Theory of Money

Because physically it always requires 9.8 joules for moving 1 kilo gram 1 meter higher, ideally money is a metric of energy with respect to store of value and media of exchange. 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.

Let $m_i$ be 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 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. The i-th row and j-th column is the required amount of j-th factor in the production of one unit of i-th factor, metric: $\frac{m_j}{m_i}$. Pardon me for the same English character of matrix transpose.

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: $m_i$

Therefore,

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

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

Note that $(I-T)^{-1}$ is dimensionless.

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 arbitrage so people have strong interest to expand $F (I-T)^{-1}E$ till $c_0=1$. Say, input $1_0$ one net $\frac{1}{c_0}$, apply $n$ times again then $\frac{1}{c_0^n}$, leading to energy arbitrage. 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 uncontroled atomic bomb rather than a power plant. Scientists and technicians design a power plant to the extent that it is resonably safe and operational. Therefore only leverage up to 2 times 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:

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

• $C$: 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 fun facts about $K$.

• it is irrelevant to $F$, nothing to do with energy harvest
• this equaition resembles the no-cash flow condition when people buys and sells 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 $2E$ is $2K$
• if $T$ is replaced with $A T A^{-1}$ and $E$ is replaced with $A E$, then the solution is $A K$

It follows:

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

$C=c_0 K$

$C=c_0 E + T C$

$G c_0=1+c_0+F C$

Note that $F K$ and $F C$ and $k_0=1$ are dimensionless. 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 consumper so they enjoy total energy surplus $(k_i-c_i) 1_i$ which is also the vital blood for the beings.

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 and its factors $F$ is circulating. 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 in 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 $K=(I-T)^{-1} E$.

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

The by-product of a production can be modeled by negative input factor. Then there is also a nagetive 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

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

• $N$: 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-hydropower, 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 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 cost of $q' B$ shall be paid by someone. While the ideal method is that hoders pay $q'$ ratio of their stake in $B$ per unit of time, the method is often based on transaction instead.

The ownership of piece of $B$ could be tokenized, aka, owing "one dollar" or "one food stamp" represents the ownership of 1 unit energy in the storage pool $B$. As said, there is energy cost $q' t 1_0$ to keep 1 unit energy accross a period of time $t$. In market economics, along with the transaction of 1 i-th factor, 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. 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 seperately, 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 unit i-th factor, 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

It depends. Moving energy across time, $q B$ is also the required energy power for this protection or service.

1. in case of covering the production of $N$ without help of energy sector for time $t$, $B=t N^T K$
2. in case of keeping the system running at most time $t$ in which $J F$ per unit time and is producing and consuming, $B=t J F C$
3. in case of covering the production of $N$ with help of energy sector for time $t$ if the energy producer calls Houston, $B=t N^T C$

Case 1 is straight forward and easily understood.

Case 2 and 3 might be weird at the first glance. Let $J = \frac{B}{t}$. As long as economic energy gain factor higher than 0, $B$ is refilled from the $G J$ any way it is $N^T K + J F K + J = G J$ and this sysmem can last indefinitely. What does the $t$ here mean any way? unless it means to provide energy power of production of $N$ factors for a shorter time $c_0 t$ as $t N^T C= c_0 t \cdot N^T K$. To understand this, imagine there are two internal batteries of the energy producer, one is empty and the other is full. The full one is to commit the energy power $J$ for time $t$ until it is empty and the empty one is collecting partial of the $G J$ to the full so that when the duty battery is empty after time $t$, the energy producer swaps the two batteries and go on the energy production task indefinitely. The reason of the case $B=t N^T C$ is to insure against the odd that one of the two batteries is dead and cannot collect the partial of the $G J$, the dead battery being fire or heat waste instead of the energy collecting. The energy producer could pronounce "we have problems", though. On the the other hand, through the economic energy gain factor route, the over all inventory change of factors per unit time is $N^T - J F$. Put the $B$ into the system equation, total energy power $J =(1+q' t) N^T C$, a 2-time leverage example, is comitted for time $t$ to maintain the system:

$\frac{1}{c_0} J =\frac{1+q' t}{c_0} N^T C=\frac{1}{c_0} N^T C + \frac{q' t}{c_0} N^T C=N^T K + q t N^T C=N^T K +q B$

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

Case 2 sets an equation about $t$. When the internal battery of the energy producer is dead, the energy power $J= \frac{B}{t}$ can sustain the task $G J = J + J F C$ for time $t$. Interestingly for sustainable economy where $N$ is zero and the only meaningful production is $F$ factors, 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.

Suppose $B$ is tokenized. 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 themselve 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 perticipants, 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 tokenized 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 maintainance 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. Thanksfuly 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 virtualized to be an energy ownership maintainer in the form of trust back by phisical energy cost. Normally, the $B$ is intact and it looks dumb to keep buring energy for its input. But actually it is necessary and the virtualized $B$ 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 is to focus on case 2 economy and 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: $m_{time}$

In an economy of no need to offset the cashflow 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 $\frac{q J F M C}{J F K}=q' t$

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 of a transaction is sent right away to the buffer sector based on the price difference facing the players, 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, the difference between deal price and $c_0 E+ T P$ is sent to the energy sector directly or sent to the buffer sector seperately and there is no fee between buffer sector and energy sector.

buyers pay the fee I

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

So

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

Now the cash flow among factors is balance for any $a$. In addition it shall make sustainable equation happy:

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

$1=q' F (I-T)^{-1} a(c_0 E+ T ((I-T (I+a ))^{-1} (I+a) E c_0))$

When $a$ is a scalar, it is:

$1=q' F (I-T)^{-1} a(c_0 E+ (1+a) T ((I-(1+a) T)^{-1} E c_0))$

buyers pay the fee II

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

So

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

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

sustainable equation:

$1=q' F (I-T)^{-1} T a$

sellers pay the fee I

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

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

So

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

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

sustainable equation:

$1=q' F K a$

sellers pay the fee II

$P-a=c_0 E+ T P$

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:

$q' F (I-T)^{-1} a=1$

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.

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

So

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

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

sustainable equation:

$q' F (I-T)^{-1} t a=1$

storage for the energy portion of F's production for time t

$B=t D^T E c_0=t J F C$ indicates $M=t I$

This is like a society where all stores are government kiosk which has all the information of factors production and the goverment's concern is only a duation down time $t$ of system operation and the government pays the cost of the buffer. When a customer puts into $c_i$ money (or energy credit) into the kiosk and demands one unit of i-th product, the kiosk immediately dispatches the action $D$ and its correspondent energy cost to all sectors. Then the product is manufactured and delivered to the customer.

And accordingly, the market prices $P = C$

buffer for purchase of factors of F

The energy transaction money of i-th sector for the calculation of fee is

$t_i c_i$

Sum over all factors, it is

$J F t C$ indicates $M=t I$

buffer for only energy of F's production

The energy transaction money of i-th sector for the calculation of fee is

$t_i D_i E_i c_0$

Sum over all factors, it is

$D^T t E c_0=J F(I-T)^{-1}t E c_0$ indicates $X=(I-T)^{-1} t E c_0$

buffer for all factors of F's production

The energy transaction money of i-th sector for the calculation of fee is

$t_i D_i c_i$

Sum over all factors, it is

$D^T t C=J F(I-T)^{-1}t C$ indicates $M=(I-T)^{-1} t$

adjusting surplus to fit sustainable equation

Sustaianble 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 $S$ could induce new factors in $T$ so that $F K$ becomes bigger, like, 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 $T$, or any RD such that $G$ becomes higher. Here below describe the way with no factors class change and with no 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 quardratic equation of $F K$. The post-inflated surplus factors in $F$ then can be near $G -1$ to reach very low $q t$ or $F$ can be small so $q t$ is large. 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. 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 a sense, this is equivalent to that all the $D$ is to cause an $N$ equal to $J F$ whose energy portion is transacted to get the fee. While the $J F$ is beyond the real needed factors for energy production, then it becomes the surplus. But there is a price: $c_0$ gets higher even though $G$ is huge.

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: $\Lambda M \big(T \Lambda\big)= M \big(\Lambda T\big)\Lambda$

Take $M(T)=(I-T)^{-1} t$ for example

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

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 intersting 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 $H$ is getteing higher at the pace of $\sqrt{G}$. In addition, assuming $c_0=1$ the condition of no energy arbitrage, 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 maintainance 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 volumne, 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 $q' b$ where when power generating is on and $q b$ when power generating is off. Every month, he will be notified a debit of $q'$ even he never use his tokens. Promissing 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 $\frac{1}{c_0}$ 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 maintaiance 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 enonomic 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.

example 1

$H=J \cdot 1_{time}$, 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 arbitrage, 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

Historically many shortcomings of coin volume inflation were shown with the design of variant $V$. As $q B = L \equiv \frac{1}{c_0} J$, 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.

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: $\frac{m_0}{time}$
• $f$: scalar, the objective fee per unit of time in term of coin, metric: $\frac{coin}{m_{time}}$

If miners are seeking surplus, then because $1_0$ costs $\frac{c_0 1_0}{H}$ coin, $f$ can purchase energy power $W=\frac{f}{\frac{c_0}{H}}=\frac{f H}{c_0}$. Therefore, $H=\frac{c_0 W}{f}=\frac{c_0 q H V}{f}=\frac{q' H V}{f}$. Therefore theoretically $f=q' V$ and $H=\frac{c_0 W}{f}=\frac{c_0 W}{q' V}=\frac{W}{q V}$. Previously, it is known $f=q' J F X \frac{1}{H}$ so theoretically $V=J F X \frac{1}{H}$ and $B=J F X$

Miners or any trust providers are supposed to have zero surplus, it means $1_0$ is $\frac{1_0}{H}$ 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 $q V$

Change of $H$ and fixed constant $V$ is the same as the case of $H=1_0$ with distribution of new coins or 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 unless in technology / politics autocrat with tight central control.

All the above is demostrated in the excel here: https://drive.google.com/uc?export=download&id=176Xbb-HOy0KF761IwjETvoaFRBZa835D

While the surplus factors can be inflated in arbitrary ways to satisfy the sustainable equation, it requires some numerical procedure to find the surplus ratio. Note that the purpose of the Excel workbook and the paper is for clearness of the fundamental concept explanation. Out of curosity, I choose to inflate the surplus factors in $F$ part only rather than the whole system including $T$ part, pretty much like a society where monarch enjoys all the surplus. This way, I can solve the surplus ratio by simple quadratic equation and there is no mismatch in the audit of the paper by the Excel workbook. As explained in the section about human surplus, all entries of the column of the matrix $T$ and $F$ shall be inflated. People of all sectors, not energy sector only, shall enjoy the surplus, though. I do provide in the end of the excel a section for handling of this fair distribution with the help of goal-seek of excel on solving the sustainable equation.

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.
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 religious trust in the rulers who eventually always take advantage of monopoly money issuing and play the book of "absolute power, absolute corruption".