# Internal Prices and Optimal Exploitation of Natural Resources

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

**Property**

**1.**

**Property**

**2.**

**Property**

**3.**

**Property**

**4.**

**Property**

**5.**

**Property**

**6.**

**Property**

**7.**

**Theorem**

**1.**

**Theorem**

**2.**

**Statement**

**1.**

**Statement**

**2.**

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

**;**visualization, V.G.I.; supervision, V.G.I. All authors have read and agreed to the published version of the manuscript.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Property**

**6.**

- (1)
- $\mathsf{\epsilon}>0$. Letting in (A3) $\mathsf{\epsilon}\to 0,$ we immediately obtain$${B}^{\prime}\left(x+0\right)\ge {r}^{\prime}\left(u\right)$$
- (2)
- $\epsilon <0$. Similarly, letting in (A3) $\epsilon \to 0$, we obtain$${B}^{\prime}\left(x-0\right)\le {r}^{\prime}\left(u\right)$$

**Proof**

**of**

**Property**

**7.**

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**Figure 1.**Fishery value function (

**A**), internal price (

**B**), and optimal catch (

**C**) over the fish stock.

**Figure 2.**The dependence of internal prices ${\overline{c}}_{1}$ (solid line) and ${\overline{c}}_{2}$ (dashed line) on the fraction $\lambda $ of total food supply in the first region.

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**MDPI and ACS Style**

Il’ichev, V.G.; Rokhlin, D.B.
Internal Prices and Optimal Exploitation of Natural Resources. *Mathematics* **2022**, *10*, 1860.
https://doi.org/10.3390/math10111860

**AMA Style**

Il’ichev VG, Rokhlin DB.
Internal Prices and Optimal Exploitation of Natural Resources. *Mathematics*. 2022; 10(11):1860.
https://doi.org/10.3390/math10111860

**Chicago/Turabian Style**

Il’ichev, Vitaly G., and Dmitry B. Rokhlin.
2022. "Internal Prices and Optimal Exploitation of Natural Resources" *Mathematics* 10, no. 11: 1860.
https://doi.org/10.3390/math10111860