Blog:

18.1.2021 - Economic Systems Theory 1:

Accounting Microfoundations for Economics:

Update: Ongoing research made it advisable to introduce a relevant distinction between bookkeeping and accounting. When writing this article it had been used interchangeably. In the upcoming work everything that is called accounting in this post is called bookkeeping. In it accounting implies value calculations that where not sufficiently formalized when writing this blog post. The "fundamental law of accounting" in this post is actually the fundamental law of bookkeeping. The upcoming fundamental law of accounting is its equivalent regarding values.
Update 2: The law of equivalent exchange in this paper is also not the same as in the upcoming work. There the name will also refer to an equivalent with respect the value of an exchange.

1. The economy is given as an enumeration of inventories, where each inventory is given by the amounts of goods in it.
2. Economic activity is given by exchanges of goods between inventories.
3. The evolution of the economy in time is described by a system equation, that uses the economic activity at a given time point as input. The system equation is a difference equation instead of the differential equations used in physics.
4. The accounting equations are generalized into a universal law, called the fundamental law of accounting, from which all accounting equations can be derived.
   Update/Errata: That should have said accounting identity and the law is not a formal generalization. It is the "first" accounting identity which must hold for the most general accounting situations, where only few things are assumed or defined. The full work will make this clearer. The mathematics below is not affected by this.

1. Commodity currency systems and proper derivations of aggregate prices, baskets and inflation.
2. Representative currency systems, as special form of commodity currency systems trading an intangible abstract representative good as currency.
3. Compound standard currency systems, a representative currency system, that is a generalization of the historical gold standard systems. This concept is introduced by WissTec and currently being worked on. Pure fiat currency systems are a variant of compound standard currency system, that, instead of some tangible good like gold, use tokens, that guarantee to be left alone by the state as long as you pay your taxes in the given currency.
4. A subsumption formalism to rigorously derive simpler coarser economic systems from complexer finer ones. This is useful for modeling and big-data analysis.
5. Inflation control in compound standard currency systems.
6. Implementation of an internet protocol more or less directly implementing concepts of the following formalism.

Functions yielding Functions:

The type \(Y\rightarrow Z\) of a function \(g : Y\rightarrow Z\) is identified as the set of mappings from \(Y\) to \(Z\) and \begin{equation} X \rightarrow Y \rightarrow Z := X \rightarrow ( Y \rightarrow Z ) \end{equation} is the set of mappings taking arguments in X and yield functions mapping from \(Y\) to \(Z\). Thus for \( f : X \rightarrow Y \rightarrow Z \) (or equivalently \( f \in X \rightarrow Y \rightarrow Z \)), \(x\in X\) and \(y \in Y\) the expression \begin{equation} f(x)(y) \in Z \end{equation} is the result of \(y\) applied to the function resulting from \(x\) applied to \(f\).

Sequence Notation:

The set of all families with indices in the countable set \(I\) and values in the set \(X\) is denoted as \begin{equation} X^{I} := I \rightarrow X \text{ .} \end{equation} Sequences \(x \in X^{I}\) use the usual syntax \begin{equation} x_i := x(i) \text{ .} \end{equation}

Sequence Space with Counting Norm:

To avoid formal problems, for example with handling infinite sums, the following sequence space is used: \begin{align} l^1(\mathcal G) := \left\{ x\in \mathcal \R^\mathcal G \left| \norm{x} := \sum_{i \in \mathcal G} |x_i| <\ \infty \right.\right\} \end{align} It is a common type of Banach space, whose norm counts, i.e. sums the absolute values of the sequence elements.

Power Set:

The not so common notation \begin{equation} 2^X := \{ A | A\subseteq X \} \end{equation} for the power set is used.

Number of Elements in a Set:

The number of elements in a set \(A\), that is its cardinality, is denoted as \begin{equation} \#A \text{ .} \end{equation}

Goods Classification:

A goods classification is a triple \((\mathcal G,G,\frak G)\), where \(\frak G\) is the set of all goods classified, \(\mathcal G \subseteq \N\) is a set of type ids and \begin{equation} G : \N \rightarrow 2^\frak G \end{equation} is a mapping, so that \(G(i)\) is the set of all goods that are classifed as being of the type with index i. To be a proper classification, the mapping must satisfy the following properties: \begin{equation} \forall i,j\in \N : i\ne j \Rightarrow G(i) \cap G(j) = \emptyset \end{equation} \begin{equation} \bigcup_{i\in \mathcal G} G(i) = \frak G \end{equation} \begin{equation} \forall i\in \mathcal G : G(i) \ne \emptyset \text{ .} \end{equation} If \(i\) is a type index, and \(g\in G(i)\), then the short formulation "\(g\) is of type \(i\)" is used. The set \(\mathcal G\) is intended to be used to express how to combine different goods classifications, which is not of further interest in this introduction.

Content:

Let \(C=(\mathcal G,G,\frak G)\) be a goods classification, then a content (of an inventory) is a vector \(a \in l_1(\mathcal G)\) so that \(a_i\in \R\) is the amount of the good of type \(i\) in content \(a\). The set of contents can shortly be written \(l_1(C_0)\) using that the goods classification \(C\) is a triple.

Distribution of Wealth:

Let \(C\) be a goods classification. Then a family of contents \begin{equation} w\in \W_C := \left\{\,\,w' \in l_1( C_0 )^\N \,\,\left|\,\, \norm{w'} := \sum_{i\in \N} \norm{w'_i} < \infty\,\, \right.\right\} \end{equation} is a distribution of wealth. Distributions of wealth use the matrix syntax \begin{equation} w_{gi} := (w_i)_g \text{ .} \end{equation} in the infinite matrix space \(w\in l_1( C_0 )^\N\).

Inventory:

Let \(C\) be a goods classification and \(w\in \W_C \) be a distribution of wealth, then \(w_i\) is the \(i\)-th inventory of the distribution of wealth and \(i\) is its inventory id.

Account:

Let \(C\) be a goods classification and \(w\in \W_C\) be a distribution of wealth, then \(w_{gi}\) is the account of good \(g\) of inventory \(i\) from the distribution of wealth \(w\).

Fundamental Law of Accounting:

Let \(C=(\mathcal G,G,\frak G)\) be a goods classification and \(w\) be a distribution of wealth, then the requirement \begin{align} \forall g\in \mathcal G : \sum_{i\in\N} w_{gi} = 0 \end{align} is called the fundamental law of accounting, that is that, what is added to an inventory had to be removed from others. The law can be compactly written as \begin{align} \sum_{i\in\N} w_{i} = 0 \text{ .} \end{align}

Redistribution of Wealth:

The set of distributions of wealth adhering to the fundamental law of accounting is called the set of redistributions of wealth. Formally, let \(C=(\mathcal G,G,\mathfrak G)\) be a goods classification, then the set of redistributions of wealth of \(C\) is defined by \begin{equation} \W_{C}^0 := \left\{\,\, w \in \W_C \,\,\left|\,\, \forall g\in \mathcal G : \sum_{i\in \N} w_{gi} = 0 \,\,\right.\right\} \text{ .} \end{equation}

Time:

The formalism uses the natural numbers as discrete time points denoted with the notation \begin{equation} \T := \Z \text{ .} \end{equation} to clearly distinguish time points from other natural numbers.

Economy:

Let \(C\) be a goods classification and \((\Omega,\mathcal F,\mathcal P)\) be a probability space, then an economy of \(C\) is a mapping \begin{equation} d \in \E_{C} := \T \rightarrow \Omega \rightarrow \W_{C}^0 \text{ .} \end{equation} In words: An economy is the time development of a redistribution of wealth as random variable. The following common notation is used: \begin{align} d_{i}(t)(\omega) &:= d(t)(\omega)_{i} \\ d_{gi}(t)(\omega) &:= d(t)(\omega)_{gi} \end{align}

Exchange:

Let \(C\) be a goods classification, then \begin{equation} \X_{C} := \left\{\,\,w \in \W_{C}^0 \,\,|\,\,\# \{\, i \in \N \,\vert\, w_i \ne 0 \,\} \le 2 \,\,\right\} \end{equation} is the set of exchanges on \(C\). The set \(\X_{C}\) is precisely the set of redistributions of wealth affecting two or zero inventories, as the next lemma, called law of equivalent exchange, will show.

Law of Equivalent Exchange:

For all \(w\in\X_{C}\) there exist a content \(c\in l_1({C_0})\) and inventory ids \(k,l\in \N\) so that \begin{equation} w_i= \begin{cases} c &\text{ if } i = k \\ -c &\text{ if } i = l \\ 0 &\text{ otherwise } \end{cases} \end{equation} where \(c\) is called the exchanged content. In words: In an exchange one inventory loses exactly what the other gains.

Proof: Due to the definition of exchanges there exist contents \(c,d\in l_1(C_0)\) and \(k,l\in\N\), so that \[w_i= \begin{cases} c &\text{ if } i = k \\ d &\text{ if } i = l \\ 0 &\text{ otherwise } \end{cases} \] and the fundamental law of accounting yields \[ 0 = \sum_{i\in \N}w_i = c + d \] and thus \(d=-c\).

Economic Activity:

Let \(C\) be a goods classification and \((\Omega,\mathcal F,\mathcal P)\) be a probability space, then the set of economic activities is defined as: \begin{equation} {\small \A_{C} := \left\{\,\, A : \T \rightarrow \Omega \rightarrow \X_{C}^{\N} \,\,\left|\,\, \forall \omega\in\Omega : \forall t\in\T : \norm{A(t)(\omega)}:=\sum_{k\in \N} \norm{A_k(t)(\omega)} <\infty \,\,\right.\right\} } \end{equation} An economic activity is therefore a mapping from a time point to a family of random variables yielding exchanges happening at a given time. The probability space is an implicit argument and used as source for randomness. For empirical time series analysis it usually does not need to be completely constructed, since constructing the required marginal distributions will suffice.

Economic System:

An economic system is a tuple \((C,\mathcal A)\) where \(C\) is a goods classification and \(\mathcal A \subseteq \A_{C}\) is the systems set of admissible economic activity.

Economic System Equation:

The equation \begin{equation} d(t+1)(\omega) - d(t)(\omega) = \sum_{k\in \N} A_k(t)(\omega) \end{equation} for an economic activity \(A\) and an economy \(d\) is called the economic system equation. It formally states, that the change in the (re)distribution of wealth at a given time point, is given by the accumulated exchanges of the economic activity at that time point.

Analytic Economic System Solution:

Let \((C,\mathcal A)\) be an economic system, then, given an economic activity \(A\in\mathcal A\) and the initial value \(d_0=d(t_0)(\omega)\) for \(\omega\in\Omega\), the economy is given by: \begin{equation} d(t)(\omega) = \begin{cases} d_0 + \sum_{t'=t_0}^{t-1} \sum_{k\in\N} A_k(t')(\omega) \text{ if } t\ge t_0 \\ d_0 - \sum_{t'=t}^{t_0-1} \sum_{k\in\N } A_k(t')(\omega) \text{ if } t < t_0 \end{cases} \end{equation}
Proof: Let \(\omega\in\Omega\), then for \(t \ge t_0\) it holds
$$ {\small \begin{aligned} &d(t)(\omega) - d(t_0)(\omega) \\ =& d(t_0+(t-t_0))(\omega) - d(t_0)(\omega) \\ =& \sum_{j=0}^{t-t_0-1} (d(t_0+j+1)(\omega) - d(t_0+j)(\omega)) \phantom{xxx} \text{ (as telescoping sum) }\\ =& \sum_{j=0}^{t-t_0-1} \sum_{k\in\N} A_k(t_0+j)(\omega) \phantom{xxx} \text{ (inserting economic systems equation) }\\ =& \sum_{t'=t_0}^{t-1} \sum_{k\in\N} A_k(t')(\omega) \end{aligned}}$$ and for \(t < t_0\) equivalently $${\small \begin{aligned} & d(t_0)(\omega) - d(t)(\omega) \\ =& d(t+(t_0-t))(\omega) - d(t)(\omega) \\ =& \sum_{j=0}^{t_0-t-1} (d(t+j+1)(\omega) - d(t+j)(\omega)) \phantom{xxx} \text{ (as telescoping sum) }\\ =& \sum_{j=0}^{t_0-t-1} \sum_{k\in\N} A_k(t+j)(\omega) \phantom{xxx} \text{ (inserting economic systems equation) }\\ =& \sum_{t'=t}^{t_0-1} \sum_{k\in\N} A_k(t')(\omega) \text{ .} \end{aligned}}$$ Rearranged and combined these equations yield the analytic solution.

Economic System Setting:

For an economic system \( (C,\mathcal A) \) a system setting is given by \begin{equation} \sigma=(A,t_0,d_0) \in \mathcal A\times \T \times \W_C^0 \end{equation} where \(t_0\) is the initial time and \(d_0\) is the initial state.

Economic System Solution Notation:

Let \( (C,\mathcal A) \) be an economic system and \( \sigma=(A,t_0,d_0) \) a system setting. The the corresponding system solver is denoted as \begin{equation} \ec : \A_{C} \times \T\times \W_{C}^0 \rightarrow \E_{C} \end{equation} and the solution \begin{equation} \ec(A,t_0,d_0) := \ec(\sigma) := t \mapsto \omega \mapsto \begin{cases} d_0 + \sum_{t'=t_0}^{t-1} \sum_{k\in\N} A_k(t')(\omega) \text{ if } t\ge t_0 \\ d_0 - \sum_{t'=t}^{t_0-1} \sum_{k\in\N } A_k(t')(\omega) \text{ if } t < t_0 \end{cases} \end{equation} is the solution of the initial value problem for economic activity \(A\) with intitial value \(d(t_0)(\omega) = d_0\).

Production Circulation System:

A production circulation system is a tuple \( \mathcal P = (\mathcal S, \mathcal I_\circ) \) of an economic system \( \mathcal S=(C,\mathcal A) \) and a set of circulation inventory ids \(\mathcal I_\circ\subset\N\), that defines a projection \begin{equation} \pi_\circ^\mathcal P(d)_i := \begin{cases} d_i &\text{ if } i \in \mathcal I_\circ\\ 0 & \text{ otherwise } \end{cases} \end{equation} that is used to define the admissible set of economic activity by \begin{equation} \mathcal A = \left\{\,\,A\in \A_{C} \,\,\left|\,\, \exists (t_0,d_0) \in \T \times \W_{C}^0 : \pi_\circ^\mathcal P\left( \ec(A,t_0,d_0)\right) \ge 0 \,\,\right.\right\} \end{equation} where the inequality in the set comprehension is to be understood point wise in time, inventory, good and random event. That is, all circulation inventories have at all times only accounts with a non-negative balance.

Projection Extracting Production Inventories:

Let \(\mathcal P\) be a production circulation system, then the projection onto the pruduction inventories is defined as \begin{equation} \pi_\pm^\mathcal P(d) := d - \pi_\circ^\mathcal P(d) \end{equation} which is equivalent to the definition: \begin{equation} \pi_\pm^\mathcal P(d)_i := \begin{cases} d_i &\text{ if } i \notin \mathcal I_\circ\\ 0 & \text{ otherwise } \end{cases} \end{equation}

Projection Extracting the Distribution of one Good:

The following projection extracts the distribution of one good \begin{align} \pi_g(d)_{hi} := \begin{cases} d_{gi} &\text{ if } h = g \\ 0 &\text{ otherwise } \end{cases} \end{align} and as corollary result it follows that \begin{equation} \pi_g \circ \pi_\circ^\mathcal P = \pi_\circ^\mathcal P \circ \pi_g \end{equation} as well as that it holds point wise that: \begin{align} d \ge 0 &\Rightarrow \pi_g(d) \ge 0\\ d \ge 0 &\Rightarrow \pi_\circ^\mathcal P(d) \ge 0 \end{align}

Stationary Production of a Good:

An economic activity \(A\) has stationary production of good \(g\) at time \(t\) if \begin{equation} \pi_\pm^\mathcal P\left(\pi_g\left(\sum_{k\in\N}A_k(t)\right)\right) = 0 \end{equation}

Goods Conservation Law:

Let \(\mathcal P = (\mathcal S, \mathcal I_\circ) \) be a production circulation system, \(t\in \T\), \(A\in \mathcal A\) and \(g\) a good id, then \begin{equation} \begin{aligned} &\pi_\pm^\mathcal P \left( \pi_g\left( \sum_{k\in\N}A_k(t)\right)\right) = 0 \\ &\phantom{xxxx}\Longrightarrow \\ &\norm{\pi_\circ^\mathcal P (\pi_g( \ec(A,t_0,d_0)(t+1)))} = \norm{\pi_\circ^\mathcal P(\pi_g(\ec(A,t_0,d_0)(t)))} \text{ .} \end{aligned} \end{equation} In words: If the production of \(g\) is stationary, then its total amount in the circular inventories does not change.

Proof: Let \begin{equation} a := \sum_{k\in \N} A_k(t) \end{equation} then using the definition of \(\pi_\pm^\mathcal P\) it yields \begin{equation} \pi_\pm^\mathcal P(\pi_g(a)) = 0 \Rightarrow \pi_\circ^\mathcal P(\pi_g(a)) = \pi_g(a) \label{GoodsConservationLawProofFixPointEq} \end{equation} and the economic systems equation further yields \begin{equation} \ec(A,t_0,d_0)(t+1) = \ec(A,t_0,d_0)(t) + a \text{ .} \end{equation} This allows to conclude the proof with $$ {\small \begin{aligned} &\norm{\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t+1))\right)} \\ &= \sum_{i\in\mathcal I} \norm{\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t+1))\right)_i} \phantom{xxx} \text{ definition of the norm } \\ &= \sum_{i\in\mathcal I} \sum_{g\in \mathcal G} |\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t+1))\right)_{gi}| \phantom{xxx} \text{ definition of the norm } \\ &= \sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t+1))\right)_{gi} \phantom{xxx} \text{ since $ \ec_i(A,t_0,d_0)(t+1)\ge 0$ for $i\in \mathcal I_\circ$ } \\ &= \sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t) + a)\right)_{gi} \phantom{xxx} \text{ due to the system equation} \\ &= \sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \left(\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi} + \pi_\circ^\mathcal P\left(\pi_g(a)\right)_{gi}\right) \\ &= \sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \left(\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi} + \pi_g(a)_{gi}\right) \phantom{xxx} \text{ using the first step of the proof} \\ &= \left(\sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi}\right) + \left(\sum_{i\in\mathcal I} \sum_{g\in \mathcal G} \pi_g(a)_{gi}\right) \\ &= \left(\sum_{i\in \mathcal I}\sum_{g\in \mathcal G} \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi}\right) + \left(\sum_{i\in \mathcal I} a_{gi}\right) \\ &= \left(\sum_{i\in \mathcal I}\sum_{j\in \mathcal G} \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi}\right) \phantom{xxx} \text{ due to the fundamental law of accouting} \\ &= \left(\sum_{i\in \mathcal I} \sum_{j\in \mathcal G} | \pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{gi}| \right) \phantom{xxx} \text{ since $ \ec_i(A,t_0,d_0)(t)\ge 0$ for $i\in \mathcal I_\circ$ } \\ &= \left(\sum_{i\in \mathcal I} \norm{\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)_{i}}\right) \\ &=\norm{\pi_\circ^\mathcal P\left(\pi_g(\ec(A,t_0,d_0)(t))\right)} \text{ .} \end{aligned}}$$

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