On recursive utilities with nonaffine aggregator and conditional certainty equivalent
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Abstract
In this paper, we consider the problem of the existence and the uniqueness of a recursive utility function defined on intertemporal lotteries. The purpose of this paper is to provide the results of the existence and the uniqueness of a recursive utility function. The utility function is obtained as the limit of iterations on a nonlinear operator and is independent on initial starting points, with iterations converging at an exponential rate. We also find the maximum utility and an optimal strategy by means of iterations of the Bellman operator.
Keywords
Recursive utilities Dynamic programming Epstein–Zin preferences Certainty equivalent Solid cone Attracting propertyJEL Classification
E21 C61 C02 C65 D641 Introduction
Dynamic models in economics often assume the utility functions defined over sequences of random consumptions are represented by a timeadditive expected overall utility which discounts future temporal utilities at a constant rate. The existing literature clearly shows that this standard utility assumption is restrictive in numerous economic situations.^{1} To name just few limitations, first in the standard case, the elasticity of intertemporal substitution (EIS) is equal to the inverse of the risk aversion coefficient. As a result, the standard utility formulation cannot explain many important puzzles in the literature (e.g., the equity premium puzzle postulated by Mehra and Prescott (1985) in the literature on asset pricing). Additionally, there is strong evidence that some decision makers prefer to know the realization of uncertainty as quickly as possible, while others prefer to know the realization of uncertainty at a later date. Again, this situation cannot be captured within the standard utility framework [see Kreps and Porteus (1978), Chew and Epstein (1989) as well as Klibanoff and Ozdenoren (2007)]. Finally, it is worth mentioning that the standard utility formulation is incompatible with confirmed important “paradoxes” in the experimental economics literature (e.g., the Allais paradox and the Ellsberg paradox).
Principle of optimality in dynamic programming together with the Bellman equation assure overall utility from today onward which can be expressed as a linear transformation of today’s temporal utility and the overall utility from future periods. In the standard discounted utility model, Koopmans (1960) formulated the set of axioms on the aggregator that connected the today’s temporal utility with the utility from future (continuation) periods. Similar class of recursive utilities provide Asheim et al. (2012) by proposing another axioms of sustainable recursive preferences. The set of deterministic recursive utilities, defined as such, included the standard discounted utility model as a special case. The ideas by Koopmans (1960) were extended by Kreps and Porteus (1978) to models with uncertainty with the finite time horizon and by Epstein and Zin (1989) to models with the infinite time horizon, in each case defining recursive utilities on a set of lotteries. Whereas Kreps and Porteus (1978) and Klibanoff and Ozdenoren (2007) parametrized a utility from future periods by means of the expected value, Epstein and Zin (1989) used the most general concept of Conditional Certainty Equivalent (henceforth, CCE for short).
A large body of the literature has already established extensions of the standard discounted utility model toward nonadditive aggregators under both deterministic and stochastic settings. Deterministic utility functions based on Koopmans equations can be found in numerous papers in the literature, including works by BoydIII (2006), Bich et al. (2018), Duran (2000), LeVan and Vailakis (2005), MartinsdaRocha and Vailakis (2010), Jaśkiewicz et al. (2014), among others, while the utility function based on Epstein and Zin (1989) equations can be found in the work of Weil (1993), Skiadas (2015), Bäuerle and Jaskiewicz (2018), Marinacci and Montrucchio (2010), Ozaki and Streufert (1996), and Bloise and Vailakis (2018).
There are three fundamental questions associated with the specification of a recursive utility. First, whether the (recursive utility) function exists and is unique? Second, whether the optimal value is a fixed point of the corresponding Bellman operator and third, whether the optimal value function is a global attractor, i.e., whether a sequence of iterations defined on the Bellman operator uniformly converges to the recursive utility function regardless of the starting point? In this paper, we consider all three aforementioned questions using a nonlinear aggregator and a subhomogeneous CCE. Clearly, apart from deterministic models, the standard expected utility operator is homogenous, and a lot of quasilinear models [as in Chew (1983), for example] are subhomogenous. Moreover, a measure of risk sensitivity postulated in Weil (1993) is a special case in our formulation of CCE. Finally, it bears mentioning that the subhomogeneity here encompasses a large class of aggregators including Kreps and Porteus (1978), Klibanoff and Ozdenoren (2007) as well as most of “Thompson aggregators” used by Marinacci and Montrucchio (2010).
From a technical perspective, we reduce the question of finding a recursive utility to finding a fixed point of an appropriately defined nonlinear operator. To the study of the set of fixed points of this operator, we apply a key theorem in Guo et al. (2004) on the cone of nonnegative functions. This theorem gives sufficient conditions for the existence and the uniqueness of fixed points, as well as provides the results on the global convergence of iterations.^{2}
This approach is new in the literature. For example, some authors (e.g., Ozaki and Streufert 1996) use Knaster–Tarski Theorem (see Tarski 1955). In Jaśkiewicz et al. (2014), the authors use Matkowski Theorem (see Matkowski 1975). Becker and RincónZapatero (2016) and Bloise and Vailakis (2018) used the geometric fixed point theorem by Krasnoselski and Zabreiko (1984). Indirect methods via iterations on extreme selections have been proposed in LeVan and Vailakis (2005). The approach proposed in this paper is different. Finally, in this paper, we provide specific examples where neither Banach Fixed Point Theorem, nor its extension in Matkowski’s Theorem (Matkowski 1975), can be applied.
The rest of the paper is organized as follows: Sect. 2 contains preliminary information on cones, as well as a statement of Guo–Cho–Zhu Theorem. Section 3 contains a description of the model and the fundamental assumptions. The main results are included in Sects. 4 and 5. In Sect. 4, the existence, the uniqueness, and the global convergence results are proven. In Sect. 5, by means of Bellman equations, we obtain similar results for the optimal value of the recursive utility formulation. The proofs of all Lemmas and Propositions can be found in “Appendix”.
2 Preliminaries
2.1 Fixed point theorems on solid normal cones
Let \((V,\cdot )\) be a Banach space with \(\mathbf {0} \in V\) as its zero vector.
Definition 1

If \(v\in C\) and \(t\in \mathbb {R}_+\), then \(t v\in C\).

If \(v\in C\) and \(v\in C\), then \(v=\mathbf {0}\).
Each cone generates a relation of partial ordering \(\le _{C}\) in the following way: \(v\le _{C} w\), iff \(wv\in C\).
Definition 2
Definition 3
Let \((V,\cdot )\) be a Banach space. Let \(V_0\subset V\), and let \(T:V_0\rightarrow V_0\) be an operator. Let \(v^*\in V_0\) be a fixed point of T. The fixed point \(v^*\) is said to have a global attracting property (or global attractivity) on \( V_0\), if for all \(v_0\in V_0\), it holds \(\lim \nolimits _{n\rightarrow \infty }T^nv_0v^*=0\), where \(T^n:=T\circ \cdots \circ T\).
Let us introduce a few definitions.
Definition 4
Let \((V,\cdot )\) be a Banach space, and let C be a cone generating an order on V. A set \(V_0\) is said to be a countably chain complete subset of V if each countable chain of elements of \(V_0\) (i.e., totally ordered subset of \(V_0\)) has a supremum and an infimum in \(V_0\).
Definition 5
Let \((V,\cdot )\) be a Banach space, and let C be a cone generating an order on V. Let \(V_0\) be a countably chain complete subset of V. An operator \(T:V_0\mapsto V_0\) is monotonically suppreserving (monotonically infpreserving) if for any countable chain \(\tilde{V}\) of elements of \(V_0\), the equality \(T(\sup \tilde{V})=\sup T(\tilde{V})\) (\( T(\sup \tilde{V})=\sup T(\tilde{V})\)) holds.
We introduce the following fixed point theorem that we appeal to in Sects. 4 and 5 to prove the existence, the uniqueness, and the global attractivity of a recursive utility function (Theorem 2). Moreover, we prove the global attractivity of the optimal value function (Theorems 3 and 4).
Theorem 1
[Theorem 3.1.7. in Guo et al. (2004)]
It should be noted that the operator T in Theorem 1 maps an open set into itself. (Hence, a continuous extension of T to the closure of C can have additional fixed points.) For example, \(T:\mathbb {R} _{+}\rightarrow \mathbb {R}_{+}\) with \(T(x)=\sqrt{x}\) has two fixed points, but only one in the interior. Hence, neither Banach Contraction Theorem nor its extensions (like Matkowski 1975) are applicable in this case.
2.2 Basic notations and definitions in a space of functions and measures
 Let \(\varTheta \) be a metric space.

\(\mathcal {B}(\varTheta )\) is the collection of Borel subsets of \(\varTheta \).
 \(B(\varTheta )\) is the set of all Borel measurable, bounded, and realvalued functions on \(\varTheta \). Clearly, \(B(\varTheta )\) is a Banach space equipped with the supnorm \(\cdot _{\varTheta }\), i.e.,and by \(\rightrightarrows \), we denote the convergence in the topology induced by this norm (i.e., uniform convergence on \(\varTheta \)).$$\begin{aligned} v_{\varTheta }:=\sup \limits _{\theta \in \varTheta }v(\theta ), \end{aligned}$$
 By \(\left[ \cdot \right] _{\varTheta }\), we denote the infimum operator on \( B(\varTheta )\), i.e.,$$\begin{aligned} \left[ v\right] _{\varTheta }=\inf \limits _{\theta \in \varTheta }v(\theta ). \end{aligned}$$


If \(\varTheta \) is a Polish space, then \(\varDelta (\varTheta )\) denotes the set of all Borel probability measures on \(\varTheta \).
Definition 6
 (i)Monotonicity: for each \(f_1\in \mathcal {F}\) and \(f_2\in \mathcal {F}\)$$\begin{aligned} \text{ if } f_1(\cdot )\le f_2(\cdot ), \text{ then } \mathcal {M}(f_1)(\cdot )\le \mathcal {M}(f_2)(\cdot ). \end{aligned}$$
 (ii)
Constant preserving property: for each \(\gamma \in \mathbb { R}\), \(\mathcal {M}(\gamma )(\cdot )\equiv \gamma \).
3 The model
3.1 Description of the model

\(X\subset \mathbb {R}\) denotes the space of all possible capital levels. Suppose that \(X=[0,\bar{x}]\), where \(\bar{x}\in \mathbb {R}_+\) or \(X=[0,\infty [\).

For each \(x\in X\), let \(\varGamma (x):=[0,x]\) be the set of feasible investment levels when the current capital level is x.

\(\varOmega :=[\underline{\omega },\overline{\omega }]\) (here \(0<\underline{ \omega }\le \overline{\omega }\)) is the space of random shocks endowed with a Borel probability measure \(\rho \).

\(q:X\times \varOmega \rightarrow \mathbb {R}\) denotes a random production function.

\(u: X \rightarrow \mathbb {R}_+\) is a temporal utility function (oneperiod utility).

\(W:\mathbb {R}_+\times \mathbb {R}_+\rightarrow \mathbb {R}_+\) is called an aggregator function.
 \(\mathcal {M}:B(X)\rightarrow B(X)\) is a Conditional Certainty Equivalent (CCE) depending on q in the following way: if \(f_1\in B(X)\), \(f_2\in B(X)\), \(y\in X\), and \(f_1(q(y,\omega ))=f_2(q(y,\omega ))\) for \(\rho \)a.a. \( \omega \in \varOmega \), then$$\begin{aligned} \mathcal {M}_y(f_1)=\mathcal {M}_y(f_2). \end{aligned}$$
Let H be the set of all feasible histories. Mathematically, H is the set of all sequences \(h:=(x_{n},y_{n})_{n=1}^{\infty }\in Gr(\varGamma )^{\infty }\). Endow H with the natural Borel product \(\sigma \) algebra on \( (Gr(\varGamma ))^{\infty }\). For \(n>1\), we denote \(H_n:=Gr(\varGamma )^{n1}\) as the set of all feasible histories before step n. A policy is a sequence of jointly Borel measurable^{6} mappings such that \(\sigma _1:X\rightarrow \varDelta (Y)\), \(\sigma _1(\varGamma (x)x)=1\) for each \(x\in X\), and for \(n>1\), \(\sigma _n:H_n\times X\rightarrow \varDelta (Y)\) is such that for each \((h_n,x)\in H_n\times X \), it holds \(\sigma _n( \varGamma (x_n)h_n,x_n)=1\). Let \(\varSigma \) denote the set of all policies. A policy \(\sigma \) is said to be pure if for each \(n\in \mathbb {N}\), \(h_n\in H_n\), \(x\in X\) there is \(y\in Y\) such that \(\sigma _n(\{y\}h_n,x)=1\). A Markov policy is such that \(\sigma _n(\cdot h_n,x)=f_n(\cdot )\), where \( f_n:X\rightarrow \varDelta (Y)\) is a Borel measurable function. The Markov policy is stationary if \(\sigma _n=f\)\((n\in \mathbb {N})\) for some Borel measurable function \(f:X\rightarrow \varDelta (Y)\). The stationary Markov policy is identified with f. Let \(x\in X\) be an initial state, and let \((\sigma _n)_{n\in \mathbb {N}}\) be an arbitrary policy. By Ionescu–Tulcea Theorem (see Neveu 1965), the production function q, the initial capital \(x\in X\), and the policy \(\sigma \) induce a unique probability measure \(P_{x}^{\sigma }\) on H.
For each \(\sigma \in \varSigma \) and \(n>1\), let \(\sigma ^{n}:H_{n}\rightarrow \varSigma \) be defined as \(\sigma ^n:=\left( \sigma _{n+\tau }\right) _{\tau =0}^{\infty }\). Here, \(\sigma ^{n}\) is called a \(n\)th shift policy, i.e., the policy from the period n onward. Observe that \(\sigma \) is a Markov policy if and only if for each \( n\ge 1\), the \(\sigma ^n\) does not depend on \(H_{n}\) (i.e., is a “constant” strategy).
3.2 Construction of a recursive utility function: basic assumptions and a literature review
The purpose of this section is a construction of a recursive utility function. We start with a list of assumptions on the temporal utility, the aggregator, the production function, and CCE that are sufficient for the existence and the global attracting property of any recursive utility function.
Assumption 1
 (i)Measurability: let \(k\in \mathbb {N}\) and \(Z\in \mathcal {B}(\mathbb { R}^k)\), and suppose that \(f:X\times Z\rightarrow \mathbb {R}\) is a jointly measurable function. Then,is jointly measurable.$$\begin{aligned} (y,z)\in X\times Z\rightarrow \mathcal {M}_y(f(\cdot ,z)), \end{aligned}$$
 (ii)Subhomogeneity: for each \(y\in X\), the operator \(\mathcal {M}_{y}\) is subhomogenous, i.e.,$$\begin{aligned} \text{ if } v\in B(X) \text{ and } t\in [0,1]\text{, } \text{ then } \mathcal {M}_{y}(t v)\ge t\mathcal {M}_y(v). \end{aligned}$$
Assumption 2
(Temporal utility) Assume \(u:X\mapsto \mathbb {R}\) is a strictly increasing, bounded, and continuous function such that \(u(0)\ge 0\).
Assumption 3

W is increasing in both arguments.

W is jointly continuous.

\(W(v_1,v_2)=0\) if and only if \(v_1=v_2=0\).
 There exists a constant \(r\in ]0,1[\) such that for all \(v_1\ge u(0)\), \( v_2>0\) and \(t\in ]0,1[\), it holds$$\begin{aligned} W(v_1,t v_2)\ge t^{r}W(v_1,v_2). \end{aligned}$$(3)
Assumption 4

For each \(\omega \in \varOmega \), \(q(\cdot ,\omega )\) is a strictly increasing and continuous function such that \(q(0,\omega )=0\).

There exists an increasing function \(K:\varOmega \rightarrow \mathbb {R}_{++}\) such that \(q(x,\omega )> x\) for all \(x\in ]0, K(\omega )[\) and \(q(x,\omega )\le x\) for all \(x\in [ K(\omega ),\infty [\).
The following assumption is needed for proving that there exists an optimal policy in the finite horizon model. Moreover, with this assumption the optimal value of the function J in the infinite time horizon model can be approximated by the nstage optimal value function for sufficiently large n.
Assumption 5
 (i)
\(\mathcal {M}_{y}\) is a monotonically suppreserving operator.
 (ii)
The operator \(y\in X\mapsto \mathcal {M}_{y}(v)\) is continuous whenever v is continuous.
Example 1
The following proposition yields a list of properties of \(\phi \) that are sufficient for subhomogeneity of the quasilinear mean in (4).
Proposition 1
 (i)
\(\phi \) is strictly increasing, and the function \(\phi ^{\prime }(\phi ^{1}(\cdot ))\phi ^{1}(\cdot )\) is concave.
 (ii)
Or \(\phi \) is strictly decreasing, and the function \( \phi ^{\prime }(\phi ^{1}(\cdot ))\phi ^{1}(\cdot )\) is convex.
Example 2
Finally, let us comment on Assumption 3. Observe that the paper by Marinacci and Montrucchio (2010) includes a few aggregators which satisfy Assumption 3.
Example 3
The next aggregator is a modification of the aggregator by Koopmans et al. (1964).
Example 4
The following proposition states that Thompson aggregators in fact obey Assumption 3 whenever \(u(0)=\delta >0\).
Proposition 2
Assume W is Thompson, \(u(0)=\delta >0\) and \(u_{X}=\bar{u}\). Let \(\bar{U}\) be the least fixed point of \(W(\bar{u},\cdot )\) and let us define \(\tilde{W}\) as in Eq. (5). Then, \(\tilde{W}\) obeys Assumption 3.
Another example shows that Assumption 3 does not imply Thompson property. Namely, in this example, the concavity at 0 is violated.
Example 5
4 Existence and global attracting property of a recursive utility function
In this section, we prove the existence and the global attractivity of a recursive utility function.
Put \(\mathbf {U}_{+}\) as the set of nonnegative functions from \( \mathbf {U}\). Then, \(\mathbf {U}_{+}\) is a normal cone. Moreover, it induces the standard componentwise order, that is \( U_{1}\le U_{2}\) iff \(U_{1}(x,\sigma )\le U_{2}(x,\sigma )\) for each \( (x,\sigma )\in X\times \varSigma \).
Definition 7
Theorem 2
 (i)\(U^*\) is globally attractive on \(\mathrm{int}(\mathbf {U}_+)\), i.e.,whenever \(U\in \mathrm{int}(\mathbf {U}_+)\).$$\begin{aligned} \lim _{n\rightarrow \infty }U^*T^n_W(U)_{X\times \varSigma }=0 \end{aligned}$$
 (ii)The truncation error satisfies:whenever \(U\in \mathrm{int}(\mathbf {U}_+)\). Here, \(M=2U_{X\times \varSigma }\), \(\alpha = \frac{t_0}{s_0}\), and \(t_0\) and \(s_0\) are chosen in the following way:$$\begin{aligned} T^n_W(U)U^*_{X\times \varSigma }\le M\left( 1\alpha ^{r^n}\right) \text{ for } \text{ all } n\in \mathbb {N} \end{aligned}$$$$\begin{aligned} 0<t_0<1<s_0, \text{ and } \text{ it } \text{ holds } t_0^{1r}U(\cdot )\le T_W(U)(\cdot )\le s_0^{1r} U(\cdot ). \end{aligned}$$
 (iii)
J is a recursive utility function. Moreover, \(J(x,\sigma )\le U^*(x,\sigma )\) for each \(x\in X\) and \(\sigma \in \varSigma \), and \( J(x,\sigma )=U^*(x,\sigma )\) whenever \(x>0\) and \(\sigma \in \varSigma _{x,\delta }\).
To prove Theorem 2, we need to prove some auxiliary results, namely Lemma 1 and Lemma 2. Proofs of both Lemmas are in the “Appendix”.
Lemma 1
 (i)
\(T_W^{\delta }\) maps both \(\mathbf {U}_+\) and \(\mathrm{int}(\mathbf {U}_+)\) into itself.
 (ii)
\(T_W^{\delta }(\cdot )\) is an increasing operator, and for each \( U\in \mathbf {U}_+\) the function \(T_W^{\delta }(U)\) is increasing in \(\delta \).
 (iii)
If \(\delta >0\), then \(T_W^{\delta }(\mathbf {0})(x,\sigma )\ge W(\delta ,0)>0\).
 (iv)
J is well defined, and if U is any fixed point of \( T_W^{\delta }\), then \(J\le U.\)
Lemma 2
Let \(x>0\) and suppose that Assumptions 1, 2, 3, and 4 hold. Then, there exists \(\delta >0\) such that \(\varSigma _{x,\delta }\ne \emptyset \).
Now, we prove Theorem 2.
Proof of Theorem 2
5 Bellman equation and the existence of an optimal program
In the remaining part of this paper, we use the following notation \( B_{+}:=B_{+}(X)\) and \(B_{+}^{o}:=\mathrm{int}(B_{+}(X))\) for short.
Theorem 3
 (i)There exists a function \(\tilde{v}\) that is a unique fixed point of BP such that \(\tilde{v}\in B_{+}^{o}\), and such that for each \( v\in B_{+}^{o}\)with the truncation error satisfying$$\begin{aligned} \lim \limits _{n\rightarrow \infty }BP^{n}(v)\tilde{v}_{X}=0, \end{aligned}$$where \(M=2v_{X}\), \(\alpha =\frac{t_{0}}{s_{0}}\), and \(t_{0}\) and \(s_{0}\) are chosen in the following way:$$\begin{aligned} BP^{n}(v)\tilde{v}_{X}\le M\left( 1\alpha ^{r^{n}}\right) \text{ for } \text{ all } n\in \mathbb {N}, \end{aligned}$$$$\begin{aligned} 0<t_{0}<1<s_{0}, \text{ and } \text{ it } \text{ holds } t_{0}^{1r}v(\cdot )\le BP(v)(\cdot )\le s_{0}^{1r}v(\cdot ). \end{aligned}$$
 (ii)
\(\tilde{v}\) is increasing. Moreover, if additionally Assumption 5 holds, \(\tilde{v}\) is continuous.
 (iii)If Assumption 5 holds, then for each \( x\in X\)and there exists a stationary and pure Uoptimal policy \(\tilde{\sigma } \in \varSigma \) satisfying$$\begin{aligned} \tilde{v}(x)=\sup \limits _{\sigma \in \varSigma }U^*(x, \sigma ), \end{aligned}$$$$\begin{aligned} \tilde{v}(x)=W(x\tilde{\sigma }(x),\mathcal {M}_{\tilde{\sigma }(x)}(\tilde{v} ))=\sup \limits _{y\in \varGamma (x)}W(u(xy),\mathcal {M}_{y}(\tilde{v})). \end{aligned}$$(12)
Corollary 1
Let us first introduce the following notation. For each sequence \( (x_{n})_{n\in \mathbb {N}},\) let the sequence \(x^{(n)}:=\left( x_{k}^{(n)}\right) _{k\in \mathbb {N}}\) be defined in the following way: \( x_{k}^{(n)}=x_{k}\) for \(k\le n\) and \(x_{k}^{(n)}=0\) for \(k\ge n+1\). In turn, the sequence \(x^{(n)}\) is defined in the following way: \( x_{k}^{(n)}=x_{nk+1}\) if \(k\le n\) and \(x_{k}^{(n)}=0\) for \(k\ge n+1\). Now, consider the \(n\)period horizon model. Each policy \(\pi \) in the nstep model is identified with \(\sigma ^{(n)}\) for some policy \(\sigma \in \varSigma \). Observe if \(\pi \) is a policy in the nstep model, then \(\pi \) can be embedded into \(\varSigma \) in the canonical way. Hence, we can write \( J_{n}(x,\pi )=J(x,\sigma ^{(n)})\).
In Theorem 4, we shall study some properties of the optimal value of J, and we argue that under additional Assumption 5 the optimal values of \(U^{*}\) and J coincide with \(X\setminus \{0\}\). We also establish some results of an existence of optimal policies in the finite step model, and we show that the Joptimal policy obeys the Bellman equations. Unfortunately, Theorem 4 does not establish the existence of a Joptimal policy, since the optimal function need not be upper semicontinuous at \(x=0\).
Theorem 4
 (i)For all \(n\in \mathbb {N}\), \(v_n^*(\cdot )\) is an increasing and continuous function. Furthermore, there exists a sequence of Borel measurable functions \((\hat{\sigma }_k)_{k\in \mathbb {N}}\) that are selections of \(\varGamma \) such that \(\hat{\sigma }^{(n)}\) is a Markov optimal policy in the nstep model, for each \(x\in X\) it holdsand for \(n\ge 1\), and \(v_1^*(x)\le v_2^*(x)\le \cdots v_n^*(x)\le \cdots \). As a result, there exists a limit$$\begin{aligned} v_{n+1}^*(x)=BP(v_n^*)(x), \end{aligned}$$(14)$$\begin{aligned} v^{*}(x):=\lim \limits _{n\rightarrow \infty }v_n(x). \end{aligned}$$(15)
 (ii)
Let v be an arbitrary fixed point of BP. Then, \( v^{*}(\cdot )\le v(\cdot )\).
 (iii)For each \(x\in X\)$$\begin{aligned} v^*(x)=\lim \limits _{n\rightarrow \infty }v_n^*(x)=\sup \limits _{\sigma \in \varSigma }J(x, \sigma ). \end{aligned}$$(16)
 (iv)
For each \(x\in X\setminus \{0\}\), it holds \(\tilde{v}(x)= v^*(x)\); as a result, \(v^*\) is continuous on \(X\setminus \{0\}\).
The following corollary establishes the existence of the Joptimal policy. The following corollary is immediate from Theorem 3(iv).
Corollary 2
If \(\tilde{\sigma }\) be a stationary Joptimal policy, then it is also a Uoptimal policy.
Remark 1
The following remark justifies some problems with the uniqueness of the solution of Bellman equation.
Remark 2
It should be noted that \(BP^{n}(\mathbf {1})(\cdot )\rightrightarrows \tilde{v} (\cdot )\), while (unless \(u(0)>0\)) \(BP^{n}(\mathbf {0})(\cdot )\rightarrow v^{*}(\cdot )\) pointwise only. The reason is neither of the function \( BP^{n}(\mathbf {0})(\cdot )\) is an element of \(B_{+}^{o}\). Hence, theorem by Guo et al. (Theorem 1) does not work with \(\mathbf {0}\) as a starting point.
Theorem 4 is applicable for many aggregators known in the literature. Sometimes, a trick is needed.
Example 6
Sometimes, the trick from the previous remark is impossible. The next theorem establishes asymptotic properties of the optimal value function with varying lower bound of the temporal utility. For any \( \delta >0\), consider the dynamical system (X, \(\varGamma ,\)\(\varOmega ,\)\( u^{\delta },\)q, W, \(\mathcal {M})\) called a \(\delta \)model, i.e., the dynamical system where u is substituted by \(u^{\delta }(\cdot )\). Let \( v_n^{\delta }\) and \(v^{\delta }\) denote the optimal values in the n and, respectively, in the infinite step model. This theorem extends the applicability toward all Thompson aggregators without restrictions that \(u(0)>0\).
Theorem 5
For proving Theorems 3 and 4, we applied Theorem 1. Alternatively, we could try to apply Knaster–Tarski Theorem (Tarski 1955). But in the next remark, we indicate some problems with the applicability of Knaster–Tarski Theorem in this context.
Example 7
Finally, we prove Theorems 3, 4, and next Theorem 5.
Proof of Theorem 3
We shall prove all the points separately. Clearly, BP is an increasing operator on \(B_+\).
Proof of (ii). It is clear that \(\tilde{v}\) is increasing. Indeed, in the point (i) of this theorem, we have already established that the range of \(BP(\cdot )\) is included in the set of increasing functions.
Now, suppose Assumption 5 holds. We show that \( \tilde{v}\) is continuous. First, we need to demonstrate that \(B(v)(\cdot )\) is a continuous function whenever v is. Indeed, if v is continuous, then from Assumption 5 the function \(y\in X\mapsto \mathcal {M}_{y}(v)\) is continuous. Therefore, \((x,y)\in Gr(\varGamma )\mapsto W(u(xy),\mathcal {M} _{y}(v))\) is jointly continuous. Hence, by Berge Maximum Theorem [Theorem 17.31 in Aliprantis and Border (2006)], it follows that \(BP(v)(\cdot )\) is continuous. Consequently, \(BP(\mathbf {1})(\cdot )\) is continuous, and hence, for each \(n\in \mathbb {N}\), \(BP^{n}(\mathbf {1})(\cdot )\) is continuous. By the point (i) of this theorem, it follows that \(BP^{n}(\mathbf {1})(\cdot )\rightrightarrows \tilde{v}(\cdot )\). Hence, \(\tilde{v}(\cdot )\) must be continuous.
Proof of Theorem 4
 \(\hat{\sigma }_1(x)=0\) for each \(x\in X\), and \(\hat{\sigma }_n(\cdot )\) is a Borel measurable selection of \(\varGamma \) satisfying:$$\begin{aligned} W(u(x\hat{\sigma }_n(x)),\mathcal {M}_{\hat{\sigma }_n(x)}(v^*_{n1}))=\max \limits _{y\in \varGamma (x)}W(u(xy),\mathcal {M}_y(v^*_{n1})). \end{aligned}$$(18)

For each \(n\in \mathbb {N}\), \(\hat{\sigma }^{(n)}=(\hat{ \sigma }_n,\hat{\sigma }_{n1},\ldots ,\hat{\sigma }_1)\) is an optimal policy in the nstep model.
Observe that \((v_n^*)_{n\in \mathbb {N}}\) is an increasing sequence. Indeed, \(v_n^*(\cdot )=BP^n(\mathbf {0})(\cdot )\), BP is an increasing operator, and \(\mathbf {0}\le BP(\mathbf {0})(\cdot )\). Therefore, using the induction method we easily conclude that \(v_n^*\) is increasing in n, and therefore, the limit in (15) must exist.
Proof of (ii). If v is a fixed point of BP in \(B_+\), then \(v=BP^n(v)\), and as we have already established in part (i) of this theorem \(v^*_n=BP^n(\mathbf {0})\) for each \(n\in \mathbb {N}\). Since BP is increasing and \(v\ge \mathbf {0}\), for each \(x\in X\) it holds \(v_n(x)\le v(x)\). Consequently, \( v^{*}(x)=\lim \nolimits _{n\rightarrow \infty }v_n^*(x)\le v(x)\).
Proof of (iv). Without loss of generality suppose that \(u(0)=0\). For otherwise, all the theses are established in Corollary 1.
Therefore, by the point (ii) of Theorem 3, it follows that \(v^*\) is continuous on \( X\setminus \{0\}\).
\(\square \)
Proof of Theorem 5
6 Final conclusions
The paper contains the proof of the existence and the uniqueness of a recursive utility in the model with a nonlinear aggregator and CCE. Moreover, we consider the problem of optimization of a recursive utility by means of Bellman equations. The transition probability satisfies assumptions known of growth models. The results which follow from this paper extend the results known from Jaśkiewicz et al. (2014). Also, many Thompson’s aggregators from Marinacci and Montrucchio (2010) and Becker and RincónZapatero (2016) satisfy the conditions of my paper. Moreover, this paper contains an example of an aggregator which satisfies its conditions, yet is not a Thompson’s aggregator. Since CCE is nonlinear, the results in this paper expand the deterministic models as in Bich et al. (2018), MartinsdaRocha and Vailakis (2010), LeVan and Vailakis (2005), and Becker and RincónZapatero (2016). However, let us note that, as opposed to Bich et al. (2018), Jaśkiewicz et al. (2014), and LeVan and Vailakis (2005), Assumption 3 requires that the temporal utility u is bounded below by 0. The main results from this paper contain an iterative algorithm used to calculate recursive utility. For our results, we make extensive use of Guo–Cho–Zhu Theorem (Guo et al. 2004), which distinguishes this paper from other papers in the literature, which used standard Banach Fixed Point Theorems (e.g., in Marinacci and Montrucchio (2010) or in MartinsdaRocha and Vailakis (2010)] or the extension from Matkowski (1975) (see also Jaśkiewicz et al. 2014), as well as papers that apply Krasnoselski and Zabreiko (1984) (see Becker and RincónZapatero 2016; Bloise and Vailakis 2018).
The applications of various aggregators and CCE are possible in various theoretical problems in financial market (see asset pricing in Talarini Jr. (2000), or sovereign debt paradox in MartinsdaRocha and Vailakis (2017)), Pareto optimal allocations (see Anderson 2005) or the managing global environment (see Asheim et al. 2012).
Footnotes
 1.
The reader is referred to Chapter 20 in Miao (2014) for more arguments.
 2.
 3.
i.e., The greatest open set included in C.
 4.
In fact, Guo et al. (2004) introduce the normality of the cone as \(v\le N v\) where N is the index of normality. Let us focus attention on \(N=1\).
 5.
In the rest of the paper, the term of increasing function means an order preserving function, i.e., \(x\le y\) implies that \(f(x)\le f(y) \).
 6.
That is Borel measurable with respect to the corresponding product topology.
 7.
 8.
According to Marinacci and Montrucchio (2010) terminology, Thompson aggregator means that is increasing in both arguments, \(W(v_{1},0)>0\) whenever \(v_{1}>0\) and is concave at 0, i.e., \(W(v_{1},t v_{2})\ge t W(v_{1},v_{2})+(1t)W(v_{1},0)\) for all \( v_{1},v_{2}\ge 0\) and \(t \in [0,1]\).
 9.
Compare with \(\tilde{W}\) in Example 4.
 10.
By Lemma 3.1.3 in Guo et al. (2004), we only have a norm continuity of \(T_{W}\).
 11.
This example has an economic motivation and is inherited from Gilboa and Schmeidler (1989) and studied later by Qu (2017). It shows that the results in this paper are useful not only in Markov decision problems but also in a large literature on robust control. For a survey of the literature on this topic, the reader is referred to Hansen and Sargent (2001), Maccheroni et al. (2006), Balbus et al. (2014), Drugeon et al. (2019), and the references cited therein. Alternatively, we can consider also the optimistic point of view and change \(\inf \) into \(\sup \) by the adaptation of the idea of Saponara (2018).
Notes
Acknowledgements
The author thanks Bob Becker, Geatano Bloise, JeanPierre Drugeon, Paweł Kliber, Cuong Le Van, Piotr Maćkowiak, Andrzej S. Nowak, Kevin Reffett, JuanPablo RincónZapatero, Yiannis Vailakis, Łukasz Woźny, all participants of the seminar in Poznań University of Economics and Business, all participants of the seminar ”Methods of Economics Dynamics” in Paris School of Economics, all participants of European Workshop of General Equilibrium Theory (2016) in Glasgow, all participants of Summer Workshop in Economic Theory (2016) in Paris, and the anonymous referee for all useful comments, during the writing of this paper. This research is supported by National Science Center in Poland with a Grant No. UMO2016/23/B/HS4/02398. The author is fully responsible for the content.
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