Tsukuba Economics Working Papers
No. 2018-002

Welfare-improving Consumption Tax in the Presence of Wage Tax under
Idiosyncratic Returns from Investment and Incomplete Markets
byHisahiro Naito
February 2018

UNIVERSITY OF TSUKUBA
Department of Economics
1-1-1 Tennodai
Tsukuba, Ibaraki 305-8571
JAPANWelfare-improving Consumption Tax in the Presenceof Wage Tax under Idiosyncratic Returns from
Investment and Incomplete Markets
Hisahiro Naito∗†
Department of Economics
Graduate School of Humanities and Social Sciences
University of Tsukuba

February 2018
Forthcoming from
FinanzArchiv / Public Finance Analysis∗

Email: naito@dpipe.tsukuba.ac.jp Address: Graduate School of Humanities and Social Sciences,1-1-1, Tennodai, Tsukuba City, Ibaraki Prefecture, Japan Postal Code: 305-8573.
†
The author appreciates the comments of Professor Toshihiro Ihori of the University of Tokyo, Professor Daiji Kawaguchi of Hitotsubashi University, the seminar participants of the Canadian Public
Economics meeting, the International Institute of Public Finance and anonymous referee. The author
also appreciates very helpful comments from the editor, Professor Schöb, which improved the exposition of this paper substantially and made this paper more transparent. This work was supported
by a grant from the Japan Society for the Promotion of Science (JSPS KAKENHI Grant Number
15K03501) and the research fund of the University of Tsukuba. The author is responsible for any
errors in this paper.

1Abstract
In a standard multi-period model, consumption tax and wage tax are equivalent. I show that when a capital market is incomplete—in the sense that therates of return from risky investments are idiosyncratic and there is no insurance
for such idiosyncratic risk—the introduction of consumption tax in the presenceof wage tax improves welfare. This holds true even in the presence of optimal or
non-optimal capital income taxes. In the general equilibrium model, the optimallevel of consumption tax is determined to balance the benefits of the risk-sharing
effect and asset accumulation effect and the costs of postponing government revenue to later periods.

JEL keywords: incomplete capital market, risk sharing, consumption tax, optimal
taxation1

Introduction

In many developed countries, revenue from consumption tax tends to be substantial—
in some cases, almost matching that derived from labor income tax. Given that tax
revenue from consumption tax is important to the revenue streams of many countries,
one might ask about the optimal ratio of consumption tax to labor income tax fromthe perspective of economic efficiency.
In a simple standard multi-period model, consumption tax and wage tax induce thesame consumption and labor supply behavior and generate the same discounted valueof tax revenue as long as the after-tax wage rate is the same.1 Although the two taxes
induce different personal savings behaviors—because of their different effects on thetiming of taxation—they also lead to the same resource allocation if government savings
are taken into consideration. This means that consumption tax becomes redundantin the presence of wage tax. Thus, a simple multi-period model cannot explain the
optimal ratio of consumption and wage taxes.2
The optimal ratio between these two forms of taxes can be evaluated with respect
to the liquidity constraints of the economy. Hubbard et al. (1986) analyze the impactof fiscal policy under liquidity constraints. When consumers face liquidity constraints
due to imperfect financial markets, they prefer to pay tax in later periods rather thanin earlier ones. This finding implies that consumption tax is more efficient than wagetax, since the tax liability of the latter occurs only when consumers are working, whilethat of the former is spread over the consumer’s lifetime.
Tax evasion behavior also offers insight into the optimal tax ratio. Boadway etal. (1994) show that in the presence of tax evasion behavior that affects labor-derived
income tax, the optimal combination of consumption tax and labor income tax isimportant.
The present study demonstrates that when a capital market is incomplete—in thesense that the rates of returns from risky investments are idiosyncratic among individ1

I define the after-tax wage rate as w(1 − τw )/(1 + τc ), where w is the wage rate, τw is the wage
tax rate, and τc is the consumption tax rate.
2This conclusion does not change, even in the presence of a non-linear income tax system. The
seminal studies by Atkinson and Stiglitz (1976, 1980) show that the introduction of price distortions
between consumer goods is optimal if the preference between consumption goods and leisure is not
weakly separable in the presence of non-linear income tax. However, consumption tax does not distort
the relative price of consumption over multiple periods. Thus, although the Atkinson–Stiglitz theorem
is appropriate for explaining the use of capital income tax, it does not address the optimal combinationof consumption and wage taxes.

1uals and that there is no insurance to insure idiosyncratic returns—the introduction of
consumption tax, even in the presence of wage tax, can increase economic efficiency,
although the government may initially use capital income tax to absorb idiosyncraticrisks.
The importance of an incomplete capital market can be seen in the data and in
the literature. First, the share of non-financial assets, whose stochastic rates of returnare not likely to be insured, is not low. For example, Bucks et al. (2009) report that
according to the 2004 US Survey of Consumer Finances, only 35.7 percent of all assets
are held in the form of publicly traded financial assets, while the rest are held as nonfinancial assets such as self-owned businesses, self-occupied homes, and real estate.
Second, Wolf (2002) reports that in the 1998 Survey of Consumer Finances, the shareof residential homes, unincorporated businesses, and real estate within total assets is
56 percent. This fact implies that consumers buying those assets can face substantial
risks, presumably with the trade-off of higher expected returns. In the macroeconomics
literature, it has been shown that a model with entrepreneurs is more successful at
replicating the wealth distribution in the United States as demonstrated by Cagetti and
De Nardi (2006) and Quadrini (2000) among others. The macroeconomics literature
also shows that the implication of idiosyncratic capital income risk is different fromthe implication of idiosyncratic labor income risk. Angeletos (2007) shows that higher
idiosyncratic risk reduces the capital labor ratio, in sharp contrast to the result in thecase of idiosyncratic labor income risk. Thus, given the importance of an incomplete
capital market, exploring the implications of tax policy in the presence of such a market
is crucial.
In the presence of an incomplete capital market, the use of consumption tax can
increase risk sharing, because although the consumption tax rate is constant, consumers
must pay a higher (lower) amount of tax when in a good (bad) state. Thus, the use of
consumption tax can increase economic efficiency. However, two questions arise withrespect to such an argument. The first question is whether the private insurance market
is able to insure individual investment risks and whether social insurance throughconsumption tax is redundant. The second question is whether social insurance throughconsumption tax is redundant when the government can instead use capital income tax,
because the source of risk is the returns from risky investments, and capital income
tax can absorb such risks. For the first question, I assume that the capital market forinvestors is incomplete in the sense that no insurance insures idiosyncratic returns from2

risky investments. Insuring this type of investment risk would pose both a moral hazard
problem and an adverse selection problem. For example, if there is no pressure to finda profitable investment opportunity because the insurance market insures investment
risks, people would not work hard to find profitable investment opportunities.
The present study investigates the issue inherent in the second question. In theanalysis presented here, I demonstrate that even if the government were to use capitalincome tax to absorb idiosyncratic risks, the use of consumption tax would increase
economic efficiency. At first glance, readers might wonder why the use of consumption
tax would increase economic efficiency in the presence of the idiosyncratic risks ofrisky investments and capital income tax. Since capital income tax can be used toabsorb those idiosyncratic risks, readers might think that the use of consumption tax
is redundant as a risk-sharing mechanism.
However, I argue that the use of consumption tax increases economic efficiency, evenwhen the government initially uses wage tax and capital income tax; that is because
consumption tax and wage tax each impose a tax burden, but at different times. The
basic intuition can be understood as follows. When a capital market is incomplete,the expected rates of return from idiosyncratic risky investments are higher than thosefrom safe investments. This is because from an individual point of view, the returns
from idiosyncratic risky investments are stochastic, whereas from the macroeconomics
viewpoint, risk from idiosyncratic risky investments does not exist. Thus, from asocial planning perspective, it is desirable to increase investment in risky assets while
simultaneously sharing those risks because the expected rates of return from risky
investments are higher than those from safe investments. When the government starts
to increase the consumption tax rate and reduce the wage tax rate by small amounts,
it will shift the tax burden from working periods to all periods, because wage tax will
impose a tax burden only when an individual works, while consumption tax will impose
a tax burden when an individual consumes. Corresponding to this change in the timingof the tax burden, an individual will increase his or her savings, including investment
in risky assets. Since the expected rates of return from risky investments are higher
than those from safe assets and those risks are idiosyncratic, there is an efficiency gainfrom increasing the consumption tax rate and reducing the wage tax rate. Clearly, thiseffect of the shift from wage tax to consumption tax does not disappear, even in thepresence of capital income tax.3
3

The logic of this argument has similar flavor to the logic of the famous Domar-Musgrave effect3

A natural question to the above argument is how the optimal level of consumption
tax is determined in the presence of wage tax. As the government increases the consumption tax rate and simultaneously decreases the wage tax rate holding the after-tax
wage rate constant, investment in safe assets and risky assets increases. In the partial equilibrium model where the factor accumulation does not affect the factor prices,maximizing the consumption tax rate is optimal. This implies that if there is no institutional upper limit on the consumption tax rate, there is no optimum. However,in a general equilibrium model, the situation is different. As the government increases
the consumption tax rate and decreases the wage tax rate while holding the after-tax
wage rate constant, individuals increase investment in safe assets and risky assets. On
the other hand, in terms of government revenue, some tax revenue is postponed tolater periods because of the different timing of the tax burden of wage tax and consumption tax. This implies that the government needs to issue government bonds.
Furthermore, an increase in investment in safe assets by individuals is insufficient toabsorb the increased issue of government bond. This implies that a shift from wage tax
to consumption tax will affect the asset portfolio of the economy and affect the factor
prices. In Section 2.3, I show that at the optimum, consumption tax is determined tobalance the benefit of risk sharing and the cost due to changes in the factor prices.
This study is related to two strands of the economics literature. First, it relates
to the literature that discuss what types of taxes work as social insurance. For example, Eaton and Rosen (1980) show that wage tax acts as insurance when the returnfrom human capital is uncertain. Kopczuk (2003) demonstrates that inheritance tax
provides social security for the rich. Conesa and Krueger (2006) calculate the optimal
tax progression, focusing on the social insurance role of taxation.4 Jacobs et al. (2012)
analyze the role of labor income tax and education subsidy when human capital ac(Domar and Musgrave, 1944). In the Domar-Musgrave model, the imposition of a proportional income
tax with full loss offset increases private risk taking since the government shares the risk through theproportional income tax. In my model, the government encourages individuals to take more risk
through a change of the timing of the imposition of taxes. I appreciate the editor, Ronnie Schöb, forpointing out the similarity of my result to the famous Domar-Musgrave effect.
4
In the public finance literature, there are several other studies. Diamond (2005) analyzes various
social insurance and tax policies with respect to the life-cycle when the market is incomplete. Boadway
and Sato (2011) analyze optimal taxation in the presence of risky earnings, using the framework of
Mirrlees’s model(Mirrlees, 1971). Nishiyama and Smetters (2005) examine the effect of replacing
income tax with consumption tax under idiosyncratic earnings shocks and an uncertain lifespan.
Gottardi et al. (2016) analyze the optimal taxation of labor income and capital income in the generalequilibrium model with production.

4

cumulation involves idiosyncratic risks. However, they do not analyze the issue of the
optimal combination of wage tax and consumption tax. To the best of my knowledge,
no study has thus far analyzed the optimal combination of consumption tax and wagetax in the presence of idiosyncratic risks. This study is the first to show that in thepresence of idiosyncratic risk, consumption tax plays an important role even in thepresence of wage tax and capital income tax.5
This study also relates to the growing body of the macroeconomics literature on
incomplete capital markets. Angeletos (2007) analyzes capital accumulation in an
economy where the return from capital is idiosyncratic and there is no insurance forsuch idiosyncratic returns. Kitao (2008) examines the effect of capital income tax
in the presence of idiosyncratic capital income risk. However, none of these studies
analyzes the effect of consumption tax in the presence of wage tax and capital income
tax. I show that the use of consumption tax increases economic efficiency even in thepresence of wage tax and capital income tax when the market is incomplete. I alsocharacterize the optimal level of consumption tax in a general equilibrium model.
The remainder of this paper is organized as follows. In Section 2.1, I show thatthe introduction of consumption tax in the presence of wage tax improves welfare. In
Section 2.2, I extend the analysis to the general equilibrium model and characterize
the optimal level of consumption tax. I show that at the optimum, the benefits of
the risk-sharing effect and increasing asset accumulation effect of consumption tax andthe costs of postponing government revenue need to be balanced. In Section 3, theconclusions are presented and issues to be addressed in future research are discussed.
2

Analysis

2.1

The Effect of Consumption Tax in the Presence of Wage Tax

Consider an economy in which a continuum of agents with identical preferences live
for two periods. The agents work only in the first period and save for consumption in
the second period. The investment opportunities comprise one safe asset and several
risky assets. Each agent looking for profitable investments allocates his or her savings
between the one safe asset and one of the risky assets. In this economy, technology
5

The model of Jacobs et al. (2012) assumes that in the first period, individuals do not work andthey work in the second period. Thus, they focus primarily on the earlier stage of individuals, whilein my model I assume that individuals work in the first period and they are retired in the second
period. Thus, my model primarily focuses on life-cycle savings and consumption behavior.

5

is linear; the rate of return from the safe asset is r and the stochastic rate of returnfrom those risky assets is re. In this study, I use e to denote that a variable with e isa random variable. It is assumed that re = e
a × FI × e, where ea is a random variable
and e ∈ [1, 2] is the individual’s effort level (e.g., looking for profitable investments),
which affects the return on investment of the chosen risky asset . I assume that e
is chosen before the realization of the stochastic shock. FI , which is assumed to beconstant in this subsection, is the marginal product of the risky asset.6 I assume
that the random variable e
a is independently and identically distributed (i.i.d.) acrossindividuals and that the level of individual effort to look for a profitable investment
opportunity is not observable by the public. This lack of observability prevents private
financial institutions from sharing the investment risk. The information on profitable
investment is not available to the government. Thus, the government cannot invest in
risky assets. I assume that the expected rate of return from risky investments is always
greater than that from safe assets as long as the agent makes a positive effort to look
for profitable investment opportunities.
The preference of an agent is represented by the following additive and separable
utility function:u(c1 ) − h(l) − v(e) +

1
E[u(e
c2 )]
1+ρ

(1)

where u0 > 0, u00 < 0, h0 > 0, h00 > 0, v 0 > 0, v 00 > 0where (l, e, c1 , c2 ) represent labor supply, effort level, consumption in period 1, andconsumption in period 2, respectively. From the linear technology assumption, thereturn on investment and the wage rate are independent of labor supply and the amountof investment.
I assume that the government institutes linear consumption tax, linear wage tax,
linear capital income taxes on the return from risky investments and the return fromsafe asset. Let s and i be the amount of investment in the safe asset and the risky asset,
respectively. Let τw and τc be the rate of wage tax and the rate of consumption tax.
Let τks and τkr be the capital income tax rate on the return from the safe investment
6

In Section 2.2, I consider the case where FI is endogenous.

6and risky investment, respectively. Then, the budget constraint is as follows:(1 + τc )c = wl(1 − τw ) − s − i(2)(1 + τc )c2 = s(1 + r(1 − τks )) + i(1 + re(1 − τkr ))(3)

Let Ω be the after-tax wage rate where Ω = w(1 − τw )/(1 + τc ). Given the capitalincome tax rate, wage tax rate, and consumption tax rate, the individual optimization
problem is as follows:
max u(Ωl −s,i,l,esi
1s(1 + r(1 − τks )) + i(1 + re(1 − τkr ))−) − h(l) − v(e) +E[u()]
1 + τc 1 + τc
1+ρ
(1 + τc )where Ω = w(1 − τw )/(1 + τc )(4)

The first-order conditions of l, s, i, and e areΩu0 (c1 ) = h0 (l),(5)
1u0 (c1 ) =E[u0 (ec2 )(1 + r(1 − τks ))]
(6)
1+ρ
1
E[u0 (ec2 )(1 + re(1 − τkr ))](7)
u0 (c1 ) =
1+ρ
1
de
r i
and v 0 (e) =E[u0 (ec2 )(1 − τkr )](8)1+ρ
de 1 + tcss(1 + r(1 − τks )) + i(1 + re(1 − τkr ))
i
−
and e
c2 =
where c1 = Ωl −1 + τc 1 + τc
(1 + tc )These first-order conditions show three important facts. First, as long as the aftertax wage rate, Ω, stays constant, an increase in τc does not affect labor supply but it
increases investment in the safe asset and the risky asset by a factor of 1 + τc . Second,
as long as the after-tax wage rate stays constant, an increase in τc does not change
indirect utility. This is because i/(1 + τc ) and s/(1 + τc ) stay constant, which implies
that the value of c1 and c2 do not change.
Now, consider a situation where there is no consumption tax but only a wagetax initially with some capital income taxes. Then, suppose that the government
increases the consumption tax rate by τc from τc = 0 and decreases the wage tax rateby τc (1 − τw ). In this case, the after-tax wage rate stays constant at w(1 − τw ).7 With7

The after-tax wage rate becomes w(1 − τw + τc (1 − τw ))/(1 + τc ) = w(1 − τw ). Thus, the after-tax
wage rate is the same as before.

7

wage tax and some capital income taxes, the present value of government revenue isequal to τw wl + {τks rs + τkr µi} (1 + r)−1 . 8 Denote E[er|e] as µ. When the wage tax
rate is τw − τc (1 − τw ) and the consumption tax rate is τc , the present value of thegovernment revenue is as follows:E[c2 ]
1wl(τw − τc (1 − τw )) + τc c1 ++{τks rs + τkr µi}
1+r
1+r
1
rs
µi
τc
(µ − r)i
+ τks
+ τkr
= wlτw +1 + τc1+r
1 + τc1 + τc(9)Note that τw is the wage tax rate when the consumption tax rate is equal to zero.
τw − τc (1 − τw ) is the wage tax rate when the consumption tax rate increases from zeroto τc . From (5)-(8), labor supply and effort level do not change as long as the after-tax
wage rate stays constant when τc increases. Thus, the first term of (9) is not affectedby τc .
The third term and the fourth term of (9) stay constant since i/(1+τc ) and s/(1+τc )stay constant.
The second term of (9) is an increasing function of τc because i/(1 + τc ) stays
constant. Thus, government revenue increases. On the other hand, consumer utility
does not change from the optimization problem of (4) as long as the after-tax wage
rate stays constant. Thus, the government can increase the tax revenue while holding
the utility of the consumer constant. Hence, there is a welfare gain by increasing theconsumption tax rate and decreasing the wage tax rate with the after-tax wage rate
held constant.9
The reader might wonder where the source of the welfare gain from increasing
the consumption tax rate lies. As I demonstrated, consumption and labor supply
remain the same when the government increases the consumption tax holding theafter-tax wage rate constant. Thus, the reader might think that resource allocation inthis economy is not affected when the government increases the consumption tax rateholding the after-tax wage rate constant.
However, when the consumption tax rate is increased, resource allocation is not
the same. Note that in this economy, there is no macroeconomic risk and the return
8

There is no macroeconomic risk from the viewpoint of the government. Thus, the government
can discount the revenue from all sources in the second period by the risk-free interest rate.
9Note that we do not need to divide government revenue by 1+τc when evaluate it. The governmentis the entity that imposes the consumption tax. When the government buys consumption goods fromthe market by using its government revenue, the government pays the consumption tax to itself.

8

from the risky investment is idiosyncratic. As a result, the equilibrium level of the risky
investment is smaller than its social optimal level because the idiosyncratic return fromthe risky investment is not insured. An increase of consumption tax rate and a decreaseof wage tax rate will increase investment in the risky asset because these changes oftaxes will shift the tax burden from the first period to the second period and because
the agents attempt to pay the increased tax burden in the second period by increasing
investment in the safe asset and the risky asset. This increase in the risky investment
is the fundamental source of the welfare gain. On the surface, the effect of this change
in resource allocation shows up as an increase in tax revenue.10
The second term of (9), τc (1 + τc )−1 i(µ − r)i(1 + r)−1 , has a special economic
meaning. Let i be the amount of risky investment when the consumption tax rateis equal to τc . The amount of risky investment is i/(1 + τc ) when the consumption
tax rate is equal to zero. The increase in the risky investment is i − i/(1 + τc ) =τc (1 + τc )−1 i. Borrowing one dollar and investing it in the risky investment will resultin a social benefit equal to µ − r. Thus, the present social value of the increased
investment in the risky asset due to an increase in the consumption tax rate is equal
to τc (1 + τc )−1 i(µ − r)(1 + r)−1 .
Proposition 1: Assume that the return from investment is idiosyncratic and the capital
market is incomplete in the sense that the expected rate of return is greater than the risk-freeinterest rate. Assume that initially the government uses the wage tax and capital income tax
but no consumption tax. Now assume that starting from τc = 0, the government increases
the consumption tax rate by τc and decreases the wage tax rate to keep the real wage rate
constant. Then, tax revenue will increase while the consumer utility stays constant, thusτc (µ−r)bringing about a welfare gain equal to 1+τ
i.
c 1+r

Proposition 1 needs some clarification. First, when the capital market is complete
and µ = r, a shift in the consumption tax from the wage tax does not increase utility.
Thus, an incomplete capital market is a crucial requirement for my result. Second,
readers might argue that the above argument is based on a partial equilibrium analysis,
which assumes that factor prices are fixed. In a model where prices are fixed, as in the10

In my thought experiment, I changed the tax rates so that the utility of the agent stays constantand examined whether a change in the consumption tax can increase government revenue. Of course,
I can conduct a different thought experiment where the government changes the consumption tax ratewith a fixed government revenue constraint. I can show that the utility of the consumer will increase
when the government raises the consumption tax rate holding its revenue constraint. In either way,
the intuition is the same.

9

current model, setting the consumption tax rate as high as possible and lowering thewage tax while holding the after-tax wage rate, Ω, constant will maximize the welfare
of consumers. Thus, if the consumption tax rate has no upper limit, no optimum canbe achieved. However, the general equilibrium model, where factor prices are affectedby the factor endowment, has an optimal level of consumption tax. As the government
increases the consumption tax rate, while decreasing the wage tax rate so that the aftertax wage rate stays constant, consumers increase investment in both safe assets andrisky assets. When the government introduces consumption tax and decreases the wage
tax rate, some government revenue comes in the second period. This implies that the
government needs to issue government bonds. Since consumers increase investment inboth safe and risky assets when the government issues government bonds, the relatively
risky asset increases and the net safe asset decreases in the capital market. This willchange factor prices and government revenue. Thus, at the optimum, the optimal
consumption tax rate is determined to balance the welfare gain from risk sharing andthe decrease in government revenue due to a change in the factor prices. In Section
2.2, I will show this result by using the general equilibrium model where the factor
prices are determined endogenously.
2.2

General Equilibrium Model and the Optimum level of the Consumption Tax Rate

In this subsection, I analyze the effect of introducing consumption tax in the generalequilibrium model and characterize the optimal level of the consumption tax rate.
Consider the overlapping generation model. In this economy, individuals live fortwo periods. In the first period, individuals work. At the end of the first period, they
retire and in the second period, they do not work. In the first period, they can save inboth safe assets and risky assets. In the second period, individuals receive the returnfrom the safe and risky investments. The population growth rate is equal to n. I
assume that the utility function is the same as in the previous subsection. Regarding
government policy, I assume that the government uses wage tax, consumption tax, andcapital income tax. I also assume that it can issue government bonds. Let Dt be thegovernment debt issued in period t divided by the number of cohorts t. I assume thatin the next period, the government needs to pay (1 + rt+1 )Dt Nt .
The rate of return from the safe investment in period t is rt and the rate of return10

from the risky investment is ret . The wage rate, the rate of return from the safe asset,
and the rate of return from the risky asset depend on the total amount of labor,safe assets, risky assets, and effort level. I assume that there is no depreciation on
both types of assets. Let Y (Lt , St−1 , It−1 ) be the aggregate production function, where
Lt = Nt lt , St−1 = Nt−1 st−1 − Nt−1 Dt−1 and It−1 = Nt−1 it−1 et−1 E[ea]. I assume that
Y (Lt , St−1 , It−1 ) exhibits constant returns to scale with respect to Lt , St−1 and It−1 andthat E[ea] = 1. I assume that the factor prices are determined as follows:
wt = FL (Lt , St−1 , It−1 ), rt = FS (Lt , St−1 , It−1 ),(10)

ret = FI (Lt , St−1 , It−1 ) × et−1 × ea(11)

where E[ea] = 1 E[ ret ] = µt

Denote individuals born in period t as cohort t and the population of cohort t as Nt .
Denote wt (1 − τwt ) as w
bt , where τwt is the wage tax rate imposed on cohort t. Let τct ,τkst , and τkrt be the consumption tax rate of cohort t, capital income tax rate imposedon the return from the safe asset of cohort t, and capital income tax rate imposed on
the return from the risky asset of cohort t. I assume that tax policies depend on theindex of cohorts. My results hold even when I assume that policies are time-invariant
and when I focus on stationary policies.
Define the indirect utility function as follows:w
bt lt − it − st) − h(lt ) − v(et )lt ,st ,it ,et
1 + τct(1 + rt+1 (1 − τkst ))st + (1 + ret+1 (1 − τkrt ))it
1
E[u(
+
)
1+ρ
1 + τctV (wbt , τct , τks , τkr ) ≡ max u((12)

Note that from the first-order conditions, we have
∂V
∂V
∂lt
∂lt
∂it
∂it
=−
Ω;
=−
Ω;
=−
Ω+i ;
∂τct
∂w
bt
∂τct
∂w
bt ∂τct
∂w
bt
∂st
∂st
=−
Ω + i; where Ω = w
bt /(1 + τct )
∂τct
∂w
bt
Tax revenue in period t is
Rt ≡ (w − wbt )lt Nt + τct(w
bt lt − it − st )(1 + rt (1 − τkst−1 ))st−1 + (1 + µt (1 − τkrt−1 ))it−1
Nt + τct−1
Nt−1
1 + τct
1 + τct−1+ τkst−1 rt st−1 Nt−1 + τkrt−1 µt it−1 Nt−1

11

The government budget constraint in period t is
Rt − (1 + rt )Dt−1 Nt−1 + Dt Nt − (Nt + Nt−1 )G ≥ 0(13)where G is per capita government expenditure, which is assumed to be exogenous andfixed.
Let the intergenerational social discount factor be θ, where θ < 1. Then, theproblem of the government is to solve the following constrained optimization problem:maxw
bt ,τct ,τkst ,τksr∞
Xθt Nt V (wbt , τct , τks , τkr )t=0

s.t. Rt − (1 + rt )Dt−1 Nt−1 + Dt Nt − (Nt + Nt−1 )G ≥ 0
Note that when the government solves the above programming problem, it takes into
consideration the fact that the factor prices are endogenous and determined by equations (10) and (11). Let Q be a Lagrangian function and βt be the Lagrangian multiplier of the government budget constraint. Then, the Lagrangian function becomes asfollows:
Q=∞X

θt Nt V (wbt , τct , τks , τkr )

t=0

+

∞
X

βt (Rt − (1 + rt )Dt−1 Nt−1 + Dt Nt )t=0

The first-order condition of Dt Nt becomes as follows:
Dt Nt : βt = βt+1 (1 + rt+1 +∂Rt+1)∂(Dt Nt )where ∂Rt+1 /∂(Dt Nt ) shows the extent to which tax revenue changes through a change
in the factor prices due to a change in the safe asset level. In the above first-orderconditions, βt is the marginal utility of having additional government revenue. 1 +rt+1 is the interest payment and principal of issuing one unit of government bonds.
∂Rt+1 /∂(Dt Nt ) measures the revenue cost of issuing government bonds in terms ofthe decreased government revenue in the next period. When the government issues
government bonds, the accumulation of safe assets decreases. This will affect the12

factor prices in the next period and also reduces tax revenue in the next period. Thus,
1 + rt+1 + ∂Rt+1 /∂(Dt Nt ) measures the total cost of issuing government bonds. The
first-order condition of Dt Nt shows that the ratio of the marginal utility of additional
government revenue in period t to that in period t+1 is equal to the total cost of issuinggovernment bonds.
Now, the first-order condition of τct is
∂V
1
+ βt
(w
bt lt − it − st )Nt∂τct(1 + τct )21+ βt+1(1 + rt+1 (1 − τkst ))st + (1 + µt+1 (1 − τkrt ))it Nt(1 + τct )2∂it ∂Q∂st ∂Q∂lt ∂Q++= 0.
+∂τct ∂lt∂τct ∂it∂τct ∂st

θt Nt

(14)

By using the first-order conditions of w
bt and Dt Nt , the first-order condition of τct
can be written as follows:1µt+1 − rt+1
βt
Nt +
2
(1 + τct ) 1 + rt+1{z}|
risk sharing effect

∂Q
∂Q
+ it
st
∂s
∂it
| t {z}effect through the increase of s and i∂Rt+11(1 + rt+1 (1 − τkst ))st + (1 + µt+1 (1 − τkrt ))it
Nt = 0+βt+1∂(Dt Nt ) 1 + rt+1
(1 + τct )2
|
{z
}

(15)

the effect through increased government debt

The first term of (15) is the one we saw in equation (9) in the previous subsection. We assume that the rate of return from the risky investment is higher than r
because of the incompleteness of the capital market. On the contrary, from a macroeconomics point of view, there is no aggregate risk. Thus, when consumption tax
increases, individuals will increase investment in risky assets and safe assets to pay thetax burden in the second period. Since the rate of return from risky investments ishigher than the risk-free interest rate but there is no aggregate risk, there is a welfare
gain by increasing the consumption tax rate. The second and third terms are theeffects on tax revenue in the next period. When consumption tax increases, individuals increase investment in risky assets and safe assets. A change in the production
factors then changes the factor prices and the tax revenue of the government. The
fourth term is the government debt effect. When the government increases the consumption tax rate and decreases the wage tax rate, some revenue is postponed to13the next period. This implies that the government needs to issue government bonds.
(1+rt+1 (1−τkst ))st +(1+µt+1 (1−τkrt ))it
Nt is the revenue that can be collected in the next pe(1+τct )2
(1+r
(1−τ
))s
+(1+µ
t
t+1
t+1 (1−τkrt ))itkst
riod and 1+r1t+1
Nt is the present discounted value of this(1+τct )2

tax revenue. This is the amount of government bonds that the government needs toissue. When the government issues this amount of debt, it will decrease the net asset level and decrease tax revenue in the second period. This effect is measured by(1+rt+1 (1−τkst ))st +(1+µt+1 (1−τkrt ))it
∂Rt+1
1
Nt . Note that although individuals increase∂(Dt Nt ) 1+rt+1(1+τct )2

investment in risky assets and safe assets, this is insufficient to offset the effect of anincrease in government bonds.11 Thus, the fourth term measures the cost of shifting
tax revenue from the current period to the next period through an increase in the
consumption tax rate and a decrease in the wage tax rate. Equation (15) shows thatthe optimal consumption tax rate is determined to balance the benefit and cost ofincreasing the consumption tax rate.12
3

Conclusions

When the rate of return from risky investments is idiosyncratic and the capital market isincomplete, the expected rate of return from a risky investment is higher than that froma safe investment. However, from a macroeconomics point of view, there is no aggregate
risk in the risky investment in the presence of idiosyncratic risks. Thus, from a social
point of view, it is desirable to increase investment in risky assets and simultaneously
enhance risk sharing. The present study showed that the use of consumption tax
increases investment in risky assets, enhances risk sharing, and increases economic
efficiency. When the factor prices are fixed, the introduction of consumption tax inthe presence of wage tax increases economic efficiency, even if the government has
instituted capital income taxes initially in the presence of idiosyncratic risks. This
improved efficiency comes from the fact that consumption tax will shift the tax burden
in later periods and that the timing effect of consumption tax increases investment inrisky assets. When the factor prices are endogenous, the optimal consumption tax rateis determined to balance the benefits of risk sharing and increasing investment in safeand risky assets and the costs of issuing government bonds.
One of the key characteristics of the tax system in developed countries is the use of
11

Note that in general ∂Rt+1 /∂(Dt Nt ) is negative.
When the capital market is complete and there is no capital income tax, the sum of the second,
third, and fourth terms becomes zero. The first term also becomes zero because rt+1 = µt+1 .
12

14

consumption tax with wage tax. This study showed that when there is idiosyncratic risk
in capital income and the capital market is incomplete, there is a role for consumption
tax.
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