Discussion Papers In Economics
And Business

Risk Sharing and Growth in the Gifts Economy
Atsue Mizushima

Keiichi Koda

Discussion Paper 07-02

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRisk Sharing and Growth in the Gifts Economy
Atsue Mizushima

Keiichi Koda

Discussion Paper 07-02

February 2007

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRisk Sharing and Growth in the Gifts Economy∗
Atsue Mizushima†,

Keiichi Koda‡

February, 2007

Abstract
We consider a two-period overlapping generations model where agents face the uncertainty of intergenerational transfers from their children. To avoid this kind of risk, agents
have an incentive to share the risk within the same generation. However, there exists an
information asymmetry about the realization of the old period’s income between the insurance company and old agents. By analyzing economies with and without risk sharing, wefind that risk sharing decreases the rate of economic growth and accelerates social welfare
when the rate of social time preference is sufficiently large.

JEL Classification Numbers: D81, D82, G22, O40
Keywords: gifts economy, risk sharing, information asymmetry, economic growth,
overlapping generations∗

We are grateful to Taro Akiyama, Hiroshi Fujiu, Koichi Futagami, and Tsunao Okumura for their helpfulcomments and suggestions. All remaining errors are naturally our own.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka Osaka 560-0043, Japan;
E-mail:cg089ma@srv.econ.osaka-u.ac.jp‡
International Graduate School of Social Science Graduate School of Economics, Yokohama National University, 79-4, Tokiwadai, Hodogaya, Yokohama Kanagawa 240-8501, Japan; E-mail:koda@ynu.ac.jp

1

Introduction

It has been widely recognized that intergenerational transfers account for a significant part ofaggregate saving (e.g., Kotlikoff (1988)), and they have been modeled in a number of different
ways since the seminal works by Becker (1965) and Barro (1974), which explain transfers as
arising from altruistic behavior. For example, under fully altruistic behavior, agents’ utility
ultimately depends on the utility of all of their descendants (e.g., Becker and Barro (1988)
and Barro and Becker (1989)). Under semialtruistic behavior, agents derive utility from theconsumption of their offspring and/or the consumption of their parents, or even simply fromthe sizes of the transfers they make (e.g., Bernheim and Ray (1987)). From a no-altruism
perspective, transfers may arise out of pure self-interest because of the perceived mutual
benefits of exchange between progenitors and progeny (e.g., Cox (1987)). Alternatively, theexistence of bequests may simply be an accident as in Abel (1985), and the existence of gifts
may be the result of social norms or customs that obligate children to provide some form ofmaterial support to their parents during old age, as argued by Morand (1999).
Despite the substantial research on intergenerational transfers, the uncertainty of intergenerational transfers has not been paid much attention. This is because in the previous
research, altruism is formalized by the assumption of fully altruistic or semialtruistic behavior. As Blackburn and Cipriani (2005) showed, however, the direction of intergenerationaltransfers shifts from children to parents to parents to children according to the state of economic development.1 In this process, it is thought that uncertainty may exist as to whether
parents have received gifts from their children. In fact, the percentage of older agents who livewith their children or grandchildren decreases with economic development: it is 75% in developing countries, whereas it is only 27% in developed countries (see United Nations (2006)).
Thus, this paper tries to complement the existing literature by studying the uncertainty ofintergenerational transfers, especially of gift transfers.2
To analyze these issues, we employ a two-period overlapping generations model, and assume that young agents are divided into two types according to their attitude to their parents.
1
Blackburn and Cipriani (2005) showed that at low levels of development, fertility is high and the flow of
net intergenerational transfers is from children to parents. At high levels of development, fertility is low andthe flow of net transfers runs from parents to children.
2
Studies of gifts economies have been conducted by O’Connell and Zeldes (1993) and Wigger (2001).
O’Connell and Zeldes (1993) found that intergenerational transfers of gifts from children to parents are dynamically inefficient. However, the result of Wigger (2001)’s investigation was that if growth is endogenous, thegift economy is dynamically efficient.

1One group constitutes agents who are dutiful or “good” to their parents and the other constitutes selfish or “bad” agents. Good children are different from bad ones as they care about
their parents and derive utility from giving them a part of their wage income as a gift, whereas
bad children do not. We assume that the type of children is uncertain ex ante. Thus, income
during the old period is stochastic. If each agent is risk averse, he or she has an incentive
to share the risk relating to income; that is, he or she may not receive any intergenerationaltransfer from their children, within the same generation.3 Thus, the agents who have theability to monitor the contracts agents act as insurance companies. Each agent can insure
himself or herself against fluctuations of income by making a contract with an insurance company. Our purpose is to analyze how the risk sharing methods that cover the uncertainty ofintergenerational transfers affect the rate of economic growth and welfare.
We begin by considering the economy without risk sharing. In this regime, the intergenerational transfer is subject to uncertainty. Thus, each agent has precautionary motive forsaving, as in Leland (1968), Sandmo (1970) and Kimball (1990).
Next, we consider the economy with risk sharing. If the insurance company can observe
the income of old agents, that is, whether their children are dutiful to their parents, theinsurance company offers the full insurance to old agents. However, there exists information
asymmetry between the insurance company and old agents about the income realization of
the old agents. Thus, we first construct the schedule of risk sharing by using the costly state
verification theorem, as in Townsend (1979), Williamson (1986), Bernanke and Gertler (1989),
and Bhattacharya (1997), and verify the lifecycle activity under the risk sharing schedule.
By analyzing the model, we find the following: the economy grows at a positive constantrate under both regimes. Second, as to the effect of risk sharing, risk sharing can smooththe fluctuation of income, which decreases the precautionary motive for saving, which in turnresults in a lower level of economic growth. In addition to this, an increase in the numberof parents with selfish children decreases the rate of economic growth. Finally, risk sharing
accelerates the welfare level of the current generation, whereas that of the future generation
deteriorates. Thus, its effect on the social welfare level depends on the social time preference
rate: when the social time preference rate is sufficiently high (low), the risk sharing regime
accelerates (reduces) the social welfare level.
The remainder of this paper is organized as follows. Section 2 sets up the basic model.
3

Krueger and Kubler (2002) stated that the role of the social security system plays a role in facilitating theallocation of aggregate risk among generations.

2Section 3 derives the economic equilibrium without risk sharing. Section 4 sketches the schedule of risk sharing, and derives the economic equilibrium with risk sharing. Section 5 examines
the relation between risk sharing and economic growth. Section 6 analyzes welfare. Section 7
contains some concluding remarks.

2

The Model

Consider a discrete time economy populated by an infinite sequence of two-period-lived overlapping generations and one-period-lived nonoverlapping generations. Time is indexed by
t = 1, 2, · · · . At each date, two sets of new generations, each consisting of a continuum of
agents with unit measure, are born.
Two-period-lived agents
Agents of generation t ≥ 1 live for periods t and t + 1. They are endowed with one unit oflabor when young and no labor when old. Within each generation, agents are divided into
two types according to their attitude toward their parents. A fraction α of young agents areof type G, who are dutiful or “good” toward their parents. Type B agents, who constitute afraction 1 − α of each generation, are undutiful or “bad” toward their parents. Type G agentsdiffer from type B agents in that they care about their parents and derive satisfaction fromgiving them a gift, whereas type B agents do not. We assume that each agent of generationt ≥ 1 has a utility function of the form:u(ctt , qt , ctt+1 ; ψt ) = ψt ln ctt + (1 − ψt ) ln qt + ln ctt+1 ,(1)where ctt is first-period (date t) consumption, qt is a (date t) gift to the agent’s parents, ctt+1 issecond-period (date t+1) consumption, and ψt is an index indicating an agent’s type that takes
a value of either 0 or 1 according to whether the agent is type G or type B. ψt , which is realized
at the beginning of date t immediately after agents of generation t are born, is distributed
independently and identically (across agents and time) with the probability distribution:(
1
with Prob. α
ψt =
(2)
0
with Prob. 1 − α,where 0 < α < 1. In what follows, agents as consumers will also be referred to as “households”.
At each date there is a single consumption good. Any old agent has access to a production
technology given by:
yt = Aktθ (Kt lt )1−θ ,3(3)

where A indicates the parameter representing the technology level and Kt is the aggregate
capital stock in the economy such that there is an externality in production of the typeconsidered by Romer (1986). At each date t ≥ 1, old agents acting as producers, whom wewill call “firms”, have the accumulated capital stock kt , which is combined with lt units ofyoung labor in order to produce output yt . We assume that capital depreciates completely inthe process of production.
Finally, at date t = 1, there is an initial old generation (generation 0). These old agentsare each endowed with kt > 0 units of capital.
One-period-lived agents
At each date t ≥ 1, in addition to two-period-lived agents, agents are born who are risk neutral
and live only one period. We will call these one-period-lived agents “insurers”, for reasons to
be made clear. When risk sharing is allowed among (old) households within each generation,
insurers set up insurance companies. We assume that the type of each (young) household andwhat may be called “the state of a (young/old) household”—how each young household spends
his or her wage income and how much each old household spends on consumption—are private
information; these are costlessly observable only to himself or herself and his or her parents.
Following Townsend (1979), Williamson (1986), Bernanke and Gertler (1989), Bhattacharya(1997) and others, however, we assume that insurers have access to a costly state verification(monitoring) technology: an insurer can expend effort to learn the state of any household—inparticular, how much an old household spends for his or her old age consumption. It requirese ≥ 0 units of effort to observe the state of any one household. Note that the state of an oldhousehold is synomynous with the information about which type of child the old agent has.
Each insurer, who is endowed with an unbounded amount of effort, has a utility functionof the form:v(xt , et ) = xt − et ,(4)

where xt is consumption at date t and et is effort expended at date t. At each date t ≥ 1, whenrisk sharing is permitted among (old) households within each generation, insurance companies
are set up that are owned and operated by insurers.

4

2.1

Behavior of agents

Both producers (firms) and households behave competitively in goods and factor markets,taking the wage rate and the rental rate as given.
Firms
There is no loss of generality in assuming that, at each date t ≥ 1, there is a single competitive
firm, which produces the total output of this economy using the production technology (3)by hiring young agents at wage rate wt and renting capital at the rental rate Rt from old
agents. As perfect competition prevails in both goods and factor markets and because kt = Kt
and lt = 1 are in equilibrium, the wage rate and the rental rate are given by the marginal
productivity relations:wt = (1 − θ)AKt1−θ ktθ lt−θ = (1 − θ)Akt ,(5)

Rt = rt + 1 = θAKt1−θ ktθ−1 lt1−θ = θA,(6)where rt is the rate of interest from date t − 1 to date t.
Households
At each date t ≥ 1, each young agent maximizes the expected value of (1) subject to thebudget constraint:
maxUt ≡ Et u(ctt , qt , ctt+1 ; ψt ) = ψt ln ctt + (1 − ψt ) ln qt + Et ln ctt+1 ,ctt ,qt ,ctt+1

s.t.
ψt ctt + (1 − ψt )qt +

ctt+1
qt+1
= wt + (1 − ψt+1 ).
1 + rt+1
1 + rt+1(7)(8)

When young, each agent earns the wage income by working for a firm, spends it either
for his or her own consumption (if he or she is of type B), or gives some as a gift to his or
her parents (if the agent is of type G), and saves the rest for his or her old age. When old,an agent uses the proceeds of his or her savings and a gift (if any) from his or her child to
support his or her old age.
A few observations are in order. First, each household’s second-period income is subject
to uncertainty because whether the agent receives a gift from his or her child depends on therealization of the random variable ψt+1 ; it differs depending on which type of child is born inthe family at date t + 1. The expectation in (7) is taken with respect to ψt+1 , the probability

5distribution of which is given by (2). Second, in maximizing the expected utility in (7), eachhousehold takes as given a gift from his or her child, as well as the wage rate and the realinterest rate. Third, the first observation above also indicates that there is an incentive for risk
sharing among agents within each generation. The budget constraint (8), however, assumes no
intragenerational risk sharing. As a benchmark, we first describe a regime where risk sharing
is prohibited. We will then consider one where risk sharing is permitted.
Insurers
There is no role for insurers to play in the economy without risk sharing. Every insurer isinactive, consuming nothing and expending zero units of effort. The autarkic level of utility
for each insurer is zero.

3

Equilibrium without Risk Sharing

Let st ≡ wt − [ψt ctt + (1 − ψt )qt ]. Then, st is the saving of an agent of generation t ≥ 1. The
first-order condition for the maximization problem (7)-(8) is rearranged to give:2(1 + rt+1 )s2t + [(2 − α)qt+1 − (1 + rt+1 )wt ]st − (1 − α)qt+1 wt = 0.

(9)Note that the solution st to (9) implies that a young agent’s saving does not depend on his or
her type. In addition, as ψt ctt + (1 − ψt )qt = wt − st , we have:ctt = qt = wt − st ,(10)

for each t ≥ 1.
Definition 1 An equilibrium will satisfy the following three conditions at each date:• Households and firms optimize, taking the wage rate and the rate of interest; that is, (5),(6), and (9) hold.
• The markets for goods, capital, and labor clear; that is, st = kt and lt = 1 hold.
• The gift qt+1 from generation t + 1, which is taken as given by each agent of generationt ≥ 1 in his or her maximization problem (7)-(8), is realized.
To express equilibrium in this economy in a compact manner, we note, first, that the third
condition together with the second condition and (5) imply:
qt+1 = wt+1 − st+1 = (1 − θ)Akt+1 − kt+2 .
6(11)

We then substitute (5), (6), and (11) as well as st = kt+1 into (9) to obtain:
2[2θA + (1 − α)(1 − θ)A]kt+1− [(1 − α)(1 − θ)2 A2 + (1 − θ)θA2 ]kt kt+1−(2 − α)kt+1 kt+2 + (1 − α)(1 − θ)Akt kt+2 = 0.

(12)

An equilibrium path for kt is one in which (12) is satisfied and the resource constraint:
kt+1 ≤ (1 − θ)Akt(13)

is met for each t ≥ 1. Let gt+1 ≡ kt+1 /kt be the (gross) growth rate of the capital stock fromdate t to date t + 1. For our purposes, it will be more useful to work with a transformed
version of (12) and (13):2 − α(1 − θ)A−αθ(1 − θ)A2+,2−α
(2 − α)[(2 − α)gt+1 − (1 − α)(1 − θ)A]
gt+1 ≤ (1 − θ)A.
gt+2 =(14)(15)

The phase diagram for (14) is shown in Figure 1. The phase line is a rectangular hyperbola
with its asymptotes given by gt+1 = (1−α)(1−θ)A/(2−α) and gt+2 = [2−α(1−θ)A]/(2−α).
Setting gt+2 = gt+1 = b in (14) and rearranging yields:(2 − α)b2 − (2αθ − 2α − θ + 3)Ab + (1 − θ)(1 − α + αθ)A2 = 0.

(16)

We define the left-hand side of (16) as h(b). Then, the roots of the quadratic equationh(b) = 0 give the fixed (stationary) points of (14), which are given by:)(p
3 − θ − 2α(1 − θ) + (1 + θ)2 − 4αθ(1 − θ),b1 = A2(2 − α)()
p3 − θ − 2α(1 − θ) − (1 + θ)2 − 4αθ(1 − θ)b1 = A.
2(2 − α)

(17)
(18)

However, an examination of Figure 1 indicates that the dynamic system (14)–(15) possesses aunique stationary solution b2 , and that this stationary solution is the economy’s saddle-path
stable equilibrium. The following proposition formalizes this observation and its proof is given
in Appendix A.
[Figure 1 here]
Proposition 1 Suppose that:p3 − θ − 2α(1 − θ) + (1 + θ)2 − 4αθ(1 − θ),A>
2(1 − θ)[1 − α(1 − θ)]7(19)Then, there exists a unique equilibrium such that kt = k1 (Aγ)t−1 , Aγ > 1, for each t ≥ 1,√
3−θ−2α(1−θ)− (1+θ)2 −4αθ(1−θ)
where: γ =.
2(2−α)
Proposition 1 makes it clear that if the productivity parameter A is sufficiently large, theeconomy without risk sharing grows at a positive (net) constant rate of Aγ − 1. We cansee from (3), (5), and (10) that output yt , wages wt , first-period consumption ctt , gifts qt ,and savings st also grow at the same constant (gross) rate Aγ. The economy’s growth ratedepends on the parameters A, θ and α. In particular, the economy grows faster if it has alower fraction of type G agents (see Section 5 below).

4

Risk Sharing

As mentioned above, all households are risk averse and have an incentive to insure themselves
against the risk of having type B children, who do not care about their parents. At the
beginning of each date, before children are born, insurance companies announce insurance
contracts, letting old households choose among them. Contract announcements are madetaking the announced contracts of other insurance companies as given.

4.1

Insurance contracts

At the beginning of date t + 1, insurance companies offer old households contracts of the form
BG
(pG
t+1 , pt+1 , ϕt+1 ) such that an old agent will pay a premium pt+1 to an insurance company if

a type G child is born, in return for which he or she will be paid pB
t+1 in the event that a type
B child is born. An optimal contract between the insurance company and an old householdmust take into account the possibility that the latter may lie about the type of child who wasborn; he or she may misreport the type G child as a type B child. One can show that, in a
two-state case like ours, under the optimal contract no verification (monitoring) occurs when
it is announced that a type G child was born. Thus, the insurance company monitors only
when the old household declares that a type B child was born. ϕt+1 is the penalty imposedat date t + 1 on an old household who misreports the type of his or her child.
B
Taking rt+1 , st and qt+1 as given, each insurance company chooses (pG
t+1 , pt+1 , ϕt+1 ) to

maximize the expected utility of the old household subject to a set of constraints:
max

B
pG
t+1 ,pt+1 ,φt+1

B
Et [ln ctt+1 ] = α ln [(1 + rt+1 )st + qt+1 − pG
t+1 ] + (1 − α) ln [(1 + rt+1 )st + pt+1 ],

(20)
8

s.t.
ln [(1 + rt+1 )st + qt+1 − pGt+1 )] ≥ ln [(1 + rt+1 )st + qt+1 − ϕt+1 ],(21)

qt+1 ≥ pG
t+ ,

(22)

qt+1 ≥ ϕt+1 ,

(23)

B
αpG
r+1 − (1 − α)(pt+1 + e) = 0(24)Bα ln[qt+1 + (1 + rt+1 )st − pG
t+1 ] + (1 − α) ln[(1 + rt+1 )st + pt+1 ]≥ α ln [qt+1 + (1 + rt+1 )st ] + (1 − α) ln [(1 + rt+1 )st ].

(25)

Constraint (21) is the truth-telling or incentive constraint on the old household; it requiresthat the contract be such that the old household has no incentive to misrepresent the childas type B when he or she is in fact type G. Constraints (22) and (23) are the limited liability
constraints, which state that neither the premium nor the penalty cannot exceed the amountof the gift that the old household receives from his or her child. (24) is the zero-profit conditionon the insurance company that arises from competition among insurance companies; it takes
into account the fact that the insurance company verifies that the truth has been told whenever
it is reported that children of type B were born4 . (25) is the participation constraint of anold household, who would not be willing to accept the contract unless he or she would be noworse off by taking out insurance than he or she would be otherwise. The solution to the
maximization problem (20)–(25) yields the optimal contract.
Proposition 2 At date t + 1, an optimal insurance contract exists if and only if:e≤1{(1 + rt+1 )st + αqt+1 − [(1 + rt+1 )st + qt+1 ]α [1 + rt+1 )st ]1−α }.
1−α

(26)

B
The optimal contract (pG
t+1 , pt+1 , ϕt+1 ) is such that:pG
t+1 = (1 − α)qt+1 + (1 − α)e,

(27)

pB
t+1 = αqt+1 − (1 − α)e,(28)

ϕ ∈ [(1 − α)(qt+1 + e), qt+1 ].

(29)

4
The left-hand side of (23) represents the insurance company’s expected profits per contract. Insurancecompanies are willing to offer any number of contracts as long as they make nonnegative expected profits. Free
entry into the insurance industry means that insurance companies earn zero expected profits in equilibrium.
The law of large numbers, however, implies that an insurance company that makes a large number of contracts
makes zero profits (rather than zero expected profits) with a probability of one in equilibrium. Insurers, whoG
run insurance companies, consume αpG
t+1 − (1 − α)pt+1 units of the consumption good while supplying (1 − α)eunits of effort.

9

Proof. See Appendix B.
(26) indicates that, given rt+1 , st , and qt+1 , the verification cost must be small enough if
old households are to insure themselves at date t + 1. The insurance contract provides oldhouseholds with the opportunity to insure themselves completely against the risk of having nogifts (no old-age care) from their children. The insurance company collects premiums summingto (1 − α)(qt+1 + e) from old households with type G children and pays out compensation
amounting to αqt+1 − (1 − α)e to those with type B children. In so doing, the insurance has
the effect of redistributing the gifts from generation t + 1 equally among old households ofgeneration t. It is easy to see from (27) and (28) that, with a given level of gift, the insurance
is more costly if the fraction of type G agents is smaller and the verification cost is greater.

4.2

Equilibrium with risk sharing

At each date t ≥ 1, each young agent in the risk sharing regime maximizes the expected valueBof (1) in anticipation of the contract (pG
t+1 , pt+1 , ϕt+1 ) being placed at date t + 1. That is, heor she solves the maximization problem:
maxUt = ψt ln ctt + (1 − ψ) ln qt + Et ln ctt+1 ,ctt ,qt ,ctt+1

(30)

s.t.
ψt ctt +(1−ψt )qt +

ctt+1
qt+1
1
G
wt = (1−ψt+1 )+[ψt+1 pB
t+1 −(1−ψt+1 )pt+1 ], (31)
1 + rt+1
1 + rt+1 1 + rt+1

B
where pG
t+1 and pt+1 are given by (27) and (28), respectively.

The first-order condition for (30)–(31) can be expressed in terms of saving as:
·
¸
1
αqt+1
(1 − α)e
st =
wt −
+
.
2
1 + rt+1
1 + rt+1(32)

To find an equilibrium, we substitute the equilibrium conditions (5), (6), (11), and st =kt+1 into (32) to obtain:
αkt+2 − [2θ + α(1 − θ)]Akt+1 + (1 − θ)θA2 kt = −(1 − α)e.

(33)

An equilibrium path for kt is one in which (33) is satisfied and the resource constraint (13)is met for each t ≥ 1. The general solution to the second-order linear difference equation (33)
is given by:
kt = Z1 bt1 + Z2 bt2 + c,
10

(34)

where:()α2 (1 − θ)2 + 4θ2
b1 = A,2α()
pα(1 − θ) + 2θ − α2 (1 − θ)2 + 4θ2
b2 = A,2αα(1 − θ) + 2θ +c=p−(1 − α)e,α − [2θ + α(1 − θ)]A + (1 − θ)θA2

(35)
(36)
(37)

with Z1 and Z2 as arbitrary constants. For later reference, we define the function f (b) as:f (b) ≡ αb2 − [2θ + α(1 − θ)]Ab + (1 − θ)θA2 .

(38)

Then, f (b) = 0 is the characteristic equation of (33), and (35) and (36) are its two (distinct)roots (characteristic roots).
Before stating the main result in this subsection, we prove the following lemma.
Lemma 1 Suppose Aξ > 1, f (1) > 0, and 1 − θ > ξ. We define the function F : [0, δ̄¯] → R
by:F (δ) = ln [α(1 − ξ) + (1 − α)θ − (1 + αQ)δ] − α ln(1 − ξ − Qδ) − (1 − α) ln θ,(39)¯
and Q ≡ fAξ
where δ̄¯ ≡ α(1−θ−ξ)1+αQ(1) . Then, there exists a unique δ̄ ∈ (0, δ̄ ) with F (δ̄) = 0.
Furthermore, F (δ̄) > 0 if and only if δ < δ̄.
Proof. As
F (0) = ln[α(1 − ξ) + (1 − α)θ] − α ln(1 − ξ) − (1 − α) ln θ > 0
and

n
h
³
F (δ̄¯) = α ln θ − ln (1 − ξ) 1 −

αQ ´
αQ io
+θ
< 0,
1 + αQ
1 + αQ
it suffices to show that F 0 (δ) < 0 for each δ ∈ [0, δ̄¯]. However,F 0 (δ) =−(1 + αQ)(1 − ξ) + αQ[α(1 − ξ) + (1 − α)θ] + (1 − α)(1 + αQ)Qδ
,
D(δ)

(40)

where D(δ) ≡ [α(1 − ξ) + (1 − α)θ − (1 + αQ)δ](1 − ξ − Qδ). As D(δ) > 0 for each δ ∈ [0, δ̄¯]and because the numerator on the right-hand side of (40) is increasing in δ, it is sufficient toshow that F 0 (δ̄¯) < 0, which is certainly true because F 0 (δ̄¯) = −(1 − ξ)/D(δ̄¯) < 0.
Proposition 3 Suppose that:pα(1 − θ) + 2θ + α2 (1 − θ)2 + 4θ2,
A>
2θ(1 − θ)11(41)and that:
e≤

δ̄Ak1
.
1−α

(42)

Then, there exists a unique equilibrium such that kt = (k1 − c)(Aξ)t−1 + c, Aξ > 1, c < 0, for√α(1−θ)+2θ− α2 (1−θ)2 +4θ2
each t ≥ 1, where ξ ≡.
2αProof. See Appendix C.
Suppose that:e≤δ̄Akt+11−α(43)holds at date t + 1. Then, old households at that date have an incentive to participate in risk
sharing. When (42) is satisfied, (43) will also be satisfied at each t ≥ 1 (see the Appendix).
If the monitoring cost is high enough to violate (42), the participation condition for oldhouseholds will not be met, at least at date t = 1. The equilibrium path in such an economy
will not be the one in the proposition. However, as the right-hand side of (43)—the criticalvalue of e above which it is too costly for old households to insure themselves against the risk
at date t + 1—is increasing in the capital stock, (43) will be met (as long as e is finite) eventhough it is violated initially. From that time on, the economy will be on the path describedin the proposition.
Suppose that (42) holds. Then, the economy grows at a (gross) rate of gt+1 ≡ kt+1 /kt =[(kt − c)Aξ + c]/kt = Aξ − c(Aξ − 1)/kt at each date t ≥ 1. Note that the economy’s growthrate is no longer constant over time; because c < 0, it is higher than Aξ, but it converges
monotonically to the latter over time. It is easy to see from (3) and (5) that both yt and wt
grow at the same rate as kt , whereas ctt , qt , and st do not.
Before proceeding, it will be useful to have the following special case of Proposition 2
as another benchmark from which to evaluate the effects of private information on economic
growth and welfare.
Proposition 4 Suppose that A satisfies (41) and that e = 0. Then, there exists a uniqueequilibrium such that kt = k1 (Aξ)t−1 for each t ≥ 1.
Proposition 4 immediately follows from Proposition 3 by setting e = 0. This assumption
is equivalent with assuming that insurers have full information about the state of any (old)household. From Proposition 3, the economy grows at a positive constant rate of Aξ − 1.

12

Another benchmark regime, the no-risk-sharing regime, could also be viewed as a specialcase, a regime that will prevail if the monitoring technology is prohibitively (or infinitely)
costly (i.e., e → ∞).

5

Risk Sharing and Economic Growth

The aim in this section is twofold. One is to consider what role risk sharing plays in accelerating
or decelerating economic growth. As monitoring costs determine the structure of risk sharing,
this question amounts to asking how differing monitoring costs will cause the economy togrow faster or slower. The other aim is to examine what effects young agents’ altruism toward
their parents have on economic growth. In particular, we ask ourselves whether the economy’s
growth rate will be higher or lower if a smaller fraction of young agents care about theirparents in old age.

5.1

Risk sharing regimes and economic growth

Let superscripts n, a, and f denote, respectively, “the no-risk-sharing regime or the economy
with e → ∞”, “the risk sharing regime with asymmetric information or the economy withe < ∞”, and “the risk sharing regime with full information or the economy with e = 0”. Our
first result is somewhat surprising: monitoring costs have positive instead of negative effects
on economic growth.
Lemma 2 Let A meet the conditions in both Propositions 1 and 3 and let γ and ξ be defined
as in Propositions 1 and 3. Then, the following statements hold:(a) 1 < Aξ < Aγ < A(1 − θ);
d(Aγ)
d(Aξ)
(b)
< 0,
< 0.
dα
dα
Proof. (a) The first inequality follows from Propositions 1 and 2. For proofs of the second
and third inequalities, see Appendix D.
(b) As Aγ is the smaller root of h(Aγ; α) = 0, we use the implicit function theorem anddifferentiate h(Aγ; α) = 0 to obtain:
A[Aγ − A(1 − θ)]2
d(Aγ)
=
.
dα
(2 − α)2Aγ − (2αθ − 2α − θ + 3)

(44)

Substituting (18) into the denominator on the right-hand side of (44), the denominator becomes negative. Thus, dAγdα < 0.
13

As Aξ is the smaller root of f (Aξ; α) = 0, we also have that:d(Aξ)
Aξ[Aξ − (1 − θ)A]=.
dα
[2θ + α(1 − θ)]A − 2αAξ(45)

From the resource constraints of (15), the numerator on the right-hand side of (45) is negative.
Substituting (36) into the denominator on the right-hand side of (45) means the denominatorbecomes positive, and, thus, dAξ
dα < 0.
We define the average (gross) growth rate ḡt+1 from date t = 1 to date t + 1 by:ḡt+1 ≡ (kt+1 /kt )1/t = (g2 , g3 · · · gt+1 )1/t ,where, recall, g2 = k2 /k1 , g3 = k3 /k2 , · · · , gt+1 = kt+1 /kt . That is, ḡt+1 measures the geometric average of the growth rates of capital stock from the initial date through date t + 1. As
kt+1 = k1 (ḡt+1 )t , ḡt+1 also measures the level of capital stock outstanding at the beginningof date t + 1. The following proposition compares the three regimes in terms of the average
growth rates.
Proposition 5

n > ḡ f(a)ḡt+1
t+1 for each t ≥ 1;fa(b)ḡt+1> ḡt+1andadḡt+1de > 0 for each t ≥ 1;

n > ḡ ana(c)ḡt+1
t+1 and ḡt+1 − ḡt+1 is increasing with t for each t ≥ t̄, where 1 ≤ t̄ < ∞.
fnProof. (a) As Propositions 1 and 2 imply that ḡt+1= Aγ and ḡt+1= Aξ, the result follows

immediately from Lemma 2.
fa(b) It suffices to show that kt+1
> kt+1andadkt+1de > 0. As Propositions 3 and 4 imply that:

f
a
kt+1
= (k1 − c)(Aξ)t + c = k1 (Aξ)t − c[(Aξ)t − 1] > k1 (Aξ)t = kt+1,

where, recall, c = −(1 − α)e/f (1) < 0 (Proposition 3), the result follows.
n = k (Aγ)t and k at(c) As kt+11
t+1 − c = (k1 − c)(Aξ) from Propositions 1 and 3, we have:h k ³ γ ´t
i
1
n
a
a
kt+1
− kt+1= (kt+1− c)− 1 − c,k1 − c ξih k ³ γ ´t1
− 1 − c.
= (k1 − c)(Aξ)tk1 − c ξ1
As c < 0, Aξ > 1, and γ > ξ from Proposition 3 and Lemma 2, it follows that if k1k−c

³ ´t
γ
ξ

≥1

k1
n
a , and k n − k aor t ≥ t̄ ≡ ln(kln1 −c)−ln, then kt+1
> kt+1
t+1
t+1 is increasing with t for each t ≥ t̄,
γ−ln ξ

from which the result immediately follows.
Although, as noted above, the no-risk-sharing regime can be thought of as a special case of
our economy, it differs from the risk sharing regimes in that each household has an uncertain14

(or stochastic) lifetime income. The extra saving caused by future income being subject
to uncertainty is known as precautionary saving. Precautionary saving is associated withconvexity of the marginal utility function or a positive third derivative of the utility function(see, among others, Leland (1968), Sandmo (1970), and Kimball (1990)). Households with alog utility function have a positive motive for precautionary saving. A rise in uncertainty about
future income encourages private saving and capital accumulation. Given the technology withaggregate spillovers coming from firms’ production activities, this effect manifests itself as
increased economic growth. That is the mechanism behind Proposition 5(a).
However, the logic leading to part (b) of Proposition 5 differs somewhat from that behindpart (a) because no risk is involved in the risk sharing regimes. As (32) makes clear, with agiven level of gift, a higher verification cost means a lower second-period income, motivatinghouseholds to respond to reduced second-period incomes by saving more. Therefore, the mechanism underlying part (b) is not so much a precautionary- saving motive as a consumptionsmoothing motive; put differently, it is associated with a negative second derivative rather
with than a positive third derivative of the utility function.
Given the result in part (c) of Proposition 5, we are now able to rank the three regimes
fnain terms of (average) growth rates: ḡt+1> ḡt+1> ḡt+1
for each t ≥ t̄. Whatever the growth

rates are in the early stages, the no-risk-sharing regime will eventually dominate the risk
sharing regimes. We will explore some welfare implications of these differing growth rates in
the following section. Before turning to the next section, it is worthwhile for our purposes
to do one more comparative statics exercise. As it will turn out, a lower (higher) share oftype G agents leads to higher (lower) economic growth across regimes with and without risk
sharing. The intuition behind this result is as follows: households have a strictly concave
utility function, motivating them to smooth their lifetime consumption across their young andold age. A lower (higher) share of type B agents means lower (higher) transfers from theirchildren, leading to a rise (decline) in their savings to provide support into their old age.
Proposition 6 A lower (higher) share of type G(B) agents implies higher growth under any
regime; that is, for each t ≥ 1:
(a)

n
dḡ f
dḡt+1dḡ a< 0; (b) t+1 < 0; (c) t+1 < 0.
dα
dα
dα

Proof. Parts (a) and (c) follow immediately from Lemma 2. To show part (b), it suffices toprove thatadḡt+1attdα < 0. Differentiating both sides of kt+1 = k1 (Aξ) − c[(Aξ) − 1] with respect

15

to α yields:
a
dkt+1
d(Aξ)
d(Aξ)
dc
= k1 t(Aξ)t−1− ct(Aξ)t−1−[(Aξ)t − 1].
dα
dα
dα
dα

As d(Aξ)/dα < 0 (Lemma 2), c < 0 (Proposition 2), Aξ > 1 (Lemma 2), and dc/dα > 0 (seethe Appendix C), the results follow.

6

Welfare Analysis

In Section 5, we have shown that when agents can share the risk, economic growth decelerates.
In this section, we will analyze whether risk sharing is desirable for society.

6.1

The individual’s welfare

To begin, consider the members of generation 0, the old households at date t = 1. Eachmember of generation 0 lives only one period (his or her old period of life), starts his or her
life with k1 units of capital, which he or she rents to the firm to earn an income of (1 + r1 )k1 .
If, in addition, a child of type G is born into his or her family, the old household has a giftof q1 from his or her child. In the absence of risk sharing, therefore, each old household’s
expected utility at the beginning of date t = 1 is given by:
U0n = E1 (ln c01 ) = α ln[(1 + r1 )k1 + q1 ] + (1 − α) ln(1 + r1 )k1 .
As r1 = θA − 1 and q1 = wt − st = (1 − θ)Ak1 − k2 = (1 − θ)Ak1 − k1 Aγ in equilibrium, it
follows that:
U0n = ln Ak1 + α ln(1 − γ) + (1 − α) ln θ.

(46)With risk sharing, on the other hand, the expected utility of each household of generation 0is:
U0a = E1 (ln c01 ) = ln[(1 + r1 )k1 + αq1 − (1 − α)e],so that, in equilibrium:hi(1 − α)eU0a = ln Ak1 + ln α(1 − ξ) + (1 − α)θ −(1 + αQ) .
Ak1

(47)

In deriving (47), we have used the fact that k2 = (k1 − c)Aξ + c along the equilibrium pathwith risk sharing (see Proposition 3). To determine how risk sharing benefits (or hurts) the
initial old household, we subtract (46) from (47):hi(1 − α)eU0a − U0n = ln α(1 − ξ) + (1 − α)θ −(1 + αQ) − [α ln(1 − γ) + (1 − α) ln θ]. (48)
Ak1
16

As ξ < γ (see Lemma 2), the right-hand side of (48) is positive if the cost of monitoring issufficiently small relative to the initial capital stock. A necessary and sufficient condition for
the right-hand side of (48) to be positive is given by:(1 − α)eα(1 − ξ) + (1 − α)θ − (1 − γ)α θ1−α<.
Ak1
1 + αQ(49)

Provided condition (49) holds, each member of generation 0 will be better off with risk sharing.
As is evident from (48), the welfare gain from risk sharing is greater for the member ofgeneration 0 if the cost of monitoring is lower and the initial capital stock is higher, and it isgreatest when e is zero (i.e., under full information).
The welfare gain as displayed on the right-hand side of (48) may be referred to as a director “short-run” effect of risk sharing.
Consider now the member of generation t ≥ 1. The welfare of each young household ismeasured by his or her expected lifetime utility given by (7). As ln ctt = ln qt = ln(wt − st ) =ln Akt (1 − θ − γ) and: Et ln ctt+1 = α ln[(1 + rt+1 )st + qt+1 ] + (1 − α) ln[(1 + rt+1 )st ] = ln Akt+1 +α ln(1 − γ) + (1 − α) ln θ in equilibrium without risk sharing, substituting these values into (7)yields:n
Utn = ln Aktn (1 − θ − γ) + ln Akt+1+ α ln(1 − γ) + (1 − α) ln θ.

(50)

Similarly, the equilibrium level of welfare of each household of generation t ≥ 1 with riskhisharing can be obtained (30) by noting that ln ctt = ln qt = ln Akt (1 − θ − ξ) − (1−α)e
Q
and
Akt
i
h
(1−α)eEt ln ctt+1 = ln[(1+rt+1 )st +αqt+1 −(1−α)e] = ln Akt+1 α(1−ξ)+(1−α)θ − (Aka ) (1+αQ) :
t+1

h
h
i
(1 − α)e i(1 − α)ea
Uta = ln Akta (1−θ −ξ)−Q+lnAkα(1−ξ)+(1−α)θ
−(1+αQ). (51)
t+1
a
Akta
Akt+1
The welfare gain (or loss) from risk sharing can then be obtained by subtracting(50) from
(51):
a
n
Uta − Utn = [(t − 1)(ln ḡta − ln ḡtn ) + t(ln ḡt+1− ln ḡt+1
)]
½ h
¾
i
(1 − α)e+ln (1 − θ − ξ) −
Q
−
ln(1
−
θ
−
γ)
(52)
Akta
½ h
¾
i
(1 − α)e+ln α(1 − ξ) + (1 − α)θ −(1 + αQ) − [α ln(1 − ξ) + (1 − α) ln θ] .
a
Akt+1

The direct short-run effect of the risk sharing of each member of generation t ≥ 1, whichshows up in the third term in brackets on the right-hand side of (52), is positive for each oldhousehold at date 1. The later the generation the household belongs to, the greater is the direct17effect from risk sharing as the cost of monitoring becomes less and less important (relative tothe capital stock) with economic growth and capital accumulation. Its magnitude tends tomonotonically increase to its maximum values as t → ∞, the value that would obtain underfull information. However, in addition to its short-run effects, risk sharing has its indirect or“long-run” effects on the welfare of the members of generation t ≥ 1. As Proposition 5 makes
it clear, risk sharing implies lower (average) growth rates, having both positive and negative
implications for the welfare of generation t ≥ 1. Although for a given level of the capitalstock, lower growth implies a lower level of saving (a higher level of first-period consumption
or a gift to his or her parent), leading to a higher level of welfare. It also means a lower levelof capital stock and income, implying a lower level of welfare. The former effect shows up
in the second term in braces on the right-hand side of (52), which is unambiguously positive
under the assumption of (49), while the latter effect is associated with the first term. Asnan − ḡ aḡt+1> ḡt+1
and ḡt+1
t+1 is increasing in t for each t ≥ t̄, some t̄ ≥ 1 (Proposition 5), thefirst term in brackets is negative for large t and grows (negatively) without bound as time (andgeneration) advances. As the second and third terms in brackets are positive and increasing
with time but bounded above, it follows that there is some generation for which the first termin braces dominates the second and third terms in braces on the right-hand side of (52); thenegative long-run growth effects from risk sharing dominates the positive direct effect. Fromthat generation on, the households will be worse off under any regimes with risk sharing thanunder the no-risk-sharing regime.

6.2

Social welfare

In the previous subsection, we only analyzed the effect of an individual’s welfare. Therefore,
this subsection investigates the social welfare effect by employing the Benthamite social welfare
function; that is, the welfare level of period t is measured by the sum of the utility of generationt − 1 and generation t who live in period t. Now, let us derive a social welfare function as a
discounting sum of the welfare level of each generation:
Wt = E∞X

β t−1 [ln ctt + α ln ct−1,G+ (1 − α) ln ct−1,B
],
t
t

(53)

t=1

where β shows the rate of social time preference across generations.
From the analysis of the previous subsection, because the initial (future) generation enjoys
the short-run effect (long-run effect), the initial (future) generation experiences a welfare
improvement (deterioration) with risk sharing. Thus, social welfare under risk sharing is18

anticipated such that when the rate of social time preference is sufficiently large (small), thelong-run effect (short-run effect) is dominated by the short-run effect (long-run effect). To
see whether this forecast is correct or not, we derive the social welfare function in the samemanner as in the previous subsection. Noting the fact that kt = k1 (Aγ)t−1 , ḡtn ≡ Aγ, kta =1(k1c )(Aξ)t−1 + c and ḡta ≡ (kta /k1 ) t−1 , the social welfare function without and with risk sharing
is derived as follows:
Wn =

1
2β
N+
ln ḡtn ,1−β(1 − β)2(54)

where N ≡ 2 ln Ak1 + ln(1 − θ − γ) + α ln(1 − γ) + (1 − α) ln θ.
12βP+
ln ḡta ,
(55)
1−β
1−β
h
h
i
i
(1−α)ewhere P ≡ 2 ln Ak1 + ln (1 − θ − ξ) − (1−α)e
+
ln
α(1
−
ξ)
+
(1
−
α)θ
−
(1
+
αQ)
.
a
a
Ak
Ak
Wa =

t

t

Therefore the difference between (54) and (55) is represented as follows:
Wa − Wn2β(ln ḡta − ln ḡtn )=(1 − β)2½ h¾
β(1 − α)e i+
ln (1 − θ − ξ) −Q − ln(1 − θ − γ)
(56)
1−β
Akta
½ h
¾
i
β
(1 − α)e+ln α(1 − ξ) + (1 − α)θ −
(1
+
αQ)
−
[α
ln(1
−
ξ)
+
(1
−
α)
ln
θ]
.
1−β
Akta
As the second and third terms of (56) are pretty much the same as in (52), then the valueof the inside bracket is positive with the same reason as given in our previous subsection. In
addition, this, the first term of (56) is negative (Proposition 5). Then, there exists a unique
P −N
β ∗ such that W a − W n = 0 and W a > (<)W n if β < (>)β ∗ , where β ∗ ≡ 2(ln ḡn −lnḡ a )−(P −N ) .
tt

These results indicate that if the rate of social time preference is sufficiently large (small),then the schedule of risk sharing accelerates (deteriorates) the social welfare level. Hence,these results follow the above anticipation.

7

Conclusion

This paper has investigated how the risk sharing methods that cover the uncertainty of intergenerational transfers affect aggregate saving and welfare in a two-period overlapping generations model. For this purpose, we investigated the gifts economy and assumed that young
agents are divided into two types according to their attitude towards their parents. One groupis constituted of agents who are dutiful to their parents and transfer gifts, whereas the other19

is constituted of agents who are selfish and do not transfer gifts. If agents are risk averse,they have an incentive to share the risk within the same generation. Thus, agents who havethe ability to monitor contracts act as insurance companies. We assume that information
asymmetry exists between agents and insurance companies about the income realization.
We investigated economies both with and without risk sharing. By analyzing these economies,
we found that the rate of economic growth becomes constant in both regimes. Income fluctuations can be completely smoothed under the schedule of risk sharing, and, thus, the precautionary motive for saving decreases, resulting in a lower rate of economic growth. As to
welfare, risk sharing can enhance the welfare level of the present generation, but it deteriorates
the welfare level of the future generation. Therefore, when the social rate of time preferenceis sufficiently high (low), risk sharing accelerates (deteriorates) the welfare level.

8

Appendices

Appendix A: Proof of Proposition 1
We define the function G(g) by:G(g) ≡

2 − α(1 − θ)Aαθ(1 − θ)A2−.
2−α
(2 − α)[(2 − α)g − (1 − α)(1 − θA]

Then, (14) can be written as gt+2 = G(gt+1 ). Evaluating the derivative of G at b1 and b2 , wefind that G0 (b1 ) < 1, and G0 (b2 ) > 1, where b1 and b2 are given by (17) and (18). Thus, any
growth path {gt+1 }∞t=1 that starts with a growth rate other than b2 at date t = 1, covers to b1(see Figure 1). As h[(1 − θ)A] = −(1 − θ)θA2 < 0, we have b2 < (1 − θ)A < b1 . It follows thatthe only growth path that is consistent with the resource constraint (15) is the stationary one
with g = b2 as its initial growth rate.
Thus, the growth path must be such that gt+1 ≡ kt+1 /kt ≡ b2 for each t ≥ 1, so that theunique equilibrium path is given by kt = Zbt2 with Z to be determined by its initial condition.
The growth rate is positive if and only if b2 > 1, which in turn holds if and only if (19) holds.
Finally, the equilibrium path must satisfy its initial condition so that Z = k1 /b2 , wherek1 > 0 is the initial endowment of capital stock of generation 0. Therefore, the equilibrium
path is given by kt = k1 (b2 )t−1 = k1 (Aγ)t−1 for each t ≥ 1.

20

Appendix B: Proof of Proposition 2
To show the sufficiency part, suppose that (26) holds. Then:e−α1
qt+1 ≤{(1 + rt+1 )st − [(1 + rt+1 )st + qt+1 ]α [(1 + rt+1 )st ]1−α } < 0.
1−α
1−α

(57)

We note that (21) is equivalent to:−pG
t+1 + ϕt+1 ≥ 0.

(58)

Let λ1 , λ2 , λ3 , and λ5 be the nonnegative multipliers associated with constraints (58), (22),(23), and (25), respectively, and let λ4 be the multiplier associated with (24). Then, thefollowing Kuhn–Tucker conditions, together with (58), (22), (23), (24), and (25), are necessary
and sufficient for the maximization problem in (20)–(25); that is:

λ1 − λ3 ≤ 0with equality if ϕt+1 > 0,
(59)
α
αλ5
−
− λ1 − λ2 + αλ4 −≤0G(1 + rt+1 )st + qt+1 − pt+1
(1 + rt+1 )st + qt+1 − pG
t+1
with equality if pG
t+1 > 0,1−α(1 − α)λ5− (1 − α)λ4 +≤0(1 + rt+1 )st + pB
(1+
rt+1 )st + pB
t+1
t+1

(60)

with equality if pB
t+1 > 0,(61)

λ1 (−pG
t+1 + ϕt+1 ) = 0,(62)

λ2 (qt+1 − pGt+1 ) = 0,(63)

λ3 (qt+1 − ϕt+1 ) = 0,(64)

B
λ5 {α ln[(1 + rt+1 )st + qt+1 − pG
t+1 ] + (1 − α) ln[(1 + rt+1 )st + pt+1 ],(65)−α ln[(1 + rt+1 )st + qt+1 ] − (1 − α) ln[(1 + rt+1 )st ]}.
B
As (24) implies that pG
t+1 and pt+1 are both strictly positive, (60) and (61) hold with equality.

Then, eliminating λ4 from the equality versions of (60) and (61) yields:
λ1 + λ2 =Bα(1 + λ5 )[qt+1 − pG
t+1 − pt+1 ].
G[(1 + rt+1 )st + pB
t+1 ][(1 + rt+1 )st + qt+1 − pt+1 ](66)

We want to show that λ1 = λ2 = 0. To see that λ1 = 0, suppose that, on the contrary, λ1 > 0.
Then, (59) implies that λ3 > 0, and (62) and (64) in turn imply that:pG
t+1 = qt+1 .
21

(67)

Substituting (67) into (24) gives:
pB
t+1 =

α
qt+1 − e.
1−α

(68)

Then, (66), (67), and (68) yield:³´α(1 + λ5 )αi
λ1 + λ2 = he−qt+1 < 0,α1−α(1 + rt+1 )st + 1−α
qt+1 − e (1 + rt+1 )st(69)

resulting in a contradiction, where the strict inequality follows from (57). To see that λ2 = 0,
suppose now that λ2 > 0. Then, (63) implies (67). The same argument as above leads us toa contradiction, which establishes that λ1 = λ2 = 0. If λ1 = λ2 = 0, then (66) and (24) imply(27) and (28). Then, (22), (58), and (27) lead to (29). Finally, it is easy to see that (27) and(28) satisfy (25) under the condition shown in (26).
To show that (26) is necessary, suppose that (26) does not hold. Then:e−α1
qt+1 >{(1 + rt+1 )st − [(1 + rt+1 )st + qt+1 ]α [(1 + rt+1 )st ]1−α }.
1−α1−αHowever, as the right-hand side of this inequality is negative, there are two possible cases:αα(i) e − 1−αqt+1 ≤ 0 and (ii) e − 1−αqt+1 > 0. Suppose that case (i) holds. Then, the above

sufficiency argument applies and we are led to (27) and (28). However, we find that (27)and (28) do not satisfy (25) when (26) is violated. Consider now case (ii). We go back to
Bthe sufficiency argument to find that pG
t+1 and pt+1 are given by (67) and (68), respectively,

recalling that (70) implies that λ1 > 0 or λ2 > in this case. However, case (ii) implies thatpG
t+1 , contradicting (24), which completes the proof.
Appendix C: Proof of Proposition 3
The characteristic roots of (33) are given by (35) and (36). The general solution to (33) is thengiven by (34). As f [(1 − θ)α] = −θ(1 − θ)A2 < 0, it must be true that b2 < (1 − θ)A < b1 . As
any equilibrium path must satisfy the resource constraint (13), the particular solution to (33)
must be such that Z1 of (34) is zero; that is, the equilibrium path is given by kt = Z2 (b2 )t +c =Z2 (Aξ)t + c for each t ≥ 1, with Z2 to be determined by its initial condition. The growth rate
is positive if and only if b2 = Aξ > 1, which in turn holds if and only if (41) holds. From(37) and (38) we find that c is given by −(1 − α)/f (1). When A meets (41), it must be thecase that f (1) > 0, and, therefore, that c < 0. The equilibrium path must satisfy its initialcondition so that Z2 = (k1 − c)/b2 , which yields kt = (k1 − c)(b2 )t−1 + c = (k1 − c)(Aξ)t−1 + c.
22

Next, we check that when the condition (42) is met and (26) is met, that (26) in Proposition
2 is indeed satisfied along the equilibrium path in Proposition 3. We note that if the risk
sharing regime prevails from date t + 1 on, then:qt+1 = wt+1 − st+1 = (1 − θ)Akt+1 − kt+2 = (1 − θ)Akt+1 − [(kt+1 − c)Aξ + c].

(70)Then, we substitute (6) and (70) as well as st = kt+1 into (26) to obtain:hh(1 − α) i(1 − α) i
ln α(1−ξ)+(1−α)θ −(1+αQ)
e −α ln (1−ξ)−Qe −(1−α) ln θ ≥ 0. (71)
Akt+1
Akt+1
As Aξ < A(1 − θ) or 1 − ξ > θ, as shown above, Lemma 1 implies that (71) holds for t + 1 = 1,
provided (41) is met. As kt+1 = (k1 − c)(Aξ)t + c > k1 for t + 1 = 2, 3, · · · along the path,(49) holds for t + 1 = 2, 3, · · · as well.
Appendix D: Proof of Lemma 2(a)By Propositions 1 and 3, Aγ and Aξ are the smaller roots of h(b) = 0 and f (b) = 0, respectively.
We define the function m(b) as m(b) ≡ h(b) − f (b). Then:m(b) = 2(1 − α)b2 − 3(1 − θ)(1 − α)Ab + (1 − θ)2 (1 − α)A2 .
The quadratic equation m(b) = 0 has two distinct roots, (1 − θ)A/2 and (1 − θ)A, which
implies that the graphs of h(b) and f (b) cross at b = (1 − θ)A/2, (1 − θ)A. This also implies
that: m(b) ≡ h(b) − f (b) > 0 for each b < (1 − θ)A/2. As h[(1 − θ)A/2] = f [(1 − θ)A/2] =22A− α(1−θ)< 0 and as Aγ and Aξ are the smaller roots of h(b) = 0 and f (b) = 0, respectively,
4

it must be the case that Aγ < (1 − θ)A/2 and Aξ < (1 − θ)A/2. However, as h(b) > f (b) for
each b < (1 − θ)A/2, as shown above, we have:
0 = h(Aγ) > f (Aγ),h(Aξ) > f (Aξ) = 0,
which in turn implies that Aγ > Aξ.
Appendix E: Proof of Proposition (c)Here, we show that dc/dα > 0. Differentiating (37) with respect to α and rearranging yields:dc
e[1 − 2θA − (1 − θ)A + (1 − θ)θA2 ]=.
dα
f (1)2To see that the numerator on the right-hand side is positive, observe that f (1) = α − [2θ +α(1 − θ)]A + (1 − θ)θA2 > 0 for any α ∈ (0, 1] (see the proof of Proposition 3) and that it
holds for α = 1 in particular.
23

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Figure 1: The dynamics of the economy

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