of bequests at death and intra vivos gifts. It is the purpose of this paper to examine the effects of the estate tax on the accumulation and distribution of wealth while recognizing the fact that the burden of the estate tax can be avoided by strategically timing bequests to the children. The main difference between the present paper and those of Boskin (1976) and Adams (1978) is that here the ability of the individual to plan his bequests is affected by the existence of a competitive insurance (annuity) market. The model employed in this analysis is the two-period life-cycle model with uncertain lifespan, similar to those employed in Abel (1985), Eckstein, Eichenbaum and Peled (1985) Karni and Zilcha (1986), and Chu (1987). In this model the lifespan of the individual is uncertain and bequests can be left either in the middle or at the end of his life. Bequests are subject to the estate tax only when they are left upon death. Consequently, the estate tax may be avoided by leaving bequests early in his life. However, the feasibility of such a tax-avoidance scheme depends on whether the individual can do away with precautionary saving which provides against the uncertainty in retirement consumption that arises from uncertain lifespan. This requires the existence of an annuity market of strategic reverse bequests. Since reverse bequests are not enforceable and are subject to the usual moral-hazard considerations, we shall assume the absence of such implicit contracts and concentrate on the implications of the annuity market for the effects of the estate tax on wealth accumulation and distribution. The following section describes the model. Section 3 characterizes individual behavior. Section 4 discusses the properties of wealth accumulation and distribution. Section 5 examines the effects of the estate tax. Section 6 provides numerical illustrations. The final section summarizes the results.
2. The model In order to focus on the role of the annuity market, we consider an economy in which the population is constant (and large) and all individuals are alike except for their lifespans, the mortality history of their families and the amounts of bequests that they receive from their parents.3 They have a maximum lifespan of two periods. They die at either the beginning or the end of the second period, with probabilities of 1 -rc and ?I, respectively. ‘See Kay and King (1986) and Bernheim (1987) for further discussions of the British and U.S. estate tax systems. ‘For studies of income distribution for the case where individuals are heterogeneous, see, for example, Laitner (1979), Loury (1981), and Ioannides and Sato (1987).
S.C. Hu, Estate taxation
219
Following Karni and Zilcha (1986), we label the family history of an individual born at (the beginning of period) t by O1=(tIl,l, 82,1, . ..). where O,_,= l(2) if the individual’s kth immediate ancestor died at the beginning (end) of the second period (k= 1 for parents, k= 2 for grandparents, etc.). We also introduce a shift operator so that if the individual has a family history of 8,, then his children have a family history of 8,+ 1 = TO,, where t?,,,, 1= l(2) if he dies early (late), and O,,,, I = 8, _ 1,1, k= 2,3,. . , co. By the same token, if the individual has a family history of 01, then his parents have the family history of O1_1 = T-‘6,. Finally, the individual is said to have a family history of et= gn if the first n elements of tit all have the value of 2, that is, if his first n consecutive immediate ancestors lived long. An individual with family history 8, receives an inheritance of h,(B,) from his parents at the beginning of his first period (or his parents’ second period), and h,(O,) of inheritance at the end of his first period (or his parents’ second period). He also receives an income (endowment) of z in the first period of his life. Suppose that he consumes ci(e,) when young and cz(Or) when old, and that he holds his savings in either regular notes or annuities. If he lives both periods, his budget constraints for the two periods are given by:
c,(a + b,(e,)+ 46) + 43,) = z +( 1+ r,)h,(0,)+ hl(e,),
(1)
cm +b,(4) = (1+ 4W
(2)
+ (1 + r&VA),
where s(0,) and a(0,) are, respectively, the amounts of regular notes and annuities that he holds, while bj(e,) is the amount of bequests that he plans to leave to his children at the end of the jth period of his life.4 Regular notes pay a rate of return of rs. Annuities pay a rate of return of re conditional upon the holder’s surviving the second period, but would cancel the principal if the holder dies early. We say that the annuity market is perfect if 1 +r,=(l +r,)/n, incomplete if 1 +r,<(l +r,)/rc, and nonexistent if ra=rs. Finally, for simplicity, let us assume that the marginal product of capital is constant, and so is r,. Note that the second-period budget is conditional upon the individual’s living until the end of the second period. If he dies prematurely at the beginning of the second period, his saving held in the form of regular notes is left to his children as unplanned (or, more appropriately, conditional) bequests, while his saving in the form of annuities is forfeited. We assume that bequests left at the end of the second period (b2) are subject to an estate tax, which is a fraction T of the bequests. Unplanned/conditional bequest (s) are also taxed but planned first-period bequests (b,) are not. Therefore, the 41n order to differentiate between planned and unplanned bequests, we assume that planned first-period bequests (b,) are left at the end of the first period, while unplanned bequests (s) are left upon death at the beginning of the second period.
220
S.C. Hu, Estate taxation
total after-tax amount of bequests of the heirs’ second period is:5
b=( 0,) =
measured
, if8
bT(',)=(l+r,)Cb,(',)+(l-t)s(',)l b;(R) =
By our assumption
(I+
r,h(Q
of constant
bT(8,) = h( 7Y,)
and
+
(I-
(3)
1,t+1=
4b,(Q
population,
we have?
h(8,) = bT( T Id,),
where h(0,) is the total value of bequests receives at the end of his first period:
Suppose
in terms of its value at the end
that the lifetime utility
(4)
that a person
of the individual
with family history
8,
is of the CES form:
u = {c:+ck; + a(bT+ z,)Y}ly.
(5)
The first two terms are the utilities of own consumption in the j-period of life (j= 1,2), while the last term is the utility of bequests. YE [- co, l] is a parameter denoting the substitutability among cr, c2 and bT, with the elasticity of substitution being greater (smaller) than 1 if y >( <)O. 6 and r~ are the discount factors for utilities of second-period consumption and bequests, respectively. Implicit here is that the utility of bequests is derived from the children’s indirect lifetime utility as perceived by the individual. The latter is in turn a function of the children’s lifetime income as perceived by him, denoted z,, plus bequests, bT. Thus, the utility of bequests is taken to be an increasing and concave function of bT+z,. The individual is said to have no bequest motive if a=O. Since the probability of death at the beginning of the second period is (1 -rc), the individual’s lifetime utility maximization problem can be expressed as: (6) subject to (1) and cr, bl, s, ~120, where zz = (1 + rs)s + (1 +
r,)a
is the amount
of wealth available
for disposal
in
51f gifts are taxed at the rate of su, then b,(O,) must be multiplied by (1-zg). The remaining analysis remains unchanged provided that r9 < T. 61f the population grows at the rate q and bequests are equally distributed among children, then b’(U,) = (1+ q)h(TO,).
S.C. Hu, Estate
taxation
221
the second period. The first term in the maximand is the utility of first-period consumption, the second term is the utility of bequests if he dies early, and IV(.) is the (indirect) utility that he enjoys in the second period from own consumption and bequests if he lives long, i.e. W(z,, bl, z,)=max
{&y2
condition
for
the
above
consumption
and
maximization
determine
second-period
as functions
CZZ - cp((1+ r,h + zJl/( 1+ (I- dcp)
z,) =
0 if
Wz,, b,, 4 =
bl { 5 %(z2,z,), [(l --z)z,+(l
+r,)b,
[szE+a((l
+zJ(l +r,)b,
+(l --z)cp)‘-kly +zJy]/y
’
if b,{ 5 Ib(z2, zJ, where cpE [~/(cJ( 1 -z))]‘“’ -H and 6,(z,, z,) = (zJ(p - z,)/( 1 + r,). The above equations show that b,(z,, b,, z,) is decreasing in b,, and in the regime where second-period bequests are positive, the estate tax amounts to a tax on second-period wealth. The first-order conditions for maximizing (6) are:
(10) ~cb(z,,b,,z,)+(l+r,)(l--)b(br+z,)Y-lI~Y1-l, ~(l+~~)Wz(z2,bl,z,)~c’:-‘, where Wb=aW/dbl,
WZ~dW/dz,,
with=ifa>O, and
with=ifb,>O,
(11) (12)
222
S.C. Hu, Estate taxation
W,/W,l(l
+r,)/(l-z),
with=if
b,>O
[by (9)].
When an annuity market does not exist, the model solution setting a = 0 in (I), (10) and (1 l), while discarding (12).
(13) is obtained
by
3. Estate taxation and individual behavior From the above first-order conditions, we can see that if the tax rate is 0, then (i) according to (13), condition (10) holds with equality if and only if (11) does, and (ii) an increase in bI accompanied by a decrease in s by b, and a decrease in b, by (1 +rs)bl leaves unaltered the first-order conditions in (8), (9), and (10)412). This implies that there is perfect substitutability between first-period and second-period bequests (of equal present value) for 6, E [0,6,(z,, zJ]. In other words, the individual is indifferent to the timing of bequests. Moreover, in this region the total value of bequests, measured by (1 + r,)bl + b,, is independent of b,. Since the individual is indifferent to the timing of planned bequests, if a tax is imposed on planned bequests in the second period, the optimal decision is for him IO move planned bequests to the first period of his life. Thus, b2 =0 for r >O. As a result, the tax falls completely upon unplanned bequests, s. Of course, when the tax is initially established, the tax also falls upon second-period bequests, because the elders do not have the leeway to accelerate bequests so as to avoid the estate tax. Solving (1) and (lo)-( 12) we can derive the individual’s demands for tirstperiod consumption, bequests, annuities, and regular notes as functions of his own lifetime wealth (z +h(8,)), the imputed value of his children’s lifetime income (z,) and the estate tax rate. For T > 0, 6, =O, and the solution values of other variables can be explicitly written in the linear form as:’
Cl(&)= a-~(Q + z + z,/(1+ r,)l,
(14)
bl(W= a,Ch(W + z + GA1+ r,)l - z,NI+ rs),
(15)
s(R)= 04~,) + z + z,/(1+ r,n
(16)
46) = cd3(~,)+ z + z,/(1+ r,)l.
(17)
If a=O, the derivatives of the marginal propensity functions (c(~‘s) with respect to T, denoted by the primes over the variables, are given by: ‘See section A.1 in the appendix for derivation of eqs. (14)-(18). Note that because of our assumption that the utility function is of the CES type, if the annuity market is absent (~(,=a), the solution values for c,, s, and b can still be expressed as a linear function of h+z+z,. However, the marginal propensity functions, Q’S, can only be derived as implicit functions of rS. r,, and L
S.C. Hu. Esrafe tavation
ci;=rx:=a;=r;=o,
223
r,=O.
(18)
If 0 > 0, then we have:
9: 3 0, x$
z:>o,
LY;>o,
if y>O,
(19)
x: > 0,
C&=0,
2;>0,
if;,>,0
(20)
while 2:<0,
and x,=0.
The marginal propensity functions (xk’s) are affected by the estate tax rate and the rates of asset returns as well as whether there exists an annuity market and whether the individual has bequest motives. (18) indicates that, in the absence of bequest motives, since the individual leaves only unplanned bequests and the unplanned bequests that he leaves to his children are unaffected by the estate tax, the estate tax crowds out, dollar for dollar, the inheritance received by the children. This is so regardless of whether an annuity market exists. As indicated in (19) and (20), with positive bequest motives (CT), if the elasticity of substitution among first-period consumption, second-period consumption, and bequests is greater than one (;j>O), an increase in the estate tax reduces the fraction of lifetime wealth invested in regular notes while it increases the fractions of lifetime wealth allocated to first-period bequests and annuities. On the other hand, if the elasticity of substitution is low (i.e. when ; is a large negative value) and if an annuity market exists, the above signs could be reversed. In any case, regardless of the elasticity of substitution, the effect of estate taxation is to cause substitution of planned bequests for conditional bequests and to reduce the mean total after-tax amount of bequests left to the children as a fraction of wealth. In order for both regular notes and annuities to be held in positive quantities (a,, r. >O), it is necessary that’ G( 1 + r,) < 1 + ra < ( 1 + t-J/( n + (1 - n)r),
(21)
where G = I + [ 1 + (( 1 - r)“0/6)‘!(“~’ ‘)lym ‘(I -x)/n 2 1. The left-hand inequality implies that r, must be higher than r,. The extreme right-hand expression is but 1 plus the (tax-adjusted) actuarially-fair rate of return on annuities, which is larger the higher the estate tax rate. Since the estate tax falls entirely upon conditional bequests at death, the right-hand inequality suggests that annuities must be less than actuarially fair in order for the estate tax to be operative. If the annuity market were perfect, the individual would have held “See section
J.P.E
C
A.2 of the appendix
for proof.
224
S.C. Hu. Estate taxation
his entire saving in annuities and thereby avoided paying the estate tax completely. This conclusion generalizes the argument of Stiglitz (1985) that in a perfect capital market the astute taxpayer can avoid all taxation on capital income and possibly all taxation on wage income as well. In the same manner, the astute taxpayer can avoid all the estate tax if the annuity market is perfect.
4. Properties of wealth accumulation
and distribution
Suppose for convenience that z, = 0. Substituting (15) and (16) into (4), we can write the bequests left by the individual and thus the inheritance received by his children as:
46 +1)= B,lx~,)+ zl or
(22) A(&+ 1) = z + D,A(U
where A(8,) = h(B,) +z denotes the individual’s total lifetime wealth, including both his own income and the inheritance that he receives from his parents:
B1= B1=(1+TS)Cc(b+(1-7s)c1,1 , ifB l.Ifl I i P2 =(l +r&,
1 =
ii2 ’
and fil(pz) is the fraction of wealth bequeathed to his children if the individual dies at the beginning (end) of the second period, i.e. B1,r+ 1= l(2). From eq. (22) we can derive the mean and the coefficient of variance (square of the coefficient of variation, SCV) of A(G,+,) conditional upon A(R): (23)
(24) In the stochastic are:
steady
state,
the mean
and coefficient
of A
(25)
E(A)=z/(l -i% SCV(A) = Var(A)/(EA)‘=
of variance
Var(fi)/(l
-fl’-
Var(/I)),
(26)
225
S.C. Hu, Estate taxation
where fl=n/3,+(l-?r)j?~
=(l +rs)[(l
-7c)(l -r)cc,+dIJ
(27)
and
Var(B)=[~(82-~)2+(1-~)(jjl-_)21=~(1--)C(1+r,)(l--)~,12
(28)
are, respectively, the mean and variance of j3, and are stationary over time. (24) and (26) indicate that the coefficient of variance is affected by the wealth level only in the short run but not in the steady state. (23) and (24) together imply that since conditional mean and SCV are both increasing in A(B,), the short-run relative variability in wealth is greater for rich families than for poor ones. By assumption, the size of the population is large and all individuals receive a constant amount of income in the first period of their lives. Therefore, the steady-state mean and coefficient of variance for the representative family can be taken to be the mean and distribution of wealth for the economy. Another coarse measure of inequality is to look at the range of wealth distribution, f, defined by the ratio of the gap between the wealth held by the richest and the wealth held by the poorest families to the mean wealth of the economy [see Sen (1973)]. Solving the stochastic difference equation in (22) yields:
(29) where B,, , = nf=oPt-i, and /?-i=/Ij if 8i.l= 1,2. Because b2 5 fir, we have &, 5 B,,, $ fl:. Therefore, bounds of A(0,) are given by:
The
above
two
9Jn the absence becomes:
equations
of bequest
motives,
is similar
that
if H,=8,_,,
=z(l -/I,)“/(J-PI),
z(O,)=z This equation
indicate
to that derived
the
maximum
the upper
level
then B k.I=O for all kzn.
if H,=O,.
by Abel (1985).
and
lower
of wealth
Therefore,
is
eq. (29)
S.C. Hu, Estate taxation
226
attended by families of which all generations die upon retirement (0, =gO), while the minimum level of wealth belongs to families of which all generations live long (fY,=8,). From these two equations, the ratio f is given by:
f = (A,,, -
Arnin)lEA=(l-8)
(30)
5. Effects of the estate tax on wealth accumulation
and distribution
The steady-state effects of estate taxation on wealth accumulation and distribution for the economy can be determined by considering eqs. (25), (26), and (30). In order to gain an understanding of these effects, we assume that the revenue collected from the estate tax is redistributed back to individuals in the form of lump-sum transfers at young age, so that individual income in the first period of life equals z+g,, where g, is lump-sum transfers from the difference equation (22) now government. lo As a result, the stochastic becomes:
(224 Because individuals pay the estate tax only on unplanned estate tax collected from a family with history 8,+ 1 is:
T~,A(e,),
x(&+1)=
o i
The conditional
I
ife 1,1+1=
bequests,
the
(11 2 Ii
mean of x(0,+ r) is:
EC46+I) IA(R))= (I- To~wUW.
(31)
Since, by assumption, the population of the economy is large, the per capita estate tax collected by the government is but the expected value of the tax collected from families of all mortality histories. Therefore, it is equal to g, + 1 =
(I+ r,Fx(% +d =( 1+ r,)( 1- 4~@4%).
(32)
‘OIf transfers are received by individuals in the second rather than the first period of their lives, then similar conclusions hold. The only difference is that the transfers are g/n instead of g for each person surviving the second period. The lifetime income is now equal to income in the first period plus transfers discounted by 1 +r,. The reason that it is discounted by 1 +r, is that a transfer provided in the second period is conditional upon the individual’s living long and thus it acts like an annuity.
S.C.
Hu.
Estate
taxation
221
The reason that we derive g by multiplying Ex by (1 +Y,) is because the tax is collected and redistributed back to the youngsters at the beginning of the period, while the amount of transfers that enters into individual budget constraints is valued at the end of the period. Note that although the government budget is balanced each period, a family gains from (is hurt by) the estate taxation/transfers scheme if E(x(B,+ ,) 1A(0,)) is smaller (larger) than Ex(8,+ i), which in turn holds if A(Q,) is smaller (larger) than EA(8,). That is, poor families benefit from the tax scheme while rich families are hurt by it. Solve (22a) and (31) jointly to obtain the steady-state means of A and x:
EA = z/t1- (1 + r,)((1- n1cx.y + 41,
(33)
Ex = (1 - rr)s~(,z/( 1 - (1 +
(34)
r,)((
1 - rr)cl, + clJ).
As can be seen from (26) and (30) both the steady-state coefficient of variance and range of wealth distribution are independent of the mean level of wealth. Thus, they remain unchanged by the lump-sum transfers. Clearly, Ex=O if rt = 1. In the absence of uncertainty about the lifespan, the individual is able to plan his bequests and thereby avoid paying tax. Therefore, the collection of the estate tax is contingent upon the existence of uncertainty and the absence of a complete annuity market. This constitutes the main difference between the present model and those of Atkinson (1971) and Sheshinski (1974) both of which assume that the utility of bequests is based on bequests at death and there is no uncertainty regarding the lifespan nor tax-avoiding intra vivos gifts. For 7c< 1, in order for the government to collect a positive revenue, it is required that c(,>O for r >O, which, according to (21) holds if T <
5’ = 1 -(r,
-
r,)/((
1 - rc)( 1 + r,)).
(35)
7’ defines the upper bound of a nonprohibitive estate tax, beyond which the estate tax revenue collected by the government falls to zero. Note that (21) is derived under the assumption that the utility function is of the CES type. Therefore, for this class of utility functions, r” is independent of the parameter denoting bequest motives, but it is lower the larger is the gap between ra and rs, which represents the completeness of the annuity market. If an annuity market does not exist (i.e. ra=r,), then r”= 1, and the estate tax will bring the government a positive revenue unless it is 100 percent. If the annuity market is perfect [i.e. 1 + r,=( 1 +r,)/z], then to =O, and the estate tax can be completely avoided. As a result, it is neutral with respect to wealth accumulation and distribution. Because the government collects zero tax at r =O, the above inequality suggests that the total revenue from the estate tax follows a Laffer-curve
S.C. Hu, Estate taxation
228
phenomenon. A rise in the tax rate increases revenue when the tax rate is sufficiently low, but it brings about a smaller total revenue as the tax rate approaches z”. For O
-~)cr;](EA)~/z,
(SCV(A))‘=[(1-82)Var(S)‘+2Var(B)88’]/(1-82-Var(B))2.
(36) (37)
In the presence of an annuity market, if a,>O, then (36) is negative unless the elasticity of substitution is sufficiently close to zero (i.e. y is a large negative value).’ 1 If T 2s' so that cc,=O, since the estate tax is completely avoided, the tax effect is equal to zero. Likewise, when there are no bequest motives, since crb and a: are both equal to zero, the change in the estate tax has no effect on EA. In the absence of an annuity market, (36) is positive if the elasticity of substitution is less than or equal to 1 (i.e. y 50) or if CI,
(38)
If there are no bequest motives, since both CQ,and CI, are independent of T, it follows from (27) and (28) that both fi and Var(p) are decreasing in T. Thus, according to (37), an increase in the estate tax decreases the coefficient of variance, although, as noted above, it leaves the steady-state mean of wealth unchanged. The distributional effect of the estate tax can also be shown by looking at its effect on the ratio r in (30). Since fil >fi2 and /?;
“See section A.3 of the appendix ‘%ee section A.3 of the appendix
for proof. for proof.
S.C. Hu. Estate
Table Effects of the estate tax on wealth Presence EA
1 accumulation
of annuities
_ if CJ>O and yL;‘* Oifu=O
SCV( A)
_
l-
-
Note: y*
229
taxation
Absence
and distribution. of annuities
+ if u>O and ys:‘** Oifa=O
where y**> 1 if a,<~.
In conclusion, as summarized in table 1, the estate tax reduces the inequality of wealth distribution as measured by the coefficient of variance and the range of wealth distribution. The estate tax is also likely to reduce accumulation of wealth in the presence of positive bequest motives and an annuity market. However, the trade-off need not hold when there are no bequest motives or when the annuity market is absent.
6. Illustrative examples Tables 2 and 3 provide numerical illustrations of the effects of estate taxation for the cases where an annuity market exists and where it does not. The assumptions underlying the two tables are: r,=0.45, rs=0.2, 6 = 1, a=0.5, x =0.6, z = 1 and z,=O. In each table the solution values of the endogenous variables are given for y =0.5 (the elasticity of substitution higher than l), y =0 (Cobb-Douglas case), and y = -2 (the elasticity of substitution lower than l), respectively. In the presence of an annuity market, the range of tax rates at which there exists an interior solution for portfolio selection is larger the lower the elasticity of substitution. The range is between 40 and 57 percent for y =0.5, between 14 and 57 percent for the Cobb-Douglas case, and between 0 and 57 percent for 7 = -2. The individual holds only regular notes if the tax rate is below the lower bound. He holds only annuities if the tax rate is above the upper bound. In any case, if there exists an annuity market, the estate tax is completely avoided when the tax rate reaches 57 percent. The two tables also suggest that the tax revenue is larger in the absence than in the presence of annuities. Furthermore, when annuities are absent, the tax revenue rises with the tax rate; but when annuities are present, the tax revenue follows the Laffer-curve phenomenon. In the latter case, the revenue maximizing rate is higher the lower the elasticity of substitution. For example, the revenue is maximized at 35.5 percent for the Cobb-Douglas case and at 37.7 percent for y= - 2. Note, however, that in case where ~=0.5, the tax revenue is decreasing with the tax rate in the region where annuities are held in positive quantities (T > 40 percent), and it is increasing
230
S.C. Hu, Estate taxation Table 2 Steady-state
equilibrium
values in the presence
0.1000
0.2000
0.5282 0.3725 0.099 1 0.0000
0.5330 0.3632 0.1036 0.0000
0.2799 0.0388 1.4242 0.0440 0.6871 0.0212
0.2638 0.029 I 1.4260 0.0323 0.5564 0.0414
0.4761 0.3464 0.1773 0.0000
0.3000
of annuities.
0.4000
0.5000
0.5380 0.3530 0.1088 0.0000
0.5433 0.3338 0.1155 0.0073
0.5476 0.1665 0.1377 0.1480
0.2492 0.0211 I .4287 0.0230 0.4470 0.0605
0.2347 0.0138 0.0148 0.3438 0.0761
0.2053 0.0023 1.3250 0.0025 0.1295 0.0441
0.476 1 0.3247 0.1821 0.0169
0.4761 0.28 16 0.1895 0.0526
0.4761 0.2182 0.2004 0.1050
0.4761 0.1162 0.2180 0.1894
0.3625 0.0335 1.6107 0.0402 0.7336 0.0223
0.3432 0.0233 I .5984 0.027 1 0.5578 0.0415
0.3220 0.0134
0.3034 0.0059 1.5274 0.0065 0.2393 0.0533
0.2895 0.0011 1.4651 0.0012 0.1004 0.0340
0.4264 0.1241 0.2713 0.1780
0.4257 0.1176 0.2736 0.1829
0.4250 0.1061 0.2768 0.1920
0.4242 0.0856 0.2815 0.2084
0.4236 0.0474 0.289 1 0.2397
0.3192 0.0043 I .6266 0.0050 0.2284 0.0080
0.3735 0.0030 I .6256 0.0035 0.1884 0.0152
0.3678 0.0019 1.6211 0.0022 0.1458 0.0206
0.3625 0.0009 1.6103 0.0010 0.0988 0.0220
0.3584 0.0001 1.5867 0.0002 0.0447 0.0 150
(1) ;1=0.5
1.4262
(2) i’=o
1.5689 0.0152 0.3872 0.0530
(3) 2’= -2
with the tax rate in the region where annuities are not held (~~40 percent). Thus, the tax revenue is maximized at borderline rate, i.e. s=40 percent. The two tables confirm the conclusion reached in the preceding section that an increase in the estate tax rate reduces the coefftcient of variance and the range of wealth distribution. They show that the distributional impact of the tax is larger in the absence of annuities or as the elasticity of substitution increases. To illustrate, in the Cob&Douglas case, a rise in z from 20 to 30 percent reduces the SCV by one-fourth (from 3.05 to 2.24 percent) in the presence of annuities, but by nearly one-half (from 2.71 to 1.52 percent) in
S.C. Hu, Estate taxation
231
Table 3 Steady-state
equilibrium
values in the absence
0.1000
0.2000
UC % % %
0.5282 0.3725 0.099 1 0.0000
B
of annuities.
0.3000
0.4000
0.5000
0.5330 0.3632 0.1036 0.0000
0.5380 0.3530 0.1088 0.0000
0.5433 0.3417 0.1149 0.0000
0.5486 0.3292 0.1220 0.0000
EA SCV( A) IE-x
0.2799 0.0388 1.4242 0.0440 0.6871 0.0212
0.2638 0.029 1 1.4260 0.0323 0.5564 0.0414
0.2492 0.02 11 1.4287 0.0230 0.4470 0.0605
0.2363 0.0145 1.4325 0.0156 0.3538 0.0783
0.2254 0.0093 1.4377 0.0100 0.2732 0.0946
2‘ % % %
0.4761 0.3464 0.1773 0.0000
0.4761 0.343 1 0.1806 0.0000
0.4761 0.3393 0.1844 0.0000
0.476 1 0.3347 0.1890 0.0000
0.4761 0.3293 0.1944 0.0000
B Var(P) EA SCV( A) IE.X
0.3625 0.0335 1.6107 0.0402 0.7336 0.0223
0.3485 0.0260 1.6168 0.0305 0.6038 0.0443
0.3354 0.0194 1.6240 0.0224 0.4929 0.066 1
0.3232 0.0139 1.6327 0.0158 0.3964 0.0874
0.3123 0.0093 1.643 1 0.0104 0.3114 0.1082
% % % 2”
0.4222 0.3217 0.2559 0.0000
0.4216 0.3216 0.2566 0.0000
0.4208 0.3214 0.2576 0.0000
0.4 199 0.3210 0.2589 0.0000
0.4188 0.3203 0.2607 0.0000
B Var(b) EA SCV( A) r E.X
0.4461 0.0289 1.8573 0.0375 0.8043 0.0239
0.43 15 0.0228 1.8602 0.0289 0.6620 0.0478
0.4172 0.0174 I .8639 0.0216 0.5413 0.07 18
0.4032 0.0128 1.8687 0.0155 0.4369 0.0959
0.3898 0.0088 1.8751 0.0105 0.3449 0.1201
(1) y=o.5
VW)
(2) )‘=o
(3) y= -2
the absence of annuities. In the presence of annuities, the reduction in the SCV owing to the tax increase is only one-third (from 3.23 to 2.30 percent) when 7 = 0.5 compared with one-half in the Cobb-Douglas case. The tables also show that within the range of values of the elasticity of substitution under consideration, the estate tax lowers the steady-state mean of wealth in the presence of annuities, but it leads to an increase in the steady-state mean of wealth when an annuity market does not exist. Although the changes in EA in either direction are relatively small, they
J.P.E.
D
S.C. Hu. Estate taxation
232
nevertheless suggest that the trade-off lation is not inevitable.
between
equality
and wealth
accumu-
7. Concluding remarks This paper has shown that, except when it is initially imposed, the estate tax falls entirely upon unplanned (conditional) bequests. To the extent that unplanned bequests are responsible for unequal distribution of wealth, the effect of the estate tax is to reduce wealth inequality. If there exists an annuity market, the reduction in wealth inequality tends to be accompanied by a reduction in the steady-state mean level of wealth, especially when the elasticity of substitution is higher than 1. If an annuity market does not exist, such a trade-off between equality and wealth accumulation need not hold all the time, especially when the elasticity of substitution is lower than 1. In this model, the estate tax is operative only to the extent that uncertainty in lifespan and imperfection in the annuity market prevent individuals from planning bequests. The tax is more effective in achieving wealth equality the more imperfect is the annuity market, as measured by the differential between the rates of return on annuities and regular notes. We have assumed that the only uncertainty is about the lifespan and, consequently, the only requirement for the individual to be able to plan bequests is the existence of a perfect annuity market. Of course, a perfect annuity market alone is not sufficient to render the estate tax inoperative if there are other uncertainties which prevent the individual from moving planned bequests to the first period of his life. One such uncertainty is about the status of health. In the present model, the status of health could be parameterized by making the discount factor 6 a random variable. The greater the randomness in 6, the smaller the amount of bequests that can be moved forward to the first period. Another assumption that is crucial to our results is that the family does not act strategically and there are no reverse bequests. Kotlikoff and Spivak (1981) have shown that by behaving strategically and by instituting reverse bequests, a family can overcome the incompleteness of the annuity market. By doing so, the family can also avoid paying the estate tax. Of course, the usual moral hazard problem arises when parents rely on reverse bequests for retirement consumption.
Appendix A.1.
Derivation
of slj
Suppose first that there exists an annuity and c2 = (1 + Y,)S+ ( 1 + r&r, we have:
market.
For r >O, since b2 =0
S.C. Hu, Estate taxation w_=ck-
l,
W,=o(l
+rJ[(l
The first-order
conditions
(1 +r,){(l
+r,)b,
233
+z,-J-‘.
in (10)<12)
-~)~(bT+z,)Y-l
can be rewritten
as:
+m(b~+z,)Y-l}=c;-l,
(114 (124
Combining
(lOa), (1 la), (12a) and (l), we obtain: (14a)
u,,= a&a,, a, = (az- MW(
(154 1-T),
a,=((1 -T)al -a,+a,)R,cr,/(l
(164 -T),
(174
where Ri= l/(1 +ri) a, =(R,/(TcS))‘“~- “, (i = s, a), az = C(Rs - RJl ((1 -T)(l -7C)O)]1"ym1), and a3 = [(R,-sR,)/(( 1 --,)x,)]~‘(~‘). (18) and (19) can be obtained upon differentiating c(~with respect to z. Suppose next that an annuity market does not exist. Then cc,=0 while c(,, c(, and CX,,are jointly determined by (l), (10a) and (1 la). Differentiating these three equations with respect to T, we obtain (20).
A.2.
Conditions for positive holdings of regular notes and annuities
As can be seen from (16a), in order for c1,>0, it is required that R,> R, and a2 >a3. Expanding the latter inequality, we obtain the right-hand inequality of (21). According to (17a), ax,>0 if (1 --)a,+u,>a,. Upon substituting the values of a, and az into the above inequality and noting that a3 2 [R,/(( 1 -z))~o)]‘“‘‘), we can show that the above inequality holds if: {l+[l+((l-z)Ya/G) or
*‘(y-l)]y-l(l
-TC)/TC}R,~R,
234
S.C. Hu, Estate taxation
G( 1 + r,) 2 1 + ra.
@lb)
Thus, the left-hand inequality inequality also implies r. > rs.
in (21) is proved.
Since G> 1, the above
A.3. The effect of the estate tax on wealth accumulation Suppose first there exists an annuity market. Expanding (36) gives: (EA)‘= -{yx,[i +
-7c-((c(b+(l
-rc)a,)(R,-RJR,]
dR, - ~R,)I(R,- %)}(W2/(z( 1 - z)( 1- y)) y*,
(36a)
where y* ~0 is the value of y at which the sum of the terms inside the braces is equal to zero. Since R,> xR, and R, > tRs, the second term inside the braces is positive. According to the right-hand inequality in (21), if gs>O, then (R,-R,)/R,s(l -71)(1-z). Therefore, 1 -7c--((clb+(l -n)g)(R,-R,)/ R,z 0. This in turn implies that y* ~0. Suppose next that an annuity market does not exist. We can show upon substituting (15~) and (16~) in (36) that
x (l-lc)f~/P-~/D>O,
ify
(36b)
y** is the value of y at which the sum of the terms inside the braces is equal to zero. Since the coefficients of y are negative, y** must be positive. Moreover, since tl,+rcc~~>c~,, j3r>(1+r,)clb and R$x-~>~ocx~-~ [by (Ila)], < 7~. we can show that y** > 1 if a, < rr. Therefore, y < y** if y < 0 or if CC, We can also show that the expression inside the braces in (27b) is positive and thus p < 0, making use of the following inequalities: 6c(:-‘>a~l;-’
[by (10a) and (lla)];
and Ug-j,py,-j,
forj=1,2
(since a,
References Abel, A., 1985, Precautionary 777-791.
bequests
and accidental
bequests,
American
Economic
Review 75,
S.C. Hu, Estate
taxation
235
Adams, J.D., 1978, Equalization of true gift and estate tax rates, Journal of Public Economics 9, 59--7 1. Atkinson, A.B., 1971, Capital taxes, redistribution of wealth and individual saving, Review of Economic Studies 38. 2099228. Bernheim, B.D.. 1987, Does the estate tax raise revenue?, Tax Policy and the Economy 1, 5547. Boskin, M., 1976. Estate taxation and charitable bequests, Journal of Public Economics 5, 27-56. Chu, C.. 1987, The effect of social security on the steady state distribution of consumption, Journal of Public Economics 34, 189-210. Eckstein, Z., M. Eichenbaum and D. Peled, 1985, Uncertain lifetimes and welfare enhancing properties of annuity markets and social security, Journal of Public Economics 26, 303-326. Feldstein, MS., 1976, Perceived wealth in bonds and social security: A comment, Journal of Political Economy 84, 331-342. loannides, Y. and R. Sato. 1987, On the distribution of wealth and intergenerational transfers, Journal of Labor Economics 5, 366-385. Karni. E. and 1. Zilcha, 1986, Welfare and comparative statics implications of fair social security: A steady-state analysis, Journal of Public Economics 30, 341-357. Kay, J.A. and M.A. King, 1986, The British tax system (Oxford University Press). Kotlikoff, L. and A. Spivak, 1981, The family as an incomplete annuity market, Journal of Political Economy 84, 372-391. Laitner, J., 1979, Household bequests, perfect expectations, and the national distribution of wealth, Econometrica 47, 1175-l 193. Loury, G., 198 I. Intergenerational transfers and the distribution of earnings, Econometrica 47, 8433867. Sen. A.K., 1973, On economic inequality (Clarendon Press, Oxford). Sheshinski, E., 1974, Short and long run effects of interest-income and estate tax, Metroeconomica 26. 109-129. Stiglitz, J.E., 1978. Notes on estate taxation, redistribution, and the concept of balanced growth, Journal of Public Economics 86, Sl37-S150. Stigiitz, J.E., 1985. The general theory of tax avoidance, National Tax Journal 38, 325-337.
of Public
Economics
UNCERTAIN
40 (1989) 217-235.
North-Holland
LIFETIME, THE ANNUITY AND ESTATE TAXATION Sheng Cheng
MARKET,
HU*
Krannert Graduate School oJ Management, Purdue University, West LaJayette, IN 47907, USA Received June 1988, revised version
received July 1989
This paper extends traditional studies of estate taxation to allow for the possibility that taxpayers may avoid the burden of the estate tax by strategically timing bequests to their children. In this model, the lifespan of the individual is uncertain and bequests can be left either in the middle or at the end of the lifespan. Bequests are subject to the estate tax only when they are left upon death. It is shown that the effects of the estate tax depend on the ability of the individual to make intra vivos gifts, which in turn is affected by the existence of a competitive annuity market.
1. Introduction The estate tax has been one of the measures taken by many industrial countries, Britain and the United States included, to reduce wealth concentration caused by intergenerational transfers. However, there are ways in which individuals can minimize or entirely eliminate estate-tax liabilities, the best known of which has been to make intra vivos gifts.’ Kay and King (1986) reported that avoidance of estate duty was so easy in Britain that it was sometimes described as a ‘voluntary tax’. Bernheim (1987) estimated that the U.S. federal estate tax represented less than 1 percent of all federal revenues each year in the first half of the 1980s despite the imposition of high statutory marginal tax rates. Although the overhaul of the British estate tax system in 1975 transformed estate duty to capital transfer tax and added gifts to the base of the transfer tax, the tax rates on gifts were considerably lower than the rates applicable to transfers on death. Likewise, although the U.S. Tax Reform Act of 1976 made an attempt to unify gift and estate taxes, there *The author has benefited from discussions with J.E. Foster and H. Yu. Thanks are due to the two anonymous referees for numerous helpful comments and suggestions. ‘See Bernheim (1987) for a discussion of other tax-avoiding methods. Feldstein (1976) has suggested that for most families the voluntary transfers are not bequests at death, but the bequests that their heirs enjoyed as children. Some of this consumption may be the purchase of education. 0047~2727/89/$3.50
c
1989, Elsevier Science Publishers
B.V. (North-Holland)
218
XC. Hu, Estate taxation
is still substantial room for individuals to completely avoid the gift tax.’ Traditional studies of estate taxation, such as Atkinson (1971), Sheshinski (1974) and Stiglitz (1978), have assumed that bequests are left upon death regardless of the differential tax treatments of bequests at death and intra vivos gifts. It is the purpose of this paper to examine the effects of the estate tax on the accumulation and distribution of wealth while recognizing the fact that the burden of the estate tax can be avoided by strategically timing bequests to the children. The main difference between the present paper and those of Boskin (1976) and Adams (1978) is that here the ability of the individual to plan his bequests is affected by the existence of a competitive insurance (annuity) market. The model employed in this analysis is the two-period life-cycle model with uncertain lifespan, similar to those employed in Abel (1985), Eckstein, Eichenbaum and Peled (1985) Karni and Zilcha (1986), and Chu (1987). In this model the lifespan of the individual is uncertain and bequests can be left either in the middle or at the end of his life. Bequests are subject to the estate tax only when they are left upon death. Consequently, the estate tax may be avoided by leaving bequests early in his life. However, the feasibility of such a tax-avoidance scheme depends on whether the individual can do away with precautionary saving which provides against the uncertainty in retirement consumption that arises from uncertain lifespan. This requires the existence of an annuity market of strategic reverse bequests. Since reverse bequests are not enforceable and are subject to the usual moral-hazard considerations, we shall assume the absence of such implicit contracts and concentrate on the implications of the annuity market for the effects of the estate tax on wealth accumulation and distribution. The following section describes the model. Section 3 characterizes individual behavior. Section 4 discusses the properties of wealth accumulation and distribution. Section 5 examines the effects of the estate tax. Section 6 provides numerical illustrations. The final section summarizes the results.
2. The model In order to focus on the role of the annuity market, we consider an economy in which the population is constant (and large) and all individuals are alike except for their lifespans, the mortality history of their families and the amounts of bequests that they receive from their parents.3 They have a maximum lifespan of two periods. They die at either the beginning or the end of the second period, with probabilities of 1 -rc and ?I, respectively. ‘See Kay and King (1986) and Bernheim (1987) for further discussions of the British and U.S. estate tax systems. ‘For studies of income distribution for the case where individuals are heterogeneous, see, for example, Laitner (1979), Loury (1981), and Ioannides and Sato (1987).
S.C. Hu, Estate taxation
219
Following Karni and Zilcha (1986), we label the family history of an individual born at (the beginning of period) t by O1=(tIl,l, 82,1, . ..). where O,_,= l(2) if the individual’s kth immediate ancestor died at the beginning (end) of the second period (k= 1 for parents, k= 2 for grandparents, etc.). We also introduce a shift operator so that if the individual has a family history of 8,, then his children have a family history of 8,+ 1 = TO,, where t?,,,, 1= l(2) if he dies early (late), and O,,,, I = 8, _ 1,1, k= 2,3,. . , co. By the same token, if the individual has a family history of 01, then his parents have the family history of O1_1 = T-‘6,. Finally, the individual is said to have a family history of et= gn if the first n elements of tit all have the value of 2, that is, if his first n consecutive immediate ancestors lived long. An individual with family history 8, receives an inheritance of h,(B,) from his parents at the beginning of his first period (or his parents’ second period), and h,(O,) of inheritance at the end of his first period (or his parents’ second period). He also receives an income (endowment) of z in the first period of his life. Suppose that he consumes ci(e,) when young and cz(Or) when old, and that he holds his savings in either regular notes or annuities. If he lives both periods, his budget constraints for the two periods are given by:
c,(a + b,(e,)+ 46) + 43,) = z +( 1+ r,)h,(0,)+ hl(e,),
(1)
cm +b,(4) = (1+ 4W
(2)
+ (1 + r&VA),
where s(0,) and a(0,) are, respectively, the amounts of regular notes and annuities that he holds, while bj(e,) is the amount of bequests that he plans to leave to his children at the end of the jth period of his life.4 Regular notes pay a rate of return of rs. Annuities pay a rate of return of re conditional upon the holder’s surviving the second period, but would cancel the principal if the holder dies early. We say that the annuity market is perfect if 1 +r,=(l +r,)/n, incomplete if 1 +r,<(l +r,)/rc, and nonexistent if ra=rs. Finally, for simplicity, let us assume that the marginal product of capital is constant, and so is r,. Note that the second-period budget is conditional upon the individual’s living until the end of the second period. If he dies prematurely at the beginning of the second period, his saving held in the form of regular notes is left to his children as unplanned (or, more appropriately, conditional) bequests, while his saving in the form of annuities is forfeited. We assume that bequests left at the end of the second period (b2) are subject to an estate tax, which is a fraction T of the bequests. Unplanned/conditional bequest (s) are also taxed but planned first-period bequests (b,) are not. Therefore, the 41n order to differentiate between planned and unplanned bequests, we assume that planned first-period bequests (b,) are left at the end of the first period, while unplanned bequests (s) are left upon death at the beginning of the second period.
220
S.C. Hu, Estate taxation
total after-tax amount of bequests of the heirs’ second period is:5
b=( 0,) =
measured
, if8
bT(',)=(l+r,)Cb,(',)+(l-t)s(',)l b;(R) =
By our assumption
(I+
r,h(Q
of constant
bT(8,) = h( 7Y,)
and
+
(I-
(3)
1,t+1=
4b,(Q
population,
we have?
h(8,) = bT( T Id,),
where h(0,) is the total value of bequests receives at the end of his first period:
Suppose
in terms of its value at the end
that the lifetime utility
(4)
that a person
of the individual
with family history
8,
is of the CES form:
u = {c:+ck; + a(bT+ z,)Y}ly.
(5)
The first two terms are the utilities of own consumption in the j-period of life (j= 1,2), while the last term is the utility of bequests. YE [- co, l] is a parameter denoting the substitutability among cr, c2 and bT, with the elasticity of substitution being greater (smaller) than 1 if y >( <)O. 6 and r~ are the discount factors for utilities of second-period consumption and bequests, respectively. Implicit here is that the utility of bequests is derived from the children’s indirect lifetime utility as perceived by the individual. The latter is in turn a function of the children’s lifetime income as perceived by him, denoted z,, plus bequests, bT. Thus, the utility of bequests is taken to be an increasing and concave function of bT+z,. The individual is said to have no bequest motive if a=O. Since the probability of death at the beginning of the second period is (1 -rc), the individual’s lifetime utility maximization problem can be expressed as: (6) subject to (1) and cr, bl, s, ~120, where zz = (1 + rs)s + (1 +
r,)a
is the amount
of wealth available
for disposal
in
51f gifts are taxed at the rate of su, then b,(O,) must be multiplied by (1-zg). The remaining analysis remains unchanged provided that r9 < T. 61f the population grows at the rate q and bequests are equally distributed among children, then b’(U,) = (1+ q)h(TO,).
S.C. Hu, Estate
taxation
221
the second period. The first term in the maximand is the utility of first-period consumption, the second term is the utility of bequests if he dies early, and IV(.) is the (indirect) utility that he enjoys in the second period from own consumption and bequests if he lives long, i.e. W(z,, bl, z,)=max
{&y2
condition
for
the
above
consumption
and
maximization
determine
second-period
as functions
CZZ - cp((1+ r,h + zJl/( 1+ (I- dcp)
z,) =
0 if
Wz,, b,, 4 =
bl { 5 %(z2,z,), [(l --z)z,+(l
+r,)b,
[szE+a((l
+zJ(l +r,)b,
+(l --z)cp)‘-kly +zJy]/y
’
if b,{ 5 Ib(z2, zJ, where cpE [~/(cJ( 1 -z))]‘“’ -H and 6,(z,, z,) = (zJ(p - z,)/( 1 + r,). The above equations show that b,(z,, b,, z,) is decreasing in b,, and in the regime where second-period bequests are positive, the estate tax amounts to a tax on second-period wealth. The first-order conditions for maximizing (6) are:
(10) ~cb(z,,b,,z,)+(l+r,)(l--)b(br+z,)Y-lI~Y1-l, ~(l+~~)Wz(z2,bl,z,)~c’:-‘, where Wb=aW/dbl,
WZ~dW/dz,,
with=ifa>O, and
with=ifb,>O,
(11) (12)
222
S.C. Hu, Estate taxation
W,/W,l(l
+r,)/(l-z),
with=if
b,>O
[by (9)].
When an annuity market does not exist, the model solution setting a = 0 in (I), (10) and (1 l), while discarding (12).
(13) is obtained
by
3. Estate taxation and individual behavior From the above first-order conditions, we can see that if the tax rate is 0, then (i) according to (13), condition (10) holds with equality if and only if (11) does, and (ii) an increase in bI accompanied by a decrease in s by b, and a decrease in b, by (1 +rs)bl leaves unaltered the first-order conditions in (8), (9), and (10)412). This implies that there is perfect substitutability between first-period and second-period bequests (of equal present value) for 6, E [0,6,(z,, zJ]. In other words, the individual is indifferent to the timing of bequests. Moreover, in this region the total value of bequests, measured by (1 + r,)bl + b,, is independent of b,. Since the individual is indifferent to the timing of planned bequests, if a tax is imposed on planned bequests in the second period, the optimal decision is for him IO move planned bequests to the first period of his life. Thus, b2 =0 for r >O. As a result, the tax falls completely upon unplanned bequests, s. Of course, when the tax is initially established, the tax also falls upon second-period bequests, because the elders do not have the leeway to accelerate bequests so as to avoid the estate tax. Solving (1) and (lo)-( 12) we can derive the individual’s demands for tirstperiod consumption, bequests, annuities, and regular notes as functions of his own lifetime wealth (z +h(8,)), the imputed value of his children’s lifetime income (z,) and the estate tax rate. For T > 0, 6, =O, and the solution values of other variables can be explicitly written in the linear form as:’
Cl(&)= a-~(Q + z + z,/(1+ r,)l,
(14)
bl(W= a,Ch(W + z + GA1+ r,)l - z,NI+ rs),
(15)
s(R)= 04~,) + z + z,/(1+ r,n
(16)
46) = cd3(~,)+ z + z,/(1+ r,)l.
(17)
If a=O, the derivatives of the marginal propensity functions (c(~‘s) with respect to T, denoted by the primes over the variables, are given by: ‘See section A.1 in the appendix for derivation of eqs. (14)-(18). Note that because of our assumption that the utility function is of the CES type, if the annuity market is absent (~(,=a), the solution values for c,, s, and b can still be expressed as a linear function of h+z+z,. However, the marginal propensity functions, Q’S, can only be derived as implicit functions of rS. r,, and L
S.C. Hu. Esrafe tavation
ci;=rx:=a;=r;=o,
223
r,=O.
(18)
If 0 > 0, then we have:
9: 3 0, x$
z:>o,
LY;>o,
if y>
(19)
x: > 0,
C&=0,
2;>0,
if;,>,0
(20)
while 2:<0,
and x,=0.
The marginal propensity functions (xk’s) are affected by the estate tax rate and the rates of asset returns as well as whether there exists an annuity market and whether the individual has bequest motives. (18) indicates that, in the absence of bequest motives, since the individual leaves only unplanned bequests and the unplanned bequests that he leaves to his children are unaffected by the estate tax, the estate tax crowds out, dollar for dollar, the inheritance received by the children. This is so regardless of whether an annuity market exists. As indicated in (19) and (20), with positive bequest motives (CT), if the elasticity of substitution among first-period consumption, second-period consumption, and bequests is greater than one (;j>O), an increase in the estate tax reduces the fraction of lifetime wealth invested in regular notes while it increases the fractions of lifetime wealth allocated to first-period bequests and annuities. On the other hand, if the elasticity of substitution is low (i.e. when ; is a large negative value) and if an annuity market exists, the above signs could be reversed. In any case, regardless of the elasticity of substitution, the effect of estate taxation is to cause substitution of planned bequests for conditional bequests and to reduce the mean total after-tax amount of bequests left to the children as a fraction of wealth. In order for both regular notes and annuities to be held in positive quantities (a,, r. >O), it is necessary that’ G( 1 + r,) < 1 + ra < ( 1 + t-J/( n + (1 - n)r),
(21)
where G = I + [ 1 + (( 1 - r)“0/6)‘!(“~’ ‘)lym ‘(I -x)/n 2 1. The left-hand inequality implies that r, must be higher than r,. The extreme right-hand expression is but 1 plus the (tax-adjusted) actuarially-fair rate of return on annuities, which is larger the higher the estate tax rate. Since the estate tax falls entirely upon conditional bequests at death, the right-hand inequality suggests that annuities must be less than actuarially fair in order for the estate tax to be operative. If the annuity market were perfect, the individual would have held “See section
J.P.E
C
A.2 of the appendix
for proof.
224
S.C. Hu. Estate taxation
his entire saving in annuities and thereby avoided paying the estate tax completely. This conclusion generalizes the argument of Stiglitz (1985) that in a perfect capital market the astute taxpayer can avoid all taxation on capital income and possibly all taxation on wage income as well. In the same manner, the astute taxpayer can avoid all the estate tax if the annuity market is perfect.
4. Properties of wealth accumulation
and distribution
Suppose for convenience that z, = 0. Substituting (15) and (16) into (4), we can write the bequests left by the individual and thus the inheritance received by his children as:
46 +1)= B,lx~,)+ zl or
(22) A(&+ 1) = z + D,A(U
where A(8,) = h(B,) +z denotes the individual’s total lifetime wealth, including both his own income and the inheritance that he receives from his parents:
B1= B1=(1+TS)Cc(b+(1-7s)c1,1 , ifB l.Ifl I i P2 =(l +r&,
1 =
ii2 ’
and fil(pz) is the fraction of wealth bequeathed to his children if the individual dies at the beginning (end) of the second period, i.e. B1,r+ 1= l(2). From eq. (22) we can derive the mean and the coefficient of variance (square of the coefficient of variation, SCV) of A(G,+,) conditional upon A(R): (23)
(24) In the stochastic are:
steady
state,
the mean
and coefficient
of A
(25)
E(A)=z/(l -i% SCV(A) = Var(A)/(EA)‘=
of variance
Var(fi)/(l
-fl’-
Var(/I)),
(26)
225
S.C. Hu, Estate taxation
where fl=n/3,+(l-?r)j?~
=(l +rs)[(l
-7c)(l -r)cc,+dIJ
(27)
and
Var(B)=[~(82-~)2+(1-~)(jjl-_)21=~(1--)C(1+r,)(l--)~,12
(28)
are, respectively, the mean and variance of j3, and are stationary over time. (24) and (26) indicate that the coefficient of variance is affected by the wealth level only in the short run but not in the steady state. (23) and (24) together imply that since conditional mean and SCV are both increasing in A(B,), the short-run relative variability in wealth is greater for rich families than for poor ones. By assumption, the size of the population is large and all individuals receive a constant amount of income in the first period of their lives. Therefore, the steady-state mean and coefficient of variance for the representative family can be taken to be the mean and distribution of wealth for the economy. Another coarse measure of inequality is to look at the range of wealth distribution, f, defined by the ratio of the gap between the wealth held by the richest and the wealth held by the poorest families to the mean wealth of the economy [see Sen (1973)]. Solving the stochastic difference equation in (22) yields:
(29) where B,, , = nf=oPt-i, and /?-i=/Ij if 8i.l= 1,2. Because b2 5 fir, we have &, 5 B,,, $ fl:. Therefore, bounds of A(0,) are given by:
The
above
two
9Jn the absence becomes:
equations
of bequest
motives,
is similar
that
if H,=8,_,,
=z(l -/I,)“/(J-PI),
z(O,)=z This equation
indicate
to that derived
the
maximum
the upper
level
then B k.I=O for all kzn.
if H,=O,.
by Abel (1985).
and
lower
of wealth
Therefore,
is
eq. (29)
S.C. Hu, Estate taxation
226
attended by families of which all generations die upon retirement (0, =gO), while the minimum level of wealth belongs to families of which all generations live long (fY,=8,). From these two equations, the ratio f is given by:
f = (A,,, -
Arnin)lEA=(l-8)
(30)
5. Effects of the estate tax on wealth accumulation
and distribution
The steady-state effects of estate taxation on wealth accumulation and distribution for the economy can be determined by considering eqs. (25), (26), and (30). In order to gain an understanding of these effects, we assume that the revenue collected from the estate tax is redistributed back to individuals in the form of lump-sum transfers at young age, so that individual income in the first period of life equals z+g,, where g, is lump-sum transfers from the difference equation (22) now government. lo As a result, the stochastic becomes:
(224 Because individuals pay the estate tax only on unplanned estate tax collected from a family with history 8,+ 1 is:
T~,A(e,),
x(&+1)=
o i
The conditional
I
ife 1,1+1=
bequests,
the
(11 2 Ii
mean of x(0,+ r) is:
EC46+I) IA(R))= (I- To~wUW.
(31)
Since, by assumption, the population of the economy is large, the per capita estate tax collected by the government is but the expected value of the tax collected from families of all mortality histories. Therefore, it is equal to g, + 1 =
(I+ r,Fx(% +d =( 1+ r,)( 1- 4~@4%).
(32)
‘OIf transfers are received by individuals in the second rather than the first period of their lives, then similar conclusions hold. The only difference is that the transfers are g/n instead of g for each person surviving the second period. The lifetime income is now equal to income in the first period plus transfers discounted by 1 +r,. The reason that it is discounted by 1 +r, is that a transfer provided in the second period is conditional upon the individual’s living long and thus it acts like an annuity.
S.C.
Hu.
Estate
taxation
221
The reason that we derive g by multiplying Ex by (1 +Y,) is because the tax is collected and redistributed back to the youngsters at the beginning of the period, while the amount of transfers that enters into individual budget constraints is valued at the end of the period. Note that although the government budget is balanced each period, a family gains from (is hurt by) the estate taxation/transfers scheme if E(x(B,+ ,) 1A(0,)) is smaller (larger) than Ex(8,+ i), which in turn holds if A(Q,) is smaller (larger) than EA(8,). That is, poor families benefit from the tax scheme while rich families are hurt by it. Solve (22a) and (31) jointly to obtain the steady-state means of A and x:
EA = z/t1- (1 + r,)((1- n1cx.y + 41,
(33)
Ex = (1 - rr)s~(,z/( 1 - (1 +
(34)
r,)((
1 - rr)cl, + clJ).
As can be seen from (26) and (30) both the steady-state coefficient of variance and range of wealth distribution are independent of the mean level of wealth. Thus, they remain unchanged by the lump-sum transfers. Clearly, Ex=O if rt = 1. In the absence of uncertainty about the lifespan, the individual is able to plan his bequests and thereby avoid paying tax. Therefore, the collection of the estate tax is contingent upon the existence of uncertainty and the absence of a complete annuity market. This constitutes the main difference between the present model and those of Atkinson (1971) and Sheshinski (1974) both of which assume that the utility of bequests is based on bequests at death and there is no uncertainty regarding the lifespan nor tax-avoiding intra vivos gifts. For 7c< 1, in order for the government to collect a positive revenue, it is required that c(,>O for r >O, which, according to (21) holds if T <
5’ = 1 -(r,
-
r,)/((
1 - rc)( 1 + r,)).
(35)
7’ defines the upper bound of a nonprohibitive estate tax, beyond which the estate tax revenue collected by the government falls to zero. Note that (21) is derived under the assumption that the utility function is of the CES type. Therefore, for this class of utility functions, r” is independent of the parameter denoting bequest motives, but it is lower the larger is the gap between ra and rs, which represents the completeness of the annuity market. If an annuity market does not exist (i.e. ra=r,), then r”= 1, and the estate tax will bring the government a positive revenue unless it is 100 percent. If the annuity market is perfect [i.e. 1 + r,=( 1 +r,)/z], then to =O, and the estate tax can be completely avoided. As a result, it is neutral with respect to wealth accumulation and distribution. Because the government collects zero tax at r =O, the above inequality suggests that the total revenue from the estate tax follows a Laffer-curve
S.C. Hu, Estate taxation
228
phenomenon. A rise in the tax rate increases revenue when the tax rate is sufficiently low, but it brings about a smaller total revenue as the tax rate approaches z”. For O
-~)cr;](EA)~/z,
(SCV(A))‘=[(1-82)Var(S)‘+2Var(B)88’]/(1-82-Var(B))2.
(36) (37)
In the presence of an annuity market, if a,>O, then (36) is negative unless the elasticity of substitution is sufficiently close to zero (i.e. y is a large negative value).’ 1 If T 2s' so that cc,=O, since the estate tax is completely avoided, the tax effect is equal to zero. Likewise, when there are no bequest motives, since crb and a: are both equal to zero, the change in the estate tax has no effect on EA. In the absence of an annuity market, (36) is positive if the elasticity of substitution is less than or equal to 1 (i.e. y 50) or if CI,
(38)
If there are no bequest motives, since both CQ,and CI, are independent of T, it follows from (27) and (28) that both fi and Var(p) are decreasing in T. Thus, according to (37), an increase in the estate tax decreases the coefficient of variance, although, as noted above, it leaves the steady-state mean of wealth unchanged. The distributional effect of the estate tax can also be shown by looking at its effect on the ratio r in (30). Since fil >fi2 and /?;
“See section A.3 of the appendix ‘%ee section A.3 of the appendix
for proof. for proof.
S.C. Hu. Estate
Table Effects of the estate tax on wealth Presence EA
1 accumulation
of annuities
_ if CJ>O and yL;‘* Oifu=O
SCV( A)
_
l-
-
Note: y*
229
taxation
Absence
and distribution. of annuities
+ if u>O and ys:‘** Oifa=O
where y**> 1 if a,<~.
In conclusion, as summarized in table 1, the estate tax reduces the inequality of wealth distribution as measured by the coefficient of variance and the range of wealth distribution. The estate tax is also likely to reduce accumulation of wealth in the presence of positive bequest motives and an annuity market. However, the trade-off need not hold when there are no bequest motives or when the annuity market is absent.
6. Illustrative examples Tables 2 and 3 provide numerical illustrations of the effects of estate taxation for the cases where an annuity market exists and where it does not. The assumptions underlying the two tables are: r,=0.45, rs=0.2, 6 = 1, a=0.5, x =0.6, z = 1 and z,=O. In each table the solution values of the endogenous variables are given for y =0.5 (the elasticity of substitution higher than l), y =0 (Cobb-Douglas case), and y = -2 (the elasticity of substitution lower than l), respectively. In the presence of an annuity market, the range of tax rates at which there exists an interior solution for portfolio selection is larger the lower the elasticity of substitution. The range is between 40 and 57 percent for y =0.5, between 14 and 57 percent for the Cobb-Douglas case, and between 0 and 57 percent for 7 = -2. The individual holds only regular notes if the tax rate is below the lower bound. He holds only annuities if the tax rate is above the upper bound. In any case, if there exists an annuity market, the estate tax is completely avoided when the tax rate reaches 57 percent. The two tables also suggest that the tax revenue is larger in the absence than in the presence of annuities. Furthermore, when annuities are absent, the tax revenue rises with the tax rate; but when annuities are present, the tax revenue follows the Laffer-curve phenomenon. In the latter case, the revenue maximizing rate is higher the lower the elasticity of substitution. For example, the revenue is maximized at 35.5 percent for the Cobb-Douglas case and at 37.7 percent for y= - 2. Note, however, that in case where ~=0.5, the tax revenue is decreasing with the tax rate in the region where annuities are held in positive quantities (T > 40 percent), and it is increasing
230
S.C. Hu, Estate taxation Table 2 Steady-state
equilibrium
values in the presence
0.1000
0.2000
0.5282 0.3725 0.099 1 0.0000
0.5330 0.3632 0.1036 0.0000
0.2799 0.0388 1.4242 0.0440 0.6871 0.0212
0.2638 0.029 I 1.4260 0.0323 0.5564 0.0414
0.4761 0.3464 0.1773 0.0000
0.3000
of annuities.
0.4000
0.5000
0.5380 0.3530 0.1088 0.0000
0.5433 0.3338 0.1155 0.0073
0.5476 0.1665 0.1377 0.1480
0.2492 0.0211 I .4287 0.0230 0.4470 0.0605
0.2347 0.0138 0.0148 0.3438 0.0761
0.2053 0.0023 1.3250 0.0025 0.1295 0.0441
0.476 1 0.3247 0.1821 0.0169
0.4761 0.28 16 0.1895 0.0526
0.4761 0.2182 0.2004 0.1050
0.4761 0.1162 0.2180 0.1894
0.3625 0.0335 1.6107 0.0402 0.7336 0.0223
0.3432 0.0233 I .5984 0.027 1 0.5578 0.0415
0.3220 0.0134
0.3034 0.0059 1.5274 0.0065 0.2393 0.0533
0.2895 0.0011 1.4651 0.0012 0.1004 0.0340
0.4264 0.1241 0.2713 0.1780
0.4257 0.1176 0.2736 0.1829
0.4250 0.1061 0.2768 0.1920
0.4242 0.0856 0.2815 0.2084
0.4236 0.0474 0.289 1 0.2397
0.3192 0.0043 I .6266 0.0050 0.2284 0.0080
0.3735 0.0030 I .6256 0.0035 0.1884 0.0152
0.3678 0.0019 1.6211 0.0022 0.1458 0.0206
0.3625 0.0009 1.6103 0.0010 0.0988 0.0220
0.3584 0.0001 1.5867 0.0002 0.0447 0.0 150
(1) ;1=0.5
1.4262
(2) i’=o
1.5689 0.0152 0.3872 0.0530
(3) 2’= -2
with the tax rate in the region where annuities are not held (~~40 percent). Thus, the tax revenue is maximized at borderline rate, i.e. s=40 percent. The two tables confirm the conclusion reached in the preceding section that an increase in the estate tax rate reduces the coefftcient of variance and the range of wealth distribution. They show that the distributional impact of the tax is larger in the absence of annuities or as the elasticity of substitution increases. To illustrate, in the Cob&Douglas case, a rise in z from 20 to 30 percent reduces the SCV by one-fourth (from 3.05 to 2.24 percent) in the presence of annuities, but by nearly one-half (from 2.71 to 1.52 percent) in
S.C. Hu, Estate taxation
231
Table 3 Steady-state
equilibrium
values in the absence
0.1000
0.2000
UC % % %
0.5282 0.3725 0.099 1 0.0000
B
of annuities.
0.3000
0.4000
0.5000
0.5330 0.3632 0.1036 0.0000
0.5380 0.3530 0.1088 0.0000
0.5433 0.3417 0.1149 0.0000
0.5486 0.3292 0.1220 0.0000
EA SCV( A) IE-x
0.2799 0.0388 1.4242 0.0440 0.6871 0.0212
0.2638 0.029 1 1.4260 0.0323 0.5564 0.0414
0.2492 0.02 11 1.4287 0.0230 0.4470 0.0605
0.2363 0.0145 1.4325 0.0156 0.3538 0.0783
0.2254 0.0093 1.4377 0.0100 0.2732 0.0946
2‘ % % %
0.4761 0.3464 0.1773 0.0000
0.4761 0.343 1 0.1806 0.0000
0.4761 0.3393 0.1844 0.0000
0.476 1 0.3347 0.1890 0.0000
0.4761 0.3293 0.1944 0.0000
B Var(P) EA SCV( A) IE.X
0.3625 0.0335 1.6107 0.0402 0.7336 0.0223
0.3485 0.0260 1.6168 0.0305 0.6038 0.0443
0.3354 0.0194 1.6240 0.0224 0.4929 0.066 1
0.3232 0.0139 1.6327 0.0158 0.3964 0.0874
0.3123 0.0093 1.643 1 0.0104 0.3114 0.1082
% % % 2”
0.4222 0.3217 0.2559 0.0000
0.4216 0.3216 0.2566 0.0000
0.4208 0.3214 0.2576 0.0000
0.4 199 0.3210 0.2589 0.0000
0.4188 0.3203 0.2607 0.0000
B Var(b) EA SCV( A) r E.X
0.4461 0.0289 1.8573 0.0375 0.8043 0.0239
0.43 15 0.0228 1.8602 0.0289 0.6620 0.0478
0.4172 0.0174 I .8639 0.0216 0.5413 0.07 18
0.4032 0.0128 1.8687 0.0155 0.4369 0.0959
0.3898 0.0088 1.8751 0.0105 0.3449 0.1201
(1) y=o.5
VW)
(2) )‘=o
(3) y= -2
the absence of annuities. In the presence of annuities, the reduction in the SCV owing to the tax increase is only one-third (from 3.23 to 2.30 percent) when 7 = 0.5 compared with one-half in the Cobb-Douglas case. The tables also show that within the range of values of the elasticity of substitution under consideration, the estate tax lowers the steady-state mean of wealth in the presence of annuities, but it leads to an increase in the steady-state mean of wealth when an annuity market does not exist. Although the changes in EA in either direction are relatively small, they
J.P.E.
D
S.C. Hu. Estate taxation
232
nevertheless suggest that the trade-off lation is not inevitable.
between
equality
and wealth
accumu-
7. Concluding remarks This paper has shown that, except when it is initially imposed, the estate tax falls entirely upon unplanned (conditional) bequests. To the extent that unplanned bequests are responsible for unequal distribution of wealth, the effect of the estate tax is to reduce wealth inequality. If there exists an annuity market, the reduction in wealth inequality tends to be accompanied by a reduction in the steady-state mean level of wealth, especially when the elasticity of substitution is higher than 1. If an annuity market does not exist, such a trade-off between equality and wealth accumulation need not hold all the time, especially when the elasticity of substitution is lower than 1. In this model, the estate tax is operative only to the extent that uncertainty in lifespan and imperfection in the annuity market prevent individuals from planning bequests. The tax is more effective in achieving wealth equality the more imperfect is the annuity market, as measured by the differential between the rates of return on annuities and regular notes. We have assumed that the only uncertainty is about the lifespan and, consequently, the only requirement for the individual to be able to plan bequests is the existence of a perfect annuity market. Of course, a perfect annuity market alone is not sufficient to render the estate tax inoperative if there are other uncertainties which prevent the individual from moving planned bequests to the first period of his life. One such uncertainty is about the status of health. In the present model, the status of health could be parameterized by making the discount factor 6 a random variable. The greater the randomness in 6, the smaller the amount of bequests that can be moved forward to the first period. Another assumption that is crucial to our results is that the family does not act strategically and there are no reverse bequests. Kotlikoff and Spivak (1981) have shown that by behaving strategically and by instituting reverse bequests, a family can overcome the incompleteness of the annuity market. By doing so, the family can also avoid paying the estate tax. Of course, the usual moral hazard problem arises when parents rely on reverse bequests for retirement consumption.
Appendix A.1.
Derivation
of slj
Suppose first that there exists an annuity and c2 = (1 + Y,)S+ ( 1 + r&r, we have:
market.
For r >O, since b2 =0
S.C. Hu, Estate taxation w_=ck-
l,
W,=o(l
+rJ[(l
The first-order
conditions
(1 +r,){(l
+r,)b,
233
+z,-J-‘.
in (10)<12)
-~)~(bT+z,)Y-l
can be rewritten
as:
+m(b~+z,)Y-l}=c;-l,
(114 (124
Combining
(lOa), (1 la), (12a) and (l), we obtain: (14a)
u,,= a&a,, a, = (az- MW(
(154 1-T),
a,=((1 -T)al -a,+a,)R,cr,/(l
(164 -T),
(174
where Ri= l/(1 +ri) a, =(R,/(TcS))‘“~- “, (i = s, a), az = C(Rs - RJl ((1 -T)(l -7C)O)]1"ym1), and a3 = [(R,-sR,)/(( 1 --,)x,)]~‘(~‘). (18) and (19) can be obtained upon differentiating c(~with respect to z. Suppose next that an annuity market does not exist. Then cc,=0 while c(,, c(, and CX,,are jointly determined by (l), (10a) and (1 la). Differentiating these three equations with respect to T, we obtain (20).
A.2.
Conditions for positive holdings of regular notes and annuities
As can be seen from (16a), in order for c1,>0, it is required that R,> R, and a2 >a3. Expanding the latter inequality, we obtain the right-hand inequality of (21). According to (17a), ax,>0 if (1 --)a,+u,>a,. Upon substituting the values of a, and az into the above inequality and noting that a3 2 [R,/(( 1 -z))~o)]‘“‘‘), we can show that the above inequality holds if: {l+[l+((l-z)Ya/G) or
*‘(y-l)]y-l(l
-TC)/TC}R,~R,
234
S.C. Hu, Estate taxation
G( 1 + r,) 2 1 + ra.
@lb)
Thus, the left-hand inequality inequality also implies r. > rs.
in (21) is proved.
Since G> 1, the above
A.3. The effect of the estate tax on wealth accumulation Suppose first there exists an annuity market. Expanding (36) gives: (EA)‘= -{yx,[i +
-7c-((c(b+(l
-rc)a,)(R,-RJR,]
dR, - ~R,)I(R,- %)}(W2/(z( 1 - z)( 1- y))
(36a)
where y* ~0 is the value of y at which the sum of the terms inside the braces is equal to zero. Since R,> xR, and R, > tRs, the second term inside the braces is positive. According to the right-hand inequality in (21), if gs>O, then (R,-R,)/R,s(l -71)(1-z). Therefore, 1 -7c--((clb+(l -n)g)(R,-R,)/ R,z 0. This in turn implies that y* ~0. Suppose next that an annuity market does not exist. We can show upon substituting (15~) and (16~) in (36) that
x (l-lc)f~/P-~/D>O,
ify
(36b)
y** is the value of y at which the sum of the terms inside the braces is equal to zero. Since the coefficients of y are negative, y** must be positive. Moreover, since tl,+rcc~~>c~,, j3r>(1+r,)clb and R$x-~>~ocx~-~ [by (Ila)], < 7~. we can show that y** > 1 if a, < rr. Therefore, y < y** if y < 0 or if CC, We can also show that the expression inside the braces in (27b) is positive and thus p < 0, making use of the following inequalities: 6c(:-‘>a~l;-’
[by (10a) and (lla)];
and Ug-j,py,-j,
forj=1,2
(since a,
References Abel, A., 1985, Precautionary 777-791.
bequests
and accidental
bequests,
American
Economic
Review 75,
S.C. Hu, Estate
taxation
235
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