# On Quasi-Homogeneous Production Functions

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Main Results

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**1.**

**Theorem**

**1.**

- i.
- The production has constant elasticity ${k}_{i}$ with respect to an input ${x}_{i}$ if and only if f reduces to the one of the following:
- (a)
- a function of the form$$f({x}_{1},{x}_{2},\dots ,{x}_{n})={x}_{i}^{\frac{q}{{g}_{i}}}F({x}_{1},\dots ,{x}_{i-1},{x}_{i+1},\dots ,{x}_{n}),$$$${g}_{1}=0,\dots ,{g}_{i-1}=0,{g}_{i}=\frac{q}{{k}_{i}}\ne 0,{g}_{i+1}=0,\dots ,{g}_{n}=0;$$or
- (b)
- a function of the form$$f({x}_{1},{x}_{2},\dots ,{x}_{n})={x}_{i}^{{k}_{i}}{x}_{j}^{\frac{q-{k}_{i}{g}_{i}}{{g}_{j}}}F({u}_{1},\dots ,{u}_{n-2}),$$$$\{{u}_{1},\dots ,{u}_{n-2}\}=\left\{\frac{{x}_{k}^{{g}_{j}}}{{x}_{j}^{{g}_{k}}}|k\in \{1,\dots ,n\}\backslash \{i,j\}\right\}.$$

- ii.
- The production has constant elasticity ${k}_{i}$ with respect to all inputs ${x}_{i}$, $i\in \{1,2,\dots ,n\}$, if and only if$${g}_{1}{k}_{1}+{g}_{2}{k}_{2}+\dots +{g}_{n}{k}_{n}=q$$$$f({x}_{1},{x}_{2},\dots ,{x}_{n})=C{x}_{1}^{{k}_{1}}{x}_{2}^{{k}_{2}}\dots {x}_{n}^{{k}_{n}},$$
- iii.
- f satisfies the PMRS property if and only f reduces to the one of the following:
- (a)
- a quasi-homogeneous function of degree 0 defined by the following homothetic symmetric CD production function$$f({x}_{1},{x}_{2},\dots ,{x}_{n})=F\left({x}_{1}^{{g}_{j}}{x}_{2}^{{g}_{j}}\dots {x}_{n}^{{g}_{j}}\right),$$
- (b)
- a quasi-homogeneous function of degree $q\ne 0$ defined by the following symmetric CD production function$$f({x}_{1},{x}_{2},\dots ,{x}_{n})=C{x}_{1}^{\frac{q}{{\sum}_{i=1}^{n}{g}_{i}}}{x}_{2}^{\frac{q}{{\sum}_{i=1}^{n}{g}_{i}}}\dots {x}_{n}^{\frac{q}{{\sum}_{i=1}^{n}{g}_{i}}},$$

**Proof**

**of**

**Theorem**

**1.**

**Corollary**

**1.**

- i.
- The output elasticity with respect to capital is a constant k if and only if f reduces to the one of the following:
- (a)
- a function having the form:$$f(K,L)={K}^{\frac{q}{{g}_{K}}}F\left(L\right),$$$${g}_{K}=\frac{q}{k}\ne 0,{g}_{L}=0;$$
- (b)
- a CD production function given by:$$f(K,L)=C{K}^{k}{L}^{\frac{q-k{g}_{K}}{{g}_{L}}},$$

- ii.
- The output elasticity with respect to labor is a constant k if and only if f reduces to the one of the following:
- (a)
- a function having the form:$$f(K,L)={L}^{\frac{q}{{g}_{L}}}F\left(K\right),$$$${g}_{K}=0,{g}_{L}=\frac{q}{k}\ne 0;$$
- (b)
- a CD production function given by:$$f(K,L)=C{K}^{\frac{q-k{g}_{L}}{{g}_{K}}}{L}^{k},$$

- iii.
- The output elasticities of labor and capital are both constant, k and l, respectively, if and only if$${g}_{K}k+{g}_{L}l=q$$$$f(K,L)=C{K}^{k}{L}^{l},$$
- iv.
- f satisfies the PMRS property if and only if f reduces to the one of the following:
- (a)
- a quasi-homogeneous function of degree 0 defined by the following homothetic CD production function$$f(K,L)=F\left({K}^{{g}_{K}}{L}^{-{g}_{L}}\right),$$
- (b)
- a quasi-homogeneous function of degree $q\ne 0$ defined by the following CD production function$$f(K,L)=C{K}^{\frac{q}{{g}_{K}+{g}_{L}}}{L}^{\frac{q}{{g}_{K}+{g}_{L}}},$$

**Remark**

**4.**

## 4. Conclusions and Future Works

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Vîlcu, A.-D.; Vîlcu, G.-E.
On Quasi-Homogeneous Production Functions. *Symmetry* **2019**, *11*, 976.
https://doi.org/10.3390/sym11080976

**AMA Style**

Vîlcu A-D, Vîlcu G-E.
On Quasi-Homogeneous Production Functions. *Symmetry*. 2019; 11(8):976.
https://doi.org/10.3390/sym11080976

**Chicago/Turabian Style**

Vîlcu, Alina-Daniela, and Gabriel-Eduard Vîlcu.
2019. "On Quasi-Homogeneous Production Functions" *Symmetry* 11, no. 8: 976.
https://doi.org/10.3390/sym11080976