ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E13-8Golden Rule, Non-distortional Tax andGovernmental Transfer
byMitsuyoshi Yanagihara
Akihiko Kaneko
Ai Sakai

December 2013

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Golden Rule, Non-distortional Tax andGovernmental Transfer
Mitsuyoshi Yanagihara* , Akihiko Kaneko** and Ai Sakai***
November 25, 2013

Abstract
We consider the combination of non-distortional taxes (subsidies), a consumption tax
and a wage income tax, in an overlapping-generations model to investigate how therates of these taxes should be set to achieve a golden rule level of capital stock. We
prove that if the initial steady state level of capital is below (above) the golden rule, thewage income tax rate should be negative (positive) and the consumption tax rate should
be positive (negative) both at the steady state and along with the transitional path to the
new steady state. We then show that, in transition, individuals can obtain benefit (loss)of a positive (negative) net transfer from (to) the government. In proving, a capital
market equilibrium condition is necessary.

Keywords: Golden rule; Capital accumulation; Overlapping-generations model;
non-distortional taxes
JEL classification: D91; E22; E62; H21*

Graduate School of Economics, Nagoya University; Furo-cho, Chikusa-ku, Nagoya,
Aichi, 464-8601, Japan; Tel.: +81-52-789-5952; Fax: +81-52-789-4924; E-mail:yanagi@soec.nagoya-u.ac.jp.
**
School of Political Science and Economics, Waseda University; 1-6-1 Nishi-waseda,
Shinjuku-ku, Tokyo, 169-8050, Japan; Tel. & Fax: +81-3-3208-1197; E-mail:akaneko@waseda.jp.
***
Graduate School of Economics, Nagoya University, Nagoya, Aichi, 464-8601,
Japan; E-mail: sakai.ai@b.mbox.nagoya-u.ac.jp.

11. Introduction
We investigate what the combination of the rates of a consumption tax and awage income tax can achieve a golden rule level of capital stock in an
overlapping-generations (OLG) model.
It is no need to mention that, in theory, non-distortional taxes as represented by a
lump-sum tax can lead to the first best situation of the economy. This is supported by
the second theorem of welfare economics that “any Pareto-efficient allocation can bemade a competitive equilibrium” (Hindriks and Miles, 2006: p.35). In other words, such
adequate redistribution policies as non-distortional taxes can lead a targeted
Pareto-efficient allocation. In the context of the OLG models, the golden rule level ofcapital can be achieved by utilizing non-distortional taxes to bring about redistribution
between different generations.1
De la Croix and Michel (2002: p.129) proves that, in the neighborhood of thesteady state, the direction of the optimal intergenerational lump-sum transfer is from theold (young) to the young (old) if the economy is dynamically efficient (inefficient) in an
OLG model developed by Diamond (1965).2 In proving this, they use the property ofsavings function, which is increasing in the income of the young and decreasing in thatof the old. This property directly implies that the transfer from the old (young) to the1

In reality, the lump-sum tax has not been prevailed. Rather, most countries havedistortional taxes, e.g. a capital income tax and labor income tax which have distortional
effects on the allocation of resource and/or time. Therefore, most of previous researches,
whose pioneering works are Ramsey (1927) and Mirrless (1971), have mainly focusedon how the second best taxation (distortional taxation) is under various economic
circumstances and have little paid attention to what the first best taxation is.
2
Ihori (2003: p.63) also mentions that the golden rule level of capital stock can beachieved by lump-sum transfers. Blanchard and Fischer (1989: p.110) also show that
pay-as-you-go pension can be used for achieving the golden rule, as de la Croix andMichel (2002: p.150).

2

young (old) brings about an increase (decrease) in savings, which is required to raise(lower) the level of capital when the economy is dynamically efficient (inefficient).
Thus, the interest of De la Croix and Michel (2002) is on the direct income transfer
between individuals to control savings.
Besides the lump-sum transfer, in an overlapping-generations model, a wage
income tax works as a non-distortional, lump-sum tax as long as a labor is inelasticallysupplied. In addition, a consumption tax can act as a non-distortional tax if its rate iscommon both for the young and the old generations. These imply that an adequate
combination of these two kinds of taxes, in other words, the indirect income transfer via
a consumption tax to control the income of the young can also lead the capital level ofthe economy to the golden rule.3 However, surprisingly, this has never been proved
formally. In real, the savings function approach developed by De la Croix and Michel(2002) cannot be applied to the case of consumption tax because it is not a direct
income transfer.
This paper, therefore, proves how the wage income tax rate and the consumption
tax rate are set to achieve a golden rule level of capital both in the steady state and alongwith the transitional path from the original steady state to the new steady state. For the
consumption tax not to have a distortional effect on the individual’s utility
maximization, the consumption tax rate is common both for the young and the old in thesame period. In proving, we use the condition that the economy monotonicallyconverges to the steady state, which has its source from the capital market equilibrium
condition. Roughly speaking, in contrast to De la Croix and Michel (2002) taking the3

The indirect income transfer via a combination of these taxes, as well as the direct
income transfer, can make the income of the young more by sacrificing the resource
owned by the initial old generation.

3

savings function approach, we take a dynamic stability condition approach.
We show that (i) if the initial steady state level of capital is below (above) the
golden rule (the equilibrium of the economy is dynamically efficient), the wage income
tax rate should be negative (positive) and the consumption tax rate should be positive(negative) both at the steady state and along with the transitional path in theneighborhood of the steady state; (ii) along with the transitional path, individuals canobtain benefit (loss) from a positive (negative) net transfer from (to) the government;(iii) once the new steady state equilibrium achieves the golden rule level of capital, thevalue of the net transfer becomes zero. These findings correspond to the ones shown in
De la Croix and Michel (2002), so that our paper complements the argument of De la
Croix and Michel (2002).
This paper is organized as follows: Section 2 presents a basic framework of thisanalysis just by reviewing the standard overlapping-generations model. Based on themodel, Section 3 shows what the combination of consumption and wage income
tax/subsidy both in the steady state and along with the transitional path to the steadystate. Section 4 concludes.

2. ModelWe use the Diamond’s (1965) overlapping-generations model. The economy
starts period 1 and lasts forever. Capital stock and the number of population in period 1,
K1 and L1 are given and a population growth rate, n 4

Lt 1  Lt
 1 , is assumed toLtbe constant over time.4 In every period, there are two types of individuals, the young
and the old.

2.1. Individuals
Individuals are identical and live for two periods: the young and the old periods.
In period t, when they are young, they supply their labor inelastically and earn wages,
wt , either for their consumption in the young period, c1t , or savings, st . As thegovernment taxes both on their wages and on their consumption by tax rates  tw and tc respectively, the budget constraint in their young period can be given by:(1   tw ) wt  (1   tc )c1t  st  5(1)

In period t+1, when they become old, they retire and just consume based on theirsavings in the previous period. As the same with consumption in their young period, theconsumption in their old period is taxed by a rate of  tc1 . Denoting the interest rate on
the savings by rt 1 , the budget constraint in their old period is expressed as:(1  rt 1 ) st  c2 t 1

(2)

Together with these two budget constraints (1) and (2) in each period, the lifetime
budget constraint can be obtained as:
4

Throughout this paper, lower case letters express per capita variables and larger ones
do aggregate variables.
5
Here the tax rates are defined as positive (negative) if they are taxes (transfers fromthe government).

5(1   tc1 )(1   ) wt  (1   )c1t 
c2t 1 .
(1  rt 1 )
w
t

c
t

(3)

The utility of individuals who are young in period t is defined as a function of c1t andc2t 1 :U t  U (c1t , c2t 1 )(4)

Finally, the individuals maximize the utility, (4), subject to the lifetime budget
constraint

(3).

This

gives

consumptionfunctionsc1t ( wt , rt 1 ; tw , tc , tc1 )and

c2t 1 ( wt , rt 1 ; tw , tc , tc1 ) , leading to the savings function and an indirect utility function:st  s ( wt , rt 1 ; tw , tc , tc1 )(5)Vt  U (c1t ( wt , rt 1 ; tw , tc , tc1 ), c2 t 1 ( wt , rt 1 ; tw , tc , tc1 )) V ( wt , rt 1 ; tw , tc , tc1 ) .

(6)

As in Blanchard and Fischer (1989), we assume (1) 0  st / wt  1 and (2)0  st / rt . The former means that a marginal propensity to save is between 0 and 1,
implying that both consumption in the young and the old periods are normal. The latter,on the other hand, implies that the substitution effect dominates the income effect. In
special, like the other previous literatures, assumption (2) has a critical role for the latter
analysis.

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2.2. Firms
Using capital, K t , and labor, Lt , firms produces output. Aggregate production
function is given as Yt  F ( K t  Lt ) . If it exhibits a constant returns to scale, thendividing both sides by Lt gives a per capita production function:yt  F (Kt1)  f (kt ) ,Lt(7)

where y and k are the output and capital in per capita terms, respectively. Under aperfect competition, the first order conditions for profit maximization gives:rt  f '(kt )

(8)

wt  f (kt )  kt f '(kt )(9)2.3.Government
In our analysis, we will put the following three assumptions for the government
to achieve the golden rule level of capital stock.

Assumptions(i) The government can use consumption and wage income taxes.
(ii) Balanced budget is kept: No government bond is permitted to be issued.
(iii) The rate of consumption tax in the young period and in the old period for every
individual are set to be the same for the consumption tax to be non-distortional..

7

Assumption (iii), in turn, implies that the consumption tax rate should be kept constantover time:  tc   tc1     c   c . To the contrary, the wage income tax rate is variable
so as to keep the budget balanced.
In sum, the government budget constraint can be written as:

 tw wt (kt )   tc c1t ( w(kt ), r (kt 1 ); tw , c )  tc(1  n)c2t ( w(kt ), r (kt 1 ); tw , c )  0 . (10)2.4. Equilibrium
A capital market equilibrium is required for this economy to achieve a generalequilibrium. To put it more concrete, in period t, by using (5), (8) and (9), capital
market equilibrium condition can be written as:

kt 1 s ( w(kt ), r (kt 1 ); tw , c )
,
1 n

(11)

which exhibits the dynamics of kt and kt 1 , with dependent of the two tax rates.

3. Tax Rates for the Achievement of the Golden RuleIn this section, we will show when a combination of tax/subsidy on wage
income/consumption along the transitional path achieves the golden rule level of capitalstock in the steady state, how net transfer from/to the government becomes. Following

8De la Croix and Michel (2002), we limit our argument on the transitional path in theneighborhood of the steady state.

3.1. Net Transfer in Young from the Government
Before investigating, from the government budget constraint (10), the following
Lemma can be easily obtained.

Lemma 1. Under the Assumptions (i) to (iii), if  tw  ()0 and  c  ( )0 , then, tw wt (kt )   c c1t ( w(kt ), r (kt 1 ); tw , c )  ()0 andc(1  n)c2t ( w(kt ), r ( kt 1 ); tw , c )  ()0

necessarily holds.

This Lemma 1 means that if the wage income is subsidized (taxed), then the income of
the young necessarily rises (falls). This is because the consumption tax rate is notassumed to be differentiated between the young and the old and only the young canwork to earn wages. In other words, the absolute value of the tax on wage income
cannot be exceeded by that on consumption in young.
Here consider the situation where the economy is dynamically efficient, r  n .
Then, income should be subsidized by financing a tax on consumption, because theamount of savings, and therefore, capital, has to be more. As proved in De la Croix andMichel (2002), to save more, the income of the young should be more. Therefore,
income should be transferred from the old to the young. Together with Lemma 1, theincome level of the young can be raised by wage income subsidy, so that consumptions

9both of the young and the old should be taxed. The reverse also holds if the economy is
dynamically inefficient.

3.2. Net Transfer from the Government through the Lifetime
Again we consider that the economy is dynamically efficient, namely, the levelof capital is less than that of the golden rule. As we have seen in the previous subsection,
the income of the young should be raised. In our framework here, it directly implies thatthe lifetime income should be raised. Therefore, in this subsection, we will confirm that
individuals receive a benefit from the combination of the wage income subsidy and theconsumption tax by inducing a positive net transfer from the government.
From (3) deducting (10),wt ( kt )  c1t ( w( kt ), r ( kt 1 ); tw , c ) (1   tc1 )c2t 1 ( w(kt ), r (kt 1 ); tw , c )(1  rt 1 (kt 1 )) tc(1  n)c2t ( w(kt 1 ), r (kt ); tw1 , c )(12)Noting  tc   tc1     c   c and eliminating c2t and c2t 1 by using (2), (12) can be
rewritten as:w( kt )  c1t ( w( kt ), r ( kt 1 ); tw , c ) 1
c2t 1 ( w(kt ), r (kt 1 ); tw , c )(1  r (kt 1 ))10

(1  rt 1 (kt 1 ))cs ( w(kt 1 ), r (kt ); tw1 , c )}{s ( w(kt ), r (kt 1 ); tw , c ) 
c(1  n)(1   )Then, using a capital market equilibrium condition (11), we can obtain:

w(kt ) c(1   c )(1  n){(1  n)kt 1  (1  r (kt 1 ))kt } c1t ( w(kt ), r (kt 1 ); tw , c ) 1
c2t 1 ( w(kt ), r (kt 1 ); tw , c )(1  r (kt 1 ))(13)

The left hand side of (13) means the net lifetime income. Therefore, we will investigate
whether its second term,c{(1  n)kt 1  (1  r (kt 1 ))kt } ,(1   c )(14)

is positive or negative, in other words, exhibits the net transfer from individuals to thegovernment (a tax) or that from the government to the individuals (a subsidy).

3.3. Dynamic Stability Condition
To see the condition that the per capita capital monotonically converges to the steadystate, we firstly linearize the capital accumulation equation (12) around theneighborhood of the steady state:11kt 1  k *   '(kt  k *)(15)

where "*" represents the value of the variable evaluated in the steady state. Because
0   '  1 is the condition for the monotonic convergence, we can write this condition

for the monotonic convergence as:

0  kt 1  k *  kt  k *(16)

3.4. Positive Net Transfer under Dynamic Efficiency
Finally, we prove that (14) is positive by using the condition for the monotonic
convergence, (16), in the neighborhood of the steady state. At a glance, from the budget
constraint of the government, (10), it seems to be natural that the lifetime income also
increases in dynamic efficiency because, as De la Croix and Michel (2002) indicates,
the income of the young should be raised for increasing the savings by subsidizing thewage income. However, the net lifetime income is possible to decrease, as long as thepresent value of the net amount of consumption tax is so large that it dominates thewage income subsidy.
Interestingly, the statement above mentioned never hold. To see this, wetransform the inside of the braces in the left hand side of (14):(1  n)kt 1  (1  rt 1 )kt
 (1  n)(kt 1  k *)  (1  n) k * (1  rt 1 )(kt  k *)  (1  rt 1 )k *12 (1  n)(kt 1  k *)  (1  rt 1 )(kt  k *)  (n  rt 1 )k * .

(17)From (16) and r  n , it can be easily verify that (17) is necessarily negative. This, inturn,

implies

that

the

net

transfer

from

the

government

toindividuals,

c{(1  n)kt 1  (1  rt 1 )kt } , is positive if and only if 0   c .
c(1   )Similarly, we can see that 0   c must hold along with the transitional path in

the neighborhood of the steady state to achieve the golden rule level of capital under the
dynamic inefficient. Therefore, we can here obtain the following Proposition:

Proposition 1
Consider the combination of tax (subsidy) of wage income (consumption), and subsidy(tax) of consumption (wage income) when the economy is dynamically efficient(inefficient). Then in the transitional path in the neighborhood of the steady state, nettransfer from the government (the individuals) to the individuals (the government) is
positive (negative).

It should be noted that to obtain this Proposition 1, we utilize the condition formonotonic convergence, in contrast to the proof given by De la Croix and Michel(2002). As De la Croix and Michel (2002) focuses on the relationship between theintergenerational lump-sum transfer and savings, it requires the feature of savings
function. On the other hand, as our proof just focuses on the difference in the levels ofcapital in the neighborhood of the steady state, the condition for monotonic convergenceof the capital to the steady state level is essential.

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3.5. Zero Transfer in the Steady State
Along with the proof of Proposition 1 for transitional path, we can also have another
Proposition regarding the amount of the transfer from/to the government in the steadystate. Because r  n holds in the golden rule and kt 1  kt  k * hold in the steadystate, the net transfer (14) becomes zero, or equivalently, (13) can be written as:

w *  c1 * 

1c2 * .
(1  r *)(18)

Therefore, we can finally have:

Proposition 2
Consider the combination of tax (subsidy) of wage income (consumption), and subsidy(tax) of consumption (wage income). When the economy achieves the golden rule levelof capital, consumption should be taxed (subsidized) and wage income should besubsidized (taxed) if the economy starts from a dynamic efficient (inefficient) region.
However, the net transfer from/to the government becomes zero in the steady state.

It is certain that even if the new steady state of the golden rule can be achieved, thecombination of the signs of the wage income tax and the consumption tax in that steadystate is the same with the one along with the transitional path from the initial steadystate. However, in the new steady state, the net transfer from the government vanishes.

14

This also means that the net amount of transfer from the old to the young (or, from theyoung to the old) becomes nullified in the steady state of the golden rule.

4. Conclusion
We have shown what the combination of the rates of a consumption tax and awage income tax can achieve a golden rule level of capital stock in an
overlapping-generations model. We firstly confirmed that if the initial steady state levelof capital is below (above) the golden rule, the wage income should be subsidized andthe consumption should be taxed both at the steady state of the golden rule and alongwith the transitional path in the neighborhood of the steady state. Then, we could seethat along with the transitional path, individuals can obtain benefit (loss) from a positive(negative) net transfer from (to) the government. Finally, in a new steady state of thegolden rule, the value of the net transfer becomes zero.

Acknowledgements
We would like to thank Hideya Kato, Yuichi Morita and Tsuyoshi Shinozaki for
their valuable suggestions and helpful comments. This work is supported by JSPS
KAKENHI Grant Number 23530366. The authors assume responsibility for any
remaining errors.

References
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