Discussion Papers In Economics
And Business
The effect of inter vivos gifts taxation on wealthinequality and economic growth

Ryota Nakano

Discussion Paper 21-04

May 2021

Graduate School of Economics
Osaka University, Toyonaka, Osaka 560-0043, JAPANThe effect of inter vivos gifts taxation on wealthinequality and economic growth ∗Ryota Nakano†

Abstract
In this study, we develop a three-period overlapping generations model with inter vivos
gifts and human capital accumulation. We examine the effect of inter vivos gift taxation onwealth inequality and economic growth. The analysis shows that an increase in the tax rate
reduces inequality, and a positive tax rate maximizes the growth rate.
Keywords: economic growth, human capital accumulation, intergenerational transfer, wealthinequality, gift taxation
JEL Classiﬁcation Numbers: O11, O40, I24∗

I thank Koichi Futagami, Ryo Horii and Tatsuro Iwaisako for helpful comments and suggestions. All remaining errors are my own.
†Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka, 560-0043, JAPAN;
e-mail address:xge009nr@student.econ.osaka-u.ac.jp

1Introduction

Taxation of gifts is introduced in many countries. However, in some countries, this does not
apply to gifts that are for educational purposes. In Japan, a tax exemption system for lump-sum
gifts of educational funds was introduced in 2013. The purpose of the system is to encourage
the transfer of assets held by elder generations to young generations and to promote economicgrowth.
Some empirical research shows that in addition to income inequality, intergenerational transfer including gifts are an important factor that affects wealth inequality.1 The above system maybe harmful for wealth inequality and economic growth. In addition to the fact that gift-giving
itself can create disparities in assets, when gifts are used to fund education there may also bedifferences in educational opportunities. Differences in educational opportunities may resultin greater income inequality. Thus, gift-giving can be a source of wealth inequality in the twoways mentioned above.
In this paper, we examine how gift tax rate affects economy. The above system mainly
focus on inter vivos gifts from grandparents to grandchildren, so that this study develops a
three-period overlapping generations model with inter vivos gifts and human capital accumulation to explicitly model the gifts. The analysis shows that an increase in the tax rate reduces
inequality, and a positive tax rate maximizes the growth rate. The results suggests that rather
than exempting gift tax, raising gift tax rate to some extent reduces inequality and moreover,promote human capital accumulation and economic growth.
Many studies on human capital accumulation and inequality employing the overlapping
generations model do not consider the effect of intergenerational transfers on inequality.2 Bossmann et al. (2007) analyzed the role of bequests with the overlapping generations model as inthis study to tax bequests. They showed that using the coefﬁcient of variation as a measure of inequality, bequests diminished the inequality of wealth. Michel and Pestieau (2004) introduced
inheritance into an overlapping generations model and compared the social optimal solution
with the market equilibrium solution by taxation. In most of the literature on intergenerationaltransfers, the use of intergenerational transfer is determined by the recipient. However, in thispaper, we focus only on the use of inter vivos gifts for funding education, since the above tax
system we are analyzing is for gifts of education funds. Thus, in addition to the inequality
caused by the gift itself, we can analyze the inequality thorough education.

1

See Davies and Shorrocks (2000) and Horioka (1993).
Galor and Moav (2004), Glomm and Kaganovich (2008), De la Croix and Doepke (2003), De la Croix andDoepke (2004), and Grossmann and Mazumder (2008) are closely related to this paper in terms of analyzing therelationship between economic growth and inequality with overlapping generations framework.
21

2The ModelConsider a three-period overlapping generations model. Time is discrete. The economy
consists of households, the government, and ﬁrms under perfect competition. We assume asmall open economy, which means that the interest rate is equal to the world interest rate r̄. An
individual lives for three periods. There are two types of individuals: L and H. The difference
between these two types is initial human capital. Population of each type is one. There isno population growth. There are two families or dynasties consisting of one grandparent, one
parent, and one child. In childhood, individuals born in period t − 1 receive both public andprivate education and make no decisions. In adulthood, they supply human capital ht to thelabor market and allocate their income to consumption ct and saving st . In old age, they retire
and allocate their savings to consumptions dt+1 and gifts bt+1 for their grandchild. The budget
constraint for individuals born in period t − 1 is given bywti hit = cit + sit ,(1 + r̄)sit = dit+1 + (1 + τ )bit+1 ,(1)(2)

where wi denotes wage rate of type i and τ denotes the gift tax rate. Each individual takes into
account the effect of taxation when making decisions about giving education gifts. Tax reduces
the amount of gifts they can give. Therefore, when they allocate b as gifts to their grandchild,the gross gift becomes (1 + τ )b. Individuals draw utility from consumption and the amount ofdonations.3 The utility function of individuals born in period t − 1 is given by
Ut−1 = ln cit + β ln dit+1 + γ ln bit+1 .

(3)

The utilities from future consumption and donation are discounted by β and γ respectively.
From (1), (2), and (3), solving the maximization problem of households, we derive1
wti hit ,1+β+γβ(1 + r̄) i i
dit+1 =
wh,
1+β+γ t tγ(1 + r̄)wi hi .
bit+1 =
(1 + τ )(1 + β + γ) t tcit =

(4)
(5)
(6)

Firms produce goods by employing the efﬁciency unit of labor ht and physical capital Kt
from households. The capital depreciates fully. The production function is given by)(L 1−α.
+ahyt = Ktα a1 hH2tt3(7)We assume that the motive for giving is the joy of giving, as in Glomm and Ravikumar (1992) and Galor andZeira (1993).

2

where yt denotes output at time t. We assume a1 ≥ a2 .4 Under perfect competition, wages andinterest rates are equal to the marginal products of each input in equilibrium. According to theassumption of a small open economy, rt = rt+1 = r̄ is satisﬁed. Thus, the wage rates areα( r̄ ) α−1∂yt= (1 − α)a1 ≡ aH ,α∂hHtα( r̄ ) α−1∂ytwtL ==(1−α)
a2 ≡ aL .
α∂hLtwtH =(8)(9)

The government taxes gifts at the tax rate τ ∈ (0, ∞). All revenue are used for publiceducation. In each period, the government’s budget constraint is balanced as follows:
Let+1 = τ bH
t+1 + τ bt+1 .

(10)

Human capital accumulation depends on private education b and public education e. The
cost of private education is ﬁnanced by after-tax gifts. Public education is provided by thegovernment. When individuals make decisions, they do not consider that part of the gift that isﬁnanced by public education. Human capital production function is given by
1hit+2 = [ϵ(bt+1 )q + (1 − ϵ)(et+1 )q ] q .

(11)Here, ϵ represents the weight of each type of education. The larger the value of ϵ is, the greater
the impact of private education on human capital accumulation is. Generally, private educationand public education are substitutable; hereafter, we focus on the case where q > 0.

3Dynamics and steady state

In this economy, we deﬁne the steady state as a situation in which both types of humancapital grow at the same rate. By substituting (10) and (6) into (11), each type of human capitalfor i, j ∈ {H, L}, i ̸= j ishit+1 =()q ] 1q
Θ [ ( i )qϵ ai ht−1 + (1 − ϵ)τ q ai hit−1 + aj hjt−1.
1+τ(12)hL

We deﬁne the inequality as ϕt ≡ hHt in this model. From (12), we derivet[ϕt+1 = ϕt−1H)qϵaqL + (1 − ϵ)τ q (aL + ϕat−1ϵaqH + (1 − ϵ)τ q (aH + ϕt−1 aL )q4] 1q.

(13)

When h is sufﬁciently accumulated, the differences between each type of human capital is small, and it may
be unnatural that there exists wage difference. However, if we consider two types of non-mobile regions, we cansimulate a situation where there is a difference in wages even if the difference in the amount of human capital issmall, such as between urban and rural areas.

3In steady state, since ϕ is constant over time, we obtain[1=ϵaqL + (1 − ϵ)τ q (aL + aϕH )q
ϵaqH + (1 − ϵ)τ q (aH + ϕaL )q]≡ F (ϕ).

(14)

Lemma 1. There exists a unique steady state which satisﬁes F (ϕ) = 1
Proof. See appendix.
Lemma 2. The steady state is stable.
Proof. See appendix.
Therefore, we ﬁnd that a unique stable steady state exists in this economy. Next, we analyze
the effect of tax on inequality in the steady state.
Proposition 1. An increase in the gift tax rate reduces inequality.
Proof. See appendix.
The intuition is as follows. An increase in the gift tax rate reduces the amount of private
education but increases the amount of public education. When private and public education are
substitutable, a larger total amount implies greater accumulation. Thus, type L with less private
education will accumulate more human capital than type H when public education increases due
to higher taxes. As a result, inequality will reduce. In other words, since the amount of giving is
proportional to the income, the burden on wealthy individuals increases when taxes are raised.
Meanwhile, an increase in tax revenue is directly reﬂected in an increase in public education, so
the beneﬁts they receive are equal regardless of their type. Therefore, the relatively poor type L
is more likely to promote the accumulation of human capital, which reduces inequality.
. InNext, we analyze the effect on the growth rate. We deﬁne the growth rate as g ≡ hht+1t
the steady state, since both types of human capital grow at the same rate, we omit the subscript
i. From (12), we derive the growth rate where{(g=Θ1+τ)1[ϵaqH + (1 − ϵ)τ q (aH + ϕaL )q ] q} 12.

(15)

Proposition 2. The relationship between the growth rate and the gift tax rate is non-monotonic,
and a positive gift tax rate maximizes the growth rate.
Proof. See appendix.
The intuition is as follows. An increase in the gift tax rate reduces private education andincreases public education in the same way as the effect on inequality. Since the increase inboth types of education has a positive effect on human capital accumulation, it has positive
and negative effects on the growth rate. In addition, there exists an effect through the ratio of4

g2
1.040

1.035

1.030

1.025
2040

60

80

100

τ

Figure 1: A numerical example:(ϵ = 0.5,q = 0.8)
human capital ϕ. From Proposition 1, inequality diminishes due to the increase in the tax rate.
Declining inequality positively affects the growth rate. Thus, an increase in the tax rate hasboth positive and negative effects on the growth rate. When the tax rate is sufﬁciently small, thepositive effects dominate. When it is sufﬁciently large, a negative effect dominates. Therefore,
there exists a tax rate between 0 and ∞, which maximizes the economic growth rate. Figure 1
is a numerical example. We also ﬁnd that to operate the gift tax rate, we cannot simultaneouslyminimize inequality and maximize the growth rate.

4Appendix

Proof of lemma 1
The derivative of (14) is−q(1 − ϵ)τ q (aH + ϕaL )q−1 (ϵ(aqH /ϕq+1 + aqL ) + (1 − ϵ)τ q (aL + aH /ϕ)q+1 )< 0.
F (ϕ) =
[ϵaqH + (1 − ϵ)τ q (aH + ϕaL )q ]2(16)′Calculating the limits, we ﬁnd that limϕ→0 F (ϕ) = ∞ and F (1) < 1.

Proof of lemma 2
The dynamics of ϕ is
[ϕt+1 = ϕt−1HϵaqL + (1 − ϵ)τ q (aL + ϕat−1)q

ϵaqH + (1 − ϵ)τ q (aH + ϕt−1 aL )q

5] 1q.

(17)

Here, ϕt+1 does not depends on ϕt and only on ϕt−1 . Namely, we can treat this equation asﬁrst-order-difference equation. Differentiating (17) with respect to ϕt−1 , we obtain
ϵ2 aqH aqL + (1 − ϵ)τ q (aH + ϕaL )q−1(ϵaqH aL+ ϵaH aqLϕq−1)∂ϕt+1[()q ] .
=∂ϕt−1[ϵaqH + (1 − ϵ)τ q (aH + ϕaL )q ] ϵaqL + (1 − ϵ)τ q aL + aϕH
Comparing denominator with numerator, we obtain)
ϵaqH aLq+ ϵaH aL
ϕq−1([)q ]aHqqqqq− [ϵaH + (1 − ϵ)τ (aH + ϕaL ) ] ϵaL + (1 − ϵ)τ aL +< 0.
ϕ(ϵ2 aqH aqL + (1 − ϵ)τ q (aH + ϕaL )q−1

∂ϕt+1
Clearly, both of them are positive, then we ﬁnd 0 < ∂ϕ< 1. Note that since, in this model,
t−1
the decisions of an individual do not affect their child but their grandchild, it can be divided into
two periods: odd and even. However, because the above proof doesn’t depend on time t, bothperiods stable and converge same steady state.

Proof of proposition 1
From (14), ϕ in the steady state satisﬁes(aH(aH + ϕaL ) − aL +ϕq)qϵ(aqL − aqH )=(1 − ϵ)τ q(18)()qϵ(aqL −aqH )Deﬁne M (τ, ϕ) as M (ϕ, τ ) ≡ (aH + ϕaL )q − aL + aϕH − (1−ϵ)τ
q . Applying implicit
function theorem,Mτ (ϕ, τ )∂ϕ=−∂τ
Mϕ (ϕ, τ )ϵϕ2 (aqH − aqL )[=(1 − ϵ)τ q+1aL

ϕ2 (aH + ϕaL)q−1 + a(HaL + aϕH)q−1 ] .

(19)Because of q > 0, ϕ is an increase function of τ .

Proof of proposition 2
There is no difference in the proof between the growth rate for the one period and the growthrate for the two periods, because, in the steady state, the growth rate is constant. For simplicity,2
we use the growth rate for the two periods. First, we show limτ →0 ∂g= ∞. Differentiating∂τ6

(15) with respect to τ and taking limit, we derive
1−q∂g 2lim= Θ[ϵaqH ] q
ϕ→0 ∂τ

{−ϵaqH + limτ →0aL (aH + ϕaL )q−1 ϵϕ2 (aqH − aqL )τ [aL ϕ2 (aH + ϕaL )q−1 + aH (aL + aH /ϕ)q−1 ]= ∞.

}(20)
2is negative. Differentiating (15) with respect toNext, when τ is large enough, we show that ∂g∂ττ and substituting into (19), we obtain1−q∂g 2
Θqqq q+(1−ϵ)τ
(a+ϕa)]=[ϵaHL∂τ(1 + τ )2 H}{qq−1qqq−1 ∂ϕ× −ϵaH + (1 − ϵ)τ (aH + ϕaL ) + (1 − ϵ)(1 + τ )τ aL (aH + ϕaL )∂τ1−qΘ=[ϵaqH + (1 − ϵ)τ q (aH + ϕaL )q ] q
2(1 + τ )qq 
(1 + τ )ϵ(ah − aL )[]× −ϵaqH + (1 − ϵ)τ q−1 (aH + ϕaL )q +.
(21)
τ 1 + aaHL ϕ1−q From (18), the second term in curly brackets can be rewritten to(1 − ϵ)τ

q−1(1 − ϵ)ϵ(aqL − aqH ) (aL + aH /ϕ)q(aH + ϕaL ) =+.
τ (1 − ϵ)τ q−1qDue to ∂ϕ> 0, this is a decrease function of τ . Therefore,∂τlim (1 − ϵ)τ q−1 (aH + ϕaL )q = 0.

τ →∞(22)

We can rewrite the third term to(1 + τ )ϵ(aqh − aqL )ϵaL (aqH − aqL )ϵaL (aqH − aqL )[] =+.
aL + ϕ1−q aH
τ (aL + ϕ1−q aH )
τ 1 + aaHL ϕ1−q(23)Taking limit each term of (23), since the maximum of ϕ is 1, we obtain
ϵaL (aqh − aql )ϵaL (aqh − aql )lim=,τ →∞ aL + ϕ1−q aH
aL + aH
ϵaL (aqh − aqL )lim= 0.
τ →∞ τ (aL + ϕ1−q aH )

7(24)
(25)

From (22), (24) and (25),limτ →∞ 

−ϵaqH + (1 − ϵ)τ q−1 (aH + ϕaL )q +q+1−ϵ(aq+1H + aL )=.
aL + aH(1 + τ )ϵ(aqh − aqL ) []τ 1 + aaHL ϕ1−q(26)From the above, all terms of (21) except the ﬁrst term in curly brackets are negligibly small2
when τ is sufﬁciently large. Thus, when τ is large enough, ∂gis negative. Therefore, there∂τ
exists τ between 0 and ∞ at least which maximize the growth rate.

References
Becker, Gary S., Scott Duke Kominers, Kevin M. Murphy, and Jörg L. Spenkuch (2018) “Atheory of intergenerational mobility,” Journal of Political Economy, Vol. 126, pp. S7–S25.
Bossmann, Martin, Christian Kleiber, and Klaus Wälde (2007) “Bequests, taxation and thedistribution of wealth in a general equilibrium model,” Journal of Public Economics, Vol.
91, No. 7-8, pp. 1247–1271.
Cremer, Helmuth and Pierre Pestieau (2007) “Intergenerational Transfer of Human Capital andOptimal Education Policy,” Journal of Public Economic Theory, Vol. 84, No. August, pp.
1–30.
Davies, James B. and Anthony F. Shorrocks (2000) “Chapter 11 The distribution of wealth,” inHandbook of Income Distribution, Vol. 1, Chap. 11, pp. 605–675.
De la Croix, David and Matthias Doepke (2003) “Inequality and growth: Why differentialfertility matters,” American Economic Review, Vol. 93, No. 4, pp. 1091–1113.
De la Croix, David and Matthias Doepke (2004) “Public versus private education when differential fertility matters,” Journal of Development Economics, Vol. 73, No. 2, pp. 607–629.
De Nardi, Mariacristina Cristina (2004) “Wealth Inequality and Intergenerational Links,” The
Review of Economic Studies, Vol. 71, No. 3, pp. 743–768.
Galor, Oded and Omer Moav (2004) “From physical to human capital accumulation: Inequality
and the process of development,” Review of Economic Studies, Vol. 71, No. 4, pp. 1001–1026.
Galor, Oded and Joseph Zeira (1993) “Income Distribution and Macroeconomics,” The Reviewof Economic Studies, Vol. 60, No. 1, p. 35.

8

Glomm, Gerhard and Michael Kaganovich (2008) “Social security , public education and thegrowth – inequality relationship,” European Economic Review, Vol. 52, pp. 1009–1034.
Glomm, Gerhard and B. Ravikumar (1992) “Public versus Private Investment in Human Capital:Endogenous Growth and Income Inequality,” Journal of Political Economy, Vol. 100, No. 4,pp. 818–834.
Grossmann, Volker and Bhashkar Mazumder (2008) “Risky human capital investment, income
distribution, and macroeconomic dynamics,” Journal of Macroeconomics, Vol. 30, No. 1,
pp. 19–42.
Horioka, Charles (1993) Saving in Japan,’in A. Heertje (Ed.) World Savings: An International
Survey: Cambridge: Blackwell.
Michel, Philippe and Pierre Pestieau (2004) “Fiscal Policy in an Overlapping Generations
Model with Bequest-as-Consumption,” Journal of Public Economic Theory, Vol. 6, pp. 397–407.

9