# Organic vs. Non-Organic Food Products: Credence and Price Competition

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## Abstract

**:**

## 1. Introduction

## 2. Related Literature

## 3. Model Formulation and Analysis

#### 3.1. Model Setting

#### 3.2. Equilibrium Analysis

#### 3.2.1. Both Firms Offer Non-Organic Food Products

**Proposition**

**1.**

#### 3.2.2. Both Firms Offer Organic Food Products

**Proposition**

**2.**

#### 3.2.3. Firm 1 Offers Organic Food Products and Firm 2 Offers Non-Organic Food Products

**Lemma**

**1.**

**Proposition**

**3.**

**Corollary**

**1.**

- 1.
- Firm 1’s organic credence investment decision ${a}_{1}^{<O,N>}$ is increasing in ${u}_{o}$ and decreasing in ${c}_{\Delta}$ and ${c}_{B}$.
- 2.
- Firm 1’s equilibrium price ${p}_{1}^{<O,N>}$ is increasing in ${u}_{o}$ and decreasing in ${c}_{B}$; Firm 2’s equilibrium price ${p}_{2}^{<O,N>}$ is decreasing in ${u}_{o}$ and increasing in ${c}_{B}$.
- 3.
- Firm 1’s equilibrium profit ${\pi}_{1}^{<O,N>}$ is increasing in ${u}_{o}$ and decreasing in ${c}_{\Delta}$, ${c}_{B}$; Firm 2’s equilibrium profit ${\pi}_{2}^{<O,N>}$ is decreasing in ${u}_{o}$ and increasing in ${c}_{\Delta}$, and ${c}_{B}$.

**Corollary**

**2.**

- 1.
- ${a}_{1}^{<O,O>}={a}_{2}^{<O,O>}>{a}_{1}^{<O,N>}$.
- 2.
- the equilibrium prices in different subgames satisfy the following relationships: ${p}_{1}^{<O,N>}\ge {p}_{1}^{<O,O>}={p}_{2}^{<O,O>}>{p}_{1}^{<N,N>}={p}_{2}^{<N,N>}\ge {p}_{2}^{<O,N>}$.
- 3.
- the market shares in different subgames satisfy the following relationships: ${D}_{1}^{<O,N>}\ge {D}_{1}^{<O,O>}={D}_{2}^{<O,O>}={D}_{1}^{<N,N>}={D}_{2}^{<N,N>}\ge {D}_{2}^{<O,N>}$.

#### 3.3. Strategy Equilibrium

**Lemma**

**2.**

**Proposition**

**4.**

**Lemma**

**3.**

**Proposition**

**5.**

**Lemma**

**4.**

- (1)
- There exists a critical value ${c}_{\Delta}^{*}$ such that: when ${c}_{\Delta}\le {c}_{\Delta}^{*}$, ${\Delta}_{1}\ge 0$; when ${c}_{\Delta}>{c}_{\Delta}^{*}$, ${\Delta}_{1}<0$,
- (2)
- There exists a critical value ${c}_{\Delta}^{**}$ such that: when ${c}_{\Delta}\ge {c}_{\Delta}^{**}$, ${\Delta}_{2}\ge 0$; when ${c}_{\Delta}<{c}_{\Delta}^{**}$, ${\Delta}_{2}<0$,
- (3)
- ${c}_{\Delta}^{*}\ge {c}_{\Delta}^{**}$.

**Proposition**

**6.**

- (1)
- If ${c}_{\Delta}<{c}_{\Delta}^{**}$, ${\Delta}_{1}\ge 0$, ${\Delta}_{2}<0$, the final equilibrium is $(O,O)$, both firms offer organic food products.
- (2)
- If ${c}_{\Delta}^{**}\le {c}_{\Delta}\le {c}_{\Delta}^{*}$, ${\Delta}_{1}\ge 0$, ${\Delta}_{2}\ge 0$, the final equilibrium is $(O,N)$ or $(N,O)$, only one firm offers organic food products and the other firm offers non-organic food products.
- (3)
- If ${c}_{\Delta}>{c}_{\Delta}^{*}$, ${\Delta}_{1}<0$, ${\Delta}_{2}>0$, the final equilibrium is $(N,N)$, both firms offer non-organic food products.

## 4. The Impacts of Market Conditions on the Final Equilibrium

#### 4.1. The Impacts of Organic Food Production Cost

#### 4.2. The Impacts of Organic Food Products’ Credence Investment Cost

#### 4.3. The Impacts of Organic Food Products’ Attractiveness

## 5. Discussion and Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Lemma 1

#### Appendix A.2. Proof of Corollary 1

**Proof**

**of Corollary 1-(1).**

**Lemma**

**A1.**

**Proof**

**of**

**Lemma**

**A1.**

**Proof**

**of Corollary 1-(2).**

**Proof**

**of Corollary 1-(3).**

#### Appendix A.3. Proof of Corollary 2

**Proof**

**of Corollary 2-(1).**

**Proof**

**of Corollary 2-(2).**

**Proof**

**of Corollary 2-(3).**

#### Appendix A.4. Proof of Lemma 2

#### Appendix A.5. Proof of Lemma 3

#### Appendix A.6. Proof of Proposition 5

#### Appendix A.7. Proof of Lemma 4

**Proof**

**of Lemma 4-(2).**

- (i)
- We have shown this in the proof of Proposition 5.
- (ii)
- From the proof of Lemma 3, we know that $F\left({c}_{B}\right)={\Delta}_{2}\left(0\right)$ is increasing in ${c}_{B}$. Besides, when ${c}_{B}={c}_{B}^{*}$, ${\Delta}_{2}\left(0\right)=0$. Thus ${\Delta}_{2}\left(0\right)<0$ directly follows from assumption ${c}_{B}<{c}_{B}^{*}$.
- (iii)
- $\begin{array}{l}{\Delta}_{2}\left({u}_{o}\right)={\displaystyle \frac{M}{2t}}{\left(t-{\displaystyle \frac{1}{3}}{u}_{o}{a}_{1}^{*}+{\displaystyle \frac{1}{3}}{u}_{o}\right)}^{2}-\left[{\displaystyle \frac{Mt}{2}}-\left(\sqrt{{\displaystyle \frac{{c}_{B}{u}_{o}M}{3}}}\right)\right]\\ \ge {\displaystyle \frac{M}{2t}}{t}^{2}-\left[{\displaystyle \frac{Mt}{2}}-\left(\sqrt{{\displaystyle \frac{{c}_{B}{u}_{o}M}{3}}}\right)\right]\\ \ge 0.\end{array}$

**Proof**

**of Lemma 4-(1).**

- (i)
- Evaluating the derivatives of ${\Delta}_{1}\left({c}_{\Delta}\right)$ with respect to ${c}_{\Delta}$ we have the following equations:$$\begin{array}{l}{\displaystyle \frac{\partial {\Delta}_{1}}{\partial {c}_{\Delta}}}={\displaystyle \frac{2M}{2t}}\left(t+{\displaystyle \frac{1}{3}}{u}_{o}{a}_{1}^{*}-{\displaystyle \frac{1}{3}}{c}_{\Delta}\right)\left({\displaystyle \frac{{u}_{o}}{3}}{\displaystyle \frac{\partial {a}_{1}^{*}}{\partial {c}_{\Delta}}}-{\displaystyle \frac{1}{3}}\right)-{c}_{B}{\left(1-{a}_{1}^{*}\right)}^{-2}{\displaystyle \frac{\partial {a}_{1}^{*}}{\partial {c}_{\Delta}}}\\ =\left[{\displaystyle \frac{M}{t}}\left(t+{\displaystyle \frac{1}{3}}{u}_{o}{a}_{1}^{*}-{\displaystyle \frac{1}{3}}{c}_{\Delta}\right){\displaystyle \frac{{u}_{o}}{3}}-{c}_{B}{\left(1-{a}_{1}^{*}\right)}^{-2}\right]{\displaystyle \frac{\partial {a}_{1}^{*}}{\partial {c}_{\Delta}}}-{\displaystyle \frac{M}{3t}}\left(t+{\displaystyle \frac{1}{3}}{u}_{o}{a}_{1}^{*}-{\displaystyle \frac{1}{3}}{c}_{\Delta}\right)\\ =-{\displaystyle \frac{M}{3t}}\left(t+{\displaystyle \frac{1}{3}}{u}_{o}{a}_{1}^{*}-{\displaystyle \frac{1}{3}}{c}_{\Delta}\right)\\ <0,\end{array}$$
- (ii)
- We will show that ${\Delta}_{1}\left({c}_{\Delta}^{**}\right)\ge 0$.Note that ${\Delta}_{2}\left({c}_{\Delta}^{**}\right)=0$, thus we have $\frac{M}{2t}}{\left(t-{\displaystyle \frac{1}{3}}{u}_{o}{a}_{1}^{*}+{\displaystyle \frac{1}{3}}{c}_{\Delta}^{**}\right)}^{2}-\left[{\displaystyle \frac{Mt}{2}}-\sqrt{{\displaystyle \frac{{c}_{B}{u}_{o}M}{3}}}\right]=0$. Besides, ${a}_{1}^{*}$ is the solution of equation $\frac{\Delta M}{3t}}\left(t+{\displaystyle \frac{1}{3}}{u}_{o}{a}_{1}^{*}-{\displaystyle \frac{1}{3}}{c}_{\Delta}^{**}\right)-{c}_{B}{\left(1-{a}_{1}^{*}\right)}^{-2}=0.$ Let $S\left({c}_{\Delta}\right)={\displaystyle \frac{{\displaystyle \frac{1}{3}}{u}_{o}{a}_{1}^{*}+{\displaystyle \frac{1}{3}}{c}_{\Delta}}{t}}$. We can see that $0\le S\left({c}_{\Delta}\right)<1$.From Condition 2, $1-{a}_{1}^{*}=\sqrt{{\displaystyle \frac{3{c}_{B}}{{u}_{o}M\left(1+S\left({c}_{\Delta}^{**}\right)\right)}}}$. From Condition 1, we have $\sqrt{{\displaystyle \frac{{c}_{B}{u}_{o}M}{3}}}={\displaystyle \frac{Mt}{2}}\left(2S\left({c}_{\Delta}^{**}\right)-S{\left({c}_{\Delta}^{**}\right)}^{2}\right).$Thus we have$$\begin{array}{l}{\Delta}_{1}\left({c}_{\Delta}^{**}\right)={\displaystyle \frac{Mt}{2}}\left(2S\left({c}_{\Delta}^{**}\right)+S{\left({c}_{\Delta}^{**}\right)}^{2}\right)-\sqrt{{\displaystyle \frac{{c}_{B}{u}_{o}M\left(1+S\left({c}_{\Delta}^{**}\right)\right)}{3}}}\\ ={\displaystyle \frac{Mt}{2}}\left(2S\left({c}_{\Delta}^{**}\right)+S{\left({c}_{\Delta}^{**}\right)}^{2}\right)-\left\{\left[{\displaystyle \frac{Mt}{2}}\left(2S\left({c}_{\Delta}^{**}\right)-S{\left({c}_{\Delta}^{**}\right)}^{2}\right)\right]\sqrt{\left(1+S\left({c}_{\Delta}^{**}\right)\right)}\right\}\\ ={\displaystyle \frac{Mt}{2}}\left(2S\left({c}_{\Delta}^{**}\right)+S{\left({c}_{\Delta}^{**}\right)}^{2}\right)-\left\{\left[{\displaystyle \frac{Mt}{2}}\left(2S\left({c}_{\Delta}^{**}\right)-S{\left({c}_{\Delta}^{**}\right)}^{2}\right)\right]\sqrt{\left(1+S\left({c}_{\Delta}^{**}\right)\right)}\right\}\\ ={\displaystyle \frac{MtS\left({c}_{\Delta}^{**}\right)}{2}}\left\{\left(2+S\left({c}_{\Delta}^{**}\right)\right)-\left[\left(2-S\left({c}_{\Delta}^{**}\right)\right)\right]\sqrt{\left(1+S\left({c}_{\Delta}^{**}\right)\right)}\right\}.\end{array}$$To show ${\Delta}_{1}\left({c}_{\Delta}^{**}\right)\ge 0$, it is sufficient to show $\left(2+S\left({c}_{\Delta}^{**}\right)\right)\ge \left[\left(2-S\left({c}_{\Delta}^{**}\right)\right)\right]\sqrt{\left(1+S\left({c}_{\Delta}^{**}\right)\right)}$. Note both sides are positive, we square two sides and have $4+2S\left({c}_{\Delta}^{**}\right)+S{\left({c}_{\Delta}^{**}\right)}^{2}\ge 4+2S\left({c}_{\Delta}^{**}\right)-S{\left({c}_{\Delta}^{**}\right)}^{2}+S{\left({c}_{\Delta}^{**}\right)}^{3}$, which apparently holds for $0\le S\left({c}_{\Delta}\right)<1$. Thus ${\Delta}_{1}\left({c}_{\Delta}^{**}\right)\ge 0$
- (iii)
- $\begin{array}{l}{\Delta}_{1}\left({u}_{o}\right)={\displaystyle \frac{M}{2t}}{\left(t+{\displaystyle \frac{1}{3}}{u}_{o}{a}_{1}^{*}-{\displaystyle \frac{1}{3}}{u}_{o}\right)}^{2}-{c}_{B}{\left(1-{a}_{1}^{*}\right)}^{-1}-{\displaystyle \frac{Mt}{2}}\\ \le {\displaystyle \frac{M}{2t}}{t}^{2}-{c}_{B}{\left(1-{a}_{1}^{*}\right)}^{-1}-{\displaystyle \frac{Mt}{2}}\\ \le 0.\end{array}$

**Proof**

**of Lemma 4-(3).**

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**MDPI and ACS Style**

Wang, Y.; Zhu, Z.; Chu, F. Organic vs. Non-Organic Food Products: Credence and Price Competition. *Sustainability* **2017**, *9*, 545.
https://doi.org/10.3390/su9040545

**AMA Style**

Wang Y, Zhu Z, Chu F. Organic vs. Non-Organic Food Products: Credence and Price Competition. *Sustainability*. 2017; 9(4):545.
https://doi.org/10.3390/su9040545

**Chicago/Turabian Style**

Wang, Yi, Zhanguo Zhu, and Feng Chu. 2017. "Organic vs. Non-Organic Food Products: Credence and Price Competition" *Sustainability* 9, no. 4: 545.
https://doi.org/10.3390/su9040545