数理解析研究所講究録 1215 巻 2001 年 78-93

78Intergenerational Transfers Motivated by
Altruism from Children towards Parents
Hiroshi Fujiu*

Faculty of Economics, Chiba Keizai University,
3-59-5 Todoroki-cho, Inage-ku, Chiba 263, Japan

Abstract
This study has two ends. The first is to construct aformal modelwith altruism from children towards parents and to reveal the structure of an equilibrium, which has not been studied in the existing
literature. The second is to establish the existence of an equilibriumin this model by demonstrating the existence of adynamically consistent allocation in the concept of subgame perfect Nash equilibrium.
KEYWORDS: intergenerational altruism; intergenerational transfers; subgame perfect Nash equilibrium.
JEL Classification Numbers: C62, D64, D91.

This paper is based upon part of Chapter 3of my dissertation submitted to YokOIwould like to thank my advisers, Professors Keiichi Koda and
Makoto Yano for their encouragement. When writing and revising the paper, Iwas espe
cially indebted to Professor Yano for his guidance. Also Iwould like to thank Professors
Taro Akiyama, Kenichi Sakakibara, and Ryuhei Wakasugi for their suggestions
llan]a National University.

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1Introduction
In the recent literature, adynamically consistent allocation is characterized
in the model with intergenerational altruism. Ray (1987) focuses on theintergenerational altruism that one generation holds towards his descendants,
which Icall forward altruism. Hori (1997) focuses on the intergenerational
generation holds towards both his parents and children,
altruism thatwhich we call tw0-sided altruism. Both studies characterize dynamicallyconsistent paths and represent them in the form of apolicy function.
This study has two purposes. The first is to construct aformal modelwith the intergenerational altruism that children hold towards parents. The
structure of an equilibrium in amodel with such altruism has not been studied in the existing literature.l This type of altruism, which Icall backward
altruism, yields two types of intergenerational transfers. One type is an intergenerational transfers from children towards parents, gift. Another type is
one from parents towards children, education investments. We demonstratethat they determine the structure of an equilibrium in this model.
The second purpose is to demonstrate the existence of adynamically consistent allocation, which we call an equilibrium in this model. This conceptof equilibrium is the same as that of subgame perfect Nash equilibrium in
Ialso characterize asteady state and reveal arelaextensive-formtionship between asteady state allocation and intergenerational altruism byusing an example. By doing so, we demonstrate that the strength of intergenerational altruism as well as its direction, also play an important role foradynamic allocation. This finding is important for determining the policyof an income redistribution over times or generations by aIn the section 2, abackward altruism model is formalized, and an equilibrium in this model is defined. In the section 3we demonstrate the existenceof an equilibrium. In the section 4, we make concluding remarks.
$0\dot{\mathrm{n}}\mathrm{e}$$\Leftrightarrow \mathrm{a}\sigma \mathrm{m}\mathrm{e}\mathrm{s}.2$$\mathrm{g}\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}.3$

10’Connell and Zeldes (1993) also focus on an economy with altruism from children
towards parents. They interest in the behavior of transfers in this economy rather thanin the mechanism of determining an equilibrium.
for the concept of equilibrium, see Selten (1975). Leininger (1986) demonstrates
the existence of aperfect equilibrium in the economy that each generation is altruistic tothe other generations.
$2\mathrm{A}\mathrm{s}$$3\mathrm{T}\mathrm{h}\mathrm{i}\mathrm{s}$effect is discussed in Yano (1998)

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2The Model
Take atwo generation overlapping economy lasting both side of time. Eachgeneration is born at the end of each period and lives the sequential two
periods. Assume that each generation is altruistic towards the previous generation, which means that children are altruistic towards parents. As aresult, children will make gifts for parents. Assume that each generation has
an income only in the first period and has no income in the second period.
Moreover, assume that an economy has no financial asset that is available forindividuals, and that all the goods are perishable. Thus, in order to consume
in the second period, parents intend to make children have more incomethat yields more gifts from children. Thus, parents make such choices, e.g.,
education investments in children.
Let generation be ageneration born at the end of period $-1$ . Notegeneration lives two periods, period and period $+1$ . In period , heobtains an income and distributes it to his own consumption , education, and gifts for parents
investments towards children (generation $+1$ )(generation $-1$ ) , which equals to parents’ consumption for their old time
. In period $t+1$ , when generation is old, he makes a;That is gt
received from his children. Generationby using gifts
consumption
$t$

$t$

$t$

$t$

$t$

$t$

$c_{t}^{1}$$y_{t}$$t$$t$$g_{t}$$t$$=c_{t}^{2}$$c_{t}^{2}$$c_{t+1}^{2}$$t’ \mathrm{s}$$e_{t+1}$$g_{t+1}$

constraints are as follows.
$c_{t}^{1}+e_{t+1}+g_{t}=y_{t}$$c_{t+1}^{2}=g_{t+1}$

(1)

,

.

(2)

Generation t is altruistic towards generation t -1 for any t. Thus, in amodel with backward altruism, autility function is expressed as follows.4$4\mathrm{T}\mathrm{h}\mathrm{e}$utility form of forward altruistic generations is expressed as$U_{t}=u_{t}+\alpha U_{t+1}$,

utility obtained from their own consumption, and whereis generation
where
total utility.
generation
The utility form of tw0-sided altruistic generations is expressed as$t’ \mathrm{s}$$u_{t}$$t+1’ \mathrm{s}$$U_{t}=u_{\mathrm{t}}+\alpha U_{t+1}+\beta U_{t-1}$.

See Hori and Kanaya (1989) and Hori (1992) for tw0-sided altruism,$U_{t+1}$is81$U_{t}=v^{1}(c_{t}^{1})+v^{2}(c_{t+1}^{2})+\beta U_{t-1}$(3),

are total utilities of generation and of generation $-1$
and
where
are utility functions for the young
andrespectively, and wheretime and for the old time respectively. Note that in (3) is adiscount factor
reflecting the degree of backward altruism, and that the higher is, the morealtruistic towards generation $t-1$ generation is. Assume 4is constant over
, the same form of utility
generations and satisfies $0<\beta<1$ . As forfunction as (3) can be expressed. That is, $U_{t-1}=v^{1}(c_{t-1}^{1})+v^{2}(c_{t}^{2})+\beta U_{t-2}$ .
Since generation have no effect on choices and on utility levels determined
before the end of period $-1$ in which generation is born, he considers, we rewrite (3) intoandand
as given. By expressing them as$U_{t}$$t$$t$$U_{t-1}$$v^{2}(\cdot)$$v^{1}(\cdot)$$\beta$$\beta$$t$$U_{t-1}$

$t$

$t$

$t$

$U_{t-2}$$c_{t-1}^{1}$$\overline{U}_{t-2}$$\overline{c}_{t-1}^{1}$$U_{t}=v^{1}(c_{t}^{1})+v^{2}(c_{t+1}^{2})+\beta[v^{1}(\overline{c}_{t-1}^{1})+v^{2}(c_{t}^{2})+\beta\overline{U}_{t-2}]$

(3’)

.

Generation will intend to increase his utility by increasing gifts
in the view point of children. Generation knows that he makes more gifts
towards generation $-1$ with alarger income. Thus, he may expect thatgeneration $t+1$ also makes more gifts towards him with alarger income.
, is determined bygift towards generation ,Then, generation
is described as afunction. Suppose that
income,generation
as follows.
with respect to$g_{t}=c_{t}^{2}$

$t$

$t$

$t$

$t$

$t+1’ \mathrm{s}$$t+1’ \mathrm{s}$$g_{t+1}$$g_{t+1}$$y_{t+1}$$y_{t+1}$$g_{t+1}=\Phi_{t+1}(y_{t+1})$(4),

towards
states that generation $+1$ makes gifts
where, and therefore we call it agift function
his parents with his given incomeof generation $t+1$ . In the below, we write afunctional forms of generation.
asgift function
Generation will intend to increase his children’s income. For this purpose, he will accumulate his children’s human capital by means of education
education investments
investments towards generation $+1$ . Generation. Genaccumulates his children’s human capital
towards his children
as an input into aproduction, anderation $t+1$ makes his human capital. Assume
he obtains an output $f(h_{t+1})$ , all of which become his incomethat these relationship are as follows.
$\Phi_{t+1}(y_{t+1})$$t$$\Phi_{t+1}(y_{t+1})$$y_{t+1}$$\Phi_{t+1}(y_{t+1})$$t+1’ \mathrm{s}$$\Phi_{t+1}$$t$$t’ \mathrm{s}$$t$$h_{t+1}$$e_{t+1}$$h_{t+1}$$y_{t+1}$$h_{t+1}=e_{t+1}$,$y_{t+1}=f(f\iota_{t+1})$

.

(5)

82
Now we are ready to describe the optimization problem of generation t.
In the optimization problem, we remove the given factors in (3’), .1
andon which generation t’s choices have no effect. From (1), (2), (3’),(4), and (5), the generation t’s optimization problem is expressed as follows.
$(\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}).)$$U_{t-2_{\rangle}}$The Generation t’s Optimization Problem$v^{1}(c_{t}^{1})+v^{2}(\Phi_{t+1}(f(e_{t+1})))+\beta v^{2}(g_{t})$$\max$(6),$(c_{t}^{1},e_{t+1},g_{t})$s.t$c_{t}^{1}+e_{t+1}+g_{t}\leq y_{t}$, givenand$y_{t}$$\Phi_{t+1}$.

We will formalize an equilibrium in abackward altruism model. As
shown in (6), generation makes choices
under agiven income,, and agiven functional form of generation $+1$ ’s gift function,.
, theseoptimal choices depend on bothandSince generation
. Let
and
choices is represented as functions with respect to bothsatisfying,
be optimal choices, means generation
optimal gifts(6). The third term of them,
towards parents. Each generation solves the same optimization problem asasgift function(6). Then, generation $t-1$ regards the generation
given in his optimization problem. For dynamic consistency, it must be that$(c_{t}^{1},e_{t+1},g_{t})$

$t$

$t$

$y_{t}$$\Phi_{t+1}$$t’ \mathrm{s}$$\Phi_{t+1}$$y_{t}$$\Phi_{t+1}$$y_{t}$$(c^{1}(y_{t}, \Phi_{t+1})$$e(y_{t}, \Phi_{t+1}),g(y_{t}, \Phi_{t+1}))$$(c_{t}^{1}, e_{t+1},g_{t})$$g(y_{t}, \Phi_{t+1})$$t’ \mathrm{s}$$\Phi_{t}(y_{t})$$t’ \mathrm{s}$$\Phi_{t}(y_{t})=g(y_{t}, \Phi_{t+1})$(7),

for any .
We are ready to define an equilibrium. Let $T=$
for any
an equilibrium if
call.
astationary equilibrium if
for any$y_{t}$. We. Also we call$(-\infty, \ldots,t, \ldots, +\infty)$$\Phi_{t}(y_{t})=g(y_{t}, \Phi_{t+1})$$\{\Phi_{t}\}_{t\in T}$$\Phi_{t}=\Phi$$\{\Phi_{t}\}_{t\in T}$

$t$

$t$

$\in T$$\in T$

3Assumptions, Existence Result, and Outline of Proof of the Existence, and
be aset of . Let
Let
and : $X^{i}arrow R_{+}(i=1,2)$ , assume the follows.
F.I. is increasing.
such that $f(y)$F.2. There exists$Y\subset R_{+}$$y_{t}$$X^{1}\subset R_{+}$$X^{2}\subset R_{+}$. As for$f$: $Yarrow R_{+}$$v^{i}$$f$$\overline{y}\in Y$$\leq y,\forall y\in\{\tilde{y}\in Y:\tilde{y}\geq\overline{y}\}$.

83
F.3. $f(0)=0$ .
F.4. : $Yarrow R_{+}$ is continuous.
. is strictly increasing for $i=1,2$ .
is strictly concave for $i=1,2$ .
V.2.
V.3. $v^{i}(0)=0$ for $i=1,2$ .
is continuous for $i=1,2$ .
:V.4.
The main result is the following.
$f$$\mathrm{V}.\mathrm{I}$$v^{i}$$v^{i}$$v^{i}$$X^{i}arrow R_{+}$

Theorem 1Under the assumptions (F. 1), (F.2), (F.3), (F.4), (V. 1), (V.2),(V.3), and (V.4), there eists a stationary equilibrium.
The rest of this section provides an outline of the proof of this theorem.
Each generation’s optimization problem can be decomposed into twoparts: intratemporal optimization problem, which intergenerational allocation problem in one period in other words, and intertemporal optimization
problem. The first problem is that an available income $m\in M\subset Y$ in period
consumption, , and gifts towards parents,
is distributed to generation. Then, a, which equals generation -l’s consumption in period ,pair of optimal choices, and , is represented as $(c(m),g(m))$ that satisfies$t’ \mathrm{s}$$t$$c_{t}^{1}$

$t$

$t$

$c_{t-1}^{2}$$g_{t}$$c_{t}^{1}$$g_{t}$$(c(m),g(m))= \arg\max_{c(,g)}v^{1}(c)+\beta v^{2}(g)s.t$. $c+g\leq m$ , $c\geq 0,g\geq 0$ .

(8)

For the preceding proof, we define$V(m)=v^{1}(c(m))+\beta v^{2}(g(m))$ .

(9)Then, we have the following lemma.

Lemma 1 c(.) and g(.) are unique, continuous, and increasing. Moreover,V(.) is continuous, strictly increasing, and strictly concave.

Proof. Omitted.
The second problem, i.e., an intertemporal optimization problem, is thatto an available income in period ,generation distributes his income, which leads to gift
$m$ , and education investments towards children,
$t$

$t$

$y_{t}$$e_{t+1}$$\mathrm{s}$84received in period $t+1$ , so as to maximize his own utility given children’s
behaviors of gifts. We may rewrite (6) as follows.
$\ovalbox{\tt\small REJECT}!\ovalbox{\tt\small REJECT}_{*+\ovalbox{\tt\small REJECT}}(y_{*+\mathrm{X}})$$e_{t+1}\in[y_{t}]\mathrm{m}\mathrm{y}V(y_{t}-e_{t+1})+v^{2}(\Phi_{t+1}(f(e_{t+1})))$

Given afunction$\Phi$.

: $R_{+}arrow R_{+}$ , let$E_{\Phi}(y)= \arg\max 0\leq e\leq y\{V(y-e)+v^{2}(\Phi(f(e)))\}$

where$E_{\Phi}(y)$

(11)

,

is aset of optimal choices of education investment, e, and$\xi_{\Phi}(y)=\min\{e:e\in E_{\Phi}(y)\}$$(G\Phi)(\eta)=g(\eta-\xi_{\Phi}(\eta))$

where

(10)

(12)

,

(13)

,

is defined in (8), and$g(\cdot)$$(H \Phi)(y)=\max\{(G\Phi)(\eta)\}0\leq\eta\leq y$

For the following lemma, call$\varphi(x)$

.

(14)

be upper semi-continuous if$\lim\sup_{narrow\infty}\varphi(x_{n})\leq\varphi(x)$as $x_{n}arrow x$ .

By the definition of equilibrium, astationary equilibrium is\Phi =G$. Then, we have the following lemma.

$\Phi$

such thatsatisfying that $(G\Phi)(y)$ is upper semi-continuous with
Lemma 2Take
respect to , and that $\Phi=H\Phi$ . Let $\Phi^{*}(y)\equiv(G\Phi)(y)$ . Then,$\Phi$$y$(15)$\Phi^{*}=G\Phi^{*}$

Proof- See Appendix A.

$\blacksquare$

In order to show that there exists astationary equilibrium, by Lemma
2, it suffices to show that there exists
satisfying that $(G\Phi)(y)$ is upper,semi-continuous with respect to , and that $\Phi=H\Phi$ . Let
where is explained in the assumption (F.2), and $Y=[0,\tilde{y}]$ . Let be asetof endowed with the topology of pointwise convergence and satisfying thatis non-decreasing,$\Phi$$y$$C$$\overline{y}$$\Phi$$\Phi$$5\mathrm{A}\mathrm{s}$$\tilde{y}=\max(\hat{y},\overline{y})^{5}$

for , refer to the assumption (F.2)$\overline{y}$85$0\leq\Phi(y)\leq\tilde{y}$

,

(16)

,

$\forall y\in \mathrm{Y}$and$\exists\ell>0$: $|\Phi(y’)-\Phi(y’)|\leq\ell|y’-y’|$ , $\forall y’,y’\in Y$.

(17)

Note that C is anonvoid convex subset of aseparated locally convex
topological vector space. First, we will demonstrate the following lemma.

Lemma 3Let $\Phi\in C$ . Then, $(G\Phi)(y)$ is upper semi-contin tous with respect
to .
$y$

Proof. See Appendix B.

$\blacksquare$

Next we will demonstrate that there exists $\Phi\in C$ such that $\Phi=H\Phi$ .
Such a is called to afixed point of amapping $H$ . In order to show that
there exists afixed point of amapping $H$ , we use the fixed point theorem of
Schauder-Tychonoff, which is explained by Edwards (1965).
$\Phi$

Theorem 2(Shauder-Tychonoff theorem) Let $E$ be a separated locallyconvex topological vector space, $K$ be a nonvoid compact convex subset of $E$ ,any continuous map of $K$ into itself. Then admits at least one fixed
point.
$u$$u$

By the definition of , is anonvoid convex subset of aseparated locallyconvex topological vector space. In order to use the fixed point theorem inthis model, we require (i) that is compact, (ii) that $H$ maps into itself,and (iii) that $H$ : $Carrow C$ is continuous.
is compact. To this end, we introduce
We will demonstrate thatthe mathematical concept of equicontirvuous, which is explained by Edwards(1965), and use the following theorem.
$C$

$C$

$C$

$C$

$C$

Theorem 3(Ascoli’s theorem) Let $T$ be a topological space, $X$ be a unibe the set of all $X$ -valued functions on T. If $F\subset X^{T}$
forrn space, andand $F(t)$ $=\{f(t) : f\in F\}$ is relatively compact inis equicontinuous on$X$
for the topology offor each $t\in T$ , then $F$ is relatively compact incompact convergence.
$X^{T}$$T$$X^{T}$

86
By this theorem, though C is endowed with the topology of pointwise
convergence, the following lemma holds.

Lemma 4$C$

is compact for the topology of compact convergence.

Proof. See Appendix C.

$\blacksquare$

We will demonstrate the rest of the requirements by the following twolemmas.
Lemma 5Let $\Phi\in C$ . Then $H\Phi\in C$ .

Proof. See Appendix D.

$\blacksquare$

Lemma 6 $H:Carrow C$ is continuous with respect to the $\sup$ norm.

Proof. See Appendix E.

$\blacksquare$

Through the above lemmas, we may obtain (i) that $C$ is compact, (ii)into itself, and (iii) that $H$ : $Carrow C$ is continuous withthat $H$ maps
respect to the $\sup$ norm. By Shauder-Tychonoff theorem, there exists $\Phi\in C$
such that $\Phi=H\Phi$ . Thus, by Lemma 2, there exists 0’ such that $\Phi^{*}=G\Phi$ ”.
Therefore, it is establish that there exists astationary equilibrium.
$C$

4Concluding Remarks
This study has formalized amodel with backward altruism and demonstrated
that backward altruism, despite one-sided altruism, yields tw0-sided intergenerational transfers. It has also demonstrated the existence of an equilibriumin this model.
We remain two problems. One is an absence of afinancial market, whichplays an important role on redistribution between different generations andthus on adynamic allocation. The other is the stability of astationary
equilibrium. These problems are focused on by other papers

87

AppendicesA. Proof of Lemma 2By (13), $\Phi^{*}(y)=(G\Phi)(y)=g(y-\xi_{\Phi}(y))$ and$(G\Phi^{*})(y)=(G(G\Phi))(y)=g(y-\xi_{G\Phi}(y))$ . Thus, in order to show (15), it
suffices to show that for any $y\in Y$ ,$\xi_{\Phi}(y)=\xi_{G\Phi}(y)$

.

(18)

To this end, we first demonstrate the following two sublemmas.

Sublemma 1Let
Then,$\varphi(x)$be upper semi-continuous (u.s.c.) with respect to .
exists, and it is u.s.c. with respect to .
$x$$\max_{0\leq x\leq y}\varphi(x)\equiv\pi(y)$Proof. Omitted.

$y$$\blacksquare$

Sublemma 2Assume that $(G\Phi)(y)$ is upper semi-continuous (u.s.c.) withrespect to . Let $y\in Y$$y$$E_{G\Phi}(y)= \arg\max 0\leq e\leq y\{V(y -e)+v^{2}((G\Phi)(f(e)))\}$,$E_{H\Phi}(y)= \arg\max_{d}0\leq e\leq y\{V(y -e)+v^{2}((H\Phi)(f(e)))\}$

.

and

Then,

$E_{H\Phi}(y)$and$E_{G\Phi}(y)$are non-empty, and$E_{H\Phi}(y)\subset E_{G\Phi}(y)$.

Proof. Omitted. ii
In order to show (18), since $\Phi=.H\Phi$ , it suffices to show that
. By Sublemma 2, $\xi_{H\Phi}(y)\in E_{G\Phi}(y)$ . Thus, it must.hold that. Suppose. Let $e^{h}=\xi_{H\Phi}(y)$ and $e^{g}=\xi_{G\Phi}(y)$ . Then,since,$\xi_{G\Phi}(y)$$\xi_{G\Phi}(y)$$\xi_{H\Phi}(y)$$\xi_{H\Phi}(y)$$=$$\xi_{H\Phi}(y)$$\geq$$>\xi_{G\Phi}(y)$$e^{g}\not\in E_{H\Phi}(y)$$V(y-e^{g})+v^{2}((H\Phi)(f(e^{\mathit{9}})))<V(y -e^{h})+v^{2}((H\Phi)(f(e^{h})))$By (14),.

(19). By Sublemma 2, $(H\Phi)(f(e^{h}))=$, $e^{g}\in E_{G\Phi}(y)$ ,$(H\Phi)(f(e^{g}))\geq(G\Phi)(f(e^{g}))$$(G\Phi)(f(e^{h}))$. Moreover, since$e^{h}$$V(y-e^{g})+v^{2}((G\Phi)(f(e^{g})))=V(y -e^{h})+v^{2}((G\Phi)(f(e^{h})))$

Then since$v^{2}$.

is strictly increasing, it follows that

$V(y -e^{g})+v^{2}((H\Phi)(f(e^{g})))\geq V(y -e^{h})+v^{2}((H\Phi)(f(e^{h})))$ ,$\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}$, however, contradicts (19). Therefore,$\xi_{H\Phi}(y)$$=\xi_{G\Phi}(y)$.

88
B. Proof of Lemma 3For this proof, since $(G\Phi)(y)=g(y -\xi_{\Phi}(y))$ by(13), we will show $g(y-\xi_{\Phi}(y))$ is upper semi-continuous (u.s.c.) with respectisis continuous, it suffices to show thatto . To this end, since
lower semi-continuous (l.s.c.) with respect to ;That is,$\xi_{\Phi}(y)$$g(\cdot)$

$y$

$y$

as

$\lim_{arrow}.\cdot\sup_{\infty}\xi_{\Phi}(y_{\dot{1}})\geq\xi_{\Phi}(y)$$y_{i}arrow y$

.

(20)

We first show the following sublemma.
Sublemma 3(i)$e’\in E_{\Phi}(y^{JJ})$$\xi_{\Phi}(y)$exists, and (ii) $\Phi\in C$ , $y’>y’$ , $e’\in E_{\Phi}(y’)$ , andimply $e’\geq e’$ .

Proof. Omitted.
. Let
such that
Take asequence. Then, from Sublemma 3, it follows thatof$\blacksquare$$\{y_{i}\}$$\{y_{j}.\cdot\}$$y\dot{.}arrow y$$\{y_{i}\}$be adeceasing sequence.

$\lim\sup\xi_{\Phi}(y\dot{.}j)\geq\xi_{\Phi}(y)$$jarrow\infty$

is non-dec easing,. SinceofTake an increasing sequence
exists for any $\in Y$ ,for any . Since
implies.
. Then, it must hold that
we may takeis
. Since. Let
Suppose that
optimal, and since is not optimal,$i$$\xi_{\Phi}(y\dot{.})\leq\xi_{\Phi}(y)$$y_{i}\leq y$$\xi_{\Phi}(y)$$\{y_{i}\}$$\{y_{j}\dot{.}\}$$\xi_{\Phi}(y)$$y$$\lim_{:arrow\infty}\xi_{\Phi}(y\dot{.})\leq\xi_{\Phi}(y)$$\lim_{:arrow\infty}\xi_{\Phi}(y\dot{.})$$e^{*}= \lim:arrow\infty\xi_{\Phi}(y:)$$\lim_{:arrow\infty}\xi_{\Phi}(y\dot{.})<\xi_{\Phi}(y)$$\xi_{\Phi}(y)$$e^{*}$.
(21)$[V(y -\xi_{\Phi}(y))+v^{2}(\Phi(f(\xi_{\Phi}(y))))]-[V(y-e^{*})+v^{2}(\Phi(f(e^{*})))]\equiv\epsilon>0$

Since$y_{i}arrow y$and$\xi_{\Phi}(y_{i})arrow e^{*}$, for any$\epsilon>0$, there exists$i_{1}$

such that$\forall i\geq i_{1}$$V(y -e^{*})+v^{2}( \Phi(f(e^{*})))+\frac{\epsilon}{2}>V(y_{i}-\xi_{\Phi}(y_{i}))+v^{2}(\Phi(f(\xi_{\Phi}(y_{i}))))$Take asequence, there exists
any$\{e_{i}\}$$\epsilon>0$$i_{2}$

.

,

(22)

such that q. $\equiv\max\{y_{i}-y +\xi_{\Phi}(y),$0}. Then, forand,such that$\forall i\geq i_{2}$$e_{i}\geq 0$$V(y \dot{.}-e:)+v^{2}(\Phi(f(e_{i})))>V(y-\xi_{\Phi}(y))+v^{2}(\Phi(f(\xi_{\Phi}(y))))-\frac{\epsilon}{2}$

.

(23)

From (21), (22), and (23), it follows that
$V(y_{i}-e:)+v^{2}(\Phi(f(e:)))>V(y.\cdot-\xi_{\Phi}(y_{i}))+v^{2}(\Phi(f(\xi_{\Phi}(y_{i}))))$

which, however, contradicts the definition of$\xi_{\Phi}(y_{i})$. Therefore,,$\lim_{iarrow\infty}\xi_{\Phi}(y_{i})--$.
From the above argument, we may obtain that (20) holds for any.
such that$\xi_{\Phi}(y)$$y_{i}arrow y$$\blacksquare$$\{y_{i}\}$

89
C. Proof of Lemma 4Since $Y=[0,\tilde{y}]$ is ametric space, it is atopologicalspace.
is auniform space. Let
be the set of all real valued functions$C$$Y$
on . By the definition of , is anonvoid convex subset ofendowedwith the topology of pointwise convergence and satisfying the following two
conditions. The first is that
$R$

$R^{\mathrm{Y}}$$C$$\exists\ell>0$: $\forall y’,y’\in Y$,$R^{\mathrm{Y}}$$\forall\Phi\in C$, $|\Phi(y’)-\Phi(y’)|<\ell|y’-y’|$ ,, there exists $=\epsilon/\ell>0$ satisfyfrom which it follows that, for any$- \oint’|<\delta$ , and for any $\Phi\in C$ ,such thating that, for any pair$|\Phi(y’)-\Phi(y’)|<\epsilon$ . Thus, the first condition means thatis equicontinuous$Y$on . The second condition is that$\epsilon>0$$(y’, \oint’)$$\delta$$| \oint$$C$$\exists M>0$:,$\forall y\in Y$$\forall\Phi\in C$, $0\leq\Phi(y)\leq M$ ,from which it follows that $C(y)=\{\Phi(y):\Phi\in C\}$ is (relatively) sequential
compact, with respect to the same topology as , for each $y\in Y$ since
is endowed with the topology of pointwise convergence. Then, by Ascoli’s
theorem, is relatively compact infor the topology of compact convergence.
$R$

$C$

$R^{Y}$$C$

D. Proof of Lemma 5Let $\Phi\in C$ , which means thatis nondecreasing with respect to and satisfies (16) and (17). Since $(G\Phi)(y)$ is
upper semi-continuous, by Sublemma 1, $(H\Phi)(y)$ exists. By (14), we may
obtain that $(H\Phi)(y)$ is non-deceasing with respect to . Thus, in order tosatisfies
show $H\Phi\in C$ , it suffices to show that$\Phi(y)$

$y$

$y$

$H\Phi$$0\leq(H\Phi)(y)\leq\tilde{y}$,$\forall y\in Y$

,

(24)

and
$\exists\ell>0$: $|(H\Phi)(y’)-(H\Phi)(y’)|\leq\ell|y’-y’|$ ,$\forall y’$, $y’\in Y$ .

(25)

First, we will show (24). Let $\in Y$ . Since,
and since (GO) $(x)=g(x-\xi_{\Phi}(x))\leq x$ , it follows that $(H \Phi)(y)=\max 0\leq x\leq y$$y$$g(x-\xi_{\Phi}(x))\leq y$$(H\Phi)(y)=\mathrm{m}\mathrm{a}_{-}\mathrm{x}_{0\leq \mathrm{x}\leq y}(G\Phi)(x)$. Since $\tilde{y}\equiv\max\{y :y \in Y\}$ , $(H\Phi)(y)\leq\tilde{y}$ . By the-definition, $(H\Phi)(y)\geq 0$ . Therefore, (24) holds.
Next, we will show (25). Assume $y’\geq y’$ without loss of generality. If$>y’$ . If $(H \Phi)(\oint)=(H\Phi)(y’)$ , then (25)$y’=y’$ , then (25) holds. Let
holds. Note that (H$)(y) is non-decreasing with respect to . Thus, we will
demonstrate that $y’>y’$ and $(H\Phi)(y’)>(H\Phi)(y’)$ imply (25). To this end,
we first demonstrate the following sublemmaof$g(\cdot)$

$\nu$

$y$

90
Sublemma 4$\xi_{\Phi}(y)$is non-decreasing with respect to y.

Proof. Omitted.
be such that $(H\Phi)(y’)=$Let $>y’$ and $(H\Phi)(y’)>(H\Phi)(y’)$ . Take$(G\Phi)(y’)$ and$\geq y’>y’$ . Then, since $(H\Phi)(y’)\geq(G\Phi)(y^{JJ})$ , it follows
bythat $(G\Phi)(y’)>(G\Phi)(y’)$ , which means(13). Since is non-decreasing, $y’- \xi_{\Phi}(\oint’)>y’-\xi_{\Phi}(y’)$ . Since is nonis non-decreasing
decreasing, $c( \oint’-\xi_{\Phi}(\oint’))-c(\oint’-\xi_{\Phi}(y’))\geq 0$ . Since. Thus,with respect to by Sublemma 4,$\blacksquare$

$\nu$

$\nu$

\prime\prime$\nu$$\tilde{g}(y^{m}-\xi_{\Phi}(y^{\prime\mu}))>g(y’-\xi_{\Phi}(y’))$
$c$

$g$

$\xi_{\Phi}(y)$$\xi_{\Phi}(y’’’)-\xi_{\Phi}(\oint’)\geq 0$$y$$g(y’-\xi_{\Phi}(y’))-g(y^{u}-\xi_{\Phi}(y^{Jl}))$$\leq$$[c(y^{u\prime}-\xi_{\Phi}(y’))-c(y’-\xi_{\Phi}(y’))]$$+[g(y’-\xi_{\Phi}(y^{m}))-g(y’-\xi_{\Phi}(y^{JJ}))]$$+[\xi_{\Phi}(y’)-\xi_{\Phi}(y’)]$.

Then, by the budget constraint of (1), we may obtain that $g(y’-\xi_{\Phi}(y’))-$$g(y’- \xi_{\Phi}(\oint’))\leq\psi’-\oint’$ ;That is,$(G\Phi)(y’)-(G\Phi)(y’)\leq y’-y’’$ .

Therefore, since $(H \Phi)(y’)=(G\Phi)(\oint’)$ , and since$y’\geq\nu\prime\prime>y’$,$0<(H\Phi)(y’)-(H\Phi)(y^{n})\leq y’-y^{J}$

From the above argument, (25) is established.
, andsatisfying that
E. Proof of Lemma 6Take asequenceis the $\sup$that there exists $\Phi\in C$ such that $\lim_{narrow\infty}||\Phi^{n}-\Phi||=0$ where
norm. Note $\Phi\in C$ implies $H\Phi\in C$ by Lemma 5. Then, in order to show
-continuous, it suffices to showthat $H:Carrow C$ is$\Phi^{n}\in C$$\{\Phi^{n}\}$$||\cdot||$$||\cdot||$$\lim_{narrow\infty}||H\Phi^{n}-H\Phi||=0$

.

(26)

Let $y^{n} \in\arg\sup_{y\in Y}|(H\Phi^{n})(y)-(H\Phi)(y)|$ . Since is compact, asequence
has aconvergent subsequence. Let be the limit of this subsequence;. Then, it follows thatthat is,$Y$$\{y^{n}\}$$y$$\lim_{narrow\infty}y^{n}=y$$|(H\Phi^{n})(y^{n})-(H\Phi)(y^{n})|$$\leq$$|(H\Phi^{n})(y^{n})-(H\Phi^{n})(y)|+|(H\Phi^{n})(y)-(H\Phi)(y)|+|(H\Phi)(y)-(H\Phi)$ (’.

91
Since $H+6$ C, it follows from (17) thatand that$|(H+)(\mathrm{y})-(H+)(\mathrm{t}\mathrm{y}^{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}|\mathrm{y}$$|(H^{\ovalbox{\tt\small REJECT}}>^{\mathrm{n}})(\mathrm{x})-(H+)(\mathrm{y}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}$

0 as$\ovalbox{\tt\small REJECT} j/"\ovalbox{\tt\small REJECT}$$|(H^{\ovalbox{\tt\small REJECT}}>^{\mathrm{n}})(\mathrm{x})-(H^{\ovalbox{\tt\small REJECT}}!\ovalbox{\tt\small REJECT}^{*})(y\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}|\mathrm{t}\ovalbox{\tt\small REJECT}$-y|,-fI. Thus, in order to show thaty, it suffices to show that$|(H\Phi^{n})(y)-(H\Phi)(y)|arrow 0$as$y^{n}arrow y$

.

(27)

To this end, we will demonstrates the following sublemma.
, $())\in C$ ,Sublemma 5Let
. Then,, thenpoint of$\Phi^{n}$$\eta^{\mathrm{n}}$,y7 $\in Y$ for any $n$ .

$\lim_{narrow\infty}\xi_{\Phi^{n}}(\eta^{n})\in E_{\Phi}(\eta)$$\eta^{n}arrow\eta$Assumeand. Moreover, if be acontinuity.
$\Phi^{n}arrow\Phi$$\eta$$\lim_{narrow\infty}\xi_{\Phi^{n}}(\eta^{n})=\xi_{\Phi}(\eta)$$\xi_{\Phi}(\eta)$

Proof. Omitted.
We will demonstrate (27), which means (a) that $\lim_{narrow\infty}(H\Phi^{n})(y)\leq$(H$)(y), and (b) that $\lim_{narrow\infty}(H\Phi^{n})(y)\geq(H\Phi)(y)$ . First, we will demonbe such that $0\leq\eta^{n}\leq y$ and. Sincestrate (a). Takeby Sublemma 5, and sinceis minimum inin (12), it follows thatby the definition of. Takebe such that $0\leq\eta^{n}\leq y$ and $(H\Phi^{n})(y)$ $=(G\Phi^{n})(\eta^{n})$ .
by (13). Then,
Note$\blacksquare$$\lim_{narrow\infty}\eta^{n}=\eta$$\{\eta^{n}\}_{n=1}^{\infty}$$\lim_{narrow\infty}\xi_{\Phi^{n}}(\eta^{n})\in E_{\Phi}(\eta)$$\xi_{\Phi}(\eta)$$E_{\Phi}(\eta)$$\lim_{narrow\infty}\xi_{\Phi^{n}}(\eta^{n})\geq$$\xi_{\Phi}(\eta)$$\xi_{\Phi}(\eta)$$\{\eta^{n}\}_{n=1}^{\infty}$$(G\Phi^{n})(\eta^{n})=g(\eta^{n}-\xi_{\Phi^{\tau\iota}}(\eta^{n}))$$\lim_{narrow\infty}g(\eta^{n}-\xi_{\Phi^{n}}(\eta^{n}))\leq g(\eta-\xi_{\Phi}(\eta))$.

, we may obtain (a) $\lim_{narrow\infty}(H\Phi^{n})(y)\leq(H\Phi)(y)$ .
Next, we will prove (b). Letbe such that $0<\eta^{*}\leq y$ andSince (G$) (y)$=g(\eta-\xi_{\Phi}(\eta))$$\eta^{*}$$(H\Phi)(y)=(G\Phi)(\eta^{*})$

.

(28)

is lower semi-continuous as shown in Lemma 3, and sinceisis continuous from thenon-decreasing by Sublemma 4, it follows that$(G\Phi)(y)=g(y-\xi_{\Phi}(y))$
left with respect to . Then, since is continuous,is also continuous from the left with respect to . Thus, for any $>0$ , wesuch that $0\leq\eta<\eta^{*}$ andmay take acontinuity point ofSince$\xi_{\Phi}(y)$$\xi_{\Phi}(y)$$\xi_{\Phi}(y)$

$g$

$y$

$y$

$\eta$$\epsilon$$\xi_{\Phi}(\cdot)$(29)(GO) $(\eta)>(G\Phi)(\eta^{*})-\epsilon$ .

. Then, since
be asequence such that $0\leq\eta^{n}\leq y$ andLet,
by Sublemma 5. Then,is acontinuity point of;That is,since is continuous,$\eta^{n}arrow\eta$$\{\eta^{n}\}_{n=1}^{\infty}$$\xi_{\Phi}(\cdot)$$g$$\eta$$\lim_{narrow\infty}\xi_{\Phi^{n}}(\eta^{n})=\xi_{\Phi}(\eta)$$\lim_{narrow\infty}g(\eta^{n}-\xi_{\Phi^{\tau\iota}}(\eta^{n}))=g(\eta-\xi_{\Phi}(\eta))$$\lim_{narrow\infty}(G\Phi^{n})(\eta^{n})=(G\Phi)(\eta)$

.

(30)

92
y, since $(H+)(y)$ is non-decreasing in y, and since it must,
hold that (HO)(y)Since 0$\ovalbox{\tt\small REJECT}$

t7”$\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$(G^{\ovalbox{\tt\small REJECT}}!\ovalbox{\tt\small REJECT})(y)$$(H\Phi^{n})(y)\geq(H\Phi^{n})(\eta^{n})\geq(G\Phi^{n})(\eta^{n})$(31)Then, from (30) and (31), it follows that$n.arrow\infty \mathrm{h}\mathrm{m}(H\Phi^{n})(y)\geq(G\Phi)(\eta)$

.

(32)

is arbitrary. Then, by (28), (29), and (32), we may obtain (b). Thus, (27) is established. Therefore, we may$H$is-continuous.
obtain (26); That is,

In (29),$\epsilon$$\lim_{narrow\infty}(H\Phi^{n})(y)\geq(H\Phi)(y)$$||\cdot||$

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