# A Chemical Analysis of Hybrid Economic Systems—Tokens and Money

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Connection between Chemical Reactions and Market Transactions

#### 2.2. Chemical Model

- $\alpha =\beta $: X and Y enter the reactions/market in exactly the same quantity.
- $f=c$ and $g=d$: The presence of the tokens in the reactions is superfluous.
- $g=0$: In the RM2s context, this means that product Y can be bought with tokens only. This strategy can be used to reward loyal customers instead of introducing vouchers.
- $f>c$ and $g<d$: Tokens are fully integrated and play an active role in the pricing of the products.
- $\alpha >\beta $: By starting with more of product X than Y, one may end up with a surplus of tokens at the end of the reactions that will have to be transferred and accounted for in the subsequent reaction mechanism.
- $\alpha <\beta $: One might run out of tokens in this step of the sequence of reaction mechanisms simply because selling product X will not generate enough tokens to buy all of the Y products. This is not an issue in RM2, but might become one in RM2s, where selling Y is restricted to token involvement.
- One can also imagine supplementing the reaction mechanisms with a way to buy X only with tokens at later stages.

#### 2.3. Kinetic Study

#### 2.4. Stability Analysis

#### 2.5. Coefficient Adjustments According to the Supply and Demand Laws

## 3. Results

#### 3.1. The Kinetic Study

#### 3.2. The Stability Analysis

#### 3.3. Coefficient Adjustments

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DT | Digital technology |

ICO | Initial Coin Offering |

${r}_{eq.number}$ | Reaction rate |

RM1 | Reaction mechanism 1 |

RM2 | Reaction mechanism 2 |

RM2s | Reaction mechanism 2 short |

ODE | Ordinary differential equation |

Tr(J) | Trace of Jacobian |

Det(J) | Determinant of Jacobian |

S(J) | $S\left(J\right)=Tr{\left(J\right)}^{2}-4Det\left(J\right)$ |

P | Price |

D | Demand |

S | Supply |

## Appendix A. Integrated Rate Law

## Appendix B. Experimental Assumptions

**Table A1.**Experimental assumptions for $\left[X\right]$ in the non-elementary framework, $\left[Z\right]\gg \left[A\right]$.

[X] (M) | [A] (M) | Rate (M/s) | Significance |
---|---|---|---|

${10}^{-20}$ | ${10}^{-20}$ | ${10}^{-23}$ | ${10}^{4}$ units of product are sold for ${10}^{4}$ units of currency |

and the transaction speed in 10 units/s | |||

2 × ${10}^{-20}$ | ${10}^{-20}$ | ${10}^{-23}$ | doubling the produce concentration does not |

affect the transaction speed | |||

${10}^{-20}$ | 2 × ${10}^{-20}$ | 2 × ${10}^{-23}$ | doubling the currency concentration |

doubles the transaction speed |

**Table A2.**Experimental assumptions for $\left[X\right]$ in the non-elementary framework, $\left[A\right]\gg \left[Z\right]$.

[X] (M) | [A] (M) | Rate (M/s) | Significance |
---|---|---|---|

${10}^{-20}$ | ${10}^{-20}$ | ${10}^{-23}$ | ${10}^{4}$ units of product are sold for ${10}^{4}$ units of currency |

and the transaction speed in 10 units/s | |||

2 × ${10}^{-20}$ | ${10}^{-20}$ | 2 × ${10}^{-23}$ | doubling the produce concentration |

doubles the transaction speed | |||

${10}^{-20}$ | 2 × ${10}^{-20}$ | ${10}^{-23}$ | doubling the currency concentration |

does not affect the transaction speed |

## Appendix C. Scaling Factors

## Appendix D. General and Special ODE Solutions

Nr. | Mec. | Fix. | Var. | Sol. |
---|---|---|---|---|

1 | X | Y, A | $\left({\mathsf{e}}^{\frac{-\mathsf{i}d\pi \beta +d\beta Log\left(x\right)+(c-d)Log\left(z\right)}{c\beta}},{\mathsf{e}}^{\frac{\mathsf{i}\pi \beta -\beta Log\left(x\right)+Log\left(z\right)}{c\beta}}\right)$ | |

2 | RM1 | Y | X, A | $\left({\mathsf{e}}^{\frac{-\mathsf{i}d\pi \alpha -d\alpha Log\left(y\right)+(c-d)Log\left(z\right)}{(c-2d)\alpha}},{\mathsf{e}}^{\frac{\mathsf{i}\pi \alpha +\alpha Log\left(y\right)-Log\left(z\right)}{(c-2d)\alpha}}\right)$ |

3 | - | X, Y, A | solutions exist only for known powers of a | |

4 | X, Y | A, B | no sol. | |

5 | X, A | Y, B | many sol. | |

6 | X, B | Y, A | $\left({\mathsf{e}}^{\frac{-\mathsf{i}g\pi \beta -f\beta Log\left(b\right)+g\beta Log\left(x\right)+(f-g)Log\left(z\right)}{f\beta}},{\mathsf{e}}^{\frac{\mathsf{i}\pi \beta -\beta Log\left(x\right)+Log\left(z\right)}{f\beta}}\right)$ | |

7 | RM2s | Y, A | X, B | $\left({a}^{-f}{z}^{1/\alpha},\frac{{a}^{-g}{z}^{1/\alpha}}{y}\right)$ |

8 | Y, B | X, A | $\left({\mathsf{e}}^{\frac{\mathsf{i}f\pi \alpha +f\alpha Log\left(b\right)+f\alpha Log\left(y\right)+(g-f)Log\left(z\right)}{g\alpha}},{\mathsf{e}}^{\frac{-\mathsf{i}\pi \alpha -\alpha Log\left(b\right)-\alpha Log\left(y\right)+Log\left(z\right)}{g\alpha}}\right)$ | |

9 | A, B | X, Y | $\left({a}^{-f}{z}^{1/\alpha},\frac{{a}^{-g}{z}^{1/\beta}}{bc}\right)$ | |

10 | X | Y, A, B | no sol. | |

11 | Y | X, A, B | no sol. | |

12 | A | X, Y, B | no sol. (for $\alpha \ne \beta $ and $\forall z\in {\Re}_{+}^{*}$) | |

many sol. (for $\alpha =\beta $) | ||||

13 | B | X, Y, A | no sol. (for $\forall z\in {\Re}_{+}^{*}$) | |

14 | - | X, Y, A, B | no sol. | |

15 | X, Y | A, B | no sol. | |

16 | RM2 | X, A | Y, B | $\left({a}^{-d}\left({a}^{-f}x+{z}^{1/\beta}\right),\frac{{a}^{d+f-g}x}{{a}^{f}x-{z}^{1/\beta}}\right)$ |

17 | X, B | Y, A | $\left(\frac{xz}{-z+bx\sqrt{-z/x}},\sqrt{-z/x}\right)$ | |

18 | Y, A | X, B | $\left({a}^{-f}{z}^{1/\alpha},\frac{{a}^{-g}{z}^{1/\alpha}}{y}\right)$ | |

19 | Y, B | X, A | solutions exist only for known powers of a | |

20 | A, B | X, Y | $\left({a}^{-f}{z}^{1/\alpha},\frac{{z}^{1/\beta}}{{a}^{d}+{a}^{g}b}\right)$ | |

21 | X | Y, A, B | solutions exist only for known powers of a | |

22 | Y | X, A, B | solutions exist only for known powers of a | |

23 | A | X, Y, B | $\left({a}^{-f}{z}^{1/\alpha},-{a}^{-d}\left({z}^{1/\alpha}-{z}^{1/\beta}\right),-\frac{{a}^{d-g}{z}^{1/\alpha}}{{z}^{1/\alpha}-{z}^{1/\beta}}\right)$ | |

24 | B | X, Y, A | no sol. (for $\forall z\in {\Re}_{+}^{*}$) | |

25 | - | X, Y, A, B | no sol. |

**Table A4.**Table of ODE classifications and solutions considering $\alpha =2$, $\beta =1$, $c=g=1$, and $d=f=2$.

Mec. | Fix. | Var. | Sol. | Cond. | Comments |
---|---|---|---|---|---|

X | Y, A | (+, −) | |||

RM1 | Y | X, A | (i, i), (i, i) | ||

- | X, Y, A | (+, +, −) | |||

X, A | Y, B | many sol. | - ${z}^{1/\beta}=x{a}^{f}$ | consistent, dependent | |

X, B | Y, A | (i, i), (i, i) | |||

Y, B | X, A | (+, −) | |||

RM2s | X | Y, A, B | - | $\forall x,a,z\in {\Re}_{+}^{*}$ | inconsistent, independent |

Y | X, A, B | - | $\forall x,a,z\in {\Re}_{+}^{*}$ | inconsistent, independent | |

A | X, Y, B | many sol. | $\alpha =\beta $ | consistent, dependent | |

- | $\alpha \ne \beta $ | inconsistent, independent | |||

B | X, Y, A | - | $\forall z\in {\Re}_{+}^{*}$ | inconsistent, independent | |

X, B | Y, A | (i, i), (i, i) | |||

Y, B | X, A | (i, i), (i, i) | ${b}^{2}y<a\sqrt{z}$ | ||

(+/−, −), (+/−, −) | ${b}^{2}y\ge a\sqrt{z}$ | ||||

RM2 | X | Y, A, B | (−, i, i), (−, i, i) | ||

Y | X, A, B | (−, i, i), (−, i, i) | |||

A | X, Y, B | (+, +, +) | |||

B | X, Y, A | - | inconsistent, independent |

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**Figure 1.**Time dependence of (

**a**) $X\left(t\right)$ and $Y\left(t\right)$ for RM1 (left) calculated assuming the $\left[A\right]\gg \left[Z\right]$ case, (

**b**) $A\left(t\right)$ for the one-currency mechanism in RM1 (middle), and (

**c**) $A\left(t\right)$, $C\left(t\right)$ for RM2s (right), calculated considering the $\left[Z\right]\gg \left[A\right]$ case. The composite currency $C\left(t\right)$ has the same time profile as that of the conventional currency $A\left(t\right)$.

**Figure 2.**Directional fields for (

**a**) RM2 and RM2s a, b variable (upper left), (

**b**) x and b variable (upper right), (

**c**) y, b variable in RM2 (lower left), and (

**d**) x, y variable (lower right). All of the plots depict stable nodes.

**Figure 3.**Directional fields for the ODEs of RM2 with x, y, and b being variable and a being fixed. The steady state is marked with a red sphere in the middle of the 3D plot. The region of interest is the lower part of the surface where the stream lines are coming out of the sphere.

Chem. Elem. | Possible Interpretations |
---|---|

Z | Total amount of produce coming from a single supplier (e.g., vegetables), Total amount of produced goods, e.g., clothing, furniture |

X | Products not usually desired by most customers: imperfect products, e.g., tomatoes, used/not new goods, e.g., out-of-fashion clothing, second-hand furniture, recycled paper goods, e.g., notebooks |

Y | Niche or impactful products: organically sourced produce, traditional products (e.g., clothing) supporting local businesses, environmentally friendly building materials, renewable energy sources, low-power electronics |

A | Conventional currency |

B | Token(s) received for actions such as: consuming imperfect fruits and vegetables, buying second-hand goods, donating food and goods to the community, taking part in clean-up projects in nature, volunteering activities that benefit the community, minimising one’s CO${}_{2}$ footprint |

T | Market transactions |

**Table 2.**The expressions for the reaction rates in RM1 and RM2s for the two cases in which the quantity of money is much larger than the quantity of products and vice versa.

Cases | Reaction Rates in RM1—Equations (2) and (3) | Reaction Rates in RM2s—Equations (6)–(8) |
---|---|---|

${r}_{2}={k}_{1}{\left[X\right]}^{1}{\left[A\right]}^{0}$ | ${r}_{6}={k}_{3}{\left[X\right]}^{1}{\left[A\right]}^{0}$ | |

$\left[A\right]\gg \left[Z\right]$ | ${r}_{3}={k}_{2}{\left[Y\right]}^{1}{\left[A\right]}^{0}$ | ${r}_{7}={k}_{4}{\left[Y\right]}^{1}{\left[C\right]}^{0}$ |

${r}_{8}={k}_{5}{\left[Y\right]}^{1}{\left[A\right]}^{0}$ | ||

$\left[Z\right]\gg \left[A\right]$ | ${r}_{2}={k}_{1}{\left[X\right]}^{0}{\left[A\right]}^{1}$ | ${r}_{6}={k}_{3}{\left[X\right]}^{0}{\left[A\right]}^{1}$ |

${r}_{3}={k}_{2}{\left[Y\right]}^{0}{\left[A\right]}^{1}$ | ${r}_{7}={k}_{4}{\left[Y\right]}^{0}{\left[C\right]}^{1}$ |

**Table 3.**The nonlinear first-order ODE systems derived from the chemical mechanisms and the scaled versions of the same equations for the RM1, RM2, and RM2s reaction mechanisms.

Mec. | ODE System | Scaled ODE System |
---|---|---|

$\frac{d\left[X\right]}{dt}=k{\left[Z\right]}^{1/\alpha}-{k}_{1}\left[X\right]{\left[A\right]}^{c}$ | ${x}^{\prime}={z}^{1/\alpha}-x{a}^{c}$ | |

RM1 | $\frac{d\left[Y\right]}{dt}=k{\left[Z\right]}^{1/\beta}-{k}_{2}\left[Y\right]{\left[A\right]}^{d}$ | ${y}^{\prime}={z}^{1/\beta}-y{a}^{d}$ |

$\frac{d\left[A\right]}{dt}=-{k}_{1}\left[X\right]{\left[A\right]}^{c}-{k}_{2}\left[Y\right]{\left[A\right]}^{d}$ | ${a}^{\prime}=-x{a}^{c}-y{a}^{d}$ | |

$\frac{d\left[X\right]}{dt}=k{\left[Z\right]}^{1/\alpha}-{k}_{3}\left[X\right]{\left[A\right]}^{f}$ | ${x}^{\prime}={z}^{1/\alpha}-x{a}^{f}$ | |

RM2s | $\frac{d\left[Y\right]}{dt}=k{\left[Z\right]}^{1/\beta}-{k}_{4}\left[Y\right]\left[B\right]{\left[A\right]}^{g}$ | ${y}^{\prime}={z}^{1/\beta}-yb{a}^{g}$ |

$\frac{d\left[A\right]}{dt}=-{k}_{3}\left[X\right]{\left[A\right]}^{f}-{k}_{4}\left[Y\right]\left[B\right]{\left[A\right]}^{g}$ | ${a}^{\prime}=-x{a}^{f}-yb{a}^{g}$ | |

$\frac{d\left[B\right]}{dt}={k}_{3}\left[X\right]{\left[A\right]}^{f}-{k}_{4}\left[Y\right]\left[B\right]{\left[A\right]}^{g}$ | ${b}^{\prime}=x{a}^{f}-yb{a}^{g}$ | |

$\frac{d\left[X\right]}{dt}=k{\left[Z\right]}^{1/\alpha}-{k}_{3}\left[X\right]{\left[A\right]}^{f}$ | ${x}^{\prime}={z}^{1/\alpha}-x{a}^{f}$ | |

RM2 | $\frac{d\left[Y\right]}{dt}=k{\left[Z\right]}^{1/\beta}-{k}_{4}\left[Y\right]\left[B\right]{\left[A\right]}^{g}-{k}_{5}\left[Y\right]{\left[A\right]}^{d}$ | ${y}^{\prime}={z}^{1/\beta}-yb{a}^{g}-y{a}^{d}$ |

$\frac{d\left[A\right]}{dt}=-{k}_{3}\left[X\right]{\left[A\right]}^{f}-{k}_{4}\left[Y\right]\left[B\right]{\left[A\right]}^{g}-{k}_{5}\left[Y\right]{\left[A\right]}^{d}$ | ${a}^{\prime}=-x{a}^{f}-yb{a}^{g}-y{a}^{d}$ | |

$\frac{d\left[B\right]}{dt}={k}_{3}\left[X\right]{\left[A\right]}^{f}-{k}_{4}\left[Y\right]\left[B\right]{\left[A\right]}^{g}$ | ${b}^{\prime}=x{a}^{f}-yb{a}^{g}$ |

**Table 4.**Table of stability behaviours around real positive critical points. The columns are in the following order: the reaction mechanism (Mec.), the fixed variables (Fix.), the variables of the ODEs (Var.), the solutions of the ODEs (Sol.), the conditions for the solutions (Cond.), the signs for the trace Tr(J), the determinant Det(J), and $S\left(J\right)=Tr{\left(J\right)}^{2}-4Det\left(J\right)$ of the ODEs’ Jacobian, and the nature of the critical point.

Mec. | Fix. | Var. | Sol. | Cond. | Tr(J), Det(J), S(J) | Node |
---|---|---|---|---|---|---|

X, Y | A, B | (0, 0) | ${V}^{\prime}(a,b)\le 0,\forall a,b\in {\Re}_{+}$ | 0, 0, 0 | stable | |

RM2s | Y, A | X, B | (+,+) | −, +, + | stable | |

A, B | X, Y | (+,+) | −, +, + | stable | ||

X, Y | A, B | (0, 0) | ${V}^{\prime}(a,b)\le 0,\forall a,b\in {\Re}_{+}$ | 0, 0, 0 | stable | |

RM2 | Y, A | X, B | (+,+) | −, +, + | stable | |

X, A | Y, B | (+,+) | ${z}^{1/\beta}\ge x{a}^{f}$ | −, +, + | stable | |

A, B | X, Y | (+,+) | −, +, + | stable | ||

A | X, Y, B | (+,+,+) | +, −, + | saddle |

**Table 5.**Table of recalculated prices of X and Y in RM1 for the cases of product deficit and surplus. From the left to the right column, the table contains the product type, whether a surplus or a deficit of it is considered, the difference in quantity considered (by adding or subtracting $\lambda X$ or $\lambda Y$), and the newly calculated coefficient according to the difference in the product.

Product | Status | Difference | New Coefficient |
---|---|---|---|

X | surplus | $+\lambda X$ | ${c}_{new}=c(1-\lambda /\alpha )$ |

X | deficit | $-\lambda X$ | ${c}_{new}=c(1+\lambda /\alpha )$ |

Y | surplus | $+\lambda Y$ | ${d}_{new}=d(1-\lambda /\beta )$ |

Y | deficit | $-\lambda Y$ | ${d}_{new}=d(1+\lambda /\beta )$ |

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**MDPI and ACS Style**

Pardi, A.-L.; Paolucci, M.
A Chemical Analysis of Hybrid Economic Systems—Tokens and Money. *Mathematics* **2021**, *9*, 2607.
https://doi.org/10.3390/math9202607

**AMA Style**

Pardi A-L, Paolucci M.
A Chemical Analysis of Hybrid Economic Systems—Tokens and Money. *Mathematics*. 2021; 9(20):2607.
https://doi.org/10.3390/math9202607

**Chicago/Turabian Style**

Pardi, Anabele-Linda, and Mario Paolucci.
2021. "A Chemical Analysis of Hybrid Economic Systems—Tokens and Money" *Mathematics* 9, no. 20: 2607.
https://doi.org/10.3390/math9202607