Sunday, November 17, 2013

In which I think about supply and demand


Note: this post makes use of LATEX.  That means it requires javascript, and cannot be viewed in a reader.

Earlier, I was entertained by learning a little bit about monopoly and monopsony.  I know I am very ignorant of economics, and that this is pretty basic stuff.  But perhaps readers can gain some insight by watching me learn things for the first time.  Or if there is no insight to be gained by my readers, at least I benefit from the writing.

I read about monopoly and monopsony on Wikipedia.  My instinct as a physicist is to (initially) ignore all real-world details and understand the simplest and most abstract problem.  So of course I go straight to the graphs.

 
Figure 1: Monopoly
 
 Figure 2: Monopsony 

These graphs were very confusing when I saw them for the first time, for several reasons:

1. It seems the axes are backwards.  It seems like prices are the independent variable, and the quantity of trades is the dependent variable, yet prices are on the vertical axis, while quantity is on the horizontal. 
2. I thought a "competitive market" leads to the quantity (Qc) and price (Pc) at the intersection of the supply (S) and demand (D) curves.  And yet in Figure 1, it's at the intersection of marginal cost (MC) and demand (D), while in Figure 2, it's at the intersection of supply (S) and marginal revenue product (MRP).
3. Under monopoly conditions, the quantity (Qm) is instead at the intersection of the marginal cost (MC) and marginal revenue (MR).  Under monopsony conditions, the quantity is at the intersection of marginal cost (MC) and marginal revenue product (MRP).  What's confusing is that marginal cost (Fig 1) and marginal cost (Fig 2) are completely different, even though they have the same name.

Explanations from Wikipedia were not forthcoming, because for some reason there's more emphasis on real world examples rather than mathematical abstractions.  It's cool though, because I can do the math myself.

Perhaps more fundamental than the supply and demand curves, are the utility functions of the buyers and sellers, which I'll call UB and US respectively.  These are functions of both quantity (q) and price (p).  US is related to the cost of producing q units, which I'll call C(q). US(p,q)=pqC(q)
Similarly, UB is related to V(q), the value of having q units. UB(p,q)=V(q)pq
The supply curve tells you how many products the industry is willing to sell, given a certain price point.  Basically, the supply is equal to the quantity such that US is maximized.  The demand curve tells you how many products consumers are willing to buy, given a certain price point.  Basically, the demand is equal to the quantity such that UB is maximized.  In mathematical terms,
S(p)=q|US(p,q)q=0
D(p)=q|UB(p,q)q=0
from which we can derive dCdq(S(p))=p
dVdq(D(p))=p
dCdq is of course the marginal cost (MC in figure 1).  So that means the marginal cost function is the inverse of the supply function.  If you plot them on the graph, they will be the same curve.  (People who aren't math purists might say they are in fact the same function.)  dVdq is the marginal value (which in the context of employers buying labor, is called marginal revenue product, MRP in figure 2).  Since the marginal value is the inverse of the demand function they will also be the same curve on a graph. 

This solves my confusion on points 1 and 2.  While supply and demand are naturally functions of price, marginal cost and marginal value are naturally functions of quantity, so it makes sense to put quantity on the horizontal axis.  And in both figures, the competitive market rate is in fact at the intersection of the supply and demand curves, even if the curves are not labeled as such.

This post is too long, so I will break here.  Next time, I will talk about why a monopolistic market is different from a competitive market, and also "deadweight loss".