# On Extendability of the Principle of Equivalent Utility

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Preliminary Results

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Example**

**1.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

- (i)
- $$(1-g(1-p\left)\right)u(w+f(x,p)-x)+g(1-p)u(w+f(x,p\left)\right)=u\left(w\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}for\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}x\in (0,\infty ),\phantom{\rule{0.277778em}{0ex}}p\in (0,1);$$
- (ii)
- $$f(x,p)\in (0,x)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}for\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}x\in (0,\infty ),\phantom{\rule{0.277778em}{0ex}}p\in (0,1);$$
- (iii)
- For every $x\in (0,\infty )$, the function $f(x,\xb7)$ is continuous, where$$\underset{p\to {0}^{+}}{lim}f(x,p)=0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\underset{p\to {1}^{-}}{lim}f(x,p)=x;$$
- (iv)
- For every $p\in (0,1)$, the function $f(\xb7,p)$ is continuous and $li{m}_{x\to {0}^{+}}f(x,p)=0$.

**Proof.**

## 4. Main Result

**Theorem**

**3.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Chudziak, M.; Żołdak, M.
On Extendability of the Principle of Equivalent Utility. *Symmetry* **2020**, *12*, 42.
https://doi.org/10.3390/sym12010042

**AMA Style**

Chudziak M, Żołdak M.
On Extendability of the Principle of Equivalent Utility. *Symmetry*. 2020; 12(1):42.
https://doi.org/10.3390/sym12010042

**Chicago/Turabian Style**

Chudziak, Małgorzata, and Marek Żołdak.
2020. "On Extendability of the Principle of Equivalent Utility" *Symmetry* 12, no. 1: 42.
https://doi.org/10.3390/sym12010042