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.