Recently it struck me that it might be useful to have a model of how the supply of final goods impacts final demand in an in-kind economy that doesn't regulate demand using means of payment. In other words, how does one balance supply and demand when the former induces the latter, without resorting to price shenanigans ála Cockshott & Cottrell? In this post I will sketch one possibility.

The scalar case

I will introduce the basic idea for the model with the simplified, scalar case. Suppose the demand $d_i$ for some good $i$ is not constant, but depends on supply via some complicated function $d_i(s_i)$. For some goods, for example potable water, $d_i$ might be constant. For other goods, say cookies, $d_i$ might be exactly $s_i$. Let us generalize both of these cases using a linear model with a constant offset, a so-called affine function (thanks Dave for pointing this out):

$d_i(s_i)=b_i+e_i{s}_i$

Here $b_i$ is the "base demand" for good $i$, while $e_i\in [-\mathrm{\infty },+1)$ represents its "demand elasticity" with respect to supply. We have three cases:

• $e_i>0$: supply of good $i$ induces demand in excess of its base demand, for example cookies
• $e_i=0$: the demand for good $i$ is independent of its supply, for example water
• $e_i<0$: the supply of good $i$ reduces its demand, for which I can't think of a good example

We would like supply to always be greater than or equal to demand, which gives the following inequality:

$0\le s_i-d_i(s_i)=s_i-(b_i+e_i{s}_i)=(1-e_i)s_i-b_i$

Hence:

$(1-e_i)s_i\ge b_i$

Solving for $s_i$ we get:

$s_i\ge \frac{b_i}{1-e_i}$

When $e_i=0$ this is no different from the case we are familiar with, where we have some demand vector $d$ that is to be fulfilled. $e_i<0$ also isn't a huge problem - the supply merely drops the more negative $e_i$ goes. But for $e_i>0$ it is easy to see that the larger it is, the more the base demand gets amplified. In the extreme, when $e_i=1$, no amount of supply will ever satisfy demand. Consumers will eat however many cookies are produced!

The multidimensional case

The model outlined in the previous section can easily be generalized to multiple dimensions. Let demand elasticity be modelled by the square matrix $E$, base demand the vector $b$ and supply the vector $s$:

$0\le s-d(s)=s-(b+Es)=(1-E)s-b$

$(1-E)s\ge b$

Again, solving for $s$ we get:

$s\ge (I-E{)}^{-1}b$

What is the use of this kind of model? Unlike the scalar model, $E$ can have off-diagonal elements. This means we can model cases like these:

• an increase in the supply of wine will induce demand for corkscrews: $e_{\text{corkscrews},\text{wine}}>0$
• wine will also induce demand for hospital services: $e_{\text{hospitals},\text{wine}}>0$
• medicine will decrease demand for hospital services: $e_{\text{hospitals},\text{medicine}}<0$
• hospital services increase demand for medicine: $e_{\text{medicine},\text{hospitals}}>0$

Notice how the last two cases affect each other. This suggests that we'll want the spectral radius of $E$ to be less than one, much like with Leontief inverses.

We can also put temporal information in $E$. For example, supplying public transit may be expected to reduce demand for cars over time. On the other hand, building roads tends to increase the demand for cars over time. It also induces demand for road maintenance in the future.

Combining the model with joint production

Let's repeat the model of joint production by David Zachariah:

$(B-A)x\ge d$

Here $B$ represents the outputs of each production process, $A$ their inputs and $d$ final demand. Both $A$ and $B$ can be rectangular. The left hand side corresponds to supply. If we let $s=(B-A)x$ then we can combine the equations like so:

$(B-A)x\ge (I-E{)}^{-1}b$

Multiplying both sides by $(I-E)$ we get:

$(I-E)(B-A)x\ge b$

Affine production

Having presented an affine model for demand, an affine model for supply (production) is the obvious next step:

$s=(B-A)x+(\beta -\alpha )$

where $B$ and $A$ are as before, while $\beta$ and $\alpha$ represent "base outputs" and "base inputs" in production. Base outputs are those things that are supplied no matter what, and base inputs are those things that are demanded in production no matter what. Think of $\alpha$ as overhead and $\beta$ as "happy accidents" of overhead. Note that $\beta$ is not byproducts in the usual sense. Such byproducts are already handled by $B$. For example, when producing parts in a stamping process, the amount of scrap metal varies roughly linearly with the amount of output produced. $\beta$ can instead be viewed as byproducts of overhead, for example scrap paper produced by accounting, or scrap parts produced by maintenance.

If we again combine this model with the affine demand model, we arrive at the following:

$(B-A)x+(\beta -\alpha )\ge (I-E{)}^{-1}b$

After left multiplying both sides by $(I-E)$ and rearranging, we get:

$(I-E)(B-A)x\ge b+\alpha -\beta -E\alpha +E\beta$

Each of the five terms on the right hand side deserve some elaboration.

• $b$ is the usual base demand
• $+\alpha$ and $-\beta$ repesent how demand is directly affected by overhead and byproducts thereof. Maintenance needs certain inputs, but it also produces certain outputs
• $+E\beta$ represents how, while byproducts from overhead may reduce the need to manufacture certain products (say scrap), they also induce demand for other products in turn (chemicals used in refining scrap)
• By symmetry, $-E\alpha$ represents the opposite. If wine is part of the overhead of some process, then less wine is available to consumers, and so the demand for corkscrews goes down

Note also that it is possible for some workplaces to have no $A$, $B$ or $x$, but only $\alpha$ and $\beta$. In other words that $A$ and $B$ are $m\times 0$ matrices, that there are no "knobs" for the planning system to turn, or that supply and demand are treated as given:

$s=\beta -\alpha$

This is likely to be the case in the reproductive sector, such as hospitals and schools. An example of something that mainly produces $\beta$ is recycling stations. The opposite, something that mostly consumes $\alpha$, include distribution centers. Distribution, in the Marxian sense, is all about which final consumer gets what. Distribution has no "output". When a distribution center, or "shop", has so much stock that it starts to depreciate, that is a failure in distribution, a failure to predict final demand. The products thus output, products not distributed, are simply waste. Waste embodies no social labour. Therefore the concrete labour spent to produce them was also waste. Hence the importance of accurately modelling demand, because not doing so results in waste, which ultimately means working longer than necessary, putting more strain on the environment than necessary, and so on.

Discussion

Reality is obviously not a simple affine function, so why bother with this kind of modelling at all? The reason is due to something called the internal model principle, elaborated on by Bruce A. Francis and Walter M. Wonham in a series of papers in the mid 1970's^[1][2], and Wonham again in 2018^[3]. In short the principle states that in order to effect good control over some system, we must also model that system. In addition, if the system under control is only weakly nonlinear, if it is sufficiently smooth, then a linear model is enough.

While we can effect some kind of regulation with an implicit model, as is done with PID controllers, such a regulator can never be a good regulator, because good regulation requires modelling. This is a general principle that applies to all regulators that rely only on error control (feedback), including the market. The market consists of islands of planning connected by the value field (exchange), where companies do their very best to prevent other companies from learning their internal dynamics, from modelling the competition. In other words, the fact that trade secrets are upheld must result in poor regulation. Similar problems existed in the Soviet planning system, since it was a distributed control system lacking any coherent "god model".

Modelling is also a large part of human consciousness. Proprioception, the sense of knowing where one's limbs are, is not directly sensed, but is a model constructed by the brain based on inputs from various sensory organs that tell the brain things like the amount of tension in each muscle, whether we're touching something, and so on. If you close your eyes and hold your hand in front of your face, there is no sensory organ telling you directly that a hand is in front of your face. It's all thanks to modelling. Without such modelling mundane tasks like walking would look incredibly amusing.

An aside on climate

The global average surface temperature on Earth plays a huge part in production, and it depends on the CO₂ concentration in the atmosphere in a non-linear way. How well would an affine model fit with this non-linear function for the temperature range that is of relevance to climate change reversal?

Earth is in essence a spacecraft, which can only exchange heat with the universe by means of thermal radiation. By Stefan–Boltzmann's law the absolute surface temperature depends on the average absorptivity $\alpha$ and emissivity $ϵ$ of the Earth as:

$T\propto \sqrt[4]{\frac{\alpha }{ϵ}}$

Carbon dioxide lowers $ϵ$, which increases $T$, hence global warming. Let's dig deeper.

From Wikipedia's article on Earth and the solar constant article we will take the following numbers:

• average surface temperature ($T$): +15°C or 288.15 Kelvin
• average radius ($R$): 6,371.0 km
• surface area ($A_2$): 510,072,000 km²
• solar constant ($S$): 1361 W/m²

The sun-facing area $A_1$ of Earth is approximately $\pi R^2$. The total incident radiation is therefore:

$P_{\text{in}}=\alpha S{A}_1$

The amount of radiated power is:

$P_{\text{out}}=ϵ\sigma A_2{T}^4$

Assuming thermal equilibrium ($P_{\text{in}}=P_{\text{out}}$), we can solve for $\alpha /ϵ$:

$\alpha S{A}_1=ϵ\sigma A_2{T}^4$

$\frac{\alpha }{ϵ}=\frac{\sigma A_2{T}^4}{S{A}_1}\approx 1.1489$

We can also solve for T in the above:

$T=\sqrt[4]{\frac{\alpha S{A}_1}{ϵ\sigma A_2}}$

Introduce $r=\alpha /ϵ$ and $k=S{A}_1/\sigma A_2$ for convenience:

$T=\sqrt[4]{rk}$

$r=\frac{T^4}{k}$

Derive $T$ w.r.t $r$:

$\frac{dT}{dr}=\frac{k}{4(rk{)}^{3/4}}\approx 62.704\text{K}$

Compute the intersect $b$:

$b=T-r\frac{dT}{dr}\approx 216.11\text{K}$

This gives the following affine model:

$T(r)\approx b+r\frac{dT}{dr}\approx 216.11+62.704r$

How well does this function correspond to the proper non-linear one? Let's say we want to reduce Earth's temperature by 1.5°C. Solving directly for a new $r_2$:

$r_2=\frac{(T-1.5{)}^4}{k}\approx 1.1251$

If instead we solve the affine model, we get:

$r_2\approx \frac{T-1.5-b}{dT/dr}\approx 1.1249$

The difference between the two models is only 0.02%. We can therefore conclude that for this specific problem, an affine model works very well! Coming up with affine models parametrized on $\alpha$ and $ϵ$ is left as an exercise for the reader.

What can we do with the above? For one thing we know that we must reduce $r$ by 2.1%. This can be done by a combination of painting the Earth white (lowering $\alpha$) and taking CO₂ out of the atmosphere (increasing $ϵ$). But how do we go about it? The answer is planning. There is no way stretching white sheets over the entirety of the Sahara will ever be profitable. Neither is it the case that fossil fuel companies would ever say no to the ground rent resulting from petroleum extraction. But through globe-spanning planning, through a collective "god model" of the Sraffian basic sector, the necessary changes in $\alpha$ and $ϵ$ can be effected.

Sources

[1] B. A. Francis and W. M. Wonham, The Internal Model Principle for Linear Multivariable Regulators, January 1975, Applied Mathematics & Optimization, vol 2, pages 170-194, doi:10.1016/0005-1098(76)90006-6

[2] B. A. Francis and W. M. Wonham, The Internal Model Principle of Control Theory, September 1976, Automatica, vol 12, issue 5, pages 457-465, doi:10.1016/0005-1098(76)90006-6

[3] W. M. Wonham, The Internal Model Principle of Control Theory, June 2018. THE INTERNAL MODEL PRINCIPLE OF CONTROL THEORY (fetched 2025-03-06)