ENTROPIE ET ECONOMIE

ENTROPIA ED ECONOMIA

ENTROPY AND ECONOMY

ENTROPIE UND WIRTSCHAFT

ЭНТРОПИЯ И ЭКОНОМИКА

ENTROPY AND ECONOMY

by Sergei Darda

ABSTRACT

Asymmetrical nature of energy flow leads to the assumption that there shall exist thermodynamic entropy that manages this asymmetrical process.

Asymmetrical nature of money flow leads as well to the assumption that there shall exist economic entropy that manages this asymmetrical process.  Concept of Economic Entropy is based on the assumption, that money is an analogue of energy.

The concept of entropy as a measure of disorder is not inherently thermodynamical, but rather universal theoretical concept (like the concept of derivatives), which have applications in different fields of knowledge (statistics, thermodynamics, or theory of information).  This paper presents theoretical foundation for definition of Economical Entropy and offers for the first time formula for calculation of Economical Entropy.

Key words: Entropy, Thermodynamics, Economics, Economic Entropy, Price, Cost

1. THERMODYNAMIC ENTROPY

There is a fundamental asymmetry in processes connected with flow of thermal energy: movement of heat always occurs in one direction – from a hotter body to a colder body.  For example two pieces of metal with temperatures T1 and T2, where T1>T2, when brought together will generate a flow of energy in the form of heat.  Heat moves from the piece with higher temperature to the piece with lower temperature T2.  Why does the flow of heat go in that direction – from hotter to colder piece of metal?

To explain this phenomenon, scientists have developed a concept called entropy.  The major assumption of the concept is that if heat moves from a hotter to a colder body in natural or spontaneous processes, the resulting change of entropy in the system should be positive.  Otherwise the natural process of the flow of heat would not occur at all.  In natural processes entropy may only increase or at least stay constant.  Expressing the change in entropy by dS,

(1.1)    dS = dQ/T

where dQ is heat given or taken from a body with temperature T.

Figure 1.1   Movement of Heat

In Figure 1.1 two bodies with different temperatures (T1>T2) were brought together and the amount of heat dQ moved from the hotter body to the colder one.  If these two bodies were in a closed system, the resulting change of entropy should be positive.  Expressing the change in entropy by dS:

(1.2)    dS1 = -dQ/T1  and  dS2 = dQ/T2

Where dS1 and dS2 - changes of entropy in hot and cold bodies resulting from the heat flow.

The amount of heat in the first expression is negative because the heat was taken from the hot body and the entropy of the hotter body decreased by dS1.  In the second expression the amount of heat is positive because the heat was given to the cold body, the entropy of which has increased by dS2.  The resulting change in entropy in this case is positive: dS1 + dS2 >0.

Despite its pure theoretical origin (entropy was invented before it was measured), the idea of entropy was successfully used to analyze the work of heat engines, which use heat to generate work.  The conditions of work of such an engine are managed and limited by the entropy.  Any engine (Figure 1.2) schematically consists of hot reservoir, cold reservoir, and working body that is usually steam or hot gas.  Heat moves from the hot reservoir to the cold reservoir through the media of the working body and creates work due to the special design of the engine.

Figure 1.2   Work of Heat Engine

How much of the heat can be turned into the work?  It can be determined from the equation for entropy:

(1.3)    Wmax = dQ1 – dQ2

(1.4)    dS1 + dS2 = 0

(1.5)    - dQ1/T1 + dQ2/T2 = 0

(1.6)    dQ1/T1 = dQ2/T2

Solving for the only unknown dQ2,

(1.7)    dQ2 = dQ1*(T2/T1)

(1.8)    Wmax = dQ1 – dQ1*(T2/T1) = dQ1*(1 – T2/T1)

The following equation determines the maximum amount of work Wmax that can be derived from the engine of any possible design with any given temperatures of cold and hot reservoirs T1 and T2:

(1.9)    Wmax = dQ1*(1 – T2/T1)

We can not derive more work than Wmax from the engine without breaking the laws of thermodynamics or creating more entropy somewhere else.

2. ECONOMIC ENTROPY

Before turning to economic entropy, it is important to set boundaries of use of the concept – it will help to grasp the idea itself and to leave aside the areas where the concept should be used with certain reservations.

1.    Only the case of production of a good or the rendering of a service of acceptable quality in one closed and relatively stable market will be considered.  Then this market will be broken up in to two markets – market of producers and market of consumers.

2.    Only the movement of money between the market of producers and the market of consumers will be observed.

3.    Exotic cases such as the trade of antiques and markets with hyperinflation, as well as cases of trading securities will not be considered.

There is a fundamental asymmetry in processes connected with the flow of money: movement of money always occurs in one direction – from the market of consumers to the market of producers.  And a producer/vendor almost always sells a good or renders a service for a price higher than the cost to produce the good or service.  Does this mean that there is an economic concept that may quantitatively describe this directed and asymmetric flow of capital?  And if so what are the conditions of the most effective use of capital in order to generate the maximum possible income?

Figure 2.1​   Movement of “Economic” Heat

Let us assume no matter how strange may it sound, that there is an economic body (Figure 2.1) with high economic temperature T1econ and an economic body with low economic temperature T2econ, where T1>T2.  Thus, it is possible to assume that there is an economic heat that moves from the hot body to the cold body, changing the entropy of this closed economic system.  In that case, the resulting change of the economic entropy should be positive in order for the process to occur at all.  Then economic entropy may be expressed as:

(2.1)    dSecon = dQecon/Tecon

dQecon – economic heat taken from or given to a system with economic temperature Tecon.

What would happen if an economic engine, or a business company, (Fig 2.2) was placed in such a way that the economic heat would generate economic work?

Figure 2.2   Work of Economic Engine

First, let’s assume that analogue of the economic engine is a business company, the analogue of heat taken from the hot body is total revenue TR of the company, and the analogue of the heat given to the cold body is total cost TC to produce a product or a service.

Second, let us assume that the hot body is the market of consumers and the cold body is the market of producers.  With these assumptions, economic temperature of the market of consumers is the market equilibrium price of the unit of product/service and economic temperature of the market of producers is the cost of producing the unit of product/service.

To find a maximum net income which business company can generate:

(2.2)    Wecon max = dQ1econ – dQ2econ

(2.3)    dS1econ + dS2econ = 0

(2.4)    dQ1econ/T1econ = dQ2econ/T2

or putting it into business terms, if

NImax – net income of a business company (business engine) – analogue of the net work Wmax produced by the heat engine;

TR – total revenue of a company – analogue of the heat dQ1 taken from the hot reservoir in the heat engine;

Ce – equilibrium cost to produce a unit of a product/service and also the temperature of the market of producers – analogue of the temperature T2 of the cold reservoir in the heat engine;

Pe – equilibrium or market price of the product/service and also the economic temperature of the market of buyers – analogue of the temperature T1 of the hot reservoir in the heat engine;

TC – total cost to produce a product/service – analogue of the heat dQ2 given to the cold reservoir in the heat engine.

TR = n*Pe, where n is the quantity of product/service produced by a price \$Pe.

TC = ne*Ce, where ne, Ce, - equilibrium quantity and equilibrium cost correspondingly.

Then,

(2.5)    NImax = TR – TC

(2.6)    TR/Pe = TC/Ce

(2.7)    NImax = TR – TR*(Ce/Pe) = TR*(1 – Ce/Pe)

(2.8)    NImax = TR*(1-Ce/Pe)  or

Maximum Net Income = Total Revenues*(1 – Equilibrium Cost/Equilibrium Price)

And formulas for maximum work and maximum income look very similar:

(2.9)    Maximum Work = dQ1*(1 – T2/T1)

(2.10)  Maximum Net Income = TR*(1 – Ce/Pe)

Entropy used to measure the degree of disorder in a system.  An increase in economic entropy will result in an increase in economic disorder.  In other words, if all the Net Income of a company will be consumed by the market of producers/consumers and will not be put to work, then the maximum possible economic disorder will take place.

(2.11)  dSmax = NImax/Ce

Or general formula for Economic Entropy:

(2.12)  dSecon = d Financial Resources / Price for the resources

Economic entropy may measure the inefficiency of a market or an industry OR A COMPANY, and can indicate the waste of capital, human resources, and materials.  Perfect economic engine or perfect company works with the lowest possible equilibrium cost Ce under the conditions, earns the maximum possible profit and produces zero economic entropy.  Inefficient business will produce economic entropy proportionally to the difference between the lowest market equilibrium cost Ce and the cost C incurred by the inefficient business:

(2.13)  dS econ = n*(C – Ce)/Ce

The total entropy of an isolated system always increases in time if the system undergoes an irreversible process (dissipation of income).  All isolated economic systems tend toward disorder and increase in economic entropy.  One of the reasons for globalization of the business is that in closed domestic markets dT (which is T1 – T2) tends to decrease.  But the reason why domestic markets exist is that they are open to innovations, new ideas, new technologies, and new products, which creates new markets with higher dT and in this respect role of entrepreneurs becomes vital in creation of sustainable economic system.

Clausius stated second law of thermodynamics as: no thermodynamic process can occur whose only result is to transfer heat from a colder to a hotter body.  Such a process is only possible if work is done on the system.

For economics it may be rephrased as:  no economic process can occur whose only result is to transfer money from the market of producers (which have lower economic temperature) to the market of consumers (which have higher economic temperature).

And this is linked very closely with the other limitation: it is very unlikely that one is going to be paid for doing nothing in perfect economic world.  Of cause in real world this is not the case, but even in the perfect world one still could be paid for doing nothing, though only at the expenses of someone else.

Concept of economic entropy is based on the assumption that money is an analogue of energy.  Asymmetrical nature of money flow leads directly to the assumption that economic entropy really shall exist.  I believe that the concept of entropy as a measure of disorder is not inherently thermodynamical, but rather universal theoretical concept (like the concept of derivatives), which have applications in different fields of knowledge (statistics, thermodynamics, or theory of information).

3. STATISTICAL THERMODYNAMICS AND STATISTICAL ECONOMICS

Using statistical theory Boltzman described entropy in terms of probability:

(3.1)    S = k*LnW

where S is statistical entropy, k is Boltzman’s constant, and W is the probability of occurrence of the event.

Because future prices of inputs (raw materials, capital cost, labor cost) and future demand are uncertain, it is possible as well to consider future prices of inputs and future demand from the statistical point of view, using probability distribution.  The same uncertainty also exists about the expected level of stock prices.  All of that suggests that there might be a relationship between economic entropy and the probability of prices, cost or demand, similar to the relationship described by Boltzman for statistical thermodynamics:

(3.2)    Se = k*LnW

Where Se is statistical entropy, k is a constant, and W – the probability of occurrence of the event, i.e. probability of a certain level of costs, or prices or demand.