The Carnot Process of Economic Growth and Wealth Distribution

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1 Jürgen Mimkes, Physics Department, Paderborn University, Germany Yuji Aruka, Faculty of Commerce, Chuo University, Japan Coworkers: Mario Hillebrand, Christian Denk, Thorsten Fründ, Stefan Kallerhoff

2 Jürgen Mimkes, Physics Department, Paderborn University, Germany Yuji Aruka, Faculty of Commerce, Chuo University, Japan First and Second Laws of Macro Economics in Differential Forms Carnot Cycles of Economic Growth and Wealth Distribution Coworkers: Mario Hillebrand, Christian Denk, Thorsten Fründ, Stefan Kallerhoff

3 First Law of Macro Economics

4 Differential forms Exact Differential Forms: df ( x, y) = f x dx + f y dy with f y x = f x y

5 Differential forms Exact Differential Forms: df ( x, y) = f x dx + f y dy with f y x = f x y Example 1 f( x, y) = x 3 y 5 d f (x, y) = 3 x y 5 d x + 5 x 3 y 4 d y with 15 x y 4 = 15 x y 4 (exact)

6 Differential forms Exact Differential Forms: The integral of an exact differential form d f depends on the integral limits and does not depend on the path of the integral (high school math). The closed integral is zero. a df b df = df + df = df df = 0 b a a b b a b a

7 Differential forms Not Exact Differential Forms: δg ( x, y) = adx + bdy with a y b x

8 Differential forms Not Exact Differential Forms: δg ( x, y) = adx + bdy with a y b x Example δ g ( x, y) = d f( x, y) / y = ( 3 x y 5 d x + 5 x 3 y 4 d y )/ y δ g ( x, y) = 3 x y 4 d x + 5 x 3 y 3 d y with 1 x y 3 15 x y 3 (not exact)

9 Differential forms Not Exact Differential Forms: The integral of a not exact differential form δ g(x, y) depends on the path of the integral. The closed integral is not zero. Y a δg b b a b b = Y C δg ( x, y) = δg + δg1 = δg δg1 a b a a 0 C

10 Differential forms Not Exact Differential Forms: The integral of a not exact differential form δ g(x, y) depends on the path of the integral. The closed integral is not zero.

11 Differential forms Not Exact Differential Forms: The integral of a not exact differential form δ g(x, y) depends on the path of the integral. The closed integral is not zero. First Law of Thermodynamics Heat δ Q(T, p) is a not exact differential, the integral depends on the path (e.g. isothermal), the closed integral is not zero.

12 Differential forms Not Exact Differential Forms: The integral of a not exact differential form δ g(x, y) depends on the path of the integral. The closed integral is not zero. First Law of Thermodynamics Heat δ Q(T, p) is a not exact differential, the integral depends on the path (e.g. isothermal), the closed integral is not zero. First Law of Macro Economics Production δ q(k, L) is a two dim. not exact differential and depends on the specific production process. The closed cycle of production is not zero.

13 Differential forms First Law of Macro Economics b a δq = δq + q = q δ 1 δ δq1 a b b a b a = Y C = q Production leads to income (Y), consumption (C) and surplus ( q). Production δ q is a not exact differential: a) the production function q does not exist in general! or the production function may not be calculated ex ante! b) the integral of δq depends on the path of integration, or the production function q depends on the production process! Ex post, when the production process is known, q may be calculated!

14 Second Law of Macro Economics

15 Differential forms Not Exact Differential Forms: A not exact form δ g may be made exact by an integrating factor y. The exact form is given by d f = y δ g. Example δ g ( x, y) = d f( x, y) / y

16 Differential forms Not Exact Differential Forms: A not exact form δ g may be made exact by an integrating factor y. The exact form is defined by d f = y δ g. Second Law of Thermodynamics The integrating factor of the non exact form of heat δ Q is the mean energy per particle or temperature, T = E / N. The exact form leads to entropy defined by d S = δ Q / T.

17 Differential forms Not Exact Differential Forms: A not exact form δ g may be made exact by an integrating factor y. The exact form is defined by d f = y δ g. Second Law of Thermodynamics The integrating factor of the non exact form of heat δ Q is the mean energy per particle or temperature, T = E / N. The exact form leads to entropy defined by d S = δ Q / T. Second Law of Macro Economics The integrating factor of the non exact form of production δ q is a mean price level or standard of living, T = K / N. The exact form leads to entropy defined by d S = δ q / T.

18 Differential forms Second Law of Macro Economics: ds = δq / T The integrating factor T is the price level of a market or the standard of living of a society (GDP per capita) The differential form d S is exact, or the entropy function S may be calculated ex ante! 3. Entropy is related to the number of possibilities (P) of production, S = ln P

19 Application Production

20 Carnot cycle First and Second Law of Economics: Carnot cycle δ q = TdS = Y T C T = T S 8 mean energy T Y C Entropy S Y T = K + T S C T = K + T 1 S T : standard of living, price level Y T : income, price at const. T C T : costs, consumption at const. T K : capital

21 Carnot : Y T = K + T S Production (Y) is ordering = reduction of entropy ( S = ln P) Production: + + Brain work: d+i+c+n+o+o+p+r+t+u production

22 Carnot : Y T = K + T S Automobile workers: Production Y T is ordering (reducing the entropy) of many parts to build the car exactly according to the plans. Automobile engineers: Production Y T is ordering (reducing the entropy) of many ideas to create a new plan for a car.

23 Production (Y) is ordering of components in the factory Production (Y) ist entropy production in nature Kubic pn-gan LED Laser diode Au (semitransparent) p - GaN:Mg n + - GaN:Si n - GaAs

24 Production function Production functio W 0,4 0,35 0,3 0,5 0, 0,15 0,1 0,05 0 Production function d W = - T d S 0 0,05 0,1 0,15 0, 0,5 0,3 Rel. number z = Nk / N of elements Entropy production function Y / T = x ln (x) y ln (y) z ln (z) or Y / T = ln (x x y y z z ) with x + y + z = 1 This replaces the Cobb Douglas production function Y / T = A (x α y β z γ ) Production factors x, y, z with α + β + γ = 1 α, β, γ =?

25 Production costs: Material (K) and labor (T S) C T = K + T S Labor costs (w) w = T S The actual work ( S for building a car) is the same in different countries, but the wages are not. They depend on the mean standard of living T.

26 Application The Carnot Process

27 Carnot cycle of a motor, the fuel is oil Second Law of Thermodynamics: Carnot cycle 8 7 Y T T 1 mean energy T W = Q C T T Entropy S

28 Carnot cycle of industry, the fuel is oil Second Law of Economics: Business cycles Y In = 10 mean capital T T 5 T T T Y q C Order, Entropy S T 1 C In = 100

Carnot cycle of industry, energy consumption and GDP Second Law of Economics: 40 35 30 5 0 15 10 5 0 1 004 GDP per Capita

29 Carnot cycle of industry, energy consumption and GDP Second Law of Economics: GDP per Capita in k US $ USA Europa Asien World energy consumption / capita World GDP / capita

Carnot cycle of industry and economic growth Second Law of Economics: Carnot cycle 8 7 Y T T 1 mean energy T 6

30 Carnot cycle of industry and economic growth Second Law of Economics: Carnot cycle 8 7 Y T T 1 mean energy T W = Q C T T Entr opy S Carnot cycle of industry, farms, markets, banks and foreign trade

31 Carnot cycle of trade between India and Germany Business cycles Income distribution mean capital, standard of living T Y Q C Entropy S T(Germany) US$ T(India).900 US$ η = (T T 1 )/T 1 = 9,5 : 1

32 Carnot cycle of industry and economic growth Two interdependent economic systems Entwicklung des BIP von Sechs Länder im Vergleich Cap ital g ro w th Y time t a = 0,15 a = 0,15 BIP per capita in US $ (1985) D US UK CH J VRC a = 0,15 US - Japan, US - D

33 Carnot cycle of industry and economic growth Two interdependent countries US-JP GDP / person Growth of economic levels Y,5 1,5 1 0, GNP/person / US $ time t (quarters) a = 0,75 a = 0,75 US Japan a = 0,7 US - Japan,

34 Carnot cycle of industry and economic growth Two interdependent countries Standard of living in East and West Germany Growth of economic levels Y 1,5 1 0, time t GDP / p erson / a in DM Year West Est a = 1,35 D West - Ost,

35 Carnot cycle of industry and economic growth Interdependent economic systems GDP per capita time series Economic level Y(t) GDP per capita (1995 US$) time t tim e USA t (1997 to Argentina 001) a = - 0,5 US - Argentinien,

36 Carnot cycle of a heat pump Second Law of Thermodynamics: Carnot cycle 8 7 Y T T 1 mean energy T W = Q C T (warm house) T 1 (cold river) Entr opy S

37 Carnot cycle of industry and economic growth a: Profitanteil der Haushalte (Konsum C) Interdependent economic systems Interdependent economic systems Economic level Y(t) time t a =0 a =0 a = 0,10 a = 0,10 a = 0,5 a = 0,5 a = 0,50 a = 0,50 Economic level Y(t) time t a = 0 a = 0 a = 0,5 a = 0,5 a = 0,75 a = 0,75 a = 1 a = 1 0 a 0,5 0,5 a 1

38 Carnot cycle of industry and economic growth Economic Principles on Pacific Islands

39 Distribution of Wealth

40 Creation of T by work in the Carnot cycle of industry Business cycles mean capital, standard of living T Y Q C Gibbs Pareto Entropy S Income distribution

41 Creation of T by work in the Carnot cycle of industry Business cycles World GDP distribution mean capital, standard of living T Y Q C GDP in US$ / cap ita Number in millions Entropy S World income distribution

A Thermodynamic Formulation of Economics Juergen Mimkes

1 1 A Thermodynamic Formulation of Economics Juergen Mimkes The thermodynamic formulation of economics is based on the laws of calculus. Differential forms in two dimensions are generally not exact forms