Macro Economics Theory

Money is simply a measure unit of energy. Here one dollar is assumed to be one joule 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.

• G: scalar. The energy output ratio to the energy input. Note that the net energy collected shall be G-1 multiple of energy input
• m: number of sectors or production factors
• F: matrix of 1 by m. The required factors to work with one unit of energy input in the energy production
• E: matrix of m by 1. The required energy for one unit of factors' production.
• T: matrix of m by m. The entry of 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

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. Often, people don't account for some factors in production to achieve cheaper cost but this is not the way here in the macroeconomics theory.

• D: matrix of m by 1. The size of the deals or action in sectors
• N: matrix of m by 1. The net factors amount obtained by the action

Therefore,

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

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

how to get one joule

• J: scalar. The energy input amount for this task

The J, along with JF factor, can produce GJ joules. Then the net energy amount GJ - J is the wanted 1 joule and for the action D such that N happens to be JF. In other words, the equations:

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

$N^T = J F$

So, the equation about J is:

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

And therefore:

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

$c_0$ is the unit cost of energy. One can say in oral language one unit of energy costs $c_0$ dollar, or one joule's energy cost is $c_0$ joule. $\frac{1}{c_0}$ is also the economic energy gain factor.

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 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 $c_i$

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 joule.

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 required energy of factors

Note that K is irrelevant to F and it follows:

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

$C=c_0 K$

$C=c_0 E + T C$

$G c_0=1+c_0+F C$

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

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

The money inflow of the sector is

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

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.

Scenarios modeling

The number 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 (while other factors were not accountable) which helped people lose weight or got people killed. Therefore, F 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. People in the factor F 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 being zero still. Note that F cannot be over-size to cause $G-1-F K < 0$. Also, it might be weird that $c_0$ is greater than 1 although this is logically possible. Therefore F shall not be large such that $G-1-F K <1$. Unlike non-sustainable economy or central plan economy where the ruler needs to helicopter money $t N_i c_i$ from mainly energy sector to i-th sector every interval time t, 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. Nevertheless, the energy cost of the garbage is accounted already in the cost of the surplus factors.

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. 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 can be derived.

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.

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 budget

• N: m by 1 matrix. The net produced factors per unit of time
• D: m by 1 matrix. The size of action or deals per unit of time
• J: scalar. Energy power

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.

If some energy storage or buffer amount for inconvenient days is suggested,

• B: energy storage amount
• q: energy loss rate per unit of time in energy storage
• $q' \equiv c_0 q$

It follows:

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

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

The total energy cost of i-th factor (buying factors from other sectors) is $N_i c_i$

In market economics, q'B is paid by the transaction involved in the economy. 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 $\frac{D_i}{n} c_i \times \frac{q'}{n} \times n=D_i c_i \times \frac{q'}{n}$. 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 is a difference between market price and the cost faced by the sector. The difference contributes to fee and additional surplus of the said supply chain producers. With competition, the market price will be only higher than the cost by the fee of the involved transaction.

Size of buffer

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

1. in case of covering the production of N and destruction of energy sector for time t, $B=t N^T K$
2. in case of covering the factors of the energy sector for time t
3. in case of covering the production of N for time t, $B=t N^T C$

In case 2, it sets an equation about t. Interestingly for sustainable economy where N is zero, it leads to $q' t F C =1$ or some similar sustainable equation. This is unlikely 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. Case 3 is the base where $t J =(1+q' t)t N^T C$ is the "pool of energy input commitment with the help of mother nature". In all cases, the storage can be tokenized and exchangeable, aka money. Per unit of time, 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 trust the token's maintainer, people know 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. When both the cost are q', some people opt in and some people opt out. Over all, half of the buffer sector is tokenized and has the same cost q' to be trustful without authority, the other half 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 the robots who are doing routine maintenance job of a solar farm. Hypothetically, if aliens come to Earth and secretly destroy 2/3 of these economy-invisible energy storage, then the size of the economy will become 1/3 because the volume or the purchasing power of the money is reduced due to over-production before hand.

For convenience's sake, the turnover time t is assumed the same for all sectors. If they are different among sectors, then $B=N^T t C$ where t is a diagonal matrix. The fee of sectors of longer turnover time shall be higher too. The fair market price of factors shall be $(I+q' t)C$

Below is to focus on buffer design of the money flow circulating economy (case 2) and industry sector is re-defined as the production of F. As mentioned of "hiding surplus in energy sector" above in section of scenarios modeling, this suggests a way to justify a fair surplus in order to agree with the physically known t by the sustainable equation.

• M: matrix of m by m

As long as the storage is linear about J and F and C, a typical form of the storage is of the form $J F M C$. Some examples:

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

To produce F, the necessary action is

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

Therefore,

$B=t D^T E c_0=t F C$

It indicates

$M=t I$

This is like a society where any store is a kiosk which has all the information of factors production. 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.

buffer for all factors of F's production

Because

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

Suppose the time-to-sale of i-th factor is $t_i$

Then, the transaction money of i-th sector for the calculation of fee is

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

Sum over all factors, it is

$D^T \begin{bmatrix}t_1 &&& \\& t_2 && \\ && \ddots & \\&&& t_m \end{bmatrix} C=F(I-T)^{-1}t C$

It indicates

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

sustainable equation

In an economy of no need to offset the cashflow gap, the equation of sustainable economy is:

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

equivalently,

$q' F M C=1$

and is translated to

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

Note that "1" here is not dimensionless but one unit of energy per unit of time. This also relates the average turn over time and q' with the ratio of energy powers. If the average is defined as the t such that $t F C =F M C$, equivalently in eyes of energy depletion $J F C=\frac{B}{t}$, then it follows $\frac{q J F M C}{J F K}=q' t$

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:

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

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, the post-inflated surplus factors in F then can be near G -1 to reach very low qt or F can be small so qt is large. For example, G being 26 and q being 0.13 and the average t being 0.5, then post-inflated FK shall be 23.75733 and the energy cost of 1 joule shall be 0.804719 dollar. In formula:

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

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

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 JF whose energy portion is transacted to get the fee. While the JF 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 column: 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.

Estimation

In case of sustainable economy, the percentages of buffer sector, industry sector, and energy sector are indicated in this equation.

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

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

Note that the larger the buffer sector, the cheaper the energy price. In socialism, anything is planned and individuals don't have buffer. This means socialism has a higher cost of energy and almost everything than those of free commercialism because of energy's role in everything. The two parts $\sum_i {t_i E_i c_0}$ and $\sum_i {\sum_j {t_i D_i T_{i j} c_j}}$ are for the two examples in the section of size of buffer above. Obviously, size of buffer of example 2 is larger.

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 must have an objective cost aka proof-of-work whose measure unit is, as taught in middle schools, joule, and any other way which 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".