Resources on thermodynamics and economics

• Economy and Thermodynamics by Borisas Cimbleris, Emeritus Professor, Federal University of Minas Gerais
  
• A sunshine limit to growth by William Rees
  
• Integrating Economics, Ecology and Thermodynamics, a book by Matthias Ruth
  
• Ecological economics encyclopedia by the International Society for Ecological Economics
  
• Google search
  

A simple model of competition and economy of scale

In my earlier post Challenging the Law of Supply and Demand I assumed that each commodity is associated to a demand function d(p) expressing the average rate at which it will be sold at the price ''p''. The demand function being non-increasing, it can be locally inverted to give a supply function p(s) which is the price that the commodity will command on the market if it is supplied at the rate s. If, further, c(s) is the cost of bringing the commodity to market at the rate s, the profit derived from selling the commodity at the rate s will be the revenue sp(s) minus the cost, or

π(s)=sp(s)-c(s).

The optimal supply rate s₀ satisfied π'(s)=0.

Now I want to study the effect of competition on supply and prices. For simplicity, I will assume that there are two different suppliers of essentially the same product (hence, a commodity) with supply rates s and S and cost functions c(s) and C(S). Their respective profits are

Π(S;s)=Sp(S+s)-C(S)
π(s;S)=sp(s+S)-c(s)

I use the notation π(s,S) to indicate that the lower-case supplier cannot directly affect the upper-case rate of supply, and so S acts as an external parameter on π(s).


The questions one might want to answer about competition are:

• How does the presence of competition affect the behaviour of a supplier?
  
• How does the presence of competition affect the profit of the whole industry?
  
• How does the presence of competition affect the total supply or, equivalently, the market price of the commodity?
  

I will leave a detailed analysis of these questions for a later time. However, thinking about the second question leads to the an analysis of the cost functions.

Economy of scale
The combined profit of the whole industry is (s+S)p(s+S)-C(S)-c(s).
We can use this to show that there is economy of scale, that is, that the cost function must satisfy c(S+s)&lec(s)+c(S). A function satisfying this condition is called sublinear. The reason for this is that, if the cost function for a supplier were not sublinear, the supplier could divide into two smaller suppliers who together could produce the same as the larger supplier with less overall cost.
There is an important hidden assumption in this proof of economy of scale: that one supplier's production cost is independent of the production of the other suppliers. This is plainly not true when we are talking about exploiting natural resources: natural resources are extracted first from the sources that are easier (and cheaper) to exploit, and when these are exhausted production moves to harder-to-exploit (and more expensive) sources. This is presumably true of every sector of the economy: there is economy of scale only when the overall production is smaller than a certain threshold.

Why economic growth?

For a while I have been wondering why so much emphasis is placed on GDP growth above almost all other macroeconomic considerations. The conventional wisdom is that, unless the GDP grows by about 3% each year, the economy is stagnating. The question is, it possible to measure economic success in terms other than economic growth? Is it possible to organize the economy in such a way that people's standard of living can be kept up without having to increase the GDP? Keep in mind that measuring the GDP in units of "money" is relatively meaningless. What matters is the amount of labour, resources, and capital available at any given time, and if that amount does not increase exponentially there seems to be a problem mainitaining the standard of living. Why?

A partial answer came from a friend of my girlfriend who runs a company making about $1,000,000 of a year. One night we got to talking about his company and after a while I asked him why so much emphasis was placed on growth, and whether it would be possible to run a successful company without growing it. He replied that was "dangerous" and said he "did not know how to run a company without growing it". I say it is a partial answer because it is not a real explanation, but a statement that we really don't know how to manage an economy in any other way than growing it exponentially, like a tumor.

Then, in Adam Smith's Wealth of Nations I found an explanation of the relation between economic growth and general welfare (although not phrased in those terms). Smith starts with a bit on living wages:

But though in disputes with their workmen, masters must generally have the advantage, there is however a certain rate below which it seems impossible to reduce, for any considerable time, the ordinary wages even of the lowest species of labour.
A man must always live by his work, and his wages must at least be sufficient to maintain him. They must even upon most occasion be somewhat more; otherwise it would be impossible for him to bring up a family, and the race of such workmen could not last beyond the first generation.

Interestingly, Smith assumes that children will be employed in a similar occupation to their parents. His living wage, to use modern parlance, must be really modest, because today's "working poor" still manage to survive and produce families. Then, a little bit on labour relations...

When in any country the demand for those who live by wages, labourers, journeymen, servants of every kind, in continually increasing; when every year furnishes employment for a greater number than had been employed the year before, the workmen have no occasion to combine in order to raise their wages. The scarcity of labour occasions a competition among masters, who bid against one another, in order to get workmen, and thus voluntarily break through the natural combination of masters not to raise wages.

Now for the relation between minimum wages and economic growth:

The demand for those who live by wages, therefore, necessarily increases with the increase of the revenue and stock of every country, and cannot possibly increase without it. The increase of revenue and stocks is the increase of national wealth. The demand for those who live by wages, therefore, naturally increases with the increase of national wealth, and cannot possibly increase without it.
It is not the actual greatness of national wealth, but its continual increase, which occasions a rise in the wages of labour. It is not, accordingly, in the richest countries, but in the most thriving, or in those which are growing rich the fastest, that the wages of labour are highest. England is certainly, in the present times, a much richer country than any part of North America. The wages of labour, however, are much higher in North America than in any part of England.
[...]
But though North America is not yet as rich as England, it is much more thriving, and advancing with much greater rapidity to the further acquisition of riches.
[...]
Though the wealth of a country should be very great, yet it has been long stationary, we must not expect to find the wages of labour very high in it. The funds destined for the payment of wages, the revenue and stock of its inhabitants, may be of the greatest extent; but if they have continued for several centuries of the same, or very nearly of the same extent, the number of labourers employed every day could easily supply, and even more than supply, the number wanted the following year. There could seldom be any scarcity of hands, nor could the masters be obliged to bid against one another in order to get them. The hands, on the contrary, would, in this case, naturally multiply beyond their employment. There would be a constant scarcity of employment, and the labourers would be obliged to bid against one another in order to get it. If in such a country the wages of labour had ever been more than sufficient to maintain the labourer, and to enable him to bring up a family, the competition of the labourers and interest of the masters would soon reduce them to this lowest rate which is consistent with common humanity. China has been long one of the richest, that is, one of the most fertile, best cultivated, most industrious, and most populous countries in the world. It seems however, to have been long stationary. [...] The accounts of all travellers, inconsistent in many other respects, agree in the low wages of labour, and the difficulty which a labourer finds in bringing up a family in China.
[...]
China, however, though it may perhaps stand still, does not seem to go backwards.
[...]
But it would be otherwise in a country where the funds destined for the maintenance of labour were sensibly decaying. Every year the demand for servants and labourers would, in all the different classes of employments, be less than it had been the year before. Many who had been bred in the superior classes, not being able to find employment in their own business, would be glad to seek it in the lowest. The lowest class being not only overstocked with its own workmen, but with the overflowing of all the other classes, the competition for employment would be so great in it, as to reduce the wages of labour to the most miserable and scanty subsistence of the labourer. Many would not be able to find employment even upon these hard terms, but would either starve, or be driven to seek subsistence either by begging, or by the perpetrating perhaps of the greatest enormities. Want, famine and mortality would immediately prevail in that class, and from thence extend themselves to all the superior classes, till the number of inhabitants in the country was reduced to what could easily be maintained by the revenue and stock which remained in it, and which had escaped either the tyranny or calamity which had destroyed the rest. This perhaps is nearly the present state of Bengal, and of some of the other English settlements in the East Indies. In a fertile country which had before been much depopulated, where subsistence, consequently, should not be very difficult, and where, notwithstanding three or four hundred thousand people die of hunger in one year, we may be assured that the funds destined for the maintenance of the labouring poor are fast decaying.

Smith ends with the following indictment of the East India Company:

The difference between the genius of the British constitution which protects and governs North America, and that of the mercantile company which oppresses and domineers in the East Indies, cannot perhaps be better illustrated than by the different state of those countries.

Why is this important?

In modern language, I think that Adam Smith is simply saying that wages are in a direct correlation to GDP per capita, which should not be surprising. This explains the emphasis on growing the GDP. Still, what if GDP growth is not sustainable for one reason or another?

There is, in fact, every indication that our current economic model is unsustainable. Resource depletion and environmental damage are almost irrepairable. Exponential GDP growth can only go on forever if the "Production Possibility Frontier" expands exponentially. The problem is that the expansion of our economy is fuelled by nonrenewable resources, and so at some point the PPF might stop growing fast enough. Considering also that a substantial fraction of the world's population survives on less than $2 a day, and that even in the "first world" long-term unemployment is rampant and only getting worse due to outsourcing, it might be that any GDP growth is only benefitting the moneyed elite, and not the generality of the working poor.

Assume for a moment that there are limits for growth of the world's GDP. Then the only way to prevent a general decay of living conditions, even for us in the first world, will be to substantially reduce population, but this is not about to happen.

So, the question is, is there a way for "steady state economics" to escape the very unattractive depiction that Adam Smith makes of it? As the efficiency of labour increases with time even with a constant GDP, are the only alternatives either a decrease in population or in living standards, or both?

To end on a grim note, it is clear that war is a way to keep the GDP growing. You spend money on making weapons, then spend money paying people to use up those weapons destroying resource, capital, and killing people, and then use even more money in reconstruction. However, if the world reaches a point where the only way to increse the GDP is war, the relative size of the "war industry" in the world economy will increase until the whole economy is in war mode, and we end up living in a world not unlike Orwell's distopia. Is this what Eisenhower was warning about when he spoke of the dangers of the military-Industrial complex?