Estate Taxation, ocial ecurity and Annuity: the Trinity and Unity? Nick L. Guo Cagri Kumru December 8, 2016 Abstract This paper revisits the annuity role of estate tax and the optimal estate tax when bequest income is endogenized in the model. It s believed that estate tax liability could implicitly provide annuity income and that in the absence of annuity markets, the lump-sum estate tax can be used to achieve the first-best solutions for individuals with strong bequest motive. ome other literature has been devoted to characterizations of the optimal estate tax, but without including the responsive bequest income to those tax schemes. We show that once the bequest income is determined in a general equilibrium model with the introduction of estate tax, it no longer bears the annuity role. Moreover, when we consider the optimal estate tax, this responsive bequest income mechanism should be included. Key words: Estate Tax, ocial ecurity, Annuity. JEL code:. University of Wisconsin-Whitewater, nicklguo@gmail.com, (1)262.472.7023. Australian National University, cagri.kumru@anu.edu.au, (61)2 612 56126. 1
1 Introduction This paper revisits the annuity role of estate tax and the optimal estate tax when bequest income is endogenized in the model. It s believed that estate tax liability could implicitly provide annuity income and that in the absence of annuity markets, the lump-sum estate tax can be used to achieve the first-best solutions for individuals with strong bequest motive 1. ome other literature has been devoted to characterizations of the optimal estate tax, but without including the responsive bequest income to those tax schemes 2. We show that once the bequest income is determined in a general equilibrium model with the introduction of estate tax, it no longer bears the annuity role. Moreover, when we consider the optimal estate tax, this responsive bequest income mechanism should be included. 2 The Basic Model In this section we use the simplest possible model to illustrate how an estate tax and transfer program affects agents well being when private annuity markets missing,. We use a two-period OLG model with logarithmic utility, zero interest rate and no population growth. The economy is populated with both old and young generations at any point in time. The mass of the young generation is normalized 1. A typical household has at most two periods to live, the probability to survive to the second period is. The total population, at any point in time, is thus 1 +. We assume that the private annuity markets are missing. The agents have access to a storage technology with 0 net return. A government runs an estate tax and transfer program: it taxes bequest at rate τ and it gives each young T r at each point in time. The household s problem is subject to, max ln (c 1) + ln (c 2 ), (1) c 1,c 2,k c 1 + k = w + T r + B, (2) c 2 = k. (3) The transfer, T r, from the government is funded by estate tax; bequest income B, is post 1 ee for example Kopczuk (2003). 2 ee for example Blumkin and adka (2003). 2
tax. The first-order condition to this problem is solving for k, 1 w + T r + B k = k, (4) k = (w + T r + B). (5) 1 + In this setting, even though households receive identical labor income, they differ in bequest income, depending on whether their parents pass away after the first period. Let µ be the distribution of the bequest that is received by each newborn. Next we will characterize this distribution. Note that each parent survives with probability. Therefore, independent of parents assets, a fraction of newborns receive zero bequest. Let B 0 = 0, then µ (B 0 ) =. (6) uch households having savings: k 0 = (w + T r), (7) 1 + and if these households die, their assets are subject to estate tax before inherited by their children. Their children receive B 1 = (1 τ)k 0 = (1 τ) (w + T r), (8) 1 + and will have saving k 1 = = ( ) w + T r + (1 τ) (w + T r) 1 + 1 + [ 1 + ] (1 τ) k 0. (9) 1 + To continue, B t = (1 τ)k t 1, (10) [ ( ) ] t k t = 1 + + (1 τ) t k 0, (11) 1 + µ(b t ) = + (1 ) + + (1 ) t. (12) 3
The estate tax-transfer program is self-financed, and the budget of this program is τ [ (1 )k 0 + (1 ) 2 k 1 + ] = T r, (13) since that the mass of the young generation is 1, who receive the transfer. Thus we can solve for the equilibrium transfer T r = τ(1 )k 0 [ 1 + (1 ) 1 + ] [ (1 τ) + (1 ) 2 1 + ( ) ] 2 (1 τ) + (1 τ) 2 +, 1 + 1 + 1 + or T r τ(1 )k 0 = T r = 1 + (1 + τ + 2 τ 2 ) = (1 ) τw. (14) 1 + 2 Lemma 1 The value of the equilibrium transfer, T r, is strictly increasing in tax rate τ. Proof. Notice that: τ (1 ) T r = w > 0. (15) 1 + 2 After solving for the equilibrium transfer, we can find the consumption allocations for different types: c 1,t = c 2,t = 1 1 + (w + T r + B t), (16) 1 + (w + T r + B t). (17) cience the estate tax/transfer regime is lump sum, it does not alter the intertemporal choice. It s then through the wealth effect that this tax and transfer program affects consumption 4
levels. To see this wealth effect more clearly, we lay out the life-time wealth of type-t agent: W (0; τ) w + T r + B 0 = w + T r, (18) W (t; τ) w + T r + B t = w + T r + (1 τ) 1 ( ) t 1+ (1 τ) t 1 (1 τ) 1 + 1+ As a contrast, when there is no tax and transfer program, or τ = 0, (w + T r). (t 1) (19) W (0; 0) = w + B 0 = w, (20) W (t; 0) = w + 1 ( ) t ( ) 1+ 1 1 + w. (t 1) (21) 1+ Notice that the transfer does not affect agents uniformly. We have the following Lemma. Lemma 2 The estate tax improves welfare for type-t agents if and only if: W (t; τ) > W (t; 0) when τ > 0. For instance, it s obvious that W (0; 0) < W (0; τ), when τ > 0. It says that those who receive no bequest income (B 0 = 0) will unambiguously benefit from the transfer program. For type-1 agents, when W (1; τ) > W (1; 0), or ( ) w + (1 τ) (w + T r) + T r > w + 1 + 1 + w, τ < 1 3 1. (22) It s obvious then when > 1 1 3, which implies that < 0, type-1 agents would not benefit 3 1 since the above inequality is violated. Therefore in this case, the estate tax and transfer program will only benefit type-0 agent. Otherwise, if < 1, then type 1 agents will prefer 3 an estate tax and transfer program when τ is less than 1 3. 1 It s also worth noticing that we have implicitly assumed that the borrowing constraint would not bind, or that k 0 > 0. o the society as a whole is also saving. This assumption will be relaxed when we introduce heterogeneity in labor income. Assuming the social planner has the same weights as population shares, we calculate the optimal estate tax policy when the estate tax and transfer is the only policy instrument. 5
The social welfare function is: W(τ) = [ln(c 1,0 ) + ln(c 2,0 )] + (1 ) [ln(c 1,1 ) + ln(c 2,1 )] +. (23) Using different survival probabilities, [0.25, 0.95], we found that it s the best to have estate tax rate τ = 100% for maximizing the social welfare. Actually, the welfare is strictly increasing in estate tax rate for all possible survival rates. 3 ocial ecurity and Estate Tax In this section, though we still assume away private annuity markets, a self-financed social security regime coexists with the estate tax. Now the household s problem is max ln (c 1 ) + ln (c 2 ), (24) subject to, c 1 + k = (1 ψ)w + T r + B, (25) c 2 = k + b. (26) In this model, the government manages both a social security program and a self-financed estate tax-transfer program. The social security has a budget: ψw = b. Households pay social security tax during working age and later receive a social security benefit if they survive to the second period. The first-order condition is solving for k (and imposing b = ψw ) 1 (1 ψ)w + T r + B k = k + b, (27) k = (1 + ) k = (1 ψ) w + T r + B b = 1 + w 1 + 2 (1 + ) ψw + 1 + T r + B. (28) 1 + We need to characterize the distribution of bequest received by the young. The distribution would be the following, starting with the zero bequest: B 0 = 0, (29) µ(b 0 ) =. (30) 6
Those households with zero bequest income would save k 0 = 1 + w 1 + 2 (1 + ) ψw + 1 + T r = (w 1 + ) 2 ψw + T r, (31) 1 + 2 and if they die early, they leave all their assets as a bequest. o their children will receive To continue, B 1 = (1 τ)k 0 = (1 τ) (w 1 + ) 2 ψw + T r. (32) 1 + 2 B t = (1 τ)k t 1, (33) [ ( ) ] t k t = 1 + + (1 τ) t (w 1 + ) 2 ψw + T r, (34) 1 + 1 + 2 µ(b t ) = + (1 ) + (1 ) t. (35) We have already used the fact that the social security has balanced budget. On the other hand, the budget for the estate tax-transfer program is τ [ (1 )k 0 + (1 ) 2 k 1 + ] = T r. (36) Thus we can solve for the equilibrium transfer or [ T r = 1 + (1 ) 1 + ] (1 τ) +, (37) τ(1 )k 0 1 + T r = (1 ) 1 + τw 1 τψw. (38) 2 For discussion in this paper, it s only sensible when T r > 0 in equilibrium, which according to the above equation requires that: ψ < 2. This is a reasonable assumption. For 1+ 2 instance, the current U.. social security tax rate is 12%, and it should satisfy this condition with the U.. survival data. When social security tax rate is too high, on the other hand, the aggregate saving becomes negative, and thus leaving negative bequest. This scenario is beyond the scope of this paper. It s obvious that when ψ = 0, the equilibrium transfer T r is equal to what we found in the last section. Moreover, the equilibrium transfer is strictly increasing in estate tax, 7
similar to the previous section. ince: (1 ) T r = τ 1 + w 1 ψw. (39) 2 It s positive whenever ψ < 2 is satisfied. 1+ 2 On the other hand, the equilibrium transfer level is also monotone in social security tax rate. We summarize in the following Lemma. Lemma 3 The equilibrium transfer level, T r, is strictly decreasing in ψ, and strictly increasing in τ when ψ < 2 holds. 1+ 2 Proof. In the above discussion, we have shown that τ T r (2 ψ 2 ψ). Also, notice that ψ T r = 1 ψw < 0. (40) The intuition of this result is that with the introduction of social security, agents have less savings, and hence they would leave behind less bequests if they die early. Consequently, the equilibrium transfer decreases with social security. The type-0 agent s life time wealth is: W (0; ψ, τ) = w(1 ψ)+b+t r = w +wψ 1 For all other types with t > 1, w ψw + ψ w + T r + (1 τ)1 ( 1+ 1 ( 1+ W (t; ψ, τ) w(1 ψ) + b + T r + B t = ) t (1 τ) t ) (w 1 + ) 2 ψw + T r. (1 τ) 1 + 2 +T r. ince neither the social security program or the estate tax/transfer program alters the intertemporal choice, we can, again, focus our attentions to where and how these programs affect agents life time wealth. For the type-0 agent, it s obvious that their welfare is further benefited from the introduction of social security program. The following Lemma provides the details for welfare comparison for type-0 agents. Lemma 4 ince W (0; ψ, τ) > W (0; ψ = 0, τ) > W (0; ψ = 0, τ = 0), type-0 agents benefit from both the social security and estate tax-transfer program. Proof. We have shown the second inequality in Lemma 2. It s also discussed that as long as ψ < 2, the equilibrium transfer is positive, thus W (0; ψ, τ) > W (0; ψ = 0, τ) holds. 1+ 2 8
Whiles we have shown that type-0 prefers to having both social security and estate tax/transfer program, it s also important to understand how these two programs affect the other types. To make such comparison, we need to define or redefine their life time wealth in other situations: a) without either the social security program or the estate tax/transfer program: W (t; ψ = 0, τ = 0) = w + B t (ψ = 0, τ = 0) = w + 1 ( 1+ 1 ( 1+ b) with only the social security program: W (t; ψ, τ = 0) = w + 1 ψw + B t(τ = 0) = w + 1 ψw + 1 ( 1+ 1 ( 1+ and finally, c) with only the estate tax/transfer program: ) t ) ) t ) w; (41) 1 + (w 1 + ) 2 ψw ; (42) 1 + 2 W (t; ψ = 0, τ) = w + B t (ψ = 0) + T r = w + (1 τ) 1 ( ) t 1+ (1 τ) t 1 (1 τ) (w + T r) + T r. (43) 1 + 1+ For instance, type-1 agents life time wealth: W (1; ψ = 0, τ = 0) = w + 1 + w; W (1; ψ, τ = 0) = w + 1 + w 2 1 + ψw; W (1; ψ = 0, τ) = w + T r + (1 τ) (w + T r) ; 1 + W (1; ψ, τ) = w + 1 ψw + T r + (1 τ) (w 1 + ) 2 ψw + T r. 1 + 2 As benchmark, W (1; ψ = 0, τ = 0) is bigger than W (1; ψ > 0, τ = 0). Actually, for all type-t (t > 1) agents, their life time wealth, and their well being, are better off without social security program. (ee Caliendo et al (2014) for more details.) o the social security program alone only benefits the poorest group. On the other hand, W (1; ψ = 0, τ = 0) > W (1; ψ = 0, τ) when τ < 1 3 1. 9
4 Fixed Tax Revenue Requirement In this section, we alter the assumption on the estate tax and transfer program. In stead, there is a fixed tax revenue requirement, R helped by either lump sum tax and estate tax. The social security program, on the other hand, remains self-financed. The assumption imposed in this section mimics those in Kopczuk (2003). In essence, the level of lump sum tax, private saving, bequest, will be determined in the model, and be responsive to the estate tax. 4.1 Without Bequest Motive imilar to the environment in Kopczuk (2003), we assume that the government faces a revenue requirement R (with R < w), and the government cares only about the expected present value of payments. What are available to the governments include lump-sum tax and estate tax. We ask the question, how does social security affect the welfare? The household problem is: subject to, max ln (c 1) + ln (c 2 ), (44) c 1,c 2,k c 1 + k = (1 ψ)w T + B, (45) c 2 = k + b. (46) The social security program is still self-financed as in the previous sections, and thus ψw = b. The estate tax and the lump sum tax together will meet a tax revenue requirement. We first solve the household s saving problem by looking at the first order condition of k, The optimal amount of saving, k, is thus k = 1 (1 ψ)w T + B k = k + b. (47) 1 + w 1 + 2 (1 + ) ψw 1 + T + B. (48) 1 + We again need to trace the distribution of the bequest. Using µ the distribution of bequest received, we will characterize this distribution. ince there is a fraction of who receive 10
zero bequest, we have B 0 = 0, and µ(b 0 ) =. This type of households would hold savings: k 0 = 1 + w 1 + 2 (1 + ) ψw 1 + T = (w 1 + ) 2 ψw T. (49) 1 + 2 If these households die early, their assets are subject to estate tax before inherited by their children. After paying the estate tax, their children receive To continue, B 1 = (1 τ)k 0 = (1 τ) (w 1 + ) 2 ψw T. (50) 1 + 2 B t = (1 τ)k t 1, (51) [ ( ) ] t k t = 1 + + (1 τ) t (w 1 + ) 2 ψw T, (52) 1 + 1 + 2 µ(b t ) = + (1 ) + + (1 ) t. (53) The lump-sum tax, T, and the estate tax need to satisfy the tax revenue requirement, R: T + τ[(1 )k 0 + (1 ) 2 k 1 +... ] = R. (54) To find out how does the estate tax rate, we have R T τ(1 )k 0 = hence the equilibrium lump sum, T, is 1 + (1 + τ + 2 2 τ), (55) (1 )(w R) T = R τ + 1 ψτw. (56) 1 + 2 Based on the previous discussion and the finding in this section, we can see clearly that the estate tax and transfer program is a special case of the lump-sum tax and estate tax program. When the external tax revenue requirement is equal to 0, these two regimes are exactly the same. Lemma 5 The equilibrium lump sum tax, T, increases with the coexistence of social security and the estate tax. 11
Proof. The equilibrium lump sum tax, T, is affected by social security: T ψ = 1 τw, (57) and in turn, 2 T ψ τ = 1 w > 0. (58) When the social security is not in place, the equilibrium lump sum tax would decrease with the estate tax. This is intuitive. However, with the introduction of both the social security and estate tax, the lump sum tax would have to increase because of lower saving and hence lower total estate tax revenue. This result, we believe, has yet been noticed by previous studies. Finally, if we define the social welfare similar to the previous section, we again found that for all possible survival probabilities and realistic social security rate, such as ψ = 12%, the optimal estate tax rate is τ = 100%. However, we argue that this is not due to the annuity role of estate tax, but instead, it s the redistribution effect due to the estate tax. 4.2 With Bequest Motive and Altruism We introduce the altruism a la Blumkin and adka (2003) to the model. The households care how much the well-being of their offspring. We start our analysis with assuming general utility function in consumption and in bequest: he household maximizes: max c 1,c 2,B 1,B 2 u (c 1 ) + (1 )v (B 1 ) + u (c 2 ) + v (B 2 ). (59) The first best allocations can be achieved, of course, by the existence of a competitive annuity markets. The result is not new, but we want to highlight the link between bequests that s left behind and bequest income in this simple model. To elaborate, the constraints are the following: c 1 + a + B 1 = y R + B, (60) c 2 + B 2 = a + B 1. (61) A representative household receives bequest income B and labor income y. After paying a lump sum tax, R, which is determined by a tax revenue requirement, he chooses how much 12
annuities, a (at discount price ) to hold, and sets aside B 1 for bequest. If the agent passes away after the first period, B 1 is bequeathed and his annuity contract will terminate. On the other hand, if the agent survives to the second period, his resource becomes annuity payment, a, plus B 1. During this period, he will decide on consumption level, c 2, bequest B 2. The life time budget constraint is thus: c 1 + c 2 + (1 )B 1 + B 2 = y R + B. (62) To characterize the allocations: since we have u (c 1 ) = u (c 2 ) = v (B 1 ) = v (B 2 ) in the optimal, then c 1 = c 2, (63) B 1 = B 2. (64) Moreover, suppose that from the first period, there is no heterogeneity in bequest income among the initial young generations, then we have the following results in the steady state: c 1 = c 2 = 1 1 + (y R) c FB, (65) B 1 = B 2 = B = v 1 (u (c FB )) B FB. (66) Finally, we want to prove that when assuming away the annuity markets completely, like we have done in the previous sections, the first best cannot be achieved by a combination of lump sum tax and estate tax. What we want to emphasize are the following: a) neither the lump sum tax nor the estate tax alters the intertemporal decision, and b) the equilibrium level of tax is affected by estate tax. Before we introduce any tax instrument, households have budget constraints as the following: c 1 + B 1 = y + B, (67) c 2 + B 2 = B 1. (68) If we introduce a lump sum tax, T, and estate tax, E, if the agent dies early to the economy, the household faces the following budget constraint: B 1 = y T E + B c 1, (69) c 2 + B 2 = y T + B c 1. (70) 13
The most straightforward way to prove the impossibility to implement the first best allocations are done by contradiction. uppose the first best allocation is restored, namely, given the lump sum tax, T, and estate tax, E, the agent chooses c 1 = c 2 = c FB, and B 1 = B 2 = B FB. If this happens, however, it implies that the budget constraints would look like, B FB = y T E + B FB c FB, (71) c FB + B FB = y T + B FB c FB. (72) Ostensibly, the following tax arrangement may restore the first best allocations: E = c FB, (73) T = y 2c FB. (74) Given these tax arrangements, the total tax revenue would be T + (1 )E = y 2c FB + (1 )c FB. Which satisfies the tax revenue requirement since c FB = 1 (y R). 1+ Looking closely, however, when the agent leaves behind B FB after the first period, and dies early, his bequest would decrease to B FB E = B FB c FB. His child will then receive less then B FB and thus a contradiction happens. This same pattern in the equilibrium, bequest income is responsive to the estate tax, has been illustrated and calculated in different settings throughout this paper. To sum, once we introduce an estate tax, a negative wealth effect will happen to the young, and his bequest income will be lower than B FB. The result can be summarized in the following impossibility theorem. And this theorem shows that there is no role for estate to act as annuities. 5 Heterogenous Income In this section, we assume the existence of different income levels. To start, we set two possible values: w = w h or w l, each with mass q and 1 q. Without income tax, and with estate tax, the lump sum tax, and fully funded social security, each households has the following maximization problem: max ln(c 1) + ln(c 2 ), (75) c 1,c 2, 14
subject to, c 1 + k = (1 ψ)w i + B t T, (76) c 2 = k + b. (77) The joint distribution of the households bequest income and labor income is: and to continue, µ(w h, B h,0 ) = q, (78) µ(w l, B l,0 ) = (1 q). (79) where B h,0 = B l,0 = 0, for i {h, l} and t 1, µ(w h, B h,t ) = q(1 ) t, (80) µ(w l, B l,t ) = (1 q)(1 ) t, (81) k i,0 = (w i 1 + ) 2 ψw 1 + 2 i T, (82) B i,t = (1 τ)k i,t 1, (83) [ ( ) ] t k i,t = 1 + + (1 τ) t k i,0. (84) 1 + Finally, assume the lump-sum tax, T, and the estate tax need to satisfy the tax revenue requirement, R: T + qτ [ (1 )k h,0 + (1 ) 2 k h,1 + ] + (1 q)τ [ (1 )k l,0 + (1 ) 2 k l,1 + ] = R.(85) The equilibrium level of tax is thus determined by: R T τ(1 )[qk h,0 + (1 q)k l,0 ] = hence the equilibrium tax level is, 1 + (1 + τ + 2 2 τ), (86) T = R (1 )[qw h + (1 q)w l R] 1 + 2 τ + 1 ψτ[qw h + (1 q)w l ]. (87) 15
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