Towards a mathematical model of the economy that moves away from extractive business approaches

If business as a discipline is to develop away from extractive practices we need to develop a mathematical language to help economists and policy makers model alternative approaches. Modelling – using both standard business calculations and simulation tools like those developed by Steve Keen – can help decision making at the level of the individual firm and policy level. We propose that adaptations of the Cobb-Douglas Equation can be used to help those doing macro-economic modelling of the sustainable economy. We hope this article contributes to knowledge.

The view of the firm

A generalized view of the firm is one where energy and materials come in. Using infrastructure (capital) and labor, a product and or service is produced that is sold, the money from the sale covering the purchase of labor and materials as well as compensating those who put the money up for the capital infrastructure.

A simplified approach using the Cobb-Douglas equation could be:

\ Y = L\ ^\alpha\cdotp K\ ^b

Where Y = output and L= Labour and K = capital

A certain amount of waste from material use is generated that goes to waste handling services or in the worst case, goes uncontrolled into the eco-system or as landfill.

An energy- and material aware equation would be

\ Y = Labour\cdotp Capital\cdotp Energy\cdotp Materials ;\cdotp Waste


Extractive business practices are ones that either rely on inputs of finite materials in their operations or ones that deplete capital in the process. Waste can be seen as a depletion of capital or measured separately. In this section we see waste as a separate variable.

Where o = original and r = remaining after the cycle (year, decade etc) of the business process:

And extraction = E

\ E = \sum Capital_o-Capital_r\cdotp\sum Materials_o-Material_r


Regenerative business practice are ones that leave capital intact or in better functionality and which do not draw down resources.

In this case we can represent it thus:

\sum Capital_o-Capital_r =0 

\sum Materials_o-Material_r =0 

For the firm to go from extractive to regenerative, the energy sources need to be switched to renewable, and the capital infrastructure needs to be adapted to use renewable energy and renewable materials too.

To better represent this we take the following:

KRenewable  (KR) to mean the Capital infrastructure – machines and other operating equipment – that operates to handle materials is a circular way.
For example, machines that use recycled materials, or produce packaging that can be recycled, can be designated KR.

The opposite is KLinear (KL) – where machinery uses extracted material and produces waste that cannot be recycled.

ERenewable represents the energy used in production from renewable sources (ER). The opposite is Non-renewable energy (EN)

M reCycled represents the material incorporated into the product from recycled sources and goes to back to reuse (MC). Non-recycled material is designated (ML) where L represents the linear material use.

MWaste  (MW) represents the amount of waste produced in the output process

Linear, non-renewable output can be represented thus:

\ Y_Linear = L\cdotp K_L \cdotp  E_N\cdotp M_L\cdotp M_W

Circular, renewable energy-driven production can be represented as

\ Y_Circular = L\cdotp  K_C \cdotp  E_R \cdotp M_C

Where (MW)= zero

In the transitional phase firms will probably have a combined production

\ Y = L\cdotp ( K_L+K_C )\cdotp(E_R +E_N)\cdotp(M_L+M_C)\cdotp(M_W)

A measure of circularity can be created from calculating the fraction of renewable – running infrastructure, the fraction of renewable energy and the fraction of recycled materials.

\ Y = L\cdotp \frac{K_C}{( K_L+K_C )}\cdotp\frac{E_R}{( E_R+E_N )}\cdotp\frac{M_C}{( M_L+M_C )}

To measure the percentage invested in circular economy the following may be  useful guideline

\ Percentage Circular = 100\cdotp \frac{K_C}{( K_L+K_C )}

Indeed a calculation can be made of the percentage of waste for the total materials employed, whether they are from linear or circular sources

\ Percentage Waste = 100\cdotp \frac{M_W}{( M_L+M_C )}


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