## Thursday, March 3, 2016

### The Fallacy of Perfect Competition

I've written extensively about perfect competition and the fact, that the claims made by economists cannot be reproduced in a computer simulation a.k.a by evidence -- incidentally, the only thing that matters in a discipline that wants to be taken seriously as a science.

The solution to this apparent riddle rests on a contradiction: the claims made about perfect competition hold only, if the firms are not profit maximizers. If they are, the quantities and prices are the same as in a monopoly, or more precisely (and less misleadingly) something I hereby define as Total Market Level.

To understand why, one must first understand the concept of maximum. A monotonous function has a maximum, if and only if its value "goes up", and then beyond its maximum position "goes down" again. So, if there is a maximum profit, this implies that there's a certain quantity, below and above which profits are smaller than at that quantity

A bit more formally: Let's assume that the profit function has a maximum Pmax at quantity Qmax. This implies that for all quantities q < Qmax or q > Qmax profits will be less than Pmax (otherwise Pmax would not be a maximum). Notice that this implies that if q < Qmax, increasing quantities produced will increase profits. But notice also, that for quantities q > Qmax, profits can only be increased by decreasing quantities produced.

This profit function depends only on the market demand curve. Indeed, above discussion did not involve any mentioning of the number of firms. Now, let's assume we have K profit-maximizing producers that together produce QK items:
QK = Σi=1..K q(i)
If QK is less that Qmax, any of the K firms that produces one additional item will make a positive profit. Being a profit maximizer, it will do that, eventually increasing QK to equal Qmax.

If, on the other hand, QK is bigger than Qmax, producing additional items will reduce profits for it (and, incidentally, for all other firms as well). However, producing even one fewer item will positively increase profits (because to the "right" of the maximum, profits are increased by reducing the number of items). Being a profit maximizer, the firm will therefore reduce the number of items produced until QK equals Qmax. This holds for any K (number of firms).

But, perhaps by producing one additional item, the firm may reduce profits for everybody else (because of the falling demand curve) but still increase profits for itself? After all, there's one additional item to sell! Well, let's check, shall we. The firm increases profits for itself if and only if
P' = (Q+1)*p(Q+1) ≥ P = Q*p(Q)
(Q+1)*p(Q+1) - Q*p(Q) ≥ 0
that is, if one more item sold at the new, lower price is still more than one less item at the old, higher price (for simplicity we assume no cost, i.e. price equals profit).

But notice, that this is the difference between two profit maximization curves, the first term being "one step" (i.e. one item) ahead. So the first term will pass the maximum profit Pmax at Qmax first, and from then on will always be smaller than the second term. So you can increase profits by selling additional items, but only until the maximum quantity is reached. After having passed Qmax, profits will fall for anyone who produces additional items.

If firms are profit maximizers, none of them will produce that one item (or any further ones), and the total number of items will be Qmax. After all, any that did would shrink its profit because of this one item (and more so because of all further ones), contradicting its being a profit maximizer.

Perfect Competition

Economists will counter that all of the above is irrelevant as an argument against the concept of perfect competition, because of the fact that perfect competition requires that firms have no market power. In other words, the offered quantities of any of the firms are so small so as not to affect market prices.

It is true that the concept of perfect competition entails this requirement, but notice that this also implies the following two statements:
• Firms are not subject to the "law of demand"
• Firms are not profit maximizers
The first of these statements is obvious: if selling a hundred units less or a thousand more does not affect market prices, then the law of demand does not hold. After all, it is the whole point of the demand curve that -- as a continuous and monotonous function of quantity -- it assigns a specific price variation to any specific variation of quantity.

The second point is less obvious, but equally cogent: if any variation of quantities doesn't also change prices, there can't be any variation of profits (which, after all is price minus cost), and the whole concept of profit maximization disappears. Poof! Gone! This last point exemplifies the mendacity of economics: by stating innocuously that firms have no market power, they underhandedly change the rules of the game: firms stop being profit maximizers. And indeed, if you look closely, you will see that at P=MC firms maximize quantity, and not profit!

Perfect competition is a singularity where the "laws" of economics don't apply. Like with the event horizon around a black hole, where the laws of physics and the concept of time stop applying, there is kind of an event horizon around perfect competition. Inside, there's no price variation with respect to variation of quantity (i.e. no law of demand) and hence no profit variation/maximization. Note in particular that you cannot "reach" perfect competition by increasing the number of firms. The whole discussion above did not involve any particular number of firms. Prices and profits are at total market price and quantity levels (what economists misleadingly call "collusive" or "monopoly" levels) for any number of firms. You have to change the rules in order to get to perfect competition.

So firms maximize quantity instead of profits inside the singularity called perfect competition. But why would they do even that? If there's no profit to be made and revenue covers only cost of production (what P=MC actually means), why would they bother maximizing even quantity? Sometimes economists claim that there's some "intrinsic" profit contained in the cost, so that they actually do make "some" profit. But there's the obvious problem that a producer could then just reduce this "intrinsic" profit, lower prices and thus attract more business, ultimately driving the intrinsic profit down to zero. Ha! economists will yell, now we gotcha! They won't do that, because they're profit maximizers! Yes, but if they are, they'd maximize profits even further, and go all the way to Pmax at quantity Qmax, wouldn't they? It's a contradiction! Either they're profit maximizers, then they end up at Pmax. Or they operate at P=MC, but then they are not profit maximizers. They cannot be/do both.

Cournot-Nash formula

The Cournot-Nash formula is a continuous function in N, the number of firms, that approaches perfect competition as the number of N is increased. Yes, this is true, but only if the firms are not profit maximizers (Note that we have seen above that for any number of profit maximizing firms, quantities and prices will be at total market levels. If there's a contradiction with Cournot-Nash it must be with respect to profit maximization).

Indeed, if they are profit maximizers, prices and quantities will end up at total market levels. You don't even have to work through the mathematics of the Cournot-Nash formula on the wikipedia page about the Cournot competition. It suffices to jump to Implications to see that...
According to this model the firms have an incentive to form a cartel, effectively turning the Cournot model into a Monopoly. Cartels are usually illegal, so firms might instead tacitly collude using self-imposing strategies to reduce output which, ceteris paribus will raise the price and thus increase profits for all firms involved.
This wording would make any sleazy politician proud of himself, had he come up with it. In short and cleared of all the misleading double talk, it means the following:
If firms are profit maximizers, prices and quantities will end up at total market levels.
If you don't believe it, here's the same paragraph with some "special typography" to assist in your grasping the core of it:
According to this model the firms have an incentive to form a cartel, effectively turning the Cournot model into a Monopoly. Cartels are usually illegal, so firms might instead tacitly collude [be] using self-imposing strategies to reduce output which, ceteris paribus will raise the price and thus increase profits for all firms involved.
There's a technical term for these "strategies to reduce output and thus increase profits": it's called profit maximization. Note that economists call it collusion (indeed, tacit collusion! Why not call it insidious?) when firms ruin the claims made by economists about perfect competition -- just by being profit maximizers.

The best for last

The concept of competition in general and our discussion at the beginning rests on the (implicit) assumption that firms have complete knowledge. If we relax that requirement we can make an interesting discovery.

If firms don't have total knowledge, they know that they have passed the maximum profit level only after having produced that one item that has reduced their profit. In other words, they don't know quantity Qmax (which will maximize profits for them), but they'll notice that profits start falling after having produced item number Qmax+1. Sorry, too late!

This is of course true for any and all of them. Each firm will therefore produce one item too many. They don't "collude" so they don't share this information. Every firm will have to discover on its own.

So K firms together will produce Qmax+K items. The interesting bit is that this is exactly what we discover if we run a computer simulation of this kind of competition between profit maximizers! Quantities are slightly higher and prices/profits are slightly lower than what would be expected at actual total market levels, because some firms (statistically half of them) have overshot while attempting to guess Qmax.

Real evidence in the form of a computer simulation thus confirms our result -- even down to its practical limitation, namely that we don't have total knowledge. Wow, if we give up unsubstantiated ideological dogmas and rely on evidence instead, economics can be a science, after all!