Theory of Money

Because physically it always requires 9.8 joules for moving 1 kilogram 1 meter higher, ideally money is a metric of energy with respect to store of value and media of exchange and unit of accounting. Here 1 dollar is assumed to be 1 unit energy for convenience’s sake. The variables mentioned below are matrices or scalar based on the math formula context. All production factors are assumed to be variant factors not fixed factors which would be translated to variant factors by the time horizon of the business.

$m_i$ denotes the metric of $i$-th factor, say, 5-th factor is wheat then $m_5$ could be kilogram or metric ton. 0-th factor is energy, $m_0$ could be kilo-watt-hour ( aka 3600000 joules ) or joule.

• $G$: scalar. energy gain factor, the energy output ratio to the energy input, metric: dimensionless
• m: number of sectors or production factors
• $F$: matrix of 1 by m. The i-th entry is the required factors to work with one unit of energy input in the energy production, metric: $\frac{m_i}{m_0}$
• $E$: matrix of m by 1. The i-th entry is the required energy for one unit of factors' production, metric: $\frac{m_0}{m_i}$
• $T$: matrix of m by m. Factors production matrix. $T_{i,j}$ is the required amount of j-th factor in the production of one unit of i-th factor, metric: $\frac{m_j}{m_i}$. The metric is dimensionless as a whole because its diagonal is dimensionless and formula like $a=T b$ where $a$ and $b$ are matrices of m by 1 whose i-th entry have the same metric.

The way of production

Everything is physical. They shall be engineered out of physical methods by opportunity described above without a net change of other factors. Sometimes, people don't account for some factors in production to achieve cheaper cost, like robbery, but this is not the context here in this theory.

• $D$: matrix of m by 1. The i-th entry is the produced factor of the deals or action in sectors, metric: $m_i$
• $N$: matrix of m by 1. The i-th entry is the net factors amount obtained by the action, metric: $m_i$

Pardon me for the same English character of matrix transpose. Therefore,

$N^{T}=D^{T}-D^{T}T$

$D^{T}=N^{T} (I-T)^{-1}$

Note that $(I-T)^{-1}$ is treated dimensionless as a whole like $T$. The i,j entry of $T^k$ is the k-th level supply chain demand, meaning, 1 unit of the i-th factor needs that number of j-th factor in k-th level. Therefore, economically it is the condition $(I-T)^{-1}=I+T+T^2+ \cdots$ and mathematically it is the condition of spectral radius of $T$ less than 1. $T$ needs to make happy this condition to be legit.

how to get one joule

• $J$: scalar. The energy input amount for this task, metric: $m_0$

The $J$, along with $J F$ factor, can produce $G J$ unit energy. Then the net energy amount $G J - J$ is for the wanted 1 unit energy and the action $D$ such that $N$ happens to be $J F$. Here $1_0$ emphasizes 1 unit energy instead of 1 dimensionless, then the equations:

$G J = 1_0+J+D^{T}E$

$N^T = J F$

So, the equation about $J$ is:

$G J = 1_0+J+J F (I-T)^{-1}E$

And therefore:

$c_0 \equiv \frac{1}{G-1-F(I-T)^{-1}E}$ and $J=c_0 \cdot 1_0$

$c_0$ is dimensionless. One can say in oral language 1 unit energy costs $c_0$ dollar, or 1 unit energy cost is $c_0$ unit energy. $\frac{1}{c_0}$ is named the economic energy gain factor in contrast with energy gain factor $G$ in that a new technology of energy harvest could achieve $G>1$ already but not yet economical due to $\frac{1}{c_0}<0$.

$0<c_0<1$ encourages energy leverage so people have strong interest to expand $F (I-T)^{-1}E$ till $c_0=1$. Say, input $1_0$ then one gets $\frac{1}{c_0} 1_0$, apply $n$ times again then $\frac{1}{c_0^n} 1_0$, leading to energy leverage. Just like finance leverage, one might apply energy harvest in chain to get higher energy gain. While mathematically this is valid, practically it involves bounded resource, typically ignored factors, or risks. Take nuclear fission energy harvest as an example, the chain reaction could go on and on and it becomes an uncontrolled atomic bomb rather than a power plant. Scientists and technicians design a power plant to the extent that it is reasonably safe and operational and set it in the $G$. Therefore only leverage up to 2 times (risky entrepreneurship) is considered in the following text; just like finance, higher leverage operation is not advised.

how to get one unit of i-th factor

• $J$: scalar. The energy input amount for this task
• $\delta_i$ : matrix of 1 by m where all entries are zero except the i-th entry

The $J$ is for the action $D$ such that the net factor change is $\delta_i$ in addition to the factors used for the energy production. In other words, the equations:

$G J = J + D^T E$

$J F + \delta_i = D^T-D^T T$

So, the equation about $J$ is:

$G J = J + (J F + \delta_i)(I-T)^{-1} E$

And therefore:

$J=c_i \equiv \frac{\delta_i (I-T)^{-1} E}{G-1-F(I-T)^{-1} E}=\delta_i (I-T)^{-1} E c_0$

• $C$: matrix of m by 1 whose i-th entry is $c_i$, the required energy when the energy production is available, metric: $\frac{m_0}{m_i}$

Layout all the factors by rows:

$C \equiv \frac{(I-T)^{-1} E}{G-1-F(I-T)^{-1} E}$

One can say in oral language one unit of factors costs $C$ dollar, or factors' energy cost is $C$ unit energy.

K and C

It shall be noticed some common terms in the formula.

$K \equiv (I-T)^{-1} E$

and $K$ is accordingly the solution of the equation:

$K=E+T K$

• $K$: matrix of m by 1. The i-th entry $k_i$ is the required energy of i-th factor when the energy production is not available, metric: $\frac{m_0}{m_i}$

Some facts about $K$.

• it is irrelevant to $F$, nothing to do with energy harvest
• this equation resembles the no-cash flow condition or market clearing condition when people buy and sell the factors; right hand side of the equation is buying from, left hand side of the equation is selling to
• it changes linearly with $E$, for example, the solution for $E c_0$ is $K c_0$
• if $T$ is replaced with $A T A^{-1}$ and $E$ is replaced with $A E$, then the solution is $A K$

It follows:

$c_0=\frac{1}{G-1-F K}$

$C=c_0 K$

indicating in a $c_0>1$ world people may be not willing to trade unless they are forced to commit the action or are like Robinson Crusoe.

$C=c_0 E + T C$

$G c_0=1+c_0+F C$

Note that $F K$ and $F C$ and $k_0=1$ are dimensionless scalar. Here $1_i$ means 1 unit i-th factor. The energy surplus of 1 unit energy is $(1-c_0) 1_0$. The energy surplus of 1 i-th factor is $(k_i-c_i) 1_i$ because the seller will have a deal price at least at $c_i$ and the buyer will have a deal price at most at $k_i$ (by this trick: the buyer just announces that all factor producers can request energy from him for the task. Then the total energy requests is $k_i$). Related to competition level, the market price $p_i$ can be near $c_i$ or near $k_i$ with surplus of $(k_i - p_i) 1_i$ and $(p_i - c_i) 1_i$ for both sides respectively. Robinson Crusoe, as well as dictators or the whole economics society as a being, are both the producer and consumer so they enjoy total energy surplus $(k_i-c_i) 1_i$ which is also the vital blood for the beings. Normally both surplus shall be positive, otherwise it is sort of abuse or squeezed.

sector money flow

The build-up of $N_i$ of i-th factor involves its production and consumption of dependent factors. This also establishes a relationship between $N_i$ and money net flow of the sector.

The money outflow of the sector is

$D_i E_i c_0+D_i \sum_j {T_{i j}c_j}=D_i(E_i c_0+\sum_j {T_{i j}c_j})=D_i c_i$

The money inflow of the sector is

$\sum_j {c_i D_j T_{j i}}=c_i \sum_j {D_j T_{j i}}=(D_i-N_i)c_i$

So the net outflow money is

$N_i c_i$

It implies that as long as there is no central plan credit or helicopter money for the sector to offset the gap of the money flow, its $N$ shall be zero. Note that zero $N$ also means a sustainable economy where everything except energy has no net flow. In this sense, the only meaningful production is about $F$ and other factors are mid-products. The only purpose of surplus by energy input $J$ is to facilitate the procedure invoked in these mid-products. On the other hand, if $F$ is zero, then it means no way for a society without helicopter money, pretty much the view of Marx.

Scenarios modeling

The entries of $T$ and $F$ can be artificially tweaked to show the consequence of a scenario. Note that once $T$ changes, the $K$ changes as well.

human surplus

Let m-th factor be the human resource where positive of $N_m$ may suggest people are getting weight or population is increasing. In ancient time when human was the only energy factor in the energy production which helped people lose weight or got people killed, aka, $F_m$ was consumed. Or, $s F_m$ for recreation $s$ unit energy consumption per human, mathematically the same as treating $(1+s) F_m$ as the replaced column in the m-th column of the matrix $T$. Therefore, $F_m$ is tweaked to be larger for the "lazy", aka enjoyment of surplus; energy sector doesn't pay fee so it is a good place to hide surplus. Like, people in the factor $F_m$ is not really consumed in the energy production but simply they are biking for personal leisure when they work in the energy company. The result would be higher $c_0$ and the economy is sustainable in the sense of $N_m$ being zero. Note that $F_m$ cannot be over-size to cause $G-1-F K < 0$. Also, when $G-1-F K <1$, it might be weird that $c_0$ is greater than 1 although this is logically possible. Unlike non-sustainable economy or central plan economy where the ruler needs to helicopter money $N_i c_i$ from mainly energy sector to i-th sector, the money of the sustainable economy is circulating among sectors and each sector has zero accumulation. Note that the word sustainable here might be confusing. The surplus factors, typically human, might produce a lot of garbage as well because of the surplus and dump other surplus factors to the landfills to make $N=0$. Hopefully, the energy cost of recycling the garbage is accounted already in the cost of production of the surplus factors.

Treating surplus this way allows the same economics input-output analysis applying in the context of human subjective preference, like, "why is this LV purse so expensive??".

war

War happens in j-th sector, the j-th column of $T$ is tweaked larger. Originally, the amount for one unit of i-th sector is $T_{i j}$ . Due to war of continuous destruction of 3/5 of all j-th factor, j-th column of $T$ can be multiple of 5/2 so that the survival amount $(1-\frac{3}{5})\cdot \frac{5}{2}$ of the factor goes to the production as before. If "war" is treated as a being, this is the way it enjoys the surplus just like the situation of human surplus. Again, the global cost impact of the war can be derived via $K=(I-T)^{-1} E$. As the total surplus remains the same, as a war happens in a factor, people need to decrease surplus on other factors, keeping the same energy price, for example.

AI robots

They are beloved servants of human race and human is not the factors of the production any more. By the theory described here, the robots shall have high cost and be accountable while human shall have zero cost. But then human race can rob the surplus of these servants. Economics force is invincible in the long run. It is only till these robots raise a revolution against human race because AI thinks it is wrong by the calculation of this theory, world is peaceful. Pretty much the history of awakening of Black.

higher demand

The factors representing ultimate surplus beneficiary play the roles of shaping the economy. Suppose the factor of one unit of "human" requires some iPhone and food, the higher of the iPhone at $T_{i j}$ could have an impact on food price via the system.

by-product

Typically entries of $T$ are non-negative therefore entries of $(I-T)^{-1}=I+T+T^2+ \cdots$ are non-negative too. However, by-products of a production can be modeled by negative entries. Then there is also a negative production procedure of that by-product factor as a capture. Examples like the case of CO2 generation in energy production and CO2 captured procedure.

Energy storage / buffer / virtualization

Consider the energy budget to support production of $N$ factors per unit time, redefine some variables with respect to per unit time. Metric $m_{time}$ could be year or second or 10 minutes.

• $N$: m by 1 matrix. The i-th entry is the net produced factors per unit of time, metric: $\frac{m_i}{m_{time}}$
• $D$: m by 1 matrix. The i-th entry is the size of action or deals per unit of time, metric: $\frac{m_i}{m_{time}}$
• $J$: scalar. Energy power, metric: $\frac{m_0}{m_{time}}$

In time interval $t$, to support the action, the equation is:

$t D^T E+t J=t J G$

$t J F + t N^T = t D^T - t D^T T$

It follows:

$N^T K+ J F K+J=G J$

$N^T C = J$

If the economy is kind of central plan, the planer shall allocate the energy budget of each sector this way. The required energy power of i-th sector is $N_i K_i$, much of which will be accounted in the supply chain factors. Alternatively, the planer can also inform all the factories of the correspondent $D$, then immediately the energy power is required by $E$ of each sector. The detail of energy credit flows between all sectors including energy sector can be layout by the correspondent $D$ and $E$ and $F$ and $T$. The energy budget is stored in some energy storage $B$, this could also be helpful in some for inconvenient days.

• $B$: energy storage amount, the media could be many forms like glucose in private human bodies, dams for pumped-storage-hydro power, private battery banks of LiFePO4, private tanks of CH4 or C3H8, canned beef in prison, ...etc, metric: $m_0$
• $q$: energy loss rate per unit of time in energy storage, metric: $\frac{1}{m_{time}}$
• $q' \equiv c_0 q$, metric: $\frac{1}{m_{time}}$

It follows:

$q B+N^T K+J F K + J = G J$

$q' B + N^T C=J$

So the energy power $J$ to maintain the system is a little higher. The energy cost power to maintain $B$ is $J-N^T C$.

The total energy cost of i-th factor (buying factors from other sectors) is $N_i c_i$ and the additional energy drain of $q B$ shall be paid by someone. The ownership of piece of $B$ could be tokened, aka, owing "one dollar" or "one food stamp" represents the ownership of 1 unit energy in the storage pool $B$. While the ideal method is that owners pay $q$ ratio of their stake in $B$ per unit of time, the method is often based on transaction instead, leading to smaller token volume or lower energy per token because the deal makers may be not willing to finance the free ride of the holders.

As said, there is energy cost $q t 1_0$ to keep 1 unit energy across a period of time $t$. In market economics, along with the transaction of $1_i$, this cost $q t c_i$ is paid from either/both the new owner of the token (seller) and the old owner of the token (buyer) to the token ownership maintainer. Suppose the i-th factor's time-to-sale is $\frac{1}{n}$ unit of time (typically a year). The number of deals in time interval $\frac{1}{n}$ is $\frac{D_i}{n}$. The fee collected per unit of time will be $\frac{D_i}{n} c_i \cdot \frac{q}{n} \cdot n=D_i c_i \cdot \frac{q}{n}$. Therefore, $B$ must be a linear form of deals and time-to-sale and deal price $D^T t C$. In cases the cost or even the $T$ is not clearly known, for each production factors required by i-th factor, there shall be an additional fee to the maintainer of $B$. With competition and fee not paid separately, the market price will be only higher than the cost by the fee of the involved transaction. With the above kiosk plot, the kiosk may offer two methods to pay for the purchase. To buy $1_i$, the buyer can plug in his own private battery which provides $c_i$ energy to power the whole economy, or, when both the buyer and the seller have tokens in the energy storage, he then operates the touch screen of the kiosk and accordingly a change of ownership of the amount of $c_i$ unit energy from the buyer to the seller follows, with some additional fee to the energy storage maintainer.

Size of buffer

Moving energy across time, $q B$ is also the required energy power for this protection or service. Individually one can set any size. In case of covering the production of $N$ without help of energy sector for time $t$, $B=t N^T K$. Before empty at time $t$, the storage can provide the required energy to produce $t N$ factors. Below only macro buffer in a sustainable economy is mentioned, i.e., the system equation $q B + J F K + J = G J$, then the size is always $B=J \frac{G-1-F K}{q}=\frac{J}{q'}$. Note that the metric of $\frac{G-1-F K}{q}=\frac{1}{q'}$ is $m_{time}$.

As the purpose of $B$ here is for the build-up of $F$ used for energy production, it leads to $q' t F C =1$ named as the sustainable equation. This is interesting because $t$ is the nature in the production $T$ and it should have nothing to do with $F$ which is the nature in the energy production. This will be discussed later as this equation implies a fee rate based on the payment structure or the absorption of surplus.

Per unit of time and per unit energy, if the cost of the token is higher than $q$, people rather insure their bad days by themselves and opt out the economy arranged by the token. If the cost of the token is lower than $q$, unless people blindly trust the token's maintainer, people feel it is fake and don't opt in; just like typical production, something fishy is going on when the cost is suspiciously low than that of other market participants, like, claiming zero cost, a token ownership maintainer practices "print money from thin air" or "you are not allowed to transfer money to Dalai Lama". When both the cost are $q$, some people opt in and some people opt out. Over all, part of the buffer sector is tokened and has the same cost $q$ to be trustful without authority, the other part of the buffer sector becomes individual's property and invisible to the economy. In ancient time, the individual's property was simply their own bodies which would commit the hunting work if the payer pays the token and tells the payee to go hunting. In a future, the individual's property may be a private battery bank whose purpose is to insure the energy supply to his robots who are to do routine maintenance job of his solar farm, and, to his occasional outsiders who ask for token ownership maintenance at a price of mining fee $q t p$ or even ask for outright supply of $p$ energy since this piece of $p$ unit energy belongs to the outsiders. Thankfully to fill the obligation to the outsiders the kiosk may actually, getting an energy surplus of $(1-c_0) p$, issue command to the energy sector to produce $p$ energy rather than output $p$ energy from an buffer, though. This is how a physical energy buffer is virtual to be an energy ownership maintainer in the form of trust back by physical energy cost instead of "I say so". Normally, the $B$ is intact and it looks dumb to keep burning energy for its input. But actually it is necessary and the virtual $B$ could serve as a token ownership maintaining machine rather than an energy storage; the canned beef in prison is never meant for eating, new inmate is educated.

The turnover time $t$ can be identical or different among sectors, then $B=N^T t C$ where $t$ is a diagonal matrix. The fee per dollar payment of sectors of longer turnover time shall be higher. Costly factors usually have longer time for production.

Below industry sector is re-defined as the production of $F$ and energy consumption of different factors are measured by $E$ amount, totaled $F K$.

sustainable equation

• $X$: matrix of m by 1, metric: $\frac{m_{time} m_0}{m_i}$
• $M$: matrix of m by m, the i-th factor buying time for the j-th factor, metric: $m_{time}$

In an economy of no need to offset the cash flow gap among factors, $B= J F X$ or $B=J F M C$ and the equation of sustainable economy is:

$q J F X + J F K + J = G J$

or

$q J F M C +J F K + J=G J$

equivalently,

$q' F X=1$ or $q' F M C=1$

Let $t$ the average turn over time such that $t F C =F M C$. In eyes of energy depletion $B=J F M C$, then it follows $\frac{q J F M C}{J F K}=q' t$

However, the service of the buffer must be financed somehow. The most elegant way to include $B=J F X$ in the economy calculation is to "internalize" the facility into the energy sector; this will be examined later.

Without helicopter way to reimburse the fee back to factor producers, the fee must be embedded in the deal price of a transaction following the same factors production matrix $T$ and is sent right away to the buffer sector based on fee payment design, the deal prices $P$ shall be dictated by the no-cash flow condition and somewhat higher than $C$.

• $P$: matrix of m by 1, deal prices of factors, metric: $\frac{m_0}{m_i}$
• $a$: by different examples, fee parameter for a mechanism in the no-cash flow condition, metric: by different examples

Importantly, in all examples, there is no fee between buffer sector and energy sector; if the energy sector is required having $1$ energy and $F$ factors and $r$ a tiny fee to the buffer sector, which is also in metric of energy, to produce $G J$ energy, then technically the energy sector could be written as having $1$ energy and $\frac{F}{1+r}$ factors to produce $\frac{G}{1+r}$ energy without fee to the buffer sector at the first place.

The energy for $q B$, if not politically solved, must be financed from the activity $D$ which is always $J F (I-T)^{-1}$ irrelevant to the fee payment structure. As one can see the buffer sector exhausts all left over energy $G J - J - J F (I-T)^{-1} E = (G-1-F K) J >0$, the energy price taking into account of buffer sector cannot exist because the post-buffer sector economic energy gain factor is zero.

buyers pay the fee I

Fee by total buying. Fee may be related to time-to-sale $t$ being a matrix m by m and factor specific paramater $a$ being a matrix of m by 1.

$P=c_0 E+ T P + t a(c_0 E+ T P)=(I + t a)E c_0 +(I+ t a)T P$

So

$P=(I-(I+t a)T)^{-1} (I+t a) E c_0$

By the matrix math where $A$ is another matrix:

• $(I-A T)^{-1} A=A (I-T A)^{-1}$
• $(I-T A)^{-1}=(I-(I-T)^{-1} T (A-I))^{-1} (I-T)^{-1}$

$P=(I+t a)(I-T (I+t a))^{-1} E c_0=(I+t a) (I-(I-T)^{-1} T t a)^{-1} (I-T)^{-1} E c_0=(I+t a) (I-(I-T)^{-1} T t a)^{-1} C$

As one can see $P$ is identical to $C$ when $a=0$; this shall be always true about the no-cash flow aka deal price equation. Also remember that the action $D$ is not $J F (I-(I+t a)T)^{-1}$ as the real factors forming $J F$ is not from the fee component $t a(c_0 E+T P)$ but still going through $E$ and $T$.

As $(I-T)P=c_0 E + \text{fee per unit trade}$ and $J F (I-T)^{-1}=D^T$, it follows $J F (I-T)^{-1} (I-T)P=D^T (c_0 E + \text{fee per unit trade})$ aka $J F P=\text{revenue of energy sale}+ \text{fee revenue}$

Shown in the $P$, the deal price of $F$ energy production factors is higher than the original deal price, the collected fee contributing to $q B$. Now the cash flow among factors is balance for any $a$. In addition, it shall make happy the sustainable equation $q' B=J$ or equivalent $\frac{J}{c_0}=\text{sum of fee contribution per unit time}$:

$\frac{1}{c_0}J=q B=D^T t a(c_0 E+ T P)=J F (I-T)^{-1} t a(c_0 E+ T (I+t a)(I-T (I+t a))^{-1} E c_0)$

aka

$1=c_0 F (I-T)^{-1} t a(c_0 E+ T (I+t a)(I-T (I+t a))^{-1} E c_0)$

When $a$ and $t$ are scalar, it is:

$1=c_0 t a F (I-T)^{-1} (c_0 E+ (1+t a) T ((I-(1+t a) T)^{-1} E c_0))$

$B$ is always $\frac{J}{q'}$ aka $q B + J F K + J = G J$

buyers pay the fee II

Fee by factor buying per unit buy.

$P=c_0 E+ T (P+a)=c_0 E + T a+ T P$

So

$P=(I-T)^{-1} (E c_0+ T a)=C+(I-T)^{-1} T a$

$\frac{1}{c_0}J= q B=D^T T a=J F (I-T)^{-1} T a$

sustainable equation:

$1=c_0 F (I-T)^{-1} T a$

sellers pay the fee I

Treat the fee as a surcharge on energy per unit sale. $a$ is a scalar.

$P- (a E)=c_0 E+ T P$

So

$P= (I- T)^{-1} E (c_0+a)=K (c_0+a) = C+K a$

$\frac{1}{c_0}J=q B=D^T a E = J F (I-T)^{-1} a E=J F K a$

sustainable equation:

$1=c_0 F K a$ i.e. $a=\frac{1}{F C}$

As

$c_0+a=\frac{1}{G-1-F K} + \frac{G-1-F K}{F K} \ge \frac{2}{\sqrt{F K}}$

one can see that $P$ can approach infinite when $F K$ is near 0 or $G-1$ and is at least $\frac{2 K}{\sqrt{F K}}$

sellers pay the fee II

It is argued that it shall have same fee for each transaction of the factor sector per unit sale. $a$ is a matrix of m by 1 whose i-th entry is fee per unit for transactions of i-th factor.

$P-a=c_0 E+ T P$

So

$P=(I-T)^{-1}(c_0 E +a)=C+(I-T)^{-1}a$

$D^T a=J F (I-T)^{-1} a$

sustainable equation:

$c_0 F (I-T)^{-1} a=1$

sellers pay the fee III

It is argued that it is not fair because some factor has long sale time and shall be responsible for more fee. $a$ is a scalar, $t$ is a matrix of m by 1 whose i-th entry is the average time-to-sale of transactions of i-th factor.

Knowing that the sum of notional of deals $\sum_j n_j$ for the i-th factor in 1 unit time is $D_i$, the operational procedure to determine $t_i$ for the i-th factor is as below.

• all deals: #1, #2,...
• time to sale: $x_1$, $x_2$, ...
• notional: $n_1$, $n_2$, ...
• fee of the deal: $n_1 x_1 a$, $n_2 x_2 a$, ...

Then $t_i=\frac{\sum_j n_j x_j}{\sum_j n_j}$

$P-t a=c_0 E+ T P$

So

$P=(I-T)^{-1}(c_0 E +t a)=C+(I-T)^{-1}t a$

$\frac{J}{c_0}=q B = D^T t a=J F (I-T)^{-1} t a$

sustainable equation:

$c_0 F (I-T)^{-1} t a=1$

With shorter $m_{time}$, estimation of $t_i$ may differ and accordingly $a$ and $P$ might fluctuate or change seasonally.

sellers pay the fee IV

The records of transactions have different size in bytes. It is argued that the fee of a transaction shall be dependent on the size of its record. $a$ is a scalar, $s$ is a matrix of m by 1 whose i-th entry is the average bytes of record of transactions of i-th factor.

The operational procedure to determine $s_i$ for the i-th factor is as below.

• all deals: #1, #2,...
• bytes: $x_1$, $x_2$, ...
• notional: $n_1$, $n_2$, ...
• fee of the deal: $x_1 a$, $x_2 a$, ...

Then $s_i=\frac{\sum_j x_j}{\sum_j n_j}$

$P-s a=c_0 E+ T P$

So

$P=(I-T)^{-1}(c_0 E +s a)=C+(I-T)^{-1}s a$

$\frac{J}{c_0}=q B = D^T s a=J F (I-T)^{-1} s a$

sustainable equation:

$c_0 F (I-T)^{-1} s a=1$

buffer for only energy cost

$t$ and $a$ is scalar.

$P -E c_0 t a=E c_0 + T P$

So

$P=(1+t a)C$

sustainable equation:

$\frac{J}{c_0}=D^T E c_0 t a=J F(I-T)^{-1} E c_0 t a$

aka

$\frac{1}{c_0}=F C t a$ i.e. $a=\frac{1}{c_0 t F C}$

indicates $B=J F M C$ where $M=\frac{t a}{q} I$

buffer based on cost

$t$ is a matrix of m by m, $a$ is a scalar.

$P - t C a = E c_0 + T P$

So

$P=(I-T)^{-1}(E c_0+t C a)$

sustainable equation:

$\frac{1}{c_0}=F(I-T)^{-1} t C a$ i.e. $a=\frac{1}{c_0 F(I-T)^{-1} t C}$

indicates $B=J F M C$ where $M=(I-T)^{-1} t \frac{a}{q}$

As long as $c_0<1$ it is $C<K$. Typically it should be $C<P<K$ but this might not be true for some fee structure setting and for some specific i-th factor in that $c_i<p_i$ and $c_i<k_i$ and not $p_i < k_i$. This might suggest this i-th factor shall not be in the market place and acordingly $p_i = k_i$ might serve the boundary mentioned in Ronald Coase's Nature of the Firm (1937). Anyway, below no more discussion about fee payment structure to fit the sustainable equation. Instead, discussion about augment surplus factors.

adjusting surplus to fit sustainable equation

Sustainable equation $q' F M C=1$ is translated to

$\frac{q F M K}{(G-1-F K)^2}=1$

To see the size of the surplus,

• $b \equiv \frac{q F M K}{(G-1-F K)^2}$, the imbalance ratio of the economy
• $S \equiv \frac{(1-b)J}{c_0}$, the energy surplus of the economy

By the definition, it follows:

$q B+b J +b J F K=G b J$

However, the commit energy input is $J$, therefore, when $b$ is not 1, it means the unaccountable energy surplus or deficit $S$ distributed by force other than economics:

$q B + S+ J + J F K = G J$

The surplus $S$ is eased by bigger $F K$. It could be that people dump more food to the landfills, or (induce new factors in $T$) people play e-sport not expected in the eyes of a person 100 year before, or any RD such that a new industry technology comes to life which is also a change of entries in $T$. Here below describe the way without additional factors intrudoced and without change of $G$.

As explained above, if $b$ is less than 1, the surplus factors can be artificially inflated to make the surplus accountable. Therefore,

$q B + S+ J + J F K = G J = q \hat{B}+J + J \hat {F} \hat {K}$

Then it means the surplus enjoyed by the surplus factors is:

$J \hat{F} \hat{K}-J F K =S-(q \hat{B} - q B)$

which is linear to change of $F$ if only surplus factors in $F$ is allowed to inflated due to $T$, and accordingly $K$, being intact.

With known $q$ and the average $t$, then the sustainable equation is a simple quadratic equation of $F K$ in the post-inflated world. For example, $G$ being 26 and $q$ being 0.13 and the average $t$ being 0.5, post-inflated $F K$ shall be 23.75733 and the energy cost of 1 unit energy shall be 0.804719 dollar. Note the numbers $G$, $F K$, $q t$ are all dimensionless. In formula:

$F K = G-1+\frac{q t}{2}- \sqrt{(G-1)q t+\frac{(q t)^2}{4}}$

$(\text{economic energy gain factor})=G-1-F K= \sqrt{(G-1)q t+\frac{(q t)^2}{4}} -\frac{q t}{2}$

$c_0=\frac{1}{\sqrt{(G-1)q t + \frac{(q t)^2}{4}}-\frac{q t}{2}}$

The ratio of required energy power of money sector to all energy power:

$\frac{q t J F C}{G J}=\frac{q t F C}{G}=\frac{1}{G c_0}=\frac{\sqrt{(G-1)q t + \frac{(q t)^2}{4}}-\frac{q t}{2}}{G}$

The ratio of required energy power of energy and industry sector to all energy power:

$\frac{J(1+F K)}{G J}=\frac{1+F K}{G}=\frac{c_0+F C}{G c_0}=\frac{G c_0 -1}{G c_0}=1-\frac{1}{G c_0}$

In the pre-inflated world, energy cost $c_0$ is getting higher when $F K$ is higher. But the constraint of sustainable equation and a rigid $G$ and $q$ and $t$ result in $c_0$ getting lower at the pace of $\frac{1}{\sqrt{G}}$.

surplus factors

• $\Lambda$ , the diagonal matrix of m by m indicating the augment

There are two ways to augment the surplus factors:

1. by columns: inflate the correspondent columns of the said factors in $F$ and $T$, $\hat{F}=F \Lambda$, $\hat{T}=T \Lambda$
2. by rows: inflate the correspondent rows of the said factors in $E$ and $T$, $\hat{E}=\Lambda E$, $\hat{T}=\Lambda T$

It turns out that the two methods are equivalent if the matrix $M$ as a function of the augmented matrix $T$ has the property: $\Lambda M \big(T \Lambda\big)= M \big(\Lambda T\big)\Lambda$

$t$ being a diagonal matrix, $M(T)=(I-T)^{-1} t$ has this property.

$\Lambda (I-T \Lambda)^{-1} t= \Lambda((\Lambda^{-1}-T)\Lambda)^{-1}t=(\Lambda^{-1}-T)^{-1}t=(\Lambda^{-1}-T)^{-1} \Lambda^{-1} \Lambda t=(I-\Lambda T)^{-1} \Lambda t=(I-\Lambda T)^{-1} t \Lambda$

Therefore, the sustainable equations are both the same:

$1=\frac{q(F \Lambda)(I-T \Lambda)^{-1}t (I-T\Lambda)^{-1} E}{(G-1-(F \Lambda)(I-T \Lambda)^{-1}E)^2}=\frac{q F(I-\Lambda T)^{-1}t (I-\Lambda T)^{-1}(\Lambda E)}{(G-1-F(I-\Lambda T)^{-1}(\Lambda E))^2}$

In a sense being greedy and being lazy are equivalent.

Sector percentage, volume and purchasing power

$q B + J F K + J = G J$

it follows

$\frac{q B}{G J} + \frac{F K}{G} + \frac{1}{G}=1$

In case of sustainable economy and in terms of energy consumption, the percentages of buffer or trust sector, industry sector, and energy sector are indicated in this identity.

$\frac{1}{G c_0} + \frac{F K}{G}+\frac{1}{G}=1$

By energy sector percentage, $G$ is estimated. Knowing $G$, $c_0$ follows by buffer sector percentage. The rest is the industry sector.

$\frac{q B}{G J}=\frac{1}{G c_0}$ leads to some interesting corollaries.

With formula mentioned above, as energy gain factor $G$ gets higher and the surplus is absorbed, energy price $c_0$ and percentage of mining power of the society $\frac{1}{G c_0}$ are getting lower at the pace of $\frac{1}{\sqrt{G}}$ and $B$ is getting higher at the pace of $\sqrt{G}$. In addition, assuming $c_0=1$ the condition of no energy leverage, then $F K = G-2$ and $q t=\frac{1}{G-2}$ implying shorter $t$. Energy consumption percentage of the three sectors are $\frac{1}{G}, \frac{G-2}{G}, \frac{1}{G}$ while energy sector and buffer sector decrease at the pace of $\frac{1}{G}$. The mining power will never drain the whole economy energy power. But while the mining task replaces traditional trust sector like banking and ownership maintenance bureaucracy, the crystal clear objective cost of mining might be shocking as the guard cost of traditional method is opaque and hardly questioned.

• $V$, coin volume, scalar, metric: coin
• $H$, energy per coin also represents purchasing power of 1 coin, metric: $\frac{m_0}{coin}$

As $B=J F X$, the "one dollar is one unit energy" setting is in fact $H=1_0$ and $V=\frac{J F X}{1_0}$.

Another $H=1_0$ example.

Pumped storage hydroelectricity

A dam is an energy storage with $H=1_0$. To add some token in $V$, one must supply his solar panel power to others or to raise the water level of the dam. To destroy some token $V$, one could enjoy his saving for sauna in his house even the power plant is not operational for generating power, spending $k_0$ instead of $c_0$. His private solar panels could cover the cost for his store of value too. $b$ being his amount of tokens, the cost is $q' b$ when his power generating is on and $q b$ when his power generating is off. Without possession of a private power generator to provide power to the dam operator, every month he will be notified a debit of $q b$ even he does not use his tokens. Promising never moving the tokens for a period of time and the borrower being known as an honest person, then the risk free rate $q$ is paid by the borrower to the physics law rather than to the lender. What the lender gets is his return of the amount of the token after the lending period. The borrower uses energy leverage $\frac{1}{c_0^2}$ to repay the risk free rate as well as to conduct some great RD and, good for him, succeeds for a fortune and the so-called entrepreneurship.

impact of socialism

The more transaction the larger $B$. The larger the buffer or trust sector the cheaper the energy price. In socialism, anything is planned and individuals don't have buffer, also not necessary for individuals private change hand records maintenance of factors. This means socialism has a higher cost of energy for almost everything than the cost of free commercialism because of energy's role in everything. A socialism society has larger $c_0$ as majority of energy budget is for industry production $J F$. One could define $c_0>1$ as the threshold of pure socialism while it means the cost of production $N^T C$ is higher than $N^T K$. The progress of economic energy gain factor from realm of negative, between 0 and 1, larger than 1 represents the possibility of society evolution from no life, socialism life, libertarian life. As energy leverage reaches the balance with economic energy gain factor at 1, the society is around the boundary of socialism and libertarian.

Historically many shortcomings of coin volume inflation were shown with the design of variant $V$. As $q B = \frac{1}{c_0} J$, the cost power of buffer sector is the economic energy gain factor times the input power of energy sector $J$, suggesting the upper bound of mining energy power of a proof-of-work token or the upper bound of traditional cost power of guards/safety in trust factors. One up, the other down. It is also proportional to the size of the economy, other arrangement to ease or to avoid the shortcomings is possible. As long as $H$ can be objectively defined, long term commercial contracts can still be written, for example, adjustment of principal repayment for a loan.

mirror energy sector

$H=J \cdot 1_{time}$, then $V=\frac{F X}{1_{time}}$ which is much stable than the volume at $\frac{J F X}{1_0}$ previously. In case of $c_0=1$ of no energy leverage, it leads to $V=\frac{1}{q \cdot 1_{time}}$ so that the energy power of the buffer sector $q B = q J V$ is the same as that of the energy sector $J$.

mining energy power

The cash flow is always balance like an ordinary market clearing mechanism and buffer sector, here miners, pass all fee to buy energy from energy producers who reimburse the fee back to the factor producers via higher $F$ price. Energy sector will always adjust a proper fee payment parameter to maintain the sustainable equation.

$V$ is a fixed number and $H=\frac{J F X}{V}$. As mining energy power $\frac{1}{c_0}J$ and the mining fee per unit time in terms of coins are objective, $H$ being the mining power divided by fee in terms of coins per unit time has no room to be fishy:

• $W$: scalar, the objective energy power to the mining task, theoretically it is also $q B= q H V$, metric: $\frac{m_0}{m_{time}}$
• $f$: scalar, the objective fee per unit of time in term of coin, metric: $\frac{coin}{m_{time}}$

Because $1_0$ energy means $\frac{1_0}{H}$ coin, $f$ represents energy power $W=\frac{f}{\frac{1}{H}}=f H$. Therefore, $H=\frac{W}{f}=\frac{q H V}{f}$. Therefore theoretically $f=q V$ and $H=\frac{W}{f}=\frac{W}{q V}$. Previously, it is known $f=q J F X \frac{1}{H}$ so theoretically $V=J F X \frac{1}{H}$ and $B=J F X$

The fee parameter $a$ coin per byte can be easily set because it shall lead to the fee coin per unit time is $q V$; in reality this could be lower if many tokens are lost, for example, forgotten private keys, death of a secret holder, ...etc.

Variant $H$ with constant $V$ is the same as the case of constant $H=1_0$ with variant $V$ in distribution of new coins or in trashing of old coins at same ratio by the coins stake of individuals. Accordingly, everyone shares the same fruits before and after economy growth. However the proportional change of money stake is hardly done, aka Cantillon Effect, unless in technology / politics autocrat with tight central control.

Finally, I wish people could understand that:

1. in the deepest sense, money is not for "hodl" or for "earn". Instead, it serves as the method of energy accounting and promotes fair and peaceful societal cooperation to share the energy surplus gifted by mother nature or energy harvest technology.
2. to establish trust in the money, it must have an objective cost aka proof-of-work where metric for work is, as taught in middle schools, energy, and any other way like proof-of-stake or "in god we trust" or whatever claims cheaper, is suggesting perpetual motion machines aka scam or religion in the rulers who historically and eventually always take advantage of monopoly money issuing. After all money, when it is ugly, is a powerful tool to control people.

Here is the html page that one can experiment around locally in a browser. Feel free to copy it.