Monopolies and price stability

Ok, ok. Economics. With maths. And graphs.
Let us nerd out brothers!


(Update: Typos have been fixed. Thanks guys!)

Anyway. Here's something that at one point in grad school, when I thought about going into IO, I though about pursuing but then it never really went anywhere and I got interested in other stuff instead. I got reminded of it recently when trying to look up my electricity bill on line, having a buttload of trouble and eventually winding up at a webpage which as it's header had the sentence:

"Our purpose is to serve our customers by guaranteeing price stability"

I'm not gonna say which electricity company it was, but hey, I'm pretty sure they're all the same even if they're not as explicit about the fact that they're ripping you off. Of course "price stability" here means "stability of consistently higher prices". But there is an interesting economic question here, just from a purely positive point of view - are prices more stable in a monopolistic market or a competitive one? Is there any justification for the obviously self serving claim of a (government granted) monopolist?

And being even more generous, I can sort of see it. When prices change a lot, it's a pain. On the simplest level, you show up to the store, having had optimized your optimal money holdings (i.e. the cash in your pocket you took with you) and the fact that the actual price is way different from what you expected means you gotta make another run to the ATM machine or forgo something. Probably less explicitly but more importantly, when prices keep changing on you it means you've gotta recalculate your optimal allocations of expenditure again. You gotta set up a new Lagrangian, take the damn first order conditions and figure out if making out the adjustments in your optimal consumption basket - given your income - are worth it. What? You don't think that way? You don't set up constrained Lagrangians every time there's a change in prices and compute your Kuhn-Tucker conditions? Well, actually you do, And so do most people. It's just they do on the intuitive level what economists formalize in (semi) fancy maths (they're really not that fancy, believe me). But the "bounded rationality" people have a point here. It's a pain in the ass expanding mental computation resources on dealing with this sort of thing.

So how do we figure it out? Well, consider a simple set up with linear market demand and constant marginal costs (I worked it out with increasing, quadratic, costs which is probably the more realistic case - particularly for the oligopoly case below - and it works roughly the same way), where firms are Cournot competitors (this makes it easier to compare competitive markets and oligopolistic/monopolistic markets). In that case an individual firm's profits are given by



where

and



so profit max implies



By symmetry, all firms produce the same amount of output so that the q's are all the same. This means that sum of q's is just q*N and we can drop the i subscript. So



and



Plugging this back into our inverse demand function P(Q) we get



This means that as the number of firms gets really large (N goes to infinity) price goes to marginal cost.



which is known as the "Cournot Theorem" and here I will take this to represent the "perfect competition" case. In that case, or in the parallel case of Bertrand competition with a homogeneous good, or the textbook example of a competitive market (except here the supply curve would be a flat line at c. Like I said, this pretty much works with a quadratic cost function in which case you get the usual upward sloping market supply curve), the variance of the price will be just the variance of marginal costs:



What will the variance of prices be under a monopoly or under oligopoly? Well, first consider the simplest case where all fluctuations in price are due to cost shocks. Then



So in this particular case, yes, prices would be less volatile under Monopoly than under competition. In fact, increasing the number of the firms in the market would increase price volatility. Here's a graph of that:



Here's a graphical representation of what's up in this case. Remember from your basic econ that a monopoly will never operate on the inelastic portion of its demand since then by decreasing output it could both cut costs and raise revenues. So it's gotta be on the elastic, or "flat" portion of its demand (in % terms of course, since this is linear demand). This in turn means that the monopolist is willing to absorb some of the cost shocks rather than passing them fully onto the consumers:



But what if price fluctuations result only from changes in demand?

Well in that case in a perfectly competitive market, with constant marginal cost, there will be no change in price, so the volatility of prices under monopoly must be greater than in a competitive market (again, this generalizes to quadratic cost functions and upward sloping supply curves as long as the shocks are to the intercept and not to the slope).

So the first lesson is:
If the fluctuations in price are due to shocks to firms' costs then the fluctuations in price will be less under a monopoly then in a competitive market.
If the fluctuations in price are due to shocks to consumers' demand then the fluctuations in price will be greater under a monopoly then in a competitive market.

How much does that generalize to the case where there are shocks both to costs and demand?
The above price equation tells us that then the variance of prices will be



Again to keep things simple lets assume that the cost shocks and the demand shocks are uncorrelated so that Cov(a,c)=0. Then we have



So is the variance of price increasing or decreasing in the number of firms?
Taking the derivative with respect to N we get

vs. 0

or

N vs.

(Addition, since I screwed up last time: If V(c) is large then for price volatility to be increasing in N, you need a small N.

Or here's another way


vs.

Which means that volatility under an oligopoly is less than volatility under perfect competition if

V(a)<2NV(c) i.e. if shocks come mostly from cost side)


Which once again says that if the shocks come mostly from the cost side then the volatility of prices will decrease with number of firms - monopoly is good for price stability. But if most of the shocks come from the demand side then price volatility will decline as the number of firms in the market decreases.





Also. Neurosis after "Souls at Zero" is not worth listening to.