# Optimal Saving by Expected Utility Operators

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## Abstract

**:**

## 1. Introduction

## 2. Two Possibilistic Expected Utilities

- A weighting function $f:[0,1]\to \mathbf{R}$ (f is a non-negative and increasing function that satisfies ${\int}_{0}^{1}f\left(\gamma \right)d\gamma =1$).
- a fuzzy number A with the level sets ${\left[A\right]}^{\gamma}=[{a}_{1}\left(\gamma \right),{a}_{2}\left(\gamma \right)]$, $\gamma \in [0,1]$;
- a utility function $u:\mathbf{R}\to \mathbf{R}$ of class ${C}^{2}$.

## 3. Expected Utility Operators and D-Operators

- (${U}_{1}$) $\mathcal{U}$ contains the constant functions and first and second degree polynomial functions;
- (${U}_{2}$) If $a,b\in \mathbf{R}$ and $g,h\in \mathcal{U}$ then $ag+bh\in \mathcal{U}$.
- For any $a\in \mathbf{R}$, we denote by $\overline{a}:\mathbf{R}\to \mathbf{R}$ the function $\overline{a}\left(x\right)=a$, for any $x\in \mathbf{R}$.
- We fix a family $\mathcal{U}$ with the properties (${U}_{1}$), (${U}_{2}$) and a weighting function $f:\mathbf{R}\to \mathbf{R}$.

**Definition**

**1**

- (a)
- $T(A,{1}_{\mathbf{R}})={E}_{f}\left(A\right)$;
- (b)
- $T(A,\overline{a})=a$;
- (c)
- $T(A,ag+bh)=aT(A,g)+bT(A,h)$;
- (d)
- $g\le h$ implies $T(A,g)\le T(A,h)$.

**Example**

**1.**

**Example**

**2.**

**Proposition**

**1**

- (i)
- $g(x,\lambda )$ is continuous with respect to x and partially derivable with respect to parameter $\lambda $;
- (ii)
- For any parameter $\lambda \in \mathbf{R}$, the function $\frac{\partial g(.,\lambda )}{\partial \lambda}:\mathbf{R}\to \mathbf{R}$ is continuous.

**Definition**

**2**

**Proposition**

**3**

## 4. The $\mathbf{T}$-Model of Optimal Saving

- two utility functions $u\left(y\right)$ and $v\left(y\right)$: $u\left(y\right)$ is the consumer’s utility function in the first period 0, and $v\left(y\right)$ is the consumer’s utility function in the second period 1;
- in period 0 there is a sure income ${y}_{0}$, and in period 1 there is an uncertain income described by a random variable $\tilde{y}$;
- s is the level of saving for period 0; to face the risk in period 1, the consumer transfers to this period a part s of ${y}_{0}$;
- r is the interest rate for saving.

- (a)
- the utility $u({y}_{0}-s)$ of the amount ${y}_{0}-s$ resulted by the transfer of s in the second period;
- (b)
- the probabilistic expected value $M\left(v((1+r)s+\tilde{y})\right)$, resulted in the second period from the uncertain income $\tilde{y}$ and from the amount $(1+r)s$.

- the uncertain income from period 1 will no longer be a random variable $\tilde{y}$, but a fuzzy number A;
- to define the total utility of the model it is necessary to have a concept of “possibilistic expected utility”.

- Two utility functions u and v, verifying the conditions from the probabilistic model;
- a sure income in period 0 and an uncertain income in period 1, described by a fuzzy number A whose level sets are ${\left[A\right]}^{\gamma}=[{a}_{1}\left(\gamma \right),{a}_{2}\left(\gamma \right)]$, $\gamma \in [0,1]$.
- s is the level of saving for period 0;
- r is the interest rate for saving.

**Proposition**

**4.**

- (i)
- ${V}_{T}^{\prime}\left(s\right)=-{u}^{\prime}({y}_{0}-s)+(1+r)T(A,{v}^{\prime}((1+r)s+y))$
- (ii)
- ${V}_{T}^{\u2033}\left(s\right)={u}^{\u2033}({y}_{0}-s)+{(1+r)}^{2}T(A,{v}^{\prime \prime}((1+r)s+y))$
- (iii)
- ${V}_{T}$ is a concave function.

**Proof.**

- (i)
- (Taking into account the axiom (${D}_{2}$) from Definition 2 and condition (c) of Definition 1 the following equalities are true:$$\begin{array}{cc}{V}_{T}^{\prime}\left(s\right)\hfill & =-{u}^{\prime}({y}_{0}-s)+\frac{d}{ds}T(A,v((1+r)s+y))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =-{u}^{\prime}({y}_{0}-s)+T(A,\frac{d}{ds}v((1+r)s+y))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =-{u}^{\prime}({y}_{0}-s)+T(A,(1+r){v}^{\prime}((1+r)s+y))\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =-{u}^{\prime}({y}_{0}-s)+(1+r)T(A,{v}^{\prime}((1+r)s+y))\hfill \end{array}$$
- (ii)
- Analogous with the proof of (i).
- (iii)
- We recall the hypotheses: ${u}^{\u2033}<0$, ${v}^{\u2033}<0$. Then, by condition (d) of Definition 1 we have $T(A,{v}^{\u2033}((1+r)s+y))\le 0$, thus ${V}_{T}^{\u2033}\left(s\right)\le 0$. □

## 5. The Approximate Computation of the $\mathbf{T}$-Optimal Saving

**Proposition**

**5.**

**Proof.**

**Remark**

**1.**

**Proposition**

**6.**

**Proof.**

## 6. The $\mathbf{T}$-Precautionary Saving

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Remark**

**2.**

**Definition**

**3.**

## 7. Particular Cases and Examples

- $f\left(t\right)=2t$, for any $t\in [0,1]$;
- A is the triangular fuzzy number $(b,\alpha ,\beta )$:

**Example**

**3.**

- for $\gamma >1$:${u}^{\prime}\left(w\right)=\frac{{w}^{1-\gamma}}{1-\gamma}$, ${u}^{\prime}\left(w\right)={w}^{-\gamma}$, ${u}^{\u2033}\left(w\right)=(-\gamma ){w}^{-\gamma -1}$, ${u}^{\u2034}\left(w\right)=\gamma (\gamma +1){w}^{-\gamma -2}$, ${u}^{iv}=-\gamma (\gamma +1)(\gamma +2){w}^{-\gamma -3}$.The optimal saving levels become:$${s}_{T1}^{\ast}\approx \frac{{y}_{0}^{-\gamma}-(1+r)[{a}^{-\gamma}+\frac{1}{2}\gamma (\gamma +1){a}^{-\gamma -2}\frac{{\alpha}^{2}+{\beta}^{2}+\alpha \beta}{18}]}{(-\gamma ){y}_{0}^{-\gamma -1}+{(1+r)}^{2}[(-\gamma ){a}^{-\gamma -1}-\frac{1}{2}\gamma (\gamma +1)(\gamma +2){a}^{-\gamma -3}\frac{{\alpha}^{2}+{\beta}^{2}+\alpha \beta}{18}]};$$$${s}_{T2}^{\ast}\approx \frac{{y}_{0}^{-\gamma}-(1+r)[{a}^{-\gamma}+\frac{1}{2}\gamma (\gamma +1){a}^{-\gamma -2}\frac{{\alpha}^{2}+{\beta}^{2}}{36}]}{(-\gamma ){y}_{0}^{-\gamma -1}+{(1+r)}^{2}[(-\gamma ){a}^{-\gamma -1}-\frac{1}{2}\gamma (\gamma +1)(\gamma +2){a}^{-\gamma -3}\frac{{\alpha}^{2}+{\beta}^{2}}{36}]}$$$${s}_{U}^{\ast}\approx \frac{{y}_{0}^{-\gamma}-(1+r)[{a}^{-\gamma}+\frac{1}{2}\gamma (\gamma +1){a}^{-\gamma -2}\frac{3{\alpha}^{2}+3{\beta}^{2}+2\alpha \beta}{72}]}{(-\gamma ){y}_{0}^{-\gamma -1}+{(1+r)}^{2}[(-\gamma ){a}^{-\gamma -1}-\frac{1}{2}\gamma (\gamma +1)(\gamma +2){a}^{-\gamma -3}\frac{3{\alpha}^{2}+3{\beta}^{2}+2\alpha \beta}{72}]}$$
- for $\gamma =1$:${u}^{\prime}\left(w\right)=\frac{1}{w}$, ${u}^{\u2033}\left(w\right)=-\frac{1}{{w}^{2}}$, ${u}^{\u2034}\left(w\right)=\frac{2}{{w}^{3}}$, ${u}^{iv}\left(w\right)=\frac{-3}{{w}^{4}}$.

## 8. Concluding Remarks

- the construction of a saving model in the possibilistic framework offered by the expected utility operators;
- the use of D-operators (a class of expected utility operators) in the analysis of the solution of the optimization problem associated with the model (existence, uniqueness and its approximate calculation);
- getting some approximation formula of the optimal saving models in a few important cases.

- (a)
- To determine a class of expected utility operators allowing to establish some necessary and sufficient conditions of extra saving (in the framework of associated saving models);
- (b)
- (c)
- the study of optimal saving with several risk parameters: Risk may be represented by a possibilistic vector (the components are fuzzy numbers) or by a mixed vector (some components are fuzzy numbers, others are random variables) (by [12], Chapters 6 and 7);
- (d)
- to apply the results from the paper for possibilistic saving models built from a dataset, using the sample percentile method of Vercher et al. [26].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Georgescu, I.; Kinnunen, J.
Optimal Saving by Expected Utility Operators. *Axioms* **2020**, *9*, 17.
https://doi.org/10.3390/axioms9010017

**AMA Style**

Georgescu I, Kinnunen J.
Optimal Saving by Expected Utility Operators. *Axioms*. 2020; 9(1):17.
https://doi.org/10.3390/axioms9010017

**Chicago/Turabian Style**

Georgescu, Irina, and Jani Kinnunen.
2020. "Optimal Saving by Expected Utility Operators" *Axioms* 9, no. 1: 17.
https://doi.org/10.3390/axioms9010017