# Lagrangians of Multiannual Growth Systems

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Discussion

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Expected value of operative capital return rate as a function of grown stock and temperature sum.

**Figure 3.**Partial derivative of gross profit rate with respect to temperature sum, according to Equation (10).

**Figure 4.**Partial derivative of operative capital return rate with respect to temperature sum, according to Equation (11).

**Figure 5.**Total derivative of the operative capital return rate with respect to the gross profit rate, according to the Lagrangian of Equation (12) extremized.

**Figure 6.**Total derivative of the operative capital return rate with respect to the capitalization, according to the Lagrangian of Equation (12) extremized.

**Figure 7.**Total derivative of the capitalization with respect to the operative capital return rate, according to the Lagrangian of Equation (13) extremized.

**Figure 8.**Total derivative of the capitalization with respect to the gross profit rate, according to the Lagrangian of Equation (13) extremized.

**Figure 9.**Total derivative of the gross profit rate with respect to capitalization, according to the Lagrangian of Equation (14) extremized.

**Figure 10.**Total derivative of the gross profit rate with respect to the operative capital return rate, according to the Lagrangian of Equation (14) extremized.

Symbol | Value | Unit | Name |
---|---|---|---|

u | 0.3 | Relative estate market premium | |

k | 0.000101 | Grown stocking scale factor | |

$\alpha $ | 0.1 | Relative growth scale factor | |

$\varsigma $ | 600 | Degrees C | Thermal sum threshold value |

$\phi $ | 450 | Degrees C | Thermal sum scale factor |

$\beta $ | 0.7 | Eur/(ha*degrees C) | Bare land value scale factor |

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**MDPI and ACS Style**

Kärenlampi, P.P.
Lagrangians of Multiannual Growth Systems. *Foundations* **2023**, *3*, 115-126.
https://doi.org/10.3390/foundations3010011

**AMA Style**

Kärenlampi PP.
Lagrangians of Multiannual Growth Systems. *Foundations*. 2023; 3(1):115-126.
https://doi.org/10.3390/foundations3010011

**Chicago/Turabian Style**

Kärenlampi, Petri P.
2023. "Lagrangians of Multiannual Growth Systems" *Foundations* 3, no. 1: 115-126.
https://doi.org/10.3390/foundations3010011