1 Introduction
The literature concerning various aspects of altruism in economic models is pretty large. For a comprehensive survey with historical notes, the reader is referred to [
9,
21,
33,
35] and their references. This paper is devoted to study some mathematical issues related to existence of equilibria in a large class of economic growth models with altruism between generations. Intergenerational altruism has usually been modeled in two ways. In the
paternalistic model, the utility of the current generation depends on its own consumption and the consumptions of other generations. In other words, the generation cares about what all or some successors will consume, but it does not take into account the utilities the successors derive from the act of consumption. In the
non-paternalistic model, each generation derives utility from its own consumption and the utilities of future generations.
First strong results on the existence of Markov perfect equilibria in paternalistic economic growth models with deterministic transitions were established by Bernheim and Ray [
11] and Leininger [
28]. They assume that each generation cares only about consumption of its immediate successor. From the mathematical point of view, their proofs are rather complicated. A simpler and more direct method was used by Balbus et al. [
3]. Certain extensions of the works of [
11,
28] to models with specific stochastic production functions are surveyed in [
8,
9,
25]. However, more general results the reader may find in [
4,
5], where the transition probability function obeys a natural weak continuity condition. Such transition probabilities are extensively used in economics, since economic dynamics are typically described by some difference equations with additive or multiplicative shocks. The most general paternalistic model was considered by Balbus et al. [
5]. However, their results concern non-atomic transition probabilities. The model studied in [
4] allows to cover both deterministic production functions and the stochastic transitions that satisfy a stochastic dominance assumption. The drawback of this approach lies in the compactness of the state space and in the separability of the utility function which takes into account only an immediate descendant for each generation.
In contrast to the paternalistic case, there have been very few rigorous studies of models assuming the non-paternalistic altruism. As suggested by Ray [
35], “this framework appears to be of somewhat greater interest in the context of applications.” Examples include the works of [
10,
29]. For instance, Ray [
35] described a general model with deterministic transitions involving non-paternalistic altruism and formulated an equilibrium concept, but its existence remains an open problem. Balbus et al. [
6] showed that an equilibrium in a stochastic version of Ray’s framework exists provided that the transition probabilities are non-atomic. The other group of models were considered by Barro [
10] or Loury [
29], who dealt with only one descendant for each generation. However, the existence of an equilibrium consisting of an indirect utility and an optimal consumption (or saving) strategy in the aforementioned models can be studied by dynamic programming methods using contraction mapping theorems. It is worth mentioning that these methods were also applied to the wide class of various decision processes with recursive utilities, see for instance [
17,
18].
In this paper, we study a version of so-called mixed models with
both paternalistic and non-paternalistic components. A need for studying mixed models is expressed on page 113 in [
35]. An approach to mixed models (two-sided altruism) is given by Hori [
23]. He considers a rather specific model with pretty strong assumptions on the utility and deterministic transition functions. An equilibrium is shown to exist in this model by the Schauder fixed-point theorem. Although the approach of [
23] concerns some specific model, it is inspiring for us.
In our model, we assume that every generation considers only its immediate successor. The equilibrium problem studied in this paper is a double fixed-point problem. One fixed point is obtained for an indirect utility function via a contraction mapping with a nonlinear discount function. This is a sort of recursive utility extensively discussed in economics, see Becker and Boyd [
15]. The second one (in an appropriate strategy space) corresponds to Nash equilibrium in an intergenerational game. Our basic tool is the Schauder–Tychonoff fixed-point theorem (see e.g., Dugundji and Granas [
16]). We would like to emphasize that the indirect utility that we consider depends on both consumption and endowment of the generation. This approach is generalized in the model of [
35]. A similar mixed model is studied in the recent paper of [
2]. However, he considers risk-sensitive generations (uses different utility functions), makes much stronger assumptions on the stochastic transition function and proves existence of equilibria in the class of Borel measurable randomized strategies. Therefore, his approach is essentially different from the one applied in this paper. Our proofs are based on techniques used in [
3,
4] with some necessary modifications. We prove a general existence theorem for the mixed model with unbounded utility functions by applying the
weighted norm approach, which was well developed in dynamic programming (see Wessels [
40]; Jaśkiewicz et al. [
27]). However, instead of standard exponential discounting, we use a nonlinear discount function. As a by-product, we considerably extend our earlier result for paternalistic models given in [
4]. The most important point is that no separability condition on the utility function is imposed.
The rest of the paper is organized as follows. Section
2 contains preliminaries. Section
3 derives the utility function which incorporates the paternalistic and non-paternalistic altruism and formulates an equilibrium. Section
4 presents basic assumptions and existence theorems. Examples satisfying our conditions are given in Sect.
5. Finally, the proofs are collected in Sects.
6 and
7.
2 Preliminaries
In this section, we introduce some notation and state a few auxiliary results. As usual, \(\mathbb {R}\) stands for the set of all real numbers and \(\mathbb {N}\) is the set of all positive integers. Let \(S= \mathbb {R}_+ = [0,\infty )\), \(S_+= \mathbb {R}_+{\setminus }\{0\} \) and \(A(s):=[0,s],\)\(s\in S.\)
Let X be the vector space of all continuous from the left functions \(\phi :S\rightarrow \mathbb {R}\) such that \(\phi (0)=0\) and that the restriction of \(\phi \) to any bounded interval [0, m] (\(m\in \mathbb {N}\)) has a bounded variation. We assume that X is endowed with the topology of weak convergence. Recall that a sequence \((\phi _n)\) converges weakly to \(\phi \in X\) if and only if \(\phi _n(s)\rightarrow \phi (s)\) as \(n\rightarrow \infty \) at any continuity point \(s\in S\) of \(\phi .\) Here, we point out that \(s=0\) is considered as a continuity point of \(\phi \in X\) if \(\lim _{s\rightarrow 0^+}\phi (s)=\phi (0).\) The weak convergence of \((\phi _n)\) to \(\phi \) is denoted by \(\phi _n{\mathop {\rightarrow }\limits ^{*}} \phi \).
Let
F be the set of all continuous function from the left mappings
\(c:S\rightarrow S\) such that the function
\( y(s):=s-c(s)\) is non-decreasing and
\( c(s)\in A(s)\) for all
\(s\in S.\) Note that
\(s\rightarrow y(s)\) is lower semicontinuous. Thus,
\(c\in F \) is upper semicontinuous. Define
$$\begin{aligned} I:=\{ y\in X: y(s)=s-c(s)\ \text{ where }\ s\in S,\ c\in F \}. \end{aligned}$$
Observe that
\(I\subset X.\) Moreover,
\(s=0\) is the continuity point of every function in
F or
I.
For a more detailed discussion, consult Lemma
1 and Appendix in [
7].
In the sequel, we shall use a generalized version of the contraction mapping principle due to [
31], see also Theorem 5.2 in [
16].
We shall also assume that
\(z \rightarrow \delta (z)/z\) is non-increasing on
\(S_+.\) This assumption implies that
\(\delta \) is subadditive and hence
$$\begin{aligned} |\delta (z_1)-\delta (z_2)|\le \delta (|z_1-z_2|)\ \ \text{ for } \text{ all }\ \ z_1, z_2 \in S. \end{aligned}$$
Moreover, the fact that
\(z\rightarrow \delta (z)/z\) is non-increasing implies that
\(\delta (\kappa z)/\kappa z \le \delta (z)/z\) for any
\(z\in S_+\) and
\(\kappa \ge 1.\) Hence, it holds
$$\begin{aligned} \delta (\kappa z)\le \kappa \delta (z)\ \ \text{ for } \text{ all }\ \ \kappa \ge 1,\ z\in S. \end{aligned}$$
(1)
As in the dynamic programming literature (see Jaśkiewicz et al. [
27]), we call
\(\delta \) a
discount function.
Let
\(d>0\) and
\(\eta :[d,\infty ) \rightarrow \mathbb {R}\) be a fixed function. Following Milgrom and Shannon [
32], we say that
\(\eta \) has the
strict single crossing property on
\([d,\infty ),\) when the following holds: if there exists some
\(x\ge d\) such that
\(\eta (x) \ge 0\), then for each
\(x'>x\), we have
\(\eta (x')> 0.\) It is worth to note that
\(\eta \) need not be increasing, see Example 3 in [
3].
Let
\(u:S\times S \rightarrow \mathbb {R}_+\) be a function of the form
\(u(a,w) =g(u_o(a,w)).\) We make the following assumptions.
(U1)g is continuous, increasing, and \(g(0)=0.\)
(U2)\(u_o(0,0)=0\) and \(u_o\) is continuous on \(S\times S\) and increasing in each variable.
(U3)For any \(w_2>w_1\) in S, \(l>0\) and for each \(d>0\), the function \(D_lu_o(x):= u_o(x,w_2)-u_o(x+l,w_1)\) has the strict single crossing property on \([d,+\infty ).\)
3 Markov Perfect Equilibria in Altruistic Growth Economies
Consider an infinite sequence of generations labeled by \(t\in T=\mathbb {N}.\) There is one commodity, which may be consumed or invested. Every generation lives one period and in the paternalistic case derives utility from its own consumption and consumption of its immediate descendant. In the non-paternalistic case generation, \(t\in T\) takes into account a utility for consumption of generation \(t+1.\) In this paper, we are interested in mixed model where both paternalistic and non-paternalistic components are present. Generation \(t\in T\) receives the endowment \(s_t\in S\) and chooses consumption level \(a_t\in A(s_t)=[0,s_t].\) The investment \(i_t:=s_t-a_t\) determines the endowment of its successor according to some transition probability q from S to S, which depends on \(i_t\in A(s_t).\) Let \(\varPhi \) be the set of Borel measurable functions \(\phi : S\rightarrow S\) such that \(\phi (s)\in A(s)\) for each \(s\in S.\) A strategy or policy for generation \(t\in T\) is a Borel measurable function \(c_t:S\rightarrow S\) such that \(c_t(s_t)\in A(s_t)\) for all \(s_t\in S.\) The set of all strategies for each generation is denoted by \(\varPi .\)
Let
\(v:S \rightarrow \mathbb {R}_+\) be a continuous increasing function such that
\(v(0)=0.\) Assume that generation
\(t\in T\) consumes
\(a \in A(s_t)\) in state
\(s_t=s\) and the following generation is going to use a strategy
\(c_{t+1}=c'\in \varPi .\) Then, the term
$$\begin{aligned} \mathcal{E}_iv(c'):=\int _Sv(c'(s'))q(ds'|i) \end{aligned}$$
is a generation
t’s evaluation of consumption policy
\(c'\) of generation
\(t+1\) under investment
\(i=s-a.\) Let
\(U(c')(s')\) denote the (Borel measurable in
\(s'\)) utility for generation
\(t+1\) resulting from its consumption policy
\(c'\) in state
\(s'\in S.\) This utility can also be evaluated by generation
t under investment
\(i=s-a\) by computing the expected value with respect to the probability measure
\(q(\cdot |i).\) More formally, generation
t can consider
$$\begin{aligned} \mathcal{E}_iU(c'):=\int _SU(c')(s')q(ds'|i). \end{aligned}$$
Assume that
\(\mathcal{E}_iv(c')\) and
\(\mathcal{E}_iU(c')\) are aggregated with the aid of the function
\(W:\mathbb {R}_+\times \mathbb {R}_+\rightarrow \mathbb {R}_+,\) i.e.,
\(w= W(\mathcal{E}_iv(c'),\mathcal{E}_iU(c'))\) is calculated. Then, the
aggregated utility for generation
t is obtained by aggregating
\(a\in A(s)\) and
w by the function
u discussed in Preliminaries. More precisely, the
utility of generation
t under investment
\(i=s-a\) is defined as
$$\begin{aligned} P (a,c',U(c'))(s): = u(a,W(\mathcal{E}_{s-a}v(c'),\mathcal{E}_{s-a}U(c'))). \end{aligned}$$
(2)
Similarly as in [
35] or [
29], we can call
U an
indirect utility for generation
\(t\in T.\) However, one should note that indirect utilities in their approaches are functions depending on endowments only.
If
\(s=s_t,\)\(a=c_t(s_t),\)\(c'=c_{t+1}\),
\(i_t=s_t-c_t(s_t),\) this utility equals
$$\begin{aligned} P (c_t(s_t),c_{t+1},U(c_{t+1}))(s_t) = u(c_t(s_t),W(\mathcal{E}_{i_t}v(c_{t+1}),\mathcal{E}_{i_t}U(c_{t+1}))). \end{aligned}$$
(3)
This clearly shows that the utility of generation
t depends on its own consumption in state
\(s_t\), the expectation of its own evaluation
v of consumption of generation
\(t+1\) (paternalistic altruism component) and the utility
U of consumption of generation
\(t+1\) (non-paternalistic altruism component). In the sequel, we impose additional assumptions on functions
v and
W and the transition probability
q to cover an unbounded case.
Note that in (
4), we deal with a double fixed-point problem. The strategy
\(c^*\) is the best response for every generation
t, if its immediate successor is going to use
\(c^*\), and each generation evaluates its consumption strategy
\(c^*\) using the same function
\(U^*.\) Following Ray [
35], one can say that it is assumed in Definition 1 that “there exist an indirect utility function and a consumption strategy (policy), both depending on current endowment, such that each generation finds it optimal to adopt that consumption strategy, provided its immediate descendant uses the same policy and exhibit the given indirect utility. Moreover, the indirect utility function generated by the generations maximization problem is also the same as that announced by its descendant.”
In the pure paternalistic case, the utility for generation
t is of simpler form
$$\begin{aligned} \tilde{P} (c_t(s_t),c_{t+1} )(s_t) = u(c_t(s_t), \mathcal{E}_{i_t}v(c_{t+1})). \end{aligned}$$
The analogous form to (
2) is
$$\begin{aligned} \tilde{P} (a,c' )(s): = u(a, \mathcal{E}_{s-a}v(c')), \end{aligned}$$
(5)
where
\(a\in A(s)\) is a consumption of generation
t,
\(i=s-a\) is its investment in state
\(s\in S\) and
\(c'\) is a consumption strategy of generation
\(t+1.\)
The definition of equilibrium is similar to that given in [
11,
28,
34] or [
3,
4].
4 Basic Assumptions and Main Results
Let
\(\Pr (S)\) be the set of all probability measures on the state space
S. We recall that a sequence
\((\mu _n)\) of probability measures on
Sconverges weakly to some
\(\mu _0\in \Pr (S)\) (
\(\mu _n \Rightarrow \mu _0\) in short) if, for any bounded continuous function
\(h:S\rightarrow \mathbb {R},\) it holds that
$$\begin{aligned} \lim _{n\rightarrow \infty }\int _Sh(s)\mu _n(ds)= \int _Sh(s)\mu _0(ds). \end{aligned}$$
We already made three assumptions (U1)-(U3) on the aggregator
u. Below, we provide additional conditions on the primitive data that will be imposed in our two main results. To include unbounded from above utilities, we shall apply a weighted norm approach inspired by the papers in dynamic programming (see Wessels [
40]; Hernández-Lerma and Lasserre [
22]; Jaśkiewicz and Nowak [
24]) or recursive utility theory (see Boyd [
14]; Durán [
17,
18]).
Let
\(\omega :S\rightarrow [1,\infty )\) be a continuous non-decreasing function. Further,
\(\omega \) will be called a weight function. We now make some basic assumptions on the transition probability.
(Q1)Assume that
\(\lambda _j:S\rightarrow [0,1],\)\(j\in J:=\{1,\ldots ,N\},\) are continuous functions such that
\(\sum _{j=1}^N \lambda _j(i)=1\) for all
\(i\in S.\) In addition, suppose that there exist transition probabilities
\(q_j\) from
S to
S,
\(j\in J\), such that for each
\(i\in S\), we have
$$\begin{aligned} q(\cdot |i)=\sum _{j=1}^N\lambda _j(i)q_j(\cdot |i). \end{aligned}$$
(8)
Moreover, for every
\(j\in J,\)\(q_j(\{0\}|0)=1\), and the transition probability
\(q_j(\cdot |i)\) has the Feller property, i.e., if
\(i_n\rightarrow i_0\) in
S as
\(n\rightarrow \infty ,\) then
\(q(\cdot |i_n) \Rightarrow q(\cdot |i_0).\)
(Q2)Every transition probability
\(q_j(\cdot |y)\) in (
8) satisfies the
stochastic dominance condition, i.e., if
\(z\rightarrow Q_j(z|i):=q_j([0,z]|i)\) is the cumulative distribution function for
\(q_j(\cdot |i),\) then for any
\(i_1<i_2\) and
\(z\in S\), we have that
\(Q_j(z|i_1)\ge Q_j(z|i_2).\)
(Q3)For every \(z\in S,\) the set \(S^z :=\{i\in S:\; q(\{z\}|i)>0\}\) is countable.
(Q4)The function \(i\rightarrow \int _S \omega (s')q(ds'|i)\) is continuous on S.
(Q5)We have
$$\begin{aligned}\kappa _0 :=\sup _{s\in S}\sup _{a\in A(s)} \frac{\int _S\omega (s')q(ds'|s-a)}{\omega (s)} <\infty .\end{aligned}$$
As in preliminaries, we make the following assumption on the discount function
\(\delta .\)(D)\(\delta :S\rightarrow S\) is continuous and non-decreasing, and \(\delta (z)<z\) for all \(z\in S_+.\) (Hence, it follows that \(\delta (0)=0.\)) Moreover, the function \(z\rightarrow \delta (z)/z\) defined on \(S_+\) is non-increasing.
We can now continue our assumptions on the utility function.
(U4)The function \(v:S\rightarrow \mathbb {R}_+\) is increasing and continuous, and \(v(0)=0.\)
(U5)If v is unbounded, then the function \(i\rightarrow \int _Sv(s')q(ds'|i)\) is continuous on S.
(U6)The function \(W:\mathbb {R}_+\times \mathbb {R}_+\rightarrow \mathbb {R}_+\) is continuous and increasing in each variable, and \(W(0,0)=0.\)
(U7)There exists a constant
\(\kappa _1>0\) such that for each
\(s\in S,\) we have
$$\begin{aligned} u(s,W(\Vert v\Vert _\omega \kappa _0 \omega (s),0)\le \kappa _1\omega (s). \end{aligned}$$
Here,
\( \Vert v\Vert _\omega \) is defined as
\(\sup _{x\in S} v(x)/\omega (x)\) and is assumed to be finite.
(U8)For any
\(r_1,r_2\in \mathbb {R}_+\) and every
\(s\in S,\)\(a\in A(s),\)\(b\ge 0,\), we have
$$\begin{aligned} |u(a,W(b,r_1))-u(a,W(b,r_2))|\le \delta (|r_1-r_2|). \end{aligned}$$
For any
\(f:F\times S\rightarrow \mathbb {R}\) we define
$$\begin{aligned} \Vert f\Vert _\omega :=\sup _{s\in S}\sup _{c\in F}\frac{|f(c,s)|}{\omega (s)}. \end{aligned}$$
Let
\(C(F\times S)\) be the Banach space of all continuous functions
\(f:F\times S\rightarrow \mathbb {R}\) such that
\(\Vert f\Vert _\omega <\infty .\) Further, in some cases we shall write
f(
c)(
s) instead of
f(
c,
s) for any
\(f\in C(F\times S)\).
We can now state our main results.
In the paternalistic case, we can drop the assumptions involving the discount function \(\delta \) and weight function \(\omega .\)
6 Basic Monotonicity Result
In this section, we provide a useful result that may have applications to various models in optimization. Let
\(\xi :S\rightarrow S\) be an upper semicontinuous function. For any
\(s\in S,\) define
$$\begin{aligned} A_o^\xi (s) := \mathrm{arg} \max _{i\in A(s)}u(s-i,\xi (i))\ \ \text{ and }\ \ a^\xi _o(s):= \min A_o^\xi (s). \end{aligned}$$
(10)
Under our assumptions (U1)–(U2), the function
\(i\rightarrow u(s-i,\xi (i))\) is upper semicontinuous on
A(
s). Therefore, the set
\(A_o^\xi (s)\) is non-empty and compact. Thus,
\(a_o^\xi (s)\) is well defined. The following result is related to Proposition 1 in [
3].
The following auxiliary result is a simple modification of Lemma 3 in [
4].
7 Proofs of Theorems 1 and 2
In this section, we assume that assumptions used in Theorems
1 and
2 are satisfied, although they are not explicitly recalled.
Let
\((h_n)\) be a sequence of Borel measurable real-valued functions on
S. For each
\(s\in S,\) define
$$\begin{aligned} \liminf _{n\rightarrow \infty , \ s'\rightarrow s}h_n(s') := \inf \{\liminf _{n\rightarrow \infty } h_n(s_n):\;s_n\rightarrow s\} \end{aligned}$$
and
$$\begin{aligned} \limsup _{n\rightarrow \infty , \ s'\rightarrow s}h_n(s') := \sup \{\limsup _{n\rightarrow \infty } h_n(s_n):\;s_n\rightarrow s\}. \end{aligned}$$
Let
V be the closed subset of all nonnegative functions
f in the Banach space
\(C(F\times S) \) such that
\(f(c,0)=0.\) Let
\(f\in V\) be fixed. For any
\(c\in F,\)\(s\in S,\) define
$$\begin{aligned} A_o(c)(s):= \mathrm{arg} \max _{i\in A(s)} u(s-i,W(\mathcal{E}_iv(c),\mathcal{E}_if(c)))\quad \mathrm{and}\quad a_o(c)(s)= \min A_o(c)(s).\nonumber \\ \end{aligned}$$
(16)
By Lemmas
3 and
4, the set
\(A_o(c)(s)\) is non-empty and compact and therefore,
\(a_o(c)(s)\) is well defined. Put for any
\(c\in F\)$$\begin{aligned} (Tf)(c,s):= \max _{i\in A(s)} u(s-i,W(\mathcal{E}_{i}v(c),\mathcal{E}_{i}f(c))), \ \ s\in S. \end{aligned}$$
(17)
In the remaining part of this section, we consider only assumptions of Theorem
2. By replacing the function
\(\omega \) with
v in Lemmas
4 and
5 and using condition (U5), one can prove that the mapping
$$\begin{aligned}i\rightarrow \int _Sv(c(s'))q(ds'|i)\end{aligned}$$
is upper semicontinuous continuous from the left. Let
$$\begin{aligned} \tilde{A}^*_o(c)(s):= \mathrm{arg} \max _{i\in A(s)}u(s-i,\mathcal{E}_iv(c))\quad \mathrm{and}\quad \tilde{a}^*_o(c)(s):= \min \tilde{A}^*_o(c)(s), \ \ s\in S,\ c\in F. \end{aligned}$$
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