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). $$\text{US}(p,q) = pq - C(q)$$ Similarly, UB is related to V(q), the value of having q units. $$\text{UB}(p,q) = V(q) - pq$$ 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,
$$ S(p)=q|_{\frac{\partial \mathrm{US}(p,q) }{\partial q}=0}$$ $$D(p)=q|_{\frac{\partial \text{UB}(p,q)}{\partial q}=0}$$ from which we can derive $$\frac{dC}{dq}(S(p)) = p$$ $$\frac{dV}{dq}(D(p)) = p$$ $\frac{dC}{dq}$ 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.)  $ \frac{dV}{dq}$ 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".

6 comments:

Larry, The Barefoot Bum said...

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.

This is an interesting insight, and one not mentioned explicitly in my economics curriculum. Your investigations are yielding value already!

It should be noted that in my economics curriculum (and judging from my texts, economics curricula in general) we use a different vocabulary when talking about consumers and producers. Consumers maximize utility, and producers maximize profit. Again, the reasons are not made explicit, but consumer utility is not considered directly measurable, and economists infer the concept from introspection and behavior. Profit, on the other hand, is more directly measurable.

Larry, The Barefoot Bum said...

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 is not correct. More precisely, it abstracts away a critical point, which is the difference between the market demand (or supply) curve and the firm's demand (or supply) curve.

For simplicity, I'll talk just about monopoly.

In an industry as a whole, the demand curve is downward sloping. Under conditions of perfect competition, however, because of diseconomies of scale (or social or government intervention), no individual firm can supply a substantial fraction of the total demand, and the total supply is determined not by any individual firm's marginal cost, but rather the "social" marginal cost of adding another firm, and taking labor away from other firms producing goods in other industries. Therefore, every individual firm's demand curve is horizontal; demand is perfectly elastic, since a consumer can simply buy another firm's product.

In both perfect competition and monopoly*, it is true that the firm's optimum is where marginal cost equals marginal revenue. However, under perfect competition, this intersection is also at the social optimum, where quantity demanded by the market (marginal utility to the consumer) equals quantity supplied by the market (marginal cost to society as a whole). In contrast, under a monopoly, the firm's optimum quantity is below the social optimum, and the price is above the social optimum.

*There are also two other broad categories of supply structure, oligopoly and monopolistic competition, which are not relevant right now.

Larry, The Barefoot Bum said...

Note that under monopsony, the green marginal cost curve (labeled MC) shown on your graph is the marginal cost to the consumer (the firm); the blue supply curve (labeled S) is the marginal cost to the supplier.

Under perfect competition, from the firm's frame of reference, the marginal cost of supply would be horizontal, and thus its marginal cost would also be horizontal, since adding another unit of labor would not raise the price it had to pay to its other employees.

miller said...

The relation between an individual firm's supply/demand and the industry's supply/demand is an issue I haven't thought about much. The utility functions I discussed must emerge from the characteristics of individual producers and consumers.

UB can be obtained by determining how much consumers value the product (in dollars), and then arranging consumers in decreasing order. The assumption being that products will first be allocated to consumers who value it most. (This may be a poor assumption if the product is underpriced and undersupplied.)

As for how US is derived from individual producers' operating costs... that's too hard for me to think about. Marginal cost to a single producer follows a completely different curve due to economies of scale, and now you're saying there's a social cost too.

miller said...

After some thought, I think I understand what you're saying.

As far as each individual producing firm is concerned, there's just an optimal firm size due to economy of scale. If the demand curve is practically horizontal (eg if the quantities produced by the firm are very small relative to the size of the industry), then this is just the size at which the average cost per product is minimized.

But the size of individual firms is not related to the size of the industry, since the number of firms is variable. Like you said, the size of the industry is determined by the "social" cost of taking labor away from other industries.

This is assuming that the number of firms is large, but this is not the case in a monopoly. In a monopoly, the marginal cost to a company is derived in a very different way than the marginal cost of a competitive industry.

Larry, The Barefoot Bum said...

This is assuming that the number of firms is large, but this is not the case in a monopoly. In a monopoly, the marginal cost to a company is derived in a very different way than the marginal cost of a competitive industry.

The marginal revenue is usually different between a monopoly and a firm under perfect competition.

In some monopolies ("natural" monopolies), where there are always-increasing economies of scale, considerations of marginal cost can be different than under perfect competition.