ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E20-1

Economic Growth, Equilibrium Welfare, andPublic Goods Provision with Intergenerational
AltruismbyToshiki Tamai

February 2020

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Economic Growth, Equilibrium Welfare, and Public Goods
Provision with Intergenerational Altruism
Toshiki Tamai!
February 14, 2020

Abstract
The present study examines the government policy of public good provision and its e§ects on
economic growth and welfare under intergenerational altruism. We consider an endogenous growthmodel with altruistic overlapping generations. The preferences of the current youth exhibit a futurebias and so democratically elected governments are subject to this future bias. The optimal rule ofpublic good supply under future bias di§ers from the original Samuelson rule. Unlike the standard
growth model without any bias, the equilibrium growth rate is not independent of government size
under the optimal rule. Future bias gives young generations the dynamic incentives to invest more.
With future bias, intergenerational redistributive e§ects of public good stimulates such incentives.
Hence, the government size a§ects economic growth via intertemporal changes in their resource
allocations. The growth e§ect of the government size provides nontrivial outcomes of welfare
analysis. Our numerical analyses show the growth and welfare superiority of the democratic but
future-biased economy to the economy with nonbiased social planner.
Keywords: Intergenerational altruism; Future bias; Economic growth; Public goods
JEL classiÖcation: D71; D72; H40; O40! Address: Graduate School of Economics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi, 464-8601, Japan.
Tel: +81-52-789-4933. E-mail: tamai@soec.nagoya-u.ac.jp. This work was supported by JSPS KAKENHI Grant Number
16K03726, 18H00865, and the Nitto Foundation.

1

1

Introduction

The present study examines the government policy of public good provision under intergenerational
altruism. The study also addresses the e§ects of the existence of public goods on capital accumulationand economic growth as well as welfare. The optimal provision of public goods requires that the sum of
marginal rates of substitutions between the public and private good must be equal to the marginal rateof transformation between the two goods (Samuelson 1954). This is widely known as the Samuelson
rule in a static model. In contrast, dynamic setting engenders di§erent outcomes. The overlapping
generations model proposed by Diamond (1965) has the potential ine¢ciency in a dynamic sense. The
economy evolves along which growth paths evolve even though the Samuelson rule in the overlapping
generations economy is parallel to static models (Batina 1990). The economy on the growth path thatdeviated from the modiÖed golden rule path will be ine¢cient if it satisÖes only the Samuelson rule.
Operative bequests such as intergenerational transfers neutralize the public intergenerational transfers on dynamic resource allocation even though each generation Önitely lives (Barro 1974).1 In otherwords, the intergenerational links enable di§erent generations with Önite lifetime to be inÖnitly-lived
households. Then, the issue for the ine¢ciency is settled down by the bequests. The bequest behavior
is analyzed by numerous studies, in particular, pioneered by Strotz (1956) and developed by Phelps andPollak (1968) and others.2 These approaches have adopted speciÖcations in intergenerational altruism.
Some of them prove or suggest the existence of the time inconsistency and present bias. Intergenerational altruism may a§ect the optimal conditions for investment decision and publicly provided goods
through emerging some types of bias. Consequently, capital accumulation and economic growth should
be ináuenced by governmentsí self-control problems for Öscal policy investigated in Phelps and Pollak(1968) and Krussel et al. (2002).
To address these issues, the present study constructs an endogenous growth model with public
goods and intergenerational altruistic preference. In particular, the study includes public goods and theArrow-Romer knowledge spillover into the altruistic overlapping generations model (Kimball 1987; Hori
and Kanaya 1989). Gonzalez et al. (2018) proved that the intergenerationally altruistic preferencesof overlapping generations exhibit future bias. The present young generation discounts the utility
from the present consumption of its parent generation in comparison with the utility from the future
consumption of its own. Hence, the young generation is more willing to postpone consumption at
the time than at any future time (Gonzalez et al. 2018). The future bias signiÖcantly a§ects thepolitical demands for intergenerational redistribution. Takeuchi (2011, 2012) found future bias throughexperiments. The experimental evidences suggest the importance of studying the implication of framing
government policy under future bias.
The present study considers a government as a provider of public goods. One-period lived governments are obliged to supply public goods to maximize their democratic objectives. Such democratic
governments are delegates of the coexisting generations such as the young and old. Since young generations have future bias while old generations has none, the governments are essentially future biased.
The present study derives the extended Samuelson rule from the dynamic model with the future-biased
governments. It reveals that the sum of marginal rates of substitutions between the public and private
good must be weighted by the preference parameters that represent the degree of altruism for its ancestors and descendants. Public good provided in each period beneÖts all existing generations during
the period, while its cost is Önanced by a tax burden on the young generation. The public good serves
the intergenerational redistribution. If the young generation can control the tax or government expenditure level through political decision, it will be an e§ective device to reallocate the intergenerational
resources. Thus, the public good as the device of redistribution and investment decision with futurebias mutually a§ect each other.
Future bias motivates people to invest more in capital. Depending on the degree of future bias, apreference for the public good ináuences the investment level through the e§ective rate of return on the
1 Becker (1974) found a similar result, which is known as Rotten Kid Theorem. Andreoni (1989, 1990) demonstrates
that the Ricardian equivalence does not hold under impure altruism (warm glow).
2 For instance, Kohlberg (1976), Goldman (1979), Harris (1985), Ray (1987), Kimball (1987), Hori and Kanaya (1989).

2

investment. Hence, the equilibrium growth rate is not independent of the government expenditure level
related to the preference for the public good even if the public good is nonproductive and it is Önancedby the lump-sum tax. Surprisingly, in this case, it is likely that the government size a§ects the log-run
growth rate positively under future bias. Therefore, the overall implication of the presence of public
goods under future bias has also nontrivial consequences of the welfare properties. The present study
numerically shows that the welfare levels of the planned economy with future-biased governments andof the decentralized economy are superior to the welfare level of the planned economy with non-biased
planner. These welfare properties are parallel to those of Krusell et al. (2002) even though their model
demonstrates present bias, whereas the proposed model exhibits future bias.
The present study is organized as follows. Section 2 discusses the related literatures. Section 3
describes the settings of our theoretical framework. Section 4 derives the equilibrium policy determinedby future-biased authorities and characterizes the equilibrium outcome. Section 5 deals with someextensions such as decentralized economy and private provision of public goods. Finally, Section 6
provides conclusions.

2

Related literature

Numerous studies examine the issues of the provision of public goods. Bergstrom et al. (1986) developed a generalized static model of public goods and characterized its equilibrium properties. Atkinson
and Stern (1974), Christiansen (1981), and Boadway and Keen (1993) studied optimal provision of
public goods when the supply costs are Önanced by distorting taxation. The dynamically extended
Samuelson rules are derived from various dynamic settings of providing public goods. For instance,
Myles (1997) showed that one of the key elements is a degree of intergenerational altruism with durable
public goods. In the model, perfect depreciation leads to the original form of Samuelson rule. Pirttila
and Tuomala (2001) derived the modiÖed Samuelson rule and optimal nonlinear income tax in an
overlapping generations model with durable public goods.
The modiÖed Samuelson rule derived in this study is parallel to those shown by the extant literatures, in the sense that the intertemporal preference parameters weight the marginal beneÖt of public
goods. A planner is future biased since the planner coincides with a democratically elected government
by future-biased individuals. The optimal rule for supplying public goods di§ers from the Samuelson
rule in the original sense. In particular, the welfare weight of the old to the young in the current period
and that during the times after the period are important to determine the optimal supply of public
goods.
Turnovsky (1996) and Tamai (2010) clariÖed the theoretical interaction between public goods andeconomic growth in endogenous growth models. These studies demonstrated that the government
expenditure does not a§ect the equilibrium growth rate when the public good is optimally suppliedby its cost-Önancing coming from nondistortionary tax.3 According to our theoretical Öndings, theequilibrium growth rate is a§ected by the preference for the public good and therefore the equilibrium
government expenditure level. The government size can be associated with economic growth either
positively or negatively, depending on the degree of altruism to ancestors and descendants. The result
provides theoretical explanation of the controversial empirical Öndings.4
The present study is also relevant to the studies on time inconsistency of preferences (e.g., Strotz
1956; Phelps and Pollak 1968; Laibson 1997). Jackson and Yariv (2014) proved that utilitarian aggregation exhibits present bias if there is some heterogeneity in the population. They showed thatthree-quarters of ìsocial plannersî exhibited present biases and less than 2% were time consistent in
3 Distortionary tax Önancing and tax deduction (subsidy) ináuence the equilibrium growth rate (e.g. Turnovsky 1996;
Tamai 2010). Unsuprisingly, productive public goods a§ects economic growth rate. For example, Barro (1990) considered
productive government expenditure as the public input, and Futagami et al. (1993) formulated public capital as the
production factor.
4 See Bergh and Henrekson (2011) for a general review on the empirical relationship between government size andeconomic growth.

3

laboratory experiments. Galperti and Strulovici (2017) developed an axiomatic theory of pure and direct altruism with nonoverlapping generations setting. Their study mainly focused on forward-looking
preferences as well as treated backward-looking preferences.5 They showed that pure altruism acrossgenerations makes it di¢cult to construct a welfare criterion that includes the preferences of all generations and renders a planner time consistent, excluding Phelps and Pollackís (1968) quasi-hyperbolicmodel.
Gonzalez et al. (2018) showed that two-sided altruistic preference of overlapping generations leads
to future bias, which was experimentally suggested by Takeuchi (2011, 2012). They also demonstrated
that the governments having future bias have incentives to legislate and sustain a pay-as-you-go pension
system through self-enforcing commitment mechanism to increase future old-age beneÖts. Considering the intergenerational redistributive e§ects of public goods, the government policy of public good
provision should be investigated under future bias. The present analysis treats the issues di§erently
from Gonzalez et al. (2018) even though the present view is similar. The present study contributes toclarify how the governments should optimally provide the public goods.
Numerous studies consider Öscal policy if the government has time-inconsistent preferences withpresent bias. For instance, Krusell et al. (2002) consider a representative-agent equilibrium modelwith the consumer who has quasi-geometric discounting and cannot commit to future actions. They
compare two di§erent economic outcomes: one derived from the competitive economy and the otherfrom a planning problem. The planner is a consumer representative who, without commitment but
in a time-consistent way, maximizes the present-value utility subject to resource constraints. Their
analysis showed that the welfare in the competitive equilibrium is strictly higher than the welfare of
the planning problem whenever the discounting is not geometric.
Aronsson and Granlund (2011) investigated a provision of durable public good in an overlapping
generations model with two types of consumers under present bias, and showed the formula derived
by Pirttila and Tuomala (2001), including the self-control problem. Halac and Yared (2014) revealed
that the ex ante optimal rule such as the simple form of a renegotiated debt limit is not sequentially
optimal if the shocks to the economy are persistent. Then, the ex ante optimal rule exhibits history
dependence and brings about impecunious misery by debt accumulation.6 As a complementary to the
all existing literatures, the present study covers the issues of the government policy concerning public
goods provision in the growing economy under future bias.

3

The basic model

Consider a closed economy with altruistic overlapping generations who live for two periods. The
population of each generation is normalized to unity. In the Örst (young) period, each generation
works with and earns the labor income. Furthermore, the young generation receives the bequest fromthe parent generation. In the second (old) period, the people retire. Disposable income in the young
period is allocated among the private consumptions in the Örst and second periods with an operative
bequest. Hence, the budget equations in the young and old period are as follows:cyt = (1 ! " t ) wt + bt ! ht ! st ;cot+1 = (1 + rt+1 ) st ! bt+1 ;where cyt is the private consumption in the young period, cot+1 the private consumption in the oldperiod, wt the wage rate, rt+1 the interest rate, " t the labor income tax rate, ht the lump-sum tax, st
the saving, bt the bequest from their parent, and bt+1 the bequest for their children.
Each generation has two-sided altruism for the ancestors and descendants and beneÖts from own
consumption on private and public goods. Assuming the time additive separability, the utility functionis
Ut = u(cyt ; gt ) + ,u(cot+1 ; gt+1 ) + -Ut!1 + .Ut+1 ;5 See Bergstrom (1999) for the study on backward-looking preferences under speciÖc preferences.
6 The political economy of Öscal policy also investigates this issue (e.g., Song et al. 2012; Arai et al. 2018).

4

where gt is the public good in period t, , is the weight of the period-t generationís utility in the oldperiod (, > 0), - is the degree of altruism for their ancestors (- > 0), and . is the degree of altruism
for their descendants (. > 0). In addition, u(cit ; gt ) is speciÖed as( i 1!#(ct ) +&gt1!#
i
for 1 6= 1 and 1 > 0,1!(u(ct ; gt ) =log cit + 2 log gt for 1 = 1;
where i = (y; o) and 2 > 0.
As shown by Kimball (1987) and Hori and Kanaya (1989), - + . < 1 leads to
Ut =

1
X
s=1

#
$
7s u(cyt!s ; gt!s ) + ,u(cot!s+1 ; gt!s+1 )+ u(cyt ; gt ) + ,u(cot+1 ; gt+1 ) +

1
X
s=1

where

(1)

p
1 ! 4-.
1 ! 1 ! 4-.
2 (0; 1) ; 8 =
2 (0; 1) :2.
2and ; yields

7=
Total di§erentiation of#$
8 s u(cyt+s ; gt+s ) + ,u(cot+s+1 ; gt+s+1 ) ;

1!

p

@7
@7
@8
@8
< 0;
> 0;
> 0; and
< 0:
@.
@@.
@The discount factor of the ancestors 7 (of the descendants 8) is positively associated with the weight
for the parentsí utility - (for the kidsí utility .) while the discount factor of the descendants 8 (of theancestors) is negatively associated with the weight for the parentsí utility - (for the kidsí utility .).
During the period t, there are two di§erent generations: young and old generation. Omitting theterm of dead ancestors, equation (1) leads to the utility function for the young generation during theperiod t as
Ut ' 7,u(cot ; gt ) + u(cyt ; gt ) + ,u(cot+1 ; gt+1 ) +

1
X
s=1

#
$
8 s u(cyt+s ; gt+s ) + ,u(cot+s+1 ; gt+s+1 )

1
X
%
&
#
$
= ! 8 !1 ! 7 ,u(cot ; gt ) +8 s u(cyt+s ; gt+s ) + 8 !1 ,u(cot+s ; gt+s ) :

(2)

s=0

Similarly, the utility function for the old generation in the period t becomes
Ut!1 ' ,u(cot ; gt ) +
=8

1
X
s=0

1
X
s=1

#
$
8 s u(cyt+s!1 ; gt+s!1 ) + ,u(cot+s ; gt+s )#$8 s u(cyt+s ; gt+s ) + 8 !1 ,u(cot+s ; gt+s ) :(3)

Assume that the production technology takes the form of the Cobb-Douglas production function
with the knowledge spillover a la Romer (1986). Formally, the production function isyt = Akt* (k t lt )1!* ;where yt is the output, kt the capital input, k t the average capital stock, lt the labor input, A > 0,
and 0 < @ & 1.7 Each Örm chooses the capital and labor inputs to maximize the proÖt, taking the
average capital stock as given. Incorporating kt = k t and lt = 1, the production function becomes
yt = Akt :7 With ! = 1, the externality is perfectly internalized.

5(4)

The interest rate and wage rate with kt = k t and lt = 1 are
rt = @A and wt = (1 ! @) Akt :(5)

Following Gonzalez et al. (2018), we assume that there exists a sequence of a one-period lived
governments, which is responsible for supplying a public good. The government taxes the labor income
or imposes the lump-sum tax to Önance the government expenditure for providing public goods. We
assume that the marginal rate of transformation between private and public good is equal to unity.
The governmentís budget equation is'" t wt > 0 and ht = 0 if wt > 0;gt =(6)ht > 0 and " t wt = 0 if wt = 0:
The period-t government has the objective function is
Wt = Ut!1 + BUt ;(7)

where B > 0.8 Equation (7) can be explained as a probabilistic voting model (Lindbeck and Weibull,
1987; Grossman and Helpman, 1998). Each generation t disregards dead ancestors by the period t.
Hence, using equations (2) and (3), equation (7) can be reduced to
Wt ' 8
+B

1
X
s=0

#
$
8 s u(cyt+s ; gt+s ) + 8 !1 ,u(cot+s ; gt+s )

(1
X
s=0

)
# y
$ % !1&!1oo
8 u(ct+s ; gt+s ) + 8 ,u(ct+s ; gt+s ) ! 8 ! 7 ,u(ct ; gt )= (8 + B)s(u(cyt ; gt ) +u(cot ; gt ) +

1
X
s=1

where
'

)
# y
$
o
8 u(ct+s ; gt+s ) + ;u(ct+s ; gt+s ) ;s(8)(1 + B7) ,,and ; ' :8+B
8

In equation (8), is the relative weight of the old to the young during the period t and ; is the relative
weight of the old to the young in the periods after the period t. The period-t government faces thedi§erent weights of the old to the young. Hence, there is a time inconsistency (Krusell et al. 2002).
Furthermore, the intergenerational altruism leads to future-biased preferences (Gonzalez et al. 2018).
Since there is only one type of asset, the following equation holds:
st = kt+1 = at+1 ;(9)where at+1 denotes the asset in the period t + 1. Equation (9) is equivalent to the clearing conditionof the asset market. This equation and the budget equations in the young and old yield
at+1 = (1 + r) at + (1 ! " t ) wt ! ht ! ct :(10)Since at ' kt , equations (5), (6), (9), and (10) lead to the following resource constraint:
Akt = cyt + cot + gt + kt+1 ! kt = ct + gt + it ;

(11)

where
ct ' cyt + cot and it ' kt+1 ! kt :8 Equation (7) is functionally equivalent to the populational welfare function presented by Hori (1997). See also Aoki
and Nishimura (2017) for the formulation.

6

4

Equilibrium government policy

This section considers that the policy making by the period-t government can actualize the desired
period-t allocation. Furthermore, the study focuses on the growth and welfare properties of the planning economy. The planning problem is based on the setting developed by Krusell et al. (2002) and
Gonzalez et al. (2018). The governments directly choose both the allocation of the aggregate resources
between consumption and investment and of the aggregate consumption between the young and theold during the period t. As mentioned earlier, such governments can be regarded as democraticallyelected planners. In the present study, this economy is referred to as a democratically planned economy.
The static problem to choose the consumption allocation in the period t is formulated as9
max [u(Dc; g) + u((1 ! D) c; g)] ;

0#+#1

where

cy
:
c
The subscript t is omitted from the notation hereafter (i.e., c stands for ct ). Furthermore, the prime isused to represent the variables one period later; c0 is used for ct+1 . Solving the optimization problem,
we obtain
1
D% '
1 :
1+ #
D'

The youngís share of private consumption decreases with the relative utility weight of the old period.
The following optimization problem represents the decision-making of the period-t government:10
'
)
c1!(
g 1!(
0
V0 (k) = max
q
(D;
)
+
1
(
)
+
8V
(k
)
;
(12)
k0 ;g
1!1
1!1
with
V (k) = q (^D ; ;)
wherec = (A + 1) k ! g ! k 0 ; ; '

c1!(
g 1!(
+ 1 (;)+ 8V (k 0 );
1!1
1!1

(13)

,
1!(
; q (D; z) ' D 1!( + z (1 ! D)
; and 1 (z) ' (1 + z) 2 for z = ;; :8The parameters and ; are important to determine whether the period-t government has a present
or future bias. From the restrictions of parameters, we have 87 < 1. Hence, as shown in Gonzalez etal. (2018), the period-t government has a future bias and < ; holds.11
The Örst-order conditions are
g : !q (D; ) c!( + 1 ( ) g !( = 0;

@V (k 0 )k 0 : !q (D; ) c!( + 8
= 0:
@k 0
Equation (14) leads to*q (D; )
1( )

+ #1

g
= 1:
c

9 See Hori (1997) and Aoki and Nishimura (2017) for solving the maximization problem.
1 0 Note that the value function (12) is a reduced form: W (k) = ($ + %) V (k).
0
1 1 Since %' < 1, the di§erence between
and * is"*=(1 + $') +
+
" =
%+$
%

7

!

"
%' " 1+$ < 0:(% + $) %

(14)
(15)

(16)

The basic premise of equation (16) is equating the sum of marginal beneÖts to the marginal cost,which is equal to unity. Note that the marginal beneÖt is measured by its weighted value. The weightin equation (16) depends on the parameters related to the intertemporal concerns for ancestors anddescendants, intertemporal substitution, and so on.
Equation (16) corresponds to the extended Samuelson rule, in the sense that it optimally determines
the allocation between private and public good during the period t with .12 The extended Samuelson
rule will not be the best for the future periods because of the di§erence between ; and . Even though
the future provision of the public goods is determined by the future governments, the governments
face identical decision making for the current government. Therefore, this rule is carried over tothe governments. On the other hand, the deviation between ; andbrings about welfare loss by
consumption misallocation. It should be compensated by more investment or more consumption. Thiseconomic response ináuences the equilibrium growth rate through such an investment change.
DeÖnition 1. A Markov strategy of the period-t government is a triplet of fct (kt ) ; it (kt ) ; gt (kt )g. A
1
Markov perfect equilibrium is a set of sequences fct (kt ) ; it (kt ) ; gt (kt )gt=0 , satisfying equations (11),(12)ñ(15), and fct (kt ) ; it (kt ) ; gt (kt )g = fc (kt ) ; i (kt ) ; g (kt )g 8t.
To derive the equilibrium government policy, we assume that the period-t government has a linear
strategy and anticipates future governmentís policy as'
Hk for period t;
0
k =
(17)
H
bk for the periods after period t:
The deÖnition of investment function and equation (17) derive'(H ! 1) k for period t;i(k) '(b
H ! 1) k for the periods after period t:

Using equations (13), (15), (16) and (17), we obtain
*
+
/
0 1!#
#
%( )
.(
q (b
D ; ;) + 1 (;) q(+;
(A + 1 ! Hb))Hq (D; )((A + 1 ! Hb) =+ q (D; ) H
b: (18)
/
0 #1
8
A+1!H
%( )
1 + q(+; )Equation (18) is the best response of the period-t government for future governments.
Following the standard growth models, we impose the following assumption.
!1

Assumption 1. (1 + A)(!1

< 8 < (1 + A):

Regarding existence and uniqueness of a Markov perfect equilibrium, equations (11), (16), (17),and (18) with D = D % = D^ and H = H
b provide the following proposition (See Appendix A for the proofof Proposition 1):

Proposition 1. In a democratically planned economy, there exists a unique Markov perfect equilibrium
in linear strategies withi% (k) = (H% ! 1) k > 0;c% (k) = (A + 1 ! H% ) 4% k > 0;
g % (k) = (A + 1 ! H% ) (1 ! 4% ) k > 0;
1 2 Myles (1997) derived a similar condition by considering a durable public good and showed that the degree ofintertemporal concern (i.e., discounting based on intergenerational altruism) and long-lived nature of the public goodweight the marginal beneÖt of public goods. He showed that the Samuelson rule is independent of the discounting rate
under perfect depreciation. In contrast, the degree of intertemporal concern for ancestors and descendants is reáectedin equation (16).

8where*+111( ) #%;
I';
1 + I%
q (D % ; )and H% is given by the solution of equation')H(
q (D % ; ;)1 (;) % A + 1 ! H=+
I+ H ' D(H):8
q (D % ; ) 1 ( )1 + I%
4% 'The last equation in Proposition 1 corresponds to the consumption Euler equation in the equilibrium. If there is no di§erence between ; and (; = ), the right-hand side of the consumption Euler
equation becomes D(H) = 1 + A. Hence, D (H) captures the deviation from the gross rate of return on
investment (1 + A) by a future bias. The left-hand side of the consumption Euler equation represents
the marginal rate of substitution (MRS) of future consumption for the current consumption. Figure 1
illustrates the degree of the deviation, 8D (H), and the MRS, H( , in the consumption Euler equation.
The bias a§ects the beneÖt of investment (i.e. increasing future consumption). Hence, 8D (H) depends
on investment H. The utility weight of the Örst period in the planning horizon is less than that of theperiod afterward. On the one hand, higher weight of future utility motivates the agent to invest. On
the other hand, the current cost of investment (reducing consumption) is not su¢ciently compensatedby higher future consumption. Therefore, 8D (H) is decreasing in H, which indicates the downward
curve in Figure 1. Since H( is the upward curve, these two graphs reveal that there exists a uniqueintersection point E.
To verify the properties of the dynamic equilibrium shown in Proposition 1, we consider the planningeconomy that the period-t government can commit to all allocations from period t + 1 onward as abenchmark case (Gonzalez et al., 2018). The equilibrium in the planned economy with a nonbiased
dictator (for instance, the elderly) can be derived from a standard dynamic optimization problem.13
This benchmark economy is referred to as an elderly planned economy. The problem of choosing thesharing rule is rewritten as
max [u(Dc; g) + ;u((1 ! D) c; g)] :
0#+#1

y

Hence, the sharing rule D and the optimal growth factor become
Dy =

1
1+;

1

1
#

1

and Hy = 8 # (1 + A) # :Furthermore, the consumption and investment functions are
%
&
%
&
%
&%
&
iy (k) = Hy ! 1 k; cy (k) = A + 1 ! Hy 4y k; and g y (k) = A + 1 ! Hy 1 ! 4y k;
where

*
+1
1
1 (;) #yandI':
1 + Iyq (D y ; ;)Comparisons between the democratically planned (future-biased) case and the elderly planned case(nonbiased) characterize the properties of the Markov equilibrium derived in Proposition 1. First, thefollowing results concerning private and public consumptions are obtained (See Appendix B for the
proof of Proposition 2):4y 'Proposition 2. The sharing rules of private consumptions in the two planned economies complies
with D % > D y : The ratios of public goods consumption to private goods consumption in the two planned
economies satisfy the following properties:
I%
I%< Iy for 1 < < ;;R Iy for < 1 < ;;I%>

Iy for< ; < 1:

1 3 The derivations of key equations in the planned economy are derived from the results of Proposition 1 by substituting

* for

.

9

The Örst result of Proposition 2 is identical as that of Gonzalez et al. (2018). A present bias
induces higher share of the young consumption to aggregate consumption. The second result showsthat the size relation of I% and Io depends on the size of and ;. We have1( )q (D % ; )=(1 + ) 2 < (1 + ;) 2 = 1 (;) ;
/
0( /
0(
%
&
1
1
=
1+ #
< 1 + ;#
= q Dy ; ; :Then, 1 plays a key role of determining the size of I% and Iy . Since the private good is rival in
consumption, the intergenerational allocation of the private consumption is a§ected by the utility
weight depending on the elasticity of intertemporal substitution 1. In contrast to the private good,
the public good is nonrival in consumption. The supply of the public good is not a§ected by theintertemporal factors.
Considering 1 > 1 as an example,14 if > 1, the impact of a rise in on the utility weight ofprivate consumption is smaller than the impact on 1 ( ), because1d1 ( )
dq (D % ; )
! #
/
!
=
%
1( ) dq (D ; )d(1 + ) 1 +1#0 > 0:

Hence,> 1 leads to I% < Iy . When ; < 1, I% > Iy holds through the opposite mechanism.
; > 1 > is the intermediate situation of the former two cases. Depending on the size of and ;,each of I% < Iy and I% > Iy is possible.
Further, exploring the growth properties, the comparison between H% and Hy and the partial derivative of H% with respect to 2 lead to the following proposition (See Appendix C for the proof of
Proposition 3):
Proposition 3. The growth properties in the two planned economies are
H% > Hy and

@H%
R0,
@2

#!1
#

R 1:

Figure 1 explains the results of Proposition 3. The equilibrium in the elderly planned economy isgiven by the point F, while the equilibrium in the democratically planned economy is the point E in
Figure 1. As explained earlier, the degree of deviation, D (H), is larger than (1 + A) for any value of
H 2 [0; 1 + A). Future bias boosts investment in private capital. This implies that the growth rate in
the democratically planned economy exceeds that in the elderly planned economy.
Proposition 3 shows that a preference for public goods 2 a§ects the equilibrium growth rate.
Without any bias, the optimal growth rate is independent of 2 (e.g., Turnovsky 1996; Tamai 2010).15
In particular, 2 a§ects D (H) through a change in I if there is a future bias. The impact depends on therelative weight of the old to the young and the elasticity of intertemporal substitution 1=1. A risein 2 decreases the marginal cost of public fund by increasing the weighted marginal beneÖt of public
goods. It changes the allocation between private and public goods. Consequently, I rises. Depending#!1# , a rise in I increases or decreases the e§ective rate of return on investment, whichon the size of
induces more or less investment. The background of the mechanism is originated in nonrivalness ofthe public good in consumption.
Finally, the welfare properties are examined to make the following proposition (See Appendix forthe proof of Proposition 4):
1 4 If - < 1, the opposite e§ects work. When - = 1, there is no di§erence between .! and .y . The terms of the utility
weights are o§set each other.
1 5 Turnovsky (1996) also showed that the Önancing methods of public goods supply a§ect the growth rate because of
the distortionary taxes.

10Proposition 4. The value functions in the two planned economies are
W%
Wy

=
=

5% v(k; H% );5y v(k; Hy );

where
1!(

(8 + B) (A + 1 ! H)
k 1!(;
(1 ! 8H1!( ) (1 ! 1)
nh
i
o
1!(
1!(
1!(
'
1 ! 8 (H% )q (D % ; ) + 8 (H% )q (D % ; ;) (4% )
nh
i
o
1!(
1!(
1!(
+ 1 ! 8 (H% )1 ( ) + 8 (H% )1 (;) (1 ! 4% );% y & % y &1!(%&
1!(
' q D ;; 4
+ 1 (;) 1 ! 4y:v(k; H) '5%5y

In the present value of utility W , the coe¢cient 5 is integrated weights of private and public
consumptions and v(k; H) is the utility level measured by a unit of integrated consumptions. For agiven k, the value of H to maximize v(k; H) is H = Hy .16 Hence, v(k; H% ) < v(k; Hy ) < 0 holds. Ignoring
the di§erence between ; and , H = Hy is the best solution of maximizing the welfare, similar tothe standard AK growth model with geometrical discounting. In other words, the excess investment
reduces the social welfare. However, there is a di§erence of the utility weights ; and , so that 5% and
5y . Furthermore, H% a§ects the level of 5% . Given that the size relation of 5% and 5y is ambiguous, it
is analytically hard to compare the welfare level. Alternatively, quantitative analysis provides obvious
numerical examples.
The parameters and the initial capital stock are speciÖed as , = 0:8, B = 0:8, 1 = 2, A = 2:25, andk0 = 1. Figure 2 illustrates two curves W % and W y with respect to 2 2 [0; 2] for - = 0:4 and . = 0:5.
It demonstrates that the welfare in the democratically planned% economy& is larger than that in theelderly planned economy. Furthermore, the welfare di§erence W % ! W y increases with 2. Setting
2 = 0:5, the robustness to changes in - and . is examined. Figure 3 reveals that the welfare di§erenceis positive on the domain of - and .. These results imply that the democratically planned economy
exhibits the welfare dominance to the elderly planned economy.

5

Extensive analysis and applications

In the previous section, we have assumed that the period-t government has the policy instruments
to execute its desired period-t allocation. The assumption allows clariÖcation of the policy insights.
However, we also should consider the market economy in which the government policy decision isseparated from personal decision making as an example of Öscal policy with some limitations. Thissection characterizes the decentralized dynamic equilibrium by using the results in the previous section
as an analytical basis. Furthermore, the equilibrium outcomes with voluntary provision of international/interregional public goods are investigated.

5.1

Competitive equilibrium

The analysis developed in the previous section is applicable to the market economy with public provision of public goods if the representative chooses the consumption level subject to equation (16)with the resource constraint. Then, the outcome in the market economy is given by Proposition 1.
1 6 Partial di§erentiation of v(k; 1) with respect to 1 yields#$(A + 1 " 1)#! (1 + A) %1#! " 111@v=
R 0 , 1 Q % # (1 + A) # = 1y :@1(1 " %11#! )2

11However, the decentralized economy should be considered in the sense that the determination of thegovernment policy is separated from the decision making of the representatives. Then, the householdís
optimization problem is formulated as
'
)
c1!(
g 1!(
0
0
V0 (a; !j8) = max
q
(D;
)
+
1
(
)
+
8V
(a
;
!
j8)
;
(19)
a0
1!1
1!1
with

c1!(
g 1!(
+ 1 (;)+ 8V (a0 ; ! 0 j8) :
(20)
1!1
1!1
In equations (19) and (20), 8 denotes the information set of the current and future government policies.
The representative chooses the investment level to maximize the value function for the given government policies. Subsequently, the government determines the level of ! to maximize the value function
subject to the decision rule derived by the representative. The detail of 8 is essential to obtain theclosed-form solution of the problem. The period-t generation anticipates the tax-related variable andpublic goods,!0 = Hb! and g 0 = H
bg;(21)
V (a; !j8) = q (^D ; ;)

where Hb is the predicted growth rate and!''(1 ! " ) w if w > 0;!h if w = 0:

The Örst-order condition of the householdís optimization problem is@V (a0 ; ! 0 )
= 0:
(22)
@a0
As described in the previous section, if the period-t generation adopts a linear strategy and the person
prospects investment policies of future generations as'
Ha for period t;
a0 =
(23)
H
ba for the periods after period t:
a0 : !q (D; ) c!( + 8

The concept of our dynamic competitive equilibrium is deÖned as follows:1

DeÖnition 2. A dynamic competitive equilibrium is a set of sequences fct ; D t ; gt ; kt ; ! t gt=0 satisfying
equations (5), (6), (9), (10), (19)ñ(23), D t = D % , and arg max!t V0 , given k0 .
In addition to Assumption 1, we impose the following assumption.
!1

Assumption 2. (1 + @A)< 8:Then, equations (19)ñ(23) provide the following results (See Appendix E for the proof of Proposition
5):
Proposition 5. There exists a unique dynamic competitive equilibrium in linear strategies withi? (k) = (H? ! 1) k > 0;c? (k) = (A + 1 ! H? ) 4? k > 0;g ? (k) = (A + 1 ! H? ) (1 ! 4? ) k > 0;
where

i
8 h
9 #1
1!(
1!(
<
1 ! 8 (H? )1 ( ) + 8 (H? )1 (;) =
1i
; I? ' h4? ';: 1 ! 8 (H? )1!( q (D % ; ) + 8 (H? )1!( q (D % ; ;) ;1 + I?

and H? is given as the solution of the equationH(
q (D % ; ;) (@A + 1 ! H)=
+ H ' F (H):8
q (D % ; )12The interpretation of Proposition 5 is similar to that of Proposition 1. The di§erences between
the results in the two propositions are the impacts of an externality @ and of the separate decision
making. Unlike the planned economy, the externality through knowledge spillover is not removed in
the competitive economy without an instrument to reduce it. The allocation between private andpublic consumption is separated from the investment decision making by the household. Hence, theequilibrium growth rate in the dynamic competitive equilibrium is independent of a taste for public
good 2.
To characterize the dynamic competitive equilibrium, we compare the outcomes of Propositions 1
and 5. The youngís share of private consumption in the dynamic competitive equilibrium is equal to
D % . Hence, we obtain D % > D y . Considering to the properties of the dynamic competitive equilibrium,
we have the following proposition (See Appendix F for the proof of Proposition 6):
Proposition 6. Private and public consumptions in the dynamic competitive equilibrium satisÖes
g?
g%
R % ,
?
c
c

#!1
#

R 1:

The growth rate in the dynamic competitive equilibrium exhibits H? & H% if
the growth rate satisÖes H? R H% , @ R @% with#!1#

- 1. If

#!1
#

< 1,

@@%
< 0.
@2
#!1

#
Proposition 6 shows thatis the key element of characterizing the public good supply andgrowth properties. The intuitions of Propositions 2 and 3 explain its basic mechanism. Focusing on#!1#
the growth properties, when
- 1, the growth rate in the dynamic competitive equilibrium is lessthan the growth rate in the democratically planned economy. The capital (positive) externality leads
to less investment because it decreases the net return on investment. Future bias increases #!1
the net#return on investment and induces more investment. Furthermore, H% increases with 2 when
-1
#!1
?
%
#
(Proposition 3). Therefore, we have H & H if
- 1. In contrast, there is a critical value of @#!1#satisfying H? = H% when< 1. It implies that H? > H% occurs if 2 is su¢ciently large. Given#!1#that H% decreases with 2 when
< 1 (Proposition 3), the net return on investment in the dynamic
competitive economy dominates the net return in the democratically planned economy for a su¢ciently
low 2. Hence, H? > H% if @ > @% . In all the cases, note that there is the possibility of H? > Hy . Futurebias boosts the equilibrium growth.
As an example of the second result in Proposition 6, the growth properties are quantitatively
assessed. The same values of the parameters of ,, B, 1, and A, and the initial capital stock are used#!1#
in the previous section. The value of @ varies from 0 to 1. Focusing on< 1, we set - = 0:1 and#!1#. = 0:2;. 0:928 with , = 0:8 and B = 0:8. Figure 4a displays two curves of (H? ! H% ) with twodi§erent values of 2 on @ 2 [0:98; 1]. It shows that there exists the value of equating H? and H% . Since
the critical values are nearly equal to unity, H? < H% is plausible within realistic values of @ 2 (0:3; 0:6).
On the other hand, Figure 4b shows that H? > Hy (H? < Hy ) holds for su¢ciently large (small) @.17
Hence, H? > Hy is a realistic situation.
The welfare level in the competitive economy depends on the size of spillover e§ect. It complicates
the welfare analysis. Numerical analysis enables direct comparison of the two di§erent welfare outcomesunder di§erent equilibria. Figure 5a illustrates the two curves of (W ? ! W % ) with two di§erent valuesof 2 on @ 2 [0:85; 1]. W ? is larger than W % for @ 2 (0:86; 0:99). This result shows that there is apossibility of welfare dominance of the decentralized economy in comparison with the elderly planned
economy, even though the values of @ are far from the realistic value range in this case. In contrast,
W ? dominates W y for the values of @ around 0:5 in Figure 5b. These numerical results demonstrate
the welfare superiority of the decentralized economy as shown in Krusell et al. (2002).

1 7 The critical value is calculated as ! $ 0:317.

13

5.2

Voluntary provision

The basic model developed in Sections 3 and 4 can be extended to the model with voluntary provisionof international/interregional public goods. The setup is in line with Tamai (2010) without a redistribution policy except for the altruistic agents with future bias. Consider that the economy consistsof n symmetric regions. All regions contribute to providing interregional public goods. Let G be thesupply of the interregional public goods and g be the contribution by each region. Then, G = ng holds.
All the other setups except for the public goods in the previous sections are unchanged. Note that all
outcomes are identical to those of Section 3 and 4 when n = 1. Under these settings, the optimization
problem for the period-t regional government is
'
)
c1!(
G1!(
0
V0 (k) = max
q (D; )+ 1( )+ 8V (k ) ;
k0 ;g
1!1
1!1
with

c1!(
G1!(
+ 1 (;)+ 8V (k 0 ):1!1
1!1
Solving the optimization problem, we obtain
V (k) = q (^D ; ;)

1!#
1
G
g= n # I% and = n # I% :
c
c

(24)

When n = 1, equation (24) becomes equation (16). Equation (24) indicates that the contribution byeach region decreases with the number of the regions, while the supply of public goods increases with
the number of the regions (e.g., Andoreoni 1988).18
1!#
DeÖned I
e as n # I% , the investment and consumption functions are obtained as follows:ei(k) = (eH ! 1) k > 0;
e > 0;
ec(k) = (A + 1 ! H
e) 4k
/
0
e k > 0;ge(k) = (A + 1 ! He) 1 ! 4

where

Note that H
e is derived from

H(
=
8

'

e'
4

1
:
1+I
e

)
q (D % ; ;)
1 (;)
A+1!H
+
I
e
+ H ' H(H; n):%q (D ; ) 1 ( )1+IeThese equations shows that Proposition 1 still holds with the values of H, I, and 4 which are di§erent
from its their originals. Therefore, the basic properties of the equilibrium are similar to those of thebasic model. Given that I
e depends on n, a change in n a§ects the equilibrium outcome.
The study then focuses on the growth property concerning a change in the number of regions n.
Di§erentiating H (H; n) with respect to n leads toq(+ # ;3) -%(3)@H (H; n)%( ) ! q(+ # ; )=2@n(1 + Ie)

From equation (25), the following result is derived.

1!1
1

.

I
e
:
n

(25)

1 8 Chamberlin (1974) and McGuire (1974) studied how group size a§ects contributions to public goods, and Andreoni(1988) generalized their results. Using a repeated game, Pecorino (1999) proved the existence of admissible values of the
discount parameter such that cooperation may be maintained in the limit.

14

Proposition 7. For 1 > 1, an increase in the number of regions a§ects the equilibrium growth ratein accordance with
#!1
@e
H
#
R0,
Q 1:
@n
The number of regions n determines the value of I
e . Hence, the similar (but opposite) mechanismin the case of 2 works. Tamai (2010) also showed that the number of agents a§ects the equilibrium
growth rate under a redistributionary taxation even though such an e§ect does not exist without theredistribution. In the model, redistributional taxation is the key to derive the interdependency between
population and growth. Therefore, the importance of the future bias is demonstrated, which originatesfrom a two-sided altruism. Proposition 7 indicates that such altruistic behavior should be consideredin the dynamic analysis of public goods provision.

6

Conclusion

The present study examined the government policy of the public good provision and its e§ects on
the economic growth and welfare in an endogenous growth model with altruistic overlapping generations. In the model, the democratically elected government is subject to future bias, which has been
inherited from the existing individuals. The future bias ináuences the equilibrium government policy
and economic performance of the equilibrium. Without any bias, the government policy of the public
good provision does not a§ect the equilibrium resource allocation and therefore the economic growth.
However, the economic growth is not independent of the government policy with future bias. This
growth e§ect of the government policy provides nontrivial outcomes in welfare analysis. In particular,
the welfare in the democratically planned economy dominates that in the elderly planned economy.
Some extensions were developed to conÖrm the robustness of the results derived from the basic
analysis. In reality, the government might not have enough instruments to control the resource allocation of the economy. The in-depth analysis considered that the determination of the government
policy is separated from personal decision making. Even if separate decision making is assumed, thegovernment and individuals are still subject to future bias. Given that an externality resulting fromknowledge spillover occurs in the decentralized economy, the equilibrium growth rate and welfare in
the decentralized economy can be larger or smaller than those in the elderly planned economy depending on the degree of the knowledge spillover. Quantitative analysis shows the superiority of thedecentralized economy to the elderly planned economy in the aspects of both growth and welfare.
The analysis on voluntary provision of public goods was also examined and applied to the regional
government policy of public good provision. We supposed that there are multiple regions with theregional governments. The regional governments are elected in a similar manner to the democraticallyplanned economy. Hence, the di§erence between the planned economy and multiregions economy is
only the scale e§ect caused by the nonrivalness in consumption and the presence of the multiregions.
The number of the regions has signiÖcant e§ects on economic growth and welfare in the multiregions
economy. Furthermore, the underlying mechanisms of the impacts are based on those of the basicmodel. Therefore, the in-depth analysis indicates that the results from the basic model are robust.
Finally, future directions of this research should be described. First, incorporating distortionary-tax
Önancing with labor-leisure choice into our model is a natural way to extend our analysis. As mentioned
earlier, some studies address the similar issues with present bias. The extension of our analysis will givea di§erent policy insight, which is important to consider the policy with intergenerational conáicts.
Second, it is interesting to consider the public good in the production and allocation between two
public goods in the utility and production function. In relation to this extension, a public capital as
durable public goods will be worthwhile to investigate. These analyses will lead to important policy
implications under the democratic determination of policy and its e§ects on economic growth andwelfare. The present study provides an analytical basis of these future studies.

15Appendix
A. Proof of Proposition 1
Di§erentiating (13) with respect to k and using equations (15) and (16) yield the following:
(
)
*
+ 1!#
#
@V (k)1( )@c@V (k 0 ) @k 0
= q (^
D ; ;) + 1 (;)
c!(
+8
@k
q (D; )@k@k 0 @k
(
)
1!#
0
*
+ #
1 + A ! @k
1( )
@k 0
@k
= q (^
D ; ;) + 1 (;)
c!(
) c!(
;
1 + q (D;hiq (D; )@k%( ) #
1 + q(+;
)
where

@c
=
@k

(A1)

0

A + 1 ! @k@khi #1 :%( )1 + q(+# ; )Inserting D = D % = D^ and H = H
b into equation (18) leads to#q(+ ;3)%(3) %
H(
q(+ # ; ) + %( ) I=(A + 1 ! H) + H:
8
1 + I%(A2)

The left-hand side of this equation, H( =8, monotonically increases with H. Furthermore, we have
H( =8 = 0 (H( =8 ! 1) as H = 0 (H ! 1). The right-hand side of the equation exhibits the following
properties:
dD(H)
dH

q(+ # ;3)%(3) %
q(+ # ; ) + %( ) I
1!
< 0;
1 + I%=

q(+ # ;3)%(3) %
q(+ # ; ) + %( ) I(1 + A) > 0;
1 + I%

1(0) =
1(1 + A)=

1 + A:

These results show that there exists a unique value of H satisÖes equation (A2).
Using H = H% , equation (17) and the deÖnition of investment function provide
i% (k) = (H% ! 1)k:

H% > 1 must hold to be i% (k) > 0. By Assumption 1, we have 8 (1 + A) > 1. Then, H% > 1 holds.
Equations (11) and (16) yield
c% (k) =Ak ! i% (k)I% [Ak ! i% (k)]and g % (k) =:%1+I
1 + I%

To ensure positive consumptions, Ak ! i% (k) > 0 is needed; H% < 1 + A. We have(

H% < 1 + A ,(1 + A)(!1> 1 + A , 8 < (1 + A):
8

Assumption 1 shows that the above condition holds. Under Assumption 1, we have1!(

8 (H% )

1!(

< 8 (1 + A)

16

< 1:

(A3)

Using equations (12) and (13), we obtaini8h< 1 ! 8 (H% )1!( q (D % ; ) + 8 (H% )1!( q (D % ; ;)
1!(
V% =
(4% )
1!(
:
1 ! 8 (H% )
h
i
9
1!(
1!(
= (A + 1 ! H% )1!( k 1!(
1 ! 8 (H% )1 ( ) + 8 (H% )
1 (;)
1!(
+
(1 ! 4% )1!(;
1!1
1 ! 8 (H% )1!(

for 1 6= 1. Furthermore, the transversality condition holds if 8 (H% )to ensure the bounded lifetime utility and transversality condition.

(A4)< 1. Equation (A3) is su¢cient

B. Proof of Proposition 2
Di§erentiating D with respect to z, we obtain
1

z dD
z#
0 < 0:= !/1
D dz
1 + z# 1

By D 0 (z) < 0, D % = D( ) > D(;) = D y holds.
Using (A5), we have
*
+
0
1!#
#!1
1(z)
7D(z)1!( /
d
=
1!z #
R 0 , z # R 1:2
dz q(D; z)q(D; z)For 1 <

(A5)

(A6)

< ;, equation (A6) has a positive sign. Hence, we obtaing%gy= I% < Iy = y :%cc

In contrast, equation (A6) is negative for< ; < 1. This result leads tog%gy= I% > Iy = y :%cc
For < 1 < ;, the sign of equation (A6) changes from negative to positive. Therefore, the magnituderelation between g % =c% and g y =cy is undetermined.

C. Proof of Proposition 3
Partial di§erentiation of q(D % ; z) with respect to z is@q(D % ; z)1!(
= (1 ! D % )> 0:@zThis equation shows that q(D % ; ;) > q(D % ; ) holds. By the deÖnition of 1(z), 1(;) > 1( ) is obtained.
These two inequalities yieldq(+ # ;3)%(3) %
q(+ # ; ) + %( ) I> 1:
1 + I%
Hence, we haveD(0) =

q(+ # ;3)%(3) %
q(+ # ; ) + %( ) I(1 + A) > 1 + A:
1 + I%

17(

Since D0 (H) < 0 and D(1 + A) = 1 + A, D(H) > 1 + A = (Ho ) =8 holds for H 2 [0; 1 + A). Using
1 < H% < 1 + A; we arrive at(

D(H% ) =(H% )((Ho )>1+A=) H% > H o .
88Partial di§erentiation of D(H) with respect to 2 ishi%(3)
q(+ # ;3)%( ) ! q(+ # ; ) (A + 1 ! H) @I%
@D(H)
=
2
@2
@2
(1 + I% )
h
i
#
%(3)
q(+ ;3)%%( ) ! q(+ # ; ) (A + 1 ! H) I=
2
(1 + I% ) 21/
0
1!#(; ! ) 1 ! # (A + 1 ! H) I%
/
0(
=
R0,
1
2
(1 + ) 1 + # (1 + I% ) 21

#!1
#

R 1:

Using this equation, we obtain

@H%
@D(H)
R0,
R0,
@2
@2

#!1
#

R 1:

D. Proof of Proposition 4
The value function in the elderly planned economy is
Vy =%& % &1!(%&1!( %&1!( 1!(q D y ; ; 4y
+ 1 (;) 1 ! 4y
A + 1 ! Hy
k
;
1!(
1!1
1 ! 8 (Hy )

where
4y '

1
:
1 + IyWe have
W % = (8 + B) V % and W y = (8 + B) V y :Dividing W % by W y leads toW%
5% v(k; H% )= y:Wy
5 v(k; Hy )Note that v(k; H) < 0 (W % < 0 and W y < 0) for 1 > 1. Hence,
5% v(k; H% )W%
R1,
R 1 , W % Q W y:
5y v(k; Hy )Wy

E. Proof of Proposition 5
Di§erentiating equation (20) with respect to a and inserting equation (22) into it yields@V (a; !)@c@V (a0 ; ! 0 ) @a0= q (^
D ; ;) c!(
+8
@a
@a
@a0
@a
'
=

@a0
q (^
D ; ;) R + [q (D; ) ! q (^D ; ;)]
@a

18

)

c!( ;

(A7)

where R ' 1 + r. With a linear strategy a0 = Ha, we haveH(q (D % ; ;) (1 + @A ! H)=
+ H:
8q (D % ; )(A8)The right-hand side F (H) is monotonically decreasing in H (F 0 (H) < 0). Furthermore, we have
F (0) =

q (D % ; ;) (1 + @A)
and F (1 + @A) = 1 + @A < 1 + A:
q (D % ; )These properties show that there exists a unique solution of (A8), H? .
The value function becomes
h
i
1!(
1!(
1 ! 8 (H? )q (D % ; ) + 8 (H? )q (D % ; ;) [(R ! H? ) k + !]1!(
V0 =
1!(
1!1
1 ! 8 (H? )
h
i
1!(
? 1!(
1 ! 8 (H )1 ( ) + 8 (H? )1 (;) [(1 ! @) Ak ! !]1!(
+
:
1!(
1!1
1 ! 8 (H? )The Örst-order condition for choosing ! is
@V0
=
@!

nh
i
o
1!(
1!(
!(
1 ! 8 (H? )q (D % ; ) + 8 (H? )q (D % ; ;) [(R ! H? ) k + !]1!(!

1 ! 8 (H? )
nh
i
o
1!(
1!(
!(
1 ! 8 (H? )1 ( ) + 8 (H? )1 (;) [(1 ! @) Ak ! !]1!(

1 ! 8 (H? )= 0:This equation leads to
8
<

h

1!(

1 ! 8 (H? )i1!(

1 ( ) + 8 (H? )

1 (;)

9 #1
=

g
i
= h
= I? :
c : 1 ! 8 (H? )1!( q (D % ; ) + 8 (H? )1!( q (D % ; ;) ;In the similar way as the proof of Proposition 1, i? (k), c? (k), and g ? (k) are derived from thedeÖnition of investment and equations (9) and (11). Using these consumption and investment functions,the value function can be rewritten asi8h< 1 ! 8 (H? )1!( q (D % ; ) + 8 (H? )1!( q (D % ; ;)
1!(
V? =
(4? )
1!(
:
1 ! 8 (H? )
h
i
9
1!(
1!(
= (A + 1 ! H? )1!( k 1!(
1 ! 8 (H? )1 ( ) + 8 (H? )
1 (;)
? 1!(
+
(1
!
4
)
;
1!(
;
1!1
1 ! 8 (H? )
where

4? '

1
:
1 + I?

1 < H? < 1 + @A is needed to ensure positive values of growth rate and consumptions. The bounded
1!(
1!(
1!(
lifetime utility and transversality condition hold because 8 (H? )< 8 (1 + @A)< 8 (1 + A)<?
1 under Assumption 1. Furthermore, Assumption 2 leads to H > 1.

19

F. Proof of Proposition 6
We have(((I% ) ! (I? )

h

1!(

i

1!(

1 ( ) + 8 (H? )
1 (;)
1( )
h
i
!
%
1!(
1!(
q (D ; )1 ! 8 (H? )q (D % ; ) + 8 (H? )q (D % ; ;)=1 ! 8 (H? )

1!(

=

,

8 (H? )
[1 ( ) q (D % ; ;) ! q (D % ; ) 1 (;)]
nh
i
o R0
1!(
1!(
q (D % ; ) 1 ! 8 (H? )q (D % ; ) + 8 (H? )q (D % ; ;)q (D % ; ;)
1 (;)
R
,
q (D % ; )1( )Therefore, we obtain

#!1
#

Q 1:

g%
g?
= I% R I? = ? ,
%
c
c

#!1
#

Q 1:

Equations (A2) and (A8) yield

F (H) ! D(H) =q (D % ; ;) (1 + @A ! H)!
q (D % ; )withF (0) ! D(0) =hq(+ # ;3)%(3)
q(+ # ; ) ! %( )i'q(+ # ;3)%(3)
q(+ # ; ) + %( )

1+

%( )
q(+ # ; )hi #1 )%( )q(+ # ; )i #1

(1 + A ! H)

I% q (D % ; ) ! q (D % ; ;) (1 + I% ) (1 ! @) Aq (D % ; ) (1 + I% )andF (1 + @A) ! D(1 + @A) = !

hq(+ # ;3)%(3) %
q(+ # ; ) + %( ) I

1 + I%

Di§erentiating [F (H) ! D(H)] with respect to H leads tod [F (H) ! D(H)]dH

hq(+ # ;3)i

(1 ! @) A

%(3)

R0

< 0:

%

q (D % ; ;)q(+ # ; ) + %( ) I= !
+q (D % ; )1 + I%hi%(3)
q(+ # ;3)%#!1%( ) ! q(+ # ; ) I
#
=
R0,
R 1:
1 + I%(A9)

When @ = 1, we have
F (0) ! D(0)=F (1 + A) ! D(1 + A)=h

q(+ # ;3)%(3)
q(+ # ; ) ! %( )1 + I%

i

I%

R0,

#!1
#

Q 1;

0:

From these results and equation (A9), we obtain H? R H% ,#!1#

Q 1 when @ = 1. We have

@F (H)
> 0:
@@
#!1

#
A decrease in @ reduces [F (H) ! D(H)]. Hence, if< 1, there exist the critical value of @ that?
%change the magnitude relationship between H and H . We have

H?
H?

R H% , @ R @% for& H% for

#!1
#

20

- 1:

#!1
#

< 1;

On the other hand, we obtain@ [F (H) ! D(H)]
@D(H)
=!
Q0,
@2
@2
#!1

#!1
#

R 1:

#
When
- 1, an increase in 2 reduces [F (H) ! D(H)]. Since F (H) < D(H), it means that H? & H%#!1#holds for 2 > 0. If< 1, a raise in 2 increases [F (H) ! D(H)] for given @. Initially, @ = @% leads?
%
to H = H . Depending on 2, the critical value of @ should be changed: large 2 makes @% small. Thus,
we have
@@%
< 0:
@2

21

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24

!" # , # $
#$

E

1+( !

F
!" #

#) #∗
Figure 1. Equilibrium

1+(

#

-10

-15
W

W*
W†

-20

-25
0.0

0.5

1.0

1.5

2.0

g

Figure 2. The preference for public goods and welfare level86
W* -W†

4
2
0

0.1
0.2

0.5
0.4

0.3
0.3
m

l

0.4
0.2
0.1

0.5

Figure 3. The degree of altruism and welfare deference0.002k¯ - k*0.000-0.002g=0.5-0.004g=0.01-0.006-0.008
0.980

0.985

0.990

0.995

1.000

a

Figure 4a. Knowledge spillover and growth difference

0.3

k¯ - k†

0.2
0.1
0.0
-0.1
-0.2
0.0

0.2

0.4

0.6

0.8

a

Figure 4b. Knowledge spillover and growth difference

1.0

0.015W¯ -W*

0.010

0.005

g=0.5
g=0.010.000-0.005
0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

a

Figure 5a. Knowledge spillover and welfare difference

1.5
1.0

W¯ -W†

0.5
0.0
-0.5
-1.0
0.3

0.4

0.5

0.6

0.7

0.8

0.9

a

Figure 5b. Knowledge spillover and welfare difference

1.0