I hesitated a lot before writing this. How to write it? Should I write it at all? Then I thought: "What would Arnold do?".
I will probably fail, but I'm going to try anyway.
I was reading Steve Roth, who linked to your paper (pdf) with Russel Standish, so I started reading it.
First I'm going to try to put myself in your shoes. Suppose I figured out something I thought was wrong with economics. Something very basic, like MR=MC, that is taught to all first year students, and that would mean that a lot of the rest of economics was wrong too. And not just empirically wrong, but logically wrong. And so I wrote a paper on this, and sent it off to some top economics journals. And it got rejected. And I thought the referees were wrong. And I only managed to publish it in non-mainstream journals and books, and so my important point gets ignored by mainstream economists.
Yep. I would be peeved at the mainstream too.
But you also need to put yourself in my shoes. You need to understand why I hesitated to write this post. It's the same reason I don't start arguing with the smartly-dressed young people who knock on my door, even if I do sometimes skim the flyer they hand me. Or avoid the angry guy standing on a soapbox. I know I won't get anywhere. I don't need this. I don't need the aggro. Even if I "succeed", (which is unlikely) what's the upside?
My better judgment tells me to ignore the knock on the door, or keep on walking. Instead, I'm posting this.
I'm not going to critique your paper. Instead I'm going to tell you what I think. I've thought about what you think. I would like you to think about what I think.
Suppose there are a million small farmers, each growing wheat, and their wheat is all the same and all sells for the same price. There's a downward-sloping market demand curve where the price of wheat P depends on total output Q, which is the sum of a million little q's.
1. When we say that the individual farmer faces a "flat" demand curve for his wheat, while the market-demand curve is downward-sloping, what do we mean by "flat"? Or, better, what ought we mean by "flat"?
(Yep, I learned something from reading your paper, because I hadn't read Stigler 57 either.)
We mean, or we ought to mean, "nearly perfectly elastic". We do not mean, or we ought not mean, "nearly zero slope".
The slope of the individual farmer's (inverse) demand function, which tells us how P varies with the output of the first farmer, q1, holding all the other farmers' q's constant, is exactly the same as the slope of the market demand function. (We agree on that.)
But the elasticity of the individual farmer's demand function is much bigger than the elasticity of the market demand function. It will be one million times bigger, for the average individual farmer.
The elasticity of the market demand curve is E = (1/slope)(P/Q).
The elasticity of the individual farmer's demand curve is e = (1/slope)(P/q1).
Since the slope is the same, but Q is a million times bigger than q1 (if he is the average farmer), e will be a million times bigger than E.
When we draw the individual farmer's demand curve, we need a scale for the horizontal axis that is one million times bigger than when we draw the market demand curve. That's why it looks much flatter, even though the slopes are the same. Elasticity adjusts for scale. A 1 tonne increase in q1 will have the same effect on P as a 1 tonne increase in Q. A 1% increase in q1 will have one millionth the effect on P as a 1% increase in Q.
2. There's a relation between marginal revenue, elasticity, and price.
"Market" marginal revenue is MR = (1-(1/E))P.
"Individual" marginal revenue is mr = (1-(1/e))P.
3. An individual farmer's profit R1 is a function of {q1, q2, q3,…qm}.
When I model the individual farmer alone on his farm deciding how much wheat to grow, I take the derivative of R1 with respect to q1, dR1/dq1, assuming dq2 = dq3 =….= dqm = 0. This is equivalent to setting individual mr = marginal cost.
When I model the same individual farmer at a National Farmers Union meeting deciding whether all farmers' wheat quotas should be increased or decreased, I take the derivative of R1 with respect to q1, dR1/dq1, assuming dq1 = dq2 = dq3 = … = dqm. This is equivalent to setting market MR = marginal cost.
[Just as there is a distinction between individual marginal revenue mr and market marginal revenue MR, there is also a distinction between individual marginal cost mc and market marginal cost MC, because the supply curve of land and other inputs facing the individual farmer is much more elastic than the market supply curve facing all farmers. But I will ignore that distinction here.]
4. You say: "The error in the standard “Marshallian” formula is now obvious: it omits the number of firms in the industry from the expression for the individual firm’s marginal revenue. With this error corrected, the correct profit-maximizing rule for a competitive firm is very similar to that for a monopoly: set marginal cost equal to industry level marginal revenue." (emphasis original).
I agree that the "Marshallian" formula does indeed omit that term, but I do not think this is an "error".
I think that real world farmers likewise omit that same term, when they are alone on their farms deciding how much wheat to grow. They think about the cost of growing more wheat, and compare that to the price of wheat. This makes sense to me, because mr is very close to P, since e is very large, so they ignore the distinction between mr and P.
But those same real world farmers, speaking at an NFU meeting about wheat quotas, do not omit that same term. They know that MR is very different from P. They understand that they could maximise profits much better if they could persuade the government to set quotas where MR equals marginal cost.
The real world farmers I have spoken to do not do what you say profit-maximising firms do. They make the exact same Marshallian "error". Except at the NFU meeting, when they do something much closer to what you say profit-maximising firms will do.
And if real world farmers really did do what you say profit-maximising firms would do, they wouldn't need the government to impose quotas anyway.
5. Your simulations (I would call them "agent-based modelling") are interesting. If you had simply said "We do agent-based modelling to see whether firms will converge on the Cournot-Nash equilibrium or the cartel equilibrium" and then run the simulations, I think your paper might find a receptive audience from many economists. (I don't know this for sure, because I know little about this area and I don't know whether this has already been done.) I think I get the intuition behind your results. If we start somewhere between the Cournot-Nash and cartel equilibria, and if a slight majority of firms reduce output at random, and a minority increase output, profits will rise, and so they will all do the same again. And if a minority reduces output and a majority increases output, profits will fall, so next period they will all reverse direction. So either way we get a majority of firms reducing output towards the cartel equilibrium.
But my hunch is that your results would be very sensitive to your assumptions about "learning". I think the results would be very different in an evolutionary model, where firms with higher profits have a lower probability of exit. Or where firms with lower profits copy the strategy of firms with higher profits.
This agent-based modelling looks like interesting stuff to me. But I would guess that most economists would have stopped reading your paper long before they got to that part.
Worthwhile Canadian Initiative 
Here’s Sonnenschein, with a co-author, stating that the existence of the downward-sloping portion of firm’s total cost function is inconsistent with the limiting case (of a countably infinite number of firms each producing a quantity of zero) that Rowe wants to consider:
“Of course, global increasing returns to scale (or more modestly, situations in which efficient scale is reached at a level of output which is noninfinitesimal relative to the total size of the market) remains a problem. Also, we do not deny the descriptive reality of the latter situation.” — Duffie and Sonnenschein (1989)
Of all those who disagree with me, I want to say I like Rumplestatskin’s comment the best.
That’s because:
1. Rumplestatskin goes right to the heart of the matter, with no red herrings: will n firms go to the joint-profit maximising cartel equilibrium MR=mc regardless of how big n is (as Steve Keen says they will), or will they instead go to the Cournot equilibrium mr=mc that approaches perfect competition in the limit as n gets large (as I say they will)?
2. Rumplestatskin actually gives me a reason why I should believe that Steve Keen is right: in repeated games cooperation can be supported, and we do sometimes observe cooperation in repeated games.
I don’t agree with Rumplestatskin, (and I don’t think that his argument based on repeated games is what Steve Keen is saying), but I want to give him credit for a comment that is both bang on topic and actually puts forward an argument.
Just to restate the main point at issue here: the n firms are playing a game which is an n-person Prisoners’ Dilemma. Steve Keen is saying the players will choose the “cooperate” solution, even if n is large, so that n firms will act as if they were all one big cartel.
Patrick: The function Q(q1,q2, …) can only be Q = q1+q2+,….+qn, in this case.
Luis Enrique: “I may be missing something because I’ve only read quickly, but is he saying the standard model is wrong because it ignores the own-firm impact on market quantity/prices?”
He is saying something stronger than that. Draw the market demand curve, and the market MR curve. He is saying that n firms will set q where marginal cost equals the MR of the market demand curve, not the mr of the individual firm’s demand curve. They act just like a cartel would act.
Robert: you are simply laying down red herrings, trying to change the subject to avoid the question that is being addressed here. Steve Keen is assuming (possibly/probably for the sake of argument) that each firm has a standard marginal cost curve. This argument is about marginal revenue, not about marginal cost. If you want to argue about marginal cost curves, I wrote a post about marginal cost curves just last week. It’s off-topic here.
And Robert: don’t miss my short comment about Sraffa in that other post.
Nick,
coincidently, I happened to have just read Rubenstein’s Economic Fables (which I heartily recommend, and it’s very cheap too) which provides a nice summary of how (real world / experimental) players in non-cooperative games often play the cooperative move and don’t head to the Nash outcome.
I don’t think there’s any need to make sharp predictions here. We might imagine some industries settle down to a cosy pattern where prices are somewhere close to the collusion level, and experienced executives know better than to rock the boat, but such a situation would be vulnerable to a player deciding to rock the boat. So we might observe in reality some markets populated by cosy instinctive colluders and others by boat-rockers, with perhaps cycles of learning and price wars, or such like. But I side with you that the more firms there are, the harder I think it would be to sustain uncoordinated cooperation. Look at the restaurant business for instance, these guys compete on price despite being little monopolists and masters of their own unique brand, and put each other out of business all the time, I wouldn’t be surprised if the average return across the industry is below zero economic profit.
I should add a link: Economic Fables. Nick I suspect you’re already familiar with it, but if not I think you’ll love his comparison of a jungle economy to a market economy.
Luis Enrique: The way I read it, sometimes it is relatively easier, and sometimes relatively harder, for individual firms to cooperate to reach the cartel solution. The larger the number of firms, the harder it is for them to cooperate. And the whole point of the Competition Bureau of Industry Canada is to try to make it harder for them to cooperate, by banning communication etc. If Steve Keen were right, the Competition Bureau would have a hopeless task, and would always fail, so we might as well scrap it.
For restaurants, I much prefer the monopolistic competition model. (Actually, I prefer that model for most firms.) Bertrand-Nash equilibrium with differentiated products.
sorry, yes I segued into talking about monopolistic competition without making it clear. But somebody could equally argue that monopolistic competitors will learn to collude.
Am I right that Keen at least starts by observing that the standard model ignores the own-firm impact on market quantity/prices? Does he claim to have a theoretical proof that uncooperative profit maximization will lead individual firms to the collusive outcome, or does that only emerge from the simulation?
Luis: “Am I right that Keen at least starts by observing that the standard model ignores the own-firm impact on market quantity/prices?”
Yes, I think that’s right.
“Does he claim to have a theoretical proof that uncooperative profit maximization will lead individual firms to the collusive outcome, or does that only emerge from the simulation?”
I interpret him as saying it’s a theoretical proof. The simulations are there as supporting evidence.
BTW: I invited the Mormons into my house to talk about their pamphlet. That doesn’t mean the Hare Krishnas and Moonies can come in ringing bells and chanting and talking about their pamphlets. We are talking here about Steve Keen’s ideas, as put forward in that paper. This post is not an open house for anyone who has any sort of beef with neoclassical economics.
And by that I mean specifically the idea that individual firms will produce where market (“industry”) MR equals marginal cost, as opposed to where individual firm’s mr equals marginal cost.
Nick: sorry, I guess it’s just beyond me. For my own sake I’ll try to explain what I’m thinking, but go ahead and ignore it since it’s probably nonsense.
With e.g Q = q1 + q2 I get for i=1:
d(p*q1)/dq1 = p + q1 * dp/dQ * (1 + dq2/dq1)
(no partial derivatives)
Granting that firms are not price takers (which I think is part of Keen’s argument), we can’t chuck out the stuff on the left of the +. And dq2/dq1 depends on the path/curve quantities follow on the q1 + q2 plane. I suppose that path is determined by the individiual firms production functions and profit max or whatever. In any case, it isn’t obvious to me that they can just say it is 0 the way they do.
Patrick: Work through the standard Cornot Nash model first. Here’s an example with 2 firms by Martin Osborne at U of Toronto.
Nick: OK. Thanks.
Nick,
Nick,
“Rumplestatskin goes right to the heart of the matter, with no red herrings: will n firms go to the joint-profit maximising cartel equilibrium MR=mc regardless of how big n is (as Steve Keen says they will), or will they instead go to the Cournot equilibrium mr=mc that approaches perfect competition in the limit as n gets large (as I say they will)?”
The answer depends on whether n is fixed and constant. If n firms can go to the joint-profit maximising cartel equilibrium without concern of changes in price P affecting n (no new market entrants, no market exits), then they will. It does not matter how big n is if dn/dP = 0. ( Market Elasticity = Individual Elasticity ).
Em – Ei = P/n * dn/dP
Em – Ei = P/n * dn/dP
The standard neoclassical assumption in perfect competition is that firms do not expect other firms to change their quantity output in reaction to a change in their own quantity output. Hence, for example, dq2/dq1 = 0. In other words, Keen and Standish assume atomism, as in the textbook.
If you want to assume only two firms, and do not like Sigma notation, for some reason, Equation 0.1 follows like so:
dP/dq1 = (dP/dQ)(dQ/dq1) = (dP/dQ)( d(q1 + q2)/dq1 )
= (dP/dQ)( (dq1/dq1) + (dq2/dq1)) = (dP/dQ)( 1 + 0 )
= dP/dQ
You can do the same for dP/dq2, if you care.
Anyway, Rowe has been saying that he accepts this. The slope of the firm’s demand curve is the same as the slope of the industry demand curve.
An implication is that given atomism, a finite number of firms, and a downward-sloping market demand curve,
it cannot be the case that Marginal Revenue equals price. The combination of these assumptions and this conclusion are logically inconsistent. (Anybody talking about engineering approximations, if honest, would still gladly agree with this statement.)
For fun, I worked through the Taylor series expansion following 0.2 in the Keen and Standish paper. And they are correct. Given atomism, a finite number of firms, and a downward-sloping market demand curve, a firm will produce less output than at the level where Marginal Cost equals price. (There are no errors in this section. For example, firms are not assumed to set variables which they do not control.)
As I have pointed out above, one cannot justify the textbook case by considering a limit case with the number of firms approaching infinity. As is recognized in the professional literature, the limit case cannot be logically combined with the textbook assumptions about cost curves.
Robert: setting aside your last paragraph about cost curves (which is not part of Steve’s argument in the bit I am disagreeing with), I am with you.
But Steve, as I interpret him, is saying something much stronger than that.
Steve is saying that firms will set output where marginal cost equals industry (or market) Marginal Revenue (“MR”). That’s what I would call the “cartel equilibrium”, where all the firms get together and act like a multi-plant monopolist who owns all the firms.
We know that for the market demand curve, MR=(1-(1/E))P , where E is the elasticity of the market demand curve. And E=(1/slope)(P/Q) where Q = sum of all the q’s.
We know that for the individual firm’s demand curve, mr=(1-(1/e))P , where e is the elasticity of the individual firm’s demand curve. And e=(1/slope)(P/q) where q is the individual firm’s output.
Slope is the same in both cases. So e will be about n times bigger than E, if there are n firms, all roughly the same size, because Q will be about n times bigger than q. So mr will be bigger than MR and closer to P if n > 1.
Steve is saying firms will set MR = marginal cost.
I am saying firms will set mr = marginal cost. (Unless they can somehow enforce a cartel?)
Steve seems to be saying that profit-maximising firms will always act like a cartel, and that this follows directly from the math.
Nick, I’m glad you liked my comment. I prefer not to get caught up in the maths, because the maths is either right or wrong. It’s the assumptions of human behaviour and the interpretation of the maths that is important.
So let me respond to your response.
“It’s that “…until they all get back to the monopoly level of output” I don’t buy.”
Well, where would they stop? Somewhere in between. Probably. But closer to a monopoly level.
“each individual farmer will produce where individual mr=mc. How do you get to them reducing output from that point to where market MR=mc? At mr=mc, each farmer is maximising his profit given the output of every other farmer.”
Individual mr=mc is where MR=mc. If a monopolist increases output by one unit, it’s the same as any of the many firms increasing output by one. So they don’t. In fact, if they reduce by one they make a fraction more profit.
“why do Canadian dairy farmers care whether the government abolishes milk quotas? “
Because their cost structure must be higher that in the US. A monopoly with higher costs would produce less than a monopoly with lower costs. Just as a competitive market of high cost producers will produce less than a competitive market of low cost producers. Merging the two markets, which is essentially what removing the quota will do, is a simple redistribution from Canadian farmers and US milk consumers, to US farmers and Canadian milk consumers.
Essentially Canadian dairy farmers are being protected from low cost competitors that would not reduce their profits to ‘normal’ levels, but drive them out of business completely.
The question is, why are costs lower in the US – is a because of natural advantage, or because they have protected their industry in the past until it reached a point where it is internationally competitive?
No one wants to be gamed.
“And if the producers can always coordinate between themselves to maximise their joint producer surplus, why can’t they also coordinate with consumers to maximise the joint producer plus consumer surplus?”
Well, this only applies if you still believe in the model that we (at least I) no longer believe is correct, or even useful. Because Keen’s model doesn’t have a concept of consumer surplus, we can’t know whether it is or isn’t maximised at his optimal level of output. And we definitely can’t say that the point you believe is optimal maximised surpluses, because at that price and output no one will produce anything!
Robert:
“An implication is that given atomism, a finite number of firms…”
No one ever (or, at least, should never) assume both atomism and a finite number of firms. Atomism is a consequence of having a continuum of firms. Atomism is not an assumption in the standard models, it is a consequence of the modelling assumption regarding the number of firms.
“As I have pointed out above, one cannot justify the textbook case by considering a limit case with the number of firms approaching infinity.” This statement is incorrect. Quoting from Keen and Standish “Stigler’s convergence argument is technically correct”
Where Keen and Standish go wrong is somewhere around equation 0.4 (and possibly also that there argument regarding the Taylor expansion does not hold in the limit case-although checking that is a high effort/low reward activity). I think this is what Nick is trying to say in his OP – by claiming that the firm will maximise its profit by maximising the total derivative, Keen and Sandish are implicitly assuming the collusive outcome.
Rumplestatskin: “Well, where would they stop? Somewhere in between. Probably. But closer to a monopoly level.”
Well, we can say exactly where they stop, if we know the market demand curve, the firms’ marginal cost curves, and the number of firms.
Let E be the elasticity of the market demand curve, n the number of firms, and mc the marginal cost.
The monopoly price will be determined by P(1-(1/E))=mc
The price with n firms will be determined by P(1-(1/nE))=mc
Suppose E=2, and mc is constant and equal to 1.
The monopolist will set P=2.
Two firms (n=2) will set P=4/3
Three firms will set P=6/5
Four firms will set P=8/7.
100 firms will set P=1.05
And so on.
Ooops. I think that should be:
100 firms will set P=1.005
(Yep, I’m cr*p at math).
But it isn’t about the math, Nick. It’s about the assumptions. The math is just arm-waving.
It strikes me that Keen and Standish make exactly the same error as the undergrad who refuses to accept that the Nash Equilibrium of the prisoners dilemma is (Defect, Defect): “But they can both achieve a higher payoff if they both choose Cooperate, so they must cooperate”.
Nick:
You say: “Two firms (n=2) will set P=4/3”. This cannot be right.
Consider a family of linear demand curves corresponding to your assumptions about E of 2. The family can be expressed by the following:
P = 3 – bQ
With Cournot competition (assuming as you did MC = 1), the profit maximizing equation will be:
for two firms: 3 – 2qb – qb = 1 => q = 2/(3b)
for three firms: 3 – 2qb – qb – qb = 1 => q = 2/(4b)
…
for n firms: q = 2/((n+1)*b)
So, for n firms, the price will be P = (n+3)/(n+1) regardless of either the slope or elasticity.
Thus:
One firm: P = 2
Two firms: P = 5/3 (not 4/3)
Three firms: P = 3/2 (not 6/5)
Four firms: P = 7/5
100 firms: P = 103/101 = 1.02
The problem with your solution is elasticity interpretation that does not make sense in its(E’s) platonic universe 🙂
bankster: you are assuming a linear market demand curve. That’s OK.
I was (implicitly) assuming a constant elasticity market demand curve. Something like P = Q^-(0.5). Q = P^-2 . That’s OK too. (But I should have said so explicitly).
Evan: OK. But where in the math does that assumption/mistake appear? (I think Robert is working his way through it, and I think his math is better than mine, so maybe he will track it down.)
Nick: Equation (0.4) doesn’t make any mathematical sense. Keen treats aggregate output Q as if it’s a parameter, rather than the sum of the firm’s own choice variable and the output of all other firms. The clumsy arithmetic he does is equivalent to assuming conjectural variations of 1.0. Starting with
profit(i) = P(Q)q(i) – c(q(i))
differentiate with respect to q(i) and set the derivatives of all q’s other than i with respect to q(i) to 1.0 to find:
(n)(P'(Q))q(i) + P = c'(q(i)).
That is Keen’s equation 0.9. Notice that the left-hand side is firm i’s marginal revenue and the right is marginal cost, so shockingly enough, the firm equates the two to maximize profits. All the weirdness about MR not equalling MC seems to be grounded in some further conceptual misunderstanding of elementary microeconomics on Keen’s part.
You’d have to blow off the dust on some old IO textbook to find a rigorous characterization of when conjectural variations of unity is exactly the same assumption as collusive behavior, but note that if firm i acted to maximize industry profit rather than own-profit it would set its own output such that
P'(Q)Q + P = c'(q(i)),
which differs from the conjectural variations of unity outcome only when nq(i) is not equal to Q.
Nick:
Yes, with constant elasticity, you are right, but only with constant elasticity which is a pretty special case, and not in general where your intuition of firms seeing a fractional elasticity does not apply.
“If you let x be anything you want, well, the theory doesn’t have much content. You can always find an x that fits the facts and makes the theory “true”.”
Not that I disagree with what preceded this, but the testable hypothesis is that firms actually use this method to price items, which you can verify by asking them.
Chris: that is very helpful. Thanks.
“Conjectural variations”. That lovely old phrase takes me back a bit!
Back soon.
Evan: “It strikes me that Keen and Standish make exactly the same error as the undergrad who refuses to accept that the Nash Equilibrium of the prisoners dilemma is (Defect, Defect): “But they can both achieve a higher payoff if they both choose Cooperate, so they must cooperate”.”
The problem being, OC, that naive college freshmen and sophomores, even playing a one time game of the prisoners dilemma with strangers who are also naive, do better than the supposedly knowledgeable group who have learned that the equilibrium is to defect, playing against each other. 😉
To be sure, if you sprinkle the knowledgeable amongst the naive, the knowledgeable will do better. But if you teach the naive, as a group, to be knowledgeable, who benefits?
I’ve put up a follow-up blog post:
link here NR
Chris: so if I read your post correctly, what Steve Keen is doing in his math is the same as what I am doing in my math when I model the NFU’s problem:
I wrote: When I model the same individual farmer at a National Farmers Union meeting deciding whether all farmers’ wheat quotas should be increased or decreased, I take the derivative of R1 with respect to q1, dR1/dq1, assuming dq1 = dq2 = dq3 = … = dqm. This is equivalent to setting market MR = marginal cost.
No, Nick, because maximizing with respect to Q is not the same as maximizing with respect to q1, given fixed values for q2,q3,… The solution to the former here is saying that firm 1 is setting Q to maxmimize its objective function–but firm 1 has no control over Q, only q1! Allowing it to optimize with respect to Q is allowing it to control other firms’ production (and is thus equivalent to collusive behavior).
??? You misunderstand me. What I was doing there was modelling collusion. I wouldn’t model it that way if each farmer was alone on his farm deciding how much wheat to plant. I would max firm 1’s profits wrt q1, taking q2 and q3 etc. as given.
Nick: yeah, that’s equivalent to what Keen does. More exactly, he writes all the q’s as functions of a parameter which happens to be called “Q” but actually has nothing to do with total quantity and imposes the restrictions that the derivatives of every q, including q_i, with respect to Q are 1.0. Then he figures out what value of Q firm i would choose to maximize own profits subject to those restrictions. This weirdness is analytically equivalent to solving the optimization problem you specify.
Nick: your biggest mistake is this, and simply this:
Most markets are oligopolistic competition, or competition with differentiation, not commodity competition.
In practice, the equivalent of the NFU exists and is effective in most markets.
That’s just an empirical thing. There are commodity markets which actually behave as you describe. But they’re abnormal. Steve Keen is analyzing the common case, the NFU case. The farmers are never on their own.
To pile on a bit further. We have known since Adam Smith that the tendency of all businessmen is towards collusion. This isn’t a new idea! It is, however, empirically verified.
So why would you consider the weird case where the farmers don’t have regular NFU meetings? It doesn’t normally happen. Usually there are a fairly small number of firms in a given market and they collude. If the market has too many firms in it, then the firms differentiate so that they’re not in the same market any more.
Nick wrote: “Lord: Maybe, but that’s why we have Industrial Economics, and oligopoly theory, to try to figure out when firms will act like wheat farmers and when they will act like a cartel, and when they will do something in between. My view is that the assumption that their outputs are perfect substitutes is the assumption that needs relaxing. I normally prefer monopolistic competition.”
OK, great! Why do you study anything else? 🙂 The “free, fully competitive commodities market” is so rare as to be unworthy of study!
Nathanael: I don’t think you have actually read Steve Keen. Steve is saying it doesn’t matter whether there are 2 firsm or thousands of firms, and whether they get together into an NFU or not, they will always act like a monopolist.
As always with this chapter, I can’t shake the feeling that everyone here is missing the point.
First, we are discussing a model of perfect competition. There is no need to invoke empirical reality – if we want to do that, then abandoning perfect competition altogether would be a start.
Secondly, there is no need to invoke Cournot or collusion for Keen’s basic argument, which is that no matter how small a firm is, its actions will always have some effect on the market price. If it is truly a profit maximiser, it will recognise this and produce slightly less than where MC=MR. If it doesn’t by assumption (begging the question of if there is some central authority setting price, which is stupid), then it will not quite maximise profits and will cause a small decrease in the market price. If every firm does this, MR and demand will diverge in the same way as they do under a monopoly market.
And yet none of this is discussed in textbooks, on courses or elsewhere. The basic truth is that Keen has noticed an inconsistency somewhere. You can move the problem by assuming what you want, but the mechanics he identifies are correct within the model.
Unlearning: Did you read the OP, or either of Chris’ responses to Keen?
“Secondly, there is no need to invoke Cournot or collusion for Keen’s basic argument, which is that no matter how small a firm is, its actions will always have some effect on the market price.” This is trivial for any finite number of firms, but false for an infinite number of firms.
“If it is truly a profit maximiser, it will recognise this and produce slightly less than where MC=MR.” This statement is just plain wrong. My guess is that marginal revenue doesn’t mean what you think it means. Take a look at Chris’ link (posted above) and go to equation 3 and read the explanation directly beneath it.
Unlearning: “And yet none of this is discussed in textbooks, on courses or elsewhere.”
It wasn’t news to me. In the Cournot game, MR is below P for finite n, and MR approaches P as n goes to infinity.
I Googled “Cournot MR number of firms, and this Wiki was the first hit. See the section on the “Cournot Theorem”.
(The Bertrand Game, which is a different interpretation of perfect competition, gives a different result, BTW.)
But Steve Keen is saying something much stronger than that. He is saying that profit maximising firms will set q where Market MR = mraginal cost, regardless of the number of firms. The rest of us would say that is only true if n=1, or if all the n firms collude to form a cartel.
Nick: I’ll agree with you that Keen is wrong, but I think it doesn’t matter that much. I’m making a different assertion, because I don’t particularly care about angels dancing on the head of a pin.
Evan,
“This is trivial for any finite number of firms, but false for an infinite number of firms.”
This seems wrong to me. As Keen says, infintesimals aren’t zero. The firm will still have an inftintesimal effect on demand.
“This statement is just plain wrong. My guess is that marginal revenue doesn’t mean what you think it means. Take a look at Chris’ link (posted above) and go to equation 3 and read the explanation directly beneath it.”
I have read the links – it was badly phrased. My point is that if the firm acts as a ‘price taker,’ its own MC=MR calculation will be incorrect.
“But Steve Keen is saying something much stronger than that. He is saying that profit maximising firms will set q where Market MR = mraginal cost, regardless of the number of firms. The rest of us would say that is only true if n=1, or if all the n firms collude to form a cartel.”
Ah, OK. But is it not true that the MR and demand curves will still diverge even if they don’t behave collusively?
Hi Unlearning. That last bit you quote was from me, not Evan, so I will respond to your “Ah, OK. But is it not true that the MR and demand curves will still diverge even if they don’t behave collusively?”
Yes. In the Cournot game, the individual firm’s marginal revenue curve is always below the individual firm’s demand curve. As the number of firms increases, the individual firm’s marginal revenue curve gets closer and closer to its demand curve. In the limit, and the number of firms approaches infinity, the mr curve approaches the demand curve.
The relation between marginal revenue and price is: mr = (1-1/e)p. And the elasticity of an individual firm’s demand curve will be given by e=nE where E is the elasticity of the market demand curve, and n is the number of firms. So we can re-write it as: mr=(1-1/nE)p. As n gets big, mr approaches p.
There’s a second way to think about perfect competition: instead of Cournot, where each firm sets q taking other’s q’s as given, the Bertrand model assumes each firm sets p assuming other firms’ p’s are given, and firms have differentiated products. In the Bertrand model, as firms’ products become closer and closer substitutes, we approach perfect competition in the limit.
Off-topic: I keep meaning to write you a post, building a macro-model where; firms have horizontal mc curves, set price as a markup over mc, produce and sell however much is demanded at that price. Plus a bit of sticky prices. That’s (roughly) how you view the world, right? And I will say it’s a New Keynesian macro-model. Very very mainstream (in macro circles anyway) since about 1987, when we figured out how to build macro models with monopolistic competition.
Unlearning:
I’m going to assume that you haven’t studied any measure theory. This is a tricky concept to explain, but I’ll give it a shot (it’s something that I should be able to explain to my students, but I’m not sure I do a very good job of it).
[Nick – if you have any input on how best to explain this to students I’d be interested to hear it!]
I think it is best to start with an example/analogy. Think of a random variable that is uniformly distributed over the interval [0,1], so that it is equally likely to take on any value between 0 and 1 (inclusive). Now ask yourself, “What is the probability that the realisation of this random variable is exactly 0.4234?”
The answer to that question is that the probability is 0. In fact, the probability of any particular number occurring is 0. However, if we aggregate up the probability of all the numbers occurring it is equal to 1. This tells us that our usual notion of summing up probabilities is not valid in this context. Now, suppose that we change our distribution very slightly, so that we double the chance that the realisation of our random variable is exactly 0.4234. What happens? Is our new object still a valid probability distribution? The answer is yes, and that the aggregation of the probability of all the numbers occurring is still equal to 1.
Now, imagine that each of the points along the unit interval is a firm, and that all of the firms produce the quantity, and this quantity happens to aggregate to 1. What happens if the firm that is located at exactly 0.4234 suddenly doubles it’s production? Has the total production of all the firms changed? No, it is still exactly equal to 1. The change in quantity by an individual firm has had no change on the total production. Now, if all firms doubled their production level, then production would double to 2. But if only one firm, in the infinite mass of firms, increases its production then there is no change in the total amount produced.
I’m sorry if this isn’t very clear, but I’m not sure what the best way to explain this is without throwing a bunch of measure theory at you.
To summarise, in a fashion that is imprecise: You are correct that infinitesimals aren’t zero, but it takes an infinite number of them before they have an impact on the total quantity.