Concordian Economics, Part 3: Mathematics

Written by Carmine Gorga, The Somist Institute

The essential characteristic of the mathematics of Concordian economics is the break up of the saving-investment nexus, a well acknowledged “quagmire” in mainstream economics. Through a complex logico-mathematical operation (see Gorga, 2002 and 2009a, pp. 69-137) in Concordian economics saving is replaced with hoarding and the relationship between hoarding and investment is one of complementarity.

 This relationship is best represented with a classical Lorenz diagram, as in the following figure:

This was explained elsewhere (Gorga, 2010a):

More hoarding clearly equals less investment—hence less economic growth. Again, the relationship between hoarding and poverty is of immediate comprehension: more is accumulated by the few, less is available to the many.

The explanation of why when there is more hoarding there is more inflation resides in this simple fact. When one buys property and keeps it in a passive state, the money used to buy that wealth remains in circulation and creates an imbalance with the amount of wealth that is for sale in the market. Hence, prices rise. That is the seed of inflation.

The difficult nature of hoarding is revealed by this simple fact. At times, as at present, when financial and industrial corporations hoard cash, hoarding performs a deflationary function. This is a peculiar function that, apart from few aberrant cases (and the general case of economies with insignificant financial institutions), occurs only at the bottom of the crisis and is quantitatively miniscule when put in relation to the value of the economic system over the entire business cycle.

A Set of Nonlinear Hypotheses

As cursorily noted above, once attention is focused on the dynamics of the economic system, one observes not only solids moving in space but also the inside of a torus, customarily a section of a torus, or a Poincaré section. The torus that can be expected in economics is likely to present many bottlenecks, due not only to the irregularities in the planes of production and consumption but especially to the shape of the plane of distribution which, as seen in Slide 5 (Part 2), is that of a rhombus.

As suggested, it is likely that stroboscopic analysis will yield lines that, at least over certain portions of the phase diagram, form any one of the typical Lorenz attractors (see Slides 6 and 7, Part 2). But perhaps the analysis does not necessarily have to terminate in any one of the typical “chaotic” representations of the reality. It seems more likely that the economic process ultimately takes the shape of a Mandelbrot set (see Gleick, 1987, p. 226).

The construction of three intercompenetrating spheres outlined in the discussion of Slide 2 (Part 2) and the Mandelbrot set have one essential characteristic in common: each point of any sphere (or, in disequilibrium, each point of the smallest sphere) touches at least one point of each of the other two spheres. James Gleick (1987, p. 220) explains:

“When a pie is cut into three slices, they meet at a single point, and the boundaries between any two slices are simple. But many processes of abstract mathematics and real-world physics turn out to create boundaries that are almost unimaginably complex… The boundary (in the Mandelbrot set, for instance) has the peculiar property that every point on it borders all three regions.”

Mandelbrot (1983, Ch. 37) also provides convincing evidence that trends in economics follow a constant pattern. This pattern might eventually be proved to be a fractal set that can be simply described as follows: up to a certain point, actions and reactions reinforce each other; then both begin to fade away; a rather abrupt break in the trend generally follows; after some recovery, there is a period of relative stasis; and then the cycle starts anew. Correspondingly, a first scanning of the data suggests that the shape of the originating curve (the “initiator”), an apparent constant in economics, is as follows:

The “generator” has to be determined through analysis of historical data from combined price-quantity volumes (see Mandelbrot, 1983, pp. 31, 34b, and 119b). Some curves are likely to be discontinuous, and for certain stages of the business cycle the shape of the originating curve is likely to be turned upside down. Certain “parts” of the resulting teragon might have different topological and fractal dimensions.

Articles in this series:

• Concordian Economics, Part 1: Intoduction
• Concordian Economics, Part 2:  Geometric Representation
• Concordian Economics, Part 3:  Mathematics (this article)
• Concordian Economics, Part 4:  The Business Cycle (to be posted)
• Concordian Economics, Part 5:   Analysis and the Black Box of Saving (to be posted)

Appendix

Symbols, Meanings, and Definitions

Symbols in Concordian economics
ERs = energy-units
MUs = matter-units
VUs = value-units
H = hoarding
P = production of all real goods and services
D = distribution of ownership rights over real and monetary wealth
CG = consumer goods
KG = capital goods
GH = goods hoarded
OCG = ownership of consumer goods
OKG = ownership of capital goods
OGH = ownership of goods hoarded
EP = economic process
PED = principle of effective demand
npW = nonproductive wealth
pW = productive wealth
MY = monetary income
E = expenditure
Eh = expenditure to purchase goods to be hoarded
Ek = expenditure to purchase capital goods
Eg = expenditure to purchase consumer goods
If = expenditure on fixed capital
Iw = expenditure on working capital
Ck = expenditure on capital goods
Cg = expenditure on consumer goods
p = rate of change in total production
d = rate of change in the values of distribution of ownership rights
c = rate of change in total expenditure
r = the rate of interest
d = existing distribution of values of ownership rights
mec = marginal efficiency of capital
YL = labor income
rW = income from ownership of real and monetary wealth (capital income)
R = rent from land and natural resources
w = value of real wealth
m = value of monetary wealth.
***
Meanings in Concordian economics and mainstream economics
Y = income produced and consumed in mainstream economics
Y = income produced, distributed, and consumed in Concordian economics
C = consumption or expenditure to buy consumer goods in mainstream economics
C = consumption or any type of expenditure in Concordian economics
S = saving means literally 100,000 things in mainstream economics
S = saving means financial savings in Concordian economics
I = investment is equal to saving in mainstream economics
I = investment is all productive wealth in Concordian economics
***
Definitions in Concordian economics and mainstream economics

Note:  This series has been adapted from Beyond Keynes …. Toward Concordian Econometrics, International Journal of Applied Economics and Econometrics, Part III of the Special Issue on J.M. Keynes, Vol. 20, No. 1, Jan-March 2012, pp. 248-277.  The references for this work are listed at the end of that paper.