The basic frame of a firm: Cournot

It seems that the debate about the fundamental “theory of the firm” is going on. Now there are issues with the theory of the firm, things that economists have been busy plugging away on for a long time now – but the critique that Steve Keen has put forward is not one of these issues. In this post I will attempt to discuss his push to change the Cournot model, which I don’t agree with. This will give us scope to discuss perfect competition another time.

Fundamentally our firm is all about “maximising profit”, that is what it wants to do, that is its objective. Say that the firm (and all its competitors) only tries to do this ONCE. This is a situation where Steve said standard firm theory fails, so even though it is unrealistic it will work for our purposes.

To figure this out we need to ask “what is a measure of this objective”. For this we need our firms profit function:

$\pi_1 = P \left( q_1,q_2, . . . ,q_n \right) q_1 - C \left( q_1 \right)$

So this tells us that profit is equal to revenue minus cost, where revenue is price times quantity, and where the price is a function of the amount produced by all n firms in the industry.

Now, we want to find where profit is as high as possible right. To do this we need to realise that our firm (firm 1) chooses $q_1$ and nothing else – this is their choice variable. As a result, they want to find what value of $q_1$ will make this profit number as big as possible!

To work this out we differentiate firm 1’s profit function with respect to their choice of output. Why? Well given that we assume that the profit function is concave in the firms quantity (or that profits increase at a decreasing rate as the firms choice of quantity rises) and that profits in the industry are positive for at least 1 level of output, we know that at one of the points where the slope of the profit function is zero we have maximum profit.

First let us take a standard static Cournot game. In this we work out the firms marginal profit for given levels of output. This tells us how a firm will set output given its expectation of other firms output.

$\frac{\partial \pi_1}{\partial q_1}=P + \frac{\partial P}{\partial q_1} q_1 - \frac{\partial C}{\partial q_1}$

Setting this to zero and solving for $q_1$ gives us $q_1$ as a function of the output choices of all other firms in the industry. This is a best response function.

To proxy for our firms “expectation” of other firms output in a consistent way they conjecture the output of other firms based on this same premise (working out their best response functions). Putting them all together we get a Nash equilibrium, where all firms choose output based on their best responses.

However, Steve has posited that he prefers it when we treat the other firms choice variables as direct functions of our firms choice variable. He does not like the Cournot Conjecture (essentially that firms act as if the other firms output level is given when they choose their own).

Ok so lets do this realising that the other firms choice of output $\left( q_2, q_3, . . . , q_n \right)$ are functions of our firms choice of output $q_1$ :

$\frac{d \pi_1}{d q_1} = P \left( q_1,q_2, . . . ,q_n \right) + q_1 \left( \frac{\partial P}{\partial q_1} + \frac{\partial P}{\partial q_2} \frac{d q_2}{d q_1} + . . . + \frac{\partial P}{\partial q_n} \frac {d q_n}{d q_1} \right) - \frac{\partial C}{\partial q_1}$

Now for our purposes this makeup is unnecessarily messy. We can say that $P \left( q_1,q_2, . . . ,q_n \right) + q_1 = R \left( q_1,q_2, . . . ,q_n \right)$ where this is our “revenue function”. In that case our derivative is:

$\frac{d \pi_1}{d q_1} = \frac{\partial R}{\partial q_1} + \frac{\partial R}{\partial q_2} \frac{d q_2}{d q_1} + . . . + \frac{\partial R}{\partial q_n} \frac {d q_n}{d q_1} - \frac{\partial C}{\partial q_1}=0$

What does this tell us in english? Well the maximum profit is at a point where the additional revenue from a small increase in production is the same as the additional cost from increasing production. Here there is a change in revenue from our own output changing, and from other firms reacting to our firms change in output – this in total is our firms “marginal revenue”. This is what I meant before when I said that economists do use MR=MC and it is appropriate.

However, depending how we conjecture these reactions our results will vary.

Steven specifically conjectures our firm’s response to other firms’ output as “the choice variable”.

But what does this function stuff mean?

Now what does it mean when we go and make the other firms choice variable a function of our own when making a choice? Well, it means that more than “we are taking into account the other persons incentives”. Why? Well a normal Cournot game (the static one we first did) works out a “best response” given the other firms own “best action”. It finds the point where both our firm and other firms are acting consistently, and where we (and others) have no incentive to “deviate” from our current actions. This is a Nash equilibrium.

Explicitly stating that the other firms choice of output is a function of our own choice of output BEFORE determining our best response (which is what we did by finding the derivative above) implies that the other firms will OBSERVE our level of output and then REACT to that observation.

This matters because it implies that a firm can COMMIT to producing a level of output that would not be credible in the static game to get a competitive advantage. For this commitment to work we need the other firm to be able to see it happen – if the firms output is unobservable (a pretty damn realistic assumption) then this way of modeling is incorrect.

Economists call the type of guy that can do this commitment a Stakelberg Leader.

In the Cournot game we do not have a Stakelberg leader, no-ones output is observable, and so any push to “commit” to a level of output that doesn’t set marginal profit to zero given your expectations of other firms reactions will be a dominated strategy. It is true that if all firms reduced output their profit would increase (that is collusion) – but that doesn’t stop it being a dominated strategy in a static game.

The fact is, in a static game you can’t react to output “changes” (relative to expectations) from another firm. And in a dynamic game economics ALREADY estimate the value of these factors (eg $\frac {d q_n}{d q_1}$ ) when solving for the firm’s optimal strategy.

Conclusion

I am unsure whether Steve simply does not realise the assumptions behind the different economic models (and the degree of sophistication that has developed in a lot of these models past his “straw men”) or whether he realises but just likes insulting other economists.

Paul Walker says:

July 28, 2009 at 19:16

As a theory of the firm man can I say that the standard perfectly competitive, oligopoly and monopoly models don’t have firms in them, they are more models of industries rather than models of firms. This follows from Coase 1937. As Martins Rickets put it “If market transactions were costless there would be no rationale for firms”. In other words, in a model with zero transaction costs, which the standard models are, firms need not exist. Production takes place over the market since it is costless to organise it that way.

Ignoring that, I agree with your discussion. In a Nash equilibrium all firms will maximise profits given their beliefs, which are correct in equilibrium, about what the other firms will do. If some firms can observe other firms actions you don’t have a static game, you now have dynamics which makes things more interesting and realistic but also a lot more complicated.

Matt Nolan says:

July 28, 2009 at 19:42

@Paul Walker

I 100% agree with you about these models being “industries” and “production functions” rather than underlying firms. Originally I was going to write about that – but I felt that, even at this level there were missing issues in what Keen wrote.

Ultimately, if he is going to discuss Cournot he should stick to Cournot, and if he is going to solve games he needs to make sure that the individual strategies are consistent and that we have a Nash Equilibrium. Fundamentally, I suspect he is missing the issue of observability and timing in his work – which is no good as far as my limited understanding goes.

The basic frame of a firm: Cournot

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