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.

Nick: this has been pointed to Keen before.

link here pdf NR

I notice the version of the paper that eventually got published in the august "Real World Economic Review" omits Keen's claim that mainstream economists never use dynamic models. (He proceeded to generously show us a dynamic model of a firm might be set up, and got it brutally wrong.)

Posted by: Chris Auld | December 01, 2012 at 11:48 AM

Chris: did your paper ever get published/circulated in any form?

Posted by: Nick Rowe | December 01, 2012 at 01:11 PM

Nick, as always, an interesting post, which has actually made me curious to read Keen's paper to see what sort of ABM he was actually using.

However, (and I realize this is a very personal pet peeve): could you please stop using the NFU as a place where people chat about wheat quotas? I think I've been to at least 18 national NFU meetings, and my family has held just about every position there is to hold in the organization, and I honestly can't remember a time when people ever talked about wheat quotas. Milk quotas? yes. Actions to protect the Wheat Board as a marketing entity? Yes. But wheat quotas I don't think ever came up. In fact, a quick search of their policy site for "quota" doesn't term up a single hit. For that matter, given how international and homogeneous the wheat market is, I honestly can't picture an Canadian wheat quota system doing anything besides hurting wheat farmers.

I know you might view it as a minor point, but when you're you're using real organizations as examples, perhaps you could see if they've ever supported the positions you talk about? Unless you would feel comfortable with someone talking about the U of Carlton Econ faculty sitting down to agree to unilaterally reject heterodox papers to raise the Marginal Citation Revenue from their own papers as a teaching example.

Posted by: Eric Pedersen | December 01, 2012 at 01:12 PM

And why beholdest thou the mote that is in thy brother’s eye, but considerest not the beam that is in thine own eye?

Posted by: Sandwichman | December 01, 2012 at 01:14 PM

It probably would be a different world if all markets were like the wheat market, but where so many firms won't compete at all if they can't be one of the top two firms in their market, knowing your competitors and anticipating their response to any action of yours and they to yours can be a powerful incentive to competitive avoidance.

Posted by: Lord | December 01, 2012 at 01:33 PM

Nick:

"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. "

That cannot be right on at least two counts:

1. If we assume that the farmer sees the whole market DC, the scale is immaterial. We can look at the tiny band around the market price wiggling along the market DC, while the farmer plays with quantities, that looks to the farmer as a horizontal/flat line -- the standard model.

2. As we sort of agreed earlier, elasticity reflects a (p, q) position on the DC. Therefore assuming the same DC and the elasticity equal infinity, we get to the intercept price which is clearly absurd. Elasticity is not applicable to the entire curve only to a point on the curve except for some artificial cases.

Perhaps you meant something else, an individual farmer's DC that can be summed up to the market DC ? Not sure if this point of view is very productive.

Posted by: bankster | December 01, 2012 at 01:58 PM

Nick: Nope, I never sent it anywhere for publication. These days it would've been a blog post or series thereof.

Posted by: Chris Auld | December 01, 2012 at 02:05 PM

Also, on a more constructive note, a couple questions:

1. I find his agent-based model (ABM) interesting (currently trying to create a version to play around with), but I'm very curious to find out how sensitive its underlying assumptions. The one I'd be most interested to see the effect of would be even weak stochasticity in the demand function. Intuitively, it would seem that for a monopolist, a noisy price-demand function wouldn't matter much, but it would make the MR curve look much flatter to an individual producer.

2. Wouldn't Keen's arguments equally apply to consumers? In that case, why do his simulations all just assume that the observed price follow a simple demand curve, rather than also simulating consumers making multiple price offers? I think this may relate to question 1, as well, in that dynamic consumers would introduce endogenous stochasticity into the demand function.

Posted by: Eric Pedersen | December 01, 2012 at 02:32 PM

May I? A question.

Millions of farmers seems too theoretical to me. There are probably rather a couple of farmers in each region where trade between regions is costly. There is no MARKET curve for these farmers unless external MARKET price increases above additional costs they have to bear to transport their wheat. How do you build the MARKET curve under such conditions? And what form will it have?

Posted by: Sergei | December 01, 2012 at 02:39 PM

Without having read Keen's paper (I just can't bring myself to do it), your characterisation of the simulations reminds me of these two papers:

Steffen Huck, Hans-Theo Normann, and Jorg Oechssler. Zero-knowledge coop-

eration in dilemma games. Journal of Theoretical Biology, 220:47{54, 2003.

Steffen Huck, Hans-Theo Normann, and Jorg Oechssler. Through trial and error

to conclusion. International Economic Review, 45(1):205{224, 2004.

The basic idea is that a reasonable learning model (win-continue lose-reverse, that is valid in situations where it is not possible to calculate best responses) can lead Cournot duopolists to the collusive, rather than Cournot-Nash, outcome. Similarly, it can lead to the cooperative outcome in a tragedy of the commons environment.

Posted by: Evan | December 01, 2012 at 02:40 PM

Worth pointing out mainstream econ got stuck in its own "is game theory a superior replacement to marginalist thinking? Are most economic interactions better characterized by strategic rather than static considerations?" debate back in the 1980s. The answer was, eventually, "no'.

Posted by: david | December 01, 2012 at 02:46 PM

Eric: my apologies to the Canadian NFU. "Milk" would have worked better than "wheat", but we sold our (UK) milk quota long ago and went for cereals and beans. Yep, Canadian wheat farmers, even though large, are probably still too small a part of the world wheat market for Canadian wheat quotas to raise prices much.

Sandwich: that's from the Bible, isn't it?

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.

bankster: If the individual farmer cuts his output 1%, all other farmers' outputs staying the same, the effect on price will be one million times smaller than if all farmers cut output by 1%. That's all I need to say that his demand curve is a million times more elastic than the market demand curve.

Posted by: Nick Rowe | December 01, 2012 at 03:13 PM

Nick,

"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."

The slope is only the same at all prices P if the market demand curve is linear. If instead the market demand curve has a constant elasticity then:

Qm = P^Em = Sum (Qi) = n * P^Ei for n firms

ln n = (Em - Ei) * ln P

Em - Ei = ln n / ln P

Em = Ei + ln n / ln P

For very large n's and small P's (a lot of farmers selling wheat) the elasticity of the market is overwhelmed by either the sheer volume of farmers or the very small margins irrespective of the elasticity of any one farmer. The opposite is true of a monopoly single farmer.

And so does the individual farmer know from going to the NFU meeting what the shape of the market demand curve is, and if so what is it?

Posted by: Frank Restly | December 01, 2012 at 03:40 PM

Nick, Wasn't yours, too?

Posted by: Sandwichman | December 01, 2012 at 04:39 PM

Chris: that's why blogs are good.

Eric: 1. My hunch is that stochastic market demand wouldn't make much difference. But looking over the hedge to see what the richer/poorer farmer next door was doing would make a big difference. (My father would always look over hedges. Nowadays at universities we call it "benchmarking".)

2. Hmmm. Probably.

Evan: that does sound very similar. Just skip down to page 65.

Sergei: if firms' outputs are not perfect substitutes, it gets a little more complicated. You can still write an individual firm's quantity demanded as a function of all firms' prices, or an individual firm's price as a function of all firms' quantities (not just the sum).

Posted by: Nick Rowe | December 01, 2012 at 05:36 PM

"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."

The slope is only the same if we assume that all 999,999 other farmers hold their q's constant. Any reasonable model or simulation that includes all 1,000,000 farmers is going to violate that assumption.

Posted by: Min | December 01, 2012 at 06:53 PM

Nick:

" If the individual farmer cuts his output 1%, all other farmers' outputs staying the same, the effect on price will be one million times smaller than if all farmers cut output by 1%. "

That's correct, but the sentence below does not follow:

"demand curve is a million times more elastic than the market demand curve."

because the 'standard' elasticity does not express what you are trying to say. I do not even understand what "demand curve is more elastic" might mean since, as I've already said, the notion of elasticity is simply not applicable to the curve as a whole hence the above does not make mathematical sense.

Posted by: bankster | December 01, 2012 at 07:43 PM

Min: there's a difference between "constant" and "independent". Take a model where each farmer decides how much wheat to plant without observing other farmers' decisions. That's the Cournot-Nash model. A different model is the Stackelberg model, where 1 plants first, followed by 2, who observes 1, followed by 3, etc.

bankster: I'm talking about point elasticity.

Posted by: Nick Rowe | December 02, 2012 at 02:10 AM

Nick, do I understand you right? The outputs are perfect substitutes but trade involves costs which are like a step function. You say that such case is like firms not producing perfect substitutes?

But my question still is about the shape of the market demand curve. What is it?

Posted by: Sergei | December 02, 2012 at 02:30 AM

Sergei: here are two cases:

1. All the buyers are in one location. Sellers are spread out, and have transportation costs of getting their product to market. That case is simple. Just add the transport costs to each seller's MC curve.

2. The buyers are spread out over space, and the sellers are in fixed locations. The classic model is Hotelling's

Posted by: Nick Rowe | December 02, 2012 at 02:55 AM

Actually, not Hotelling's, because Hotelling's ice cream sellers can move. But I've seen a variant somewhere with fixed locations for sellers. Differential transportation costs make the different firm's wheat imperfect substitutes, even if the wheat is all the same.

Posted by: Nick Rowe | December 02, 2012 at 03:16 AM

Dear Nick,

Thanks for your post. You have actually engaged more with my analysis than any other Neoclassical economist to date, so I will happily write a response.

However I'm too busy with other issues to respond in detail for about a month. In the meantime, can I suggest another reading for you: Chapter 4 of Alan Blinder's "Asking About Prices":

http://www.amazon.com/Asking-about-Prices-Understanding-Stickiness/dp/0871541211

Cheers, Steve Keen

Posted by: ProfSteveKeen | December 02, 2012 at 06:32 AM

Nice post (and Chris artikel was good to).

but what is this Cournot-Nash model you are talking about? In the regular Cournot model, each firm certainly do observe the other firms decisions and adjust its output in responce to it. (And if we assume that they carry through with the calculation, instead of just responding to the other firms behavior, they set their output to a level which is optimal with respect to their competitors likely responces)

Posted by: Nemi | December 02, 2012 at 07:22 AM

Thanks Steve. Well, maybe, this post was worth writing.

I read an article by Blinder years ago, reporting the results of a survey he had done, asking firms why they had sticky prices. I think that's the same thing. I thought it was an important article. I thought about the reasons the firms had given, and how to model them. Most of them are beyond my abilities to model usefully. I did once build a macro model explaining why firms had sticky prices. But in hindsight I'm not sure I like my model. But I think (most) prices are sticky, and price stickiness matters for short run macro, and maybe for some long run macro too. I assume prices are sticky. But I wish I understood better why prices are sticky, and exactly how and when they change, and what the short run Phillips Curve really looks like. Sometimes I try to have another go at that question. Mostly I feel old and hope the next generation will do better at solving it.

Nemi: Cournot-Nash is just another, more formal, name for Cournot.

Assume 2 firms in a one-period game, where each firm only has one move. If one firm moves first, and the second firm observes the first firm's move before making its own move, we get a very different equilibrium. That's Stackelberg. If both firms move at the same time, so neither gets to see the other's move before deciding its move, that is Cournot-Nash.

In a repeated game, with multiple periods, things can get hairy. Especially if the players don't know when the game will end. They might collude. I used to follow the game theory/IO literature on this, long ago. But now I don't.

I wonder, if there are n players, moving in sequence, does Stackelberg equilibrium converge to Cournot equilibrium when n gets very large? I think it does. Does anybody know?

Posted by: Nick Rowe | December 02, 2012 at 08:44 AM

Chris: I read that paper, it was great!

Nick: Off topic, but could you pretty please do a response to this new David Glasner post? It was his contribution to MoA-MoE - he talks about how he disagrees with you and Bill on the endogeneity of inside money. Hoped that you could do a post straightening this out. Thanks in advance!

http://uneasymoney.com/2012/11/25/its-the-endogeneity-redacted/

Posted by: Saturos | December 02, 2012 at 08:50 AM

Nick Rowe: "there's a difference between "constant" and "independent".

That's my line, Nick. :)

Suppose that Q = q1 + q2. Then

ΔQ = Δq1 + Δq2 , and

ΔQ/Δq1 = 1 + Δq2/Δq1 , and

∂Q/∂q1 = 1 , and

∂q2/∂q1 = -1

The last equation is true, even if q1 and q2 are causally independent.

The next to last equation is not what you get if you let the differences approach 0 in the limit. You have to treat q2 as constant, not as causally independent.

What you seem to be saying, and where you may differ with Keen, is that if the agents do not collude, they should use the next to last equation, and assume that everybody else makes no change. That is different from assuming independence.

Posted by: Min | December 02, 2012 at 06:34 PM

Nick,

"I wonder, if there are n players, moving in sequence, does Stackelberg equilibrium converge to Cournot equilibrium when n gets very large? I think it does. Does anybody know?"

See http://en.wikipedia.org/wiki/Stackelberg_competition

Look under the heading credible and non-credible threats by the follower. The follower can punish the leader by choosing a non-optimal Stackelberg quantity that lowers the profitability of both the leader and the follower below the Cournot equilibrium.

Posted by: Frank Restly | December 02, 2012 at 06:44 PM

Nick,

"But I think (most) prices are sticky, and price stickiness matters for short run macro, and maybe for some long run macro too. I assume prices are sticky. But I wish I understood better why prices are sticky."

Prices are sticky because the cost of money (the interest rate) is limited to a 0% lower bound on a pre-tax basis. In credit based currencies, money begins as a legally binding obligation between borrower and lender. However there is no legal obligation to purchase goods and services. Hence you have a mismatched system, borrowers must repay lenders but holders of currency may shop at their leisure.

Posted by: Frank Restly | December 02, 2012 at 07:03 PM

Duck Duck Go says King James Bible (Cambridge Ed.) Matthew 7:3

And why beholdest thou the mote that is in thy brother's eye, but considerest not the beam that is in thine own eye?

Posted by: walt | December 02, 2012 at 07:17 PM

walt: 1. You seem to have missed Sandwichman's having made that exact same rather unproductive comment.

2. More importantly, you seem to have missed that what this post is doing is precisely what you say I should be doing. I am considering a very large beam that Steve Keen says is in my neoclassical eye.

Posted by: Nick Rowe | December 03, 2012 at 08:16 AM

Thanks for engaging with (and agreeing with?) Keen, and in such a civil manner. Helps us avoid the usual vitriol of heterodox versus mainstream.

Having said that, this post confuses me. Economists pride themselves on the logical consistency of their theories of profit maximisation - in fact, so much so that they seem to value it above the real world. Take a textbook I own:

"…[the student] rightly assumes that few firms can have any detailed knowledge of marginal revenue or marginal cost. However, it should be remembered that marginal analysis does not pretend to describe how firms maximise profits or revenue. It simply tells us what the output and price must be if they do succeed in maximising these items, whether by luck or judgement."

So what about if we replaced appropriate words to include Keen's qualifications? Surely the argument would still apply?

Now, you seem to imply this is a silly perspective in your post, and we should look at what farmers actually do. I agree. So why not just accept that they use cost plus pricing and do away with the whole marginalist theory of the firm?

Posted by: UnlearningEcon | December 03, 2012 at 11:00 AM

Unlearning: very briefly, because I gotta go teach. I agree (I think) with Steve on the "flatness" of the individual firm's demand curve and what we should mean by "flatness". But we disagree on the bigger stuff.

"So what about if we replaced appropriate words to include Keen's qualifications? Surely the argument would still apply?"

Sorry, you lost me there.

"Now, you seem to imply this is a silly perspective in your post,..."

I don't think I meant to imply that, or maybe you lost me.

"So why not just accept that they use cost plus pricing and do away with the whole marginalist theory of the firm?"

1. Because the (wheat) farmers I know don't do cost plus pricing. They don't really do pricing at all. They choose q, not P. And they try to set q roughly where they think MC will equal P, as far as they can figure that out.

2. Because "cost plus pricing" can mean "marginal cost plus a-factor-that-depends-on-elasticity-of-demand pricing", which is a marginalist theory of the firm, and very textbook.

With some trivial math (which I have probably screwed up), my "set q where mr=mc" can be rewritten as "set P equal to [1/(1-1/e))]mc".

Posted by: Nick Rowe | December 03, 2012 at 11:48 AM

Nick, Perhaps the Sandwichman read too much into the "angry guy standing on a soapbox" framing of the post. At least there was nothing there about foaming at the mouth. But walt clearly didn't 'miss' Sandwichman's comment. He was playing straight man to your (presumably) ironic question about the passage being from the Bible.

Youneed to put yourself inmyshoes. I don't have the deftness with the math that you do. I have to go through it step by step to make sure no one's pulling a fast one on me. But meanwhile I pick up on the dismissive verbal cues and gestures. So by the time I get to "Suppose there are a million small farmers, each growing wheat, and their wheat is all the same and all sells for the same price" what I'm hearing is "assume we have a can opener." What good is it going to do to "suppose there are a million small farmers..." when in the real world of Monsanto and Archer, Daniels, Midland, small farmers in India drink pesticide because, "the prospect of death is less of a threat to workers than going hungry and not being able to feed their families."To the Sandwichman, the beam is supposing there are a million farmers. The mote is knocking on one's door, smartly dressed.

Posted by: Sandwichman | December 03, 2012 at 11:55 AM

I apologize if I'm going into the weeds here ... I'm probably missing something. I tried reading the Keen paper, but right off I get stumped. After the quote from the Stigler paper they say (I'll use uppercase D for partial derivatives):

dQ/dq_i = dP/dQ * dQ/dq_i

= dP/dQ * (sum(j=1 to n) Dq_j/Dq_i)

= ...

by dQ/dq_t does the author mean the total derivative (since presumably Q(q_0, q_1, ... , q_n))? And if so, is it really just the sum of the partials w.r.t q_i?

Posted by: Patrick | December 03, 2012 at 12:08 PM

Off topic, but of interest to Steve Keen foes and fans - U Western Sydney's economics department is closing, and Professor Keen is in the firing line: http://www.abc.net.au/news/2012-11-06/uws-set-to-ditch-economics/4355272?section=nsw

Posted by: Frances Woolley | December 03, 2012 at 03:01 PM

Sandwichman: If you read Steve Keen's paper you will see he is making the same assumption. Lots of small farmers. That is not what this argument is about.

"I don't have the deftness with the math that you do."

I am very undeft with math.

Patrick: remember, Q = sum of all the little q's, and P is a function of that sum only. P(Q) = P(q0+q1+q2 etc.) I will have another look later. (The math formatting in that journal makes it a little trickier to read.)

Frances: yes. I think it has something to do with how the Australian government lets universities make conditional offers to students, affecting how many students apply in first and second rounds.

Posted by: Nick Rowe | December 03, 2012 at 05:38 PM

Interestingly, Nick, I think agriculture is one of the few areas that exhibits the kind of production function typically found in textbooks, so it's quite feasible that farmers are more likely to follow that kind of rule.

Having said that, a lot of evidence suggests the majority of firms use a cost-plus rule (anecdotally, all of my friends who study/have studied business or some form of industry have learned cost-plus pricing and know nothing about marginalism, unless they happened to take an economics module).

"Sorry, you lost me there."

Let me do the rewriting, then:

"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."

can turn into:

"…[the student] rightly assumes that few firms can have any detailed knowledge of the minimal effect of an increase in quantity on the market price. However, it should be remembered that marginal analysis does not pretend to describe how firms maximise profits or revenue. It simply tells us what the output and price must be if they do succeed in maximising these items, whether by luck or judgement."

"With some trivial math (which I have probably screwed up), my "set q where mr=mc" can be rewritten as "set P equal to [1/(1-1/e))]mc"."

To me this isn't the same as the usual cost-plus rule, which is AVC + x%. I know there are different versions but fundamentally there is no need to invoke marginal or even economic concepts.

Posted by: UnlearningEcon | December 03, 2012 at 06:18 PM

I've read your post, Auld's critique, Keen's article and book.

The basic story is that if the firm doesn't face a perfectly flat demand curve, and if there is any price change with their output, then the outcome of perfect competition is infeasible because all firms will simply reduce output (ever so slightly) to increase profits, until they all get back to the monopoly level of output.

While you say there are one million farmers, that means that each individual farmer only needs to reduce output by some fraction of one-millionth to get to the monopoly output.

If you take a look at it from a market-wide perspective, we have a bunch of producers essentially giving away surpluses to consumers because they can't coordinate. Yet so many repeated games show that cooperative strategies evolve and work with large groups. They are stable in the long term.

Another point (that hasn't really been addressed) is that firms don't maximise profits, but returns. You know, the % profit per unit of cost. Where there is a flat demand curve, this is at the point of minimum average total cost. Why go past this point? Since the demand curve is flat, you could build two factories/farms producing at this point and make greater returns than expanding production to MR (demand) = MC.

Posted by: Rumplestatskin | December 03, 2012 at 07:58 PM

Unlearning: One problem with the "set P = markup of x% over costs" rule is that it doesn't tell us what x is. Marginal analysis tells us that x=1/(e-1) (unless I've screwed the math up again). The more elastic is demand, the smaller the markup over mc. Those goods which have a less elastic demand (at the level of the individual firm) will have a higher markup. That's a (sort of) testable implication. And it looks right to me. The goods which do seem to have less elastic demands do seem to have higher markups. 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".

Posted by: Nick Rowe | December 03, 2012 at 08:13 PM

Patrick: which page/equations are you looking at there?

On page 62, equation 0.4, Steve Keen is maximising firm i's profit by taking the derivative of firm i's profit wrt aggregate Q, not firm i's qi. And he assumes just below equation 0.5 that dqj/dQ =1 for all j. That's where he loses me.

Posted by: Nick Rowe | December 03, 2012 at 08:31 PM

"If you read Steve Keen's paper you will see he is making the same assumption. Lots of small farmers."

I don't see "lots of small farmers," "millions of small farmers" or even "farmers." Page number?

Posted by: Sandwichman | December 03, 2012 at 08:40 PM

Rumplestatskin: "The basic story is that if the firm doesn't face a perfectly flat demand curve, and if there is any price change with their output, then the outcome of perfect competition is infeasible because all firms will simply reduce output (ever so slightly) to increase profits, until they all get back to the monopoly level of output."

It's that "...until they all get back to the monopoly level of output" I don't buy.

If the number of farmers is finite, the individual farmer's demand curve won't be perfectly elastic, so mr will be a little bit below P, so each inddividual 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.

"If you take a look at it from a market-wide perspective, we have a bunch of producers essentially giving away surpluses to consumers because they can't coordinate. Yet so many repeated games show that cooperative strategies evolve and work with large groups. They are stable in the long term."

Then why do Canadian dairy farmers care whether the government abolishes milk quotas? If you were right, they would coordinate to get the monopoly output even if there were no government quotas.

Posted by: Nick Rowe | December 03, 2012 at 08:44 PM

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?

Posted by: Nick Rowe | December 03, 2012 at 08:47 PM

I long ago put up an applet at http://www.dreamscape.com/rvien/Economics/Applets/KeenSimulation/KeenSimulation.html. The Java source, using some long-deprecated methods, is downloadable there as a tar file. This code implements the Keen and Standish simulation at some stage of their work. I forget the details.

I thought it was well known that almost everything in intro neoclassical textbooks was wrong.

The limit case that Nick wants to consider, in which a countably infinite number of firms each produce a quantity of zero, is logically inconsistent with the existence of a U-shaped cost curve. It is also rejected by Aumann.

I note that Nick, in his story about wheat farmers, never brings up the existence of whatever is the Canadian equivalent of the Chicago commodity exchange. In actually existing capitalism, a market that acts if it is perfectly competitive, in some sense, must be carefully constructed. One does not have merely firms (= plants) and consumers, but also speculators who are willing to take either side of the market, as desired. One also has traders who must "make" the market and a set of rules who matching bids and asks.

Most industrially-produced commodities do not have such markets and do have prices that are administrated. They do not face U-shaped cost curves in their plants. They are not trading-off marginal costs for known technology. They introduce new technology, usually irreversibly, over time though. The principle of substitution is both empirically and logically false.

.

Posted by: Robert | December 03, 2012 at 11:57 PM

Robert: The debate here is not about costs. It is about marginal revenue. Is the elasticity of demand for the individual wheat farmer equal to the elasticity of the market demand curve for wheat? I say it is much much greater. Do wheat farmer maximise joint profits of all wheat farmers? I say they don't.

Posted by: Nick Rowe | December 04, 2012 at 12:25 AM

Nick,

Em(P) = ( P * dQm/dP ) / Qm

Em(P) * dP / P = dQm / Qm

Integral ( Em(P) * dP / P ) = Integral ( dQm / Qm )

Integral ( Em(P) * dP / P ) = ln ( Qm )

Ei(P) = ( P * dQi/dP ) / Qi

Ei(P) * dP / P = dQi / Qi

Integral ( Ei(P) * dP / P ) = Integral ( dQi / Qi )

Integral ( Ei(P) * dP / P ) = ln ( Qi )

Assuming Qm = n * Qi

Integral ( Em(P) * dP / P ) = ln ( Qi ) + ln ( n ) = Integral ( Ei(P) * dP / P ) + ln (n)

Integral [ ( Em(P) - Ei(P) ) * dP/P ] = ln (n)

Em(P) - Ei(P) = P * d ( ln(n) ) / dP = P/n * ( dn/dP )

Em(P) - Ei(P) is nonzero only if dn/dP is nonzero. If the elasticity of the number of farmers with respect to price is zero, then the elasticity of market demand will be equal to the elasticity of individual demand.

Think of it this way. If changing the number of farmers does not lead to more or less competition (reflected in higher prices for less competition, and lower prices for more competition), then the market demand elasticity is equal to the individual demand elasticity. This is true irrespective of the shape (linear or non-linear) of the market and individual demand curves.

Posted by: Frank Restly | December 04, 2012 at 01:13 AM

Nick: You got much farther than I! Right off the bat in 0.1 where he applies the chain rule is where he looses me. Because I have a weak brain, I'm thinking of 2 producers producing q1 and q2. Q(q1,q2, ...) is what? A curve confined to lie in the q1 + q2 plane? So I'm just thinking that maybe the application of the chain rule doesn't all disappear quite like he claims. The Q curve could be 'curvy'.

To use a physical analogy, one can be at position 0 with any number of velocities (or similarly one can have 0 velocity, but very large acceleration.

But I confess that it's late and I'm probably missing something.

Posted by: Patrick | December 04, 2012 at 01:37 AM

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? If so that's not an error but a simplifying assumption. I remember being told we assum the producer is relatively small enough to ignore that term. This is very common, for example the decentralised neoclassical model with aggregate capital externalities, we assume households are sufficiently small to ignore their own impact on the aggregate. This sort of thing happens all the time because models, particularly the model of perfect competition, are simple stories not super accurate simulcra of reality, as Keen should realise because he uses models that miss out a lot himself. He presents this stuff as some sort of take down of the mainstream, no wonder he is ignored.

On the idea that firms won't necessarily compete but may learn to collude amongst themselves, we'll that's hardly a revolutionary idea, and of course true to some extent, in some cases. The predictions of perfect competition could be changed by any number of considerations

Posted by: Luis Enrique | December 04, 2012 at 02:47 AM

"fundamentally there is no need to invoke marginal or even economic concepts"

I'm going to assume you've expressed yourself poorly there, because otherwise you seem to be claiming that firms make costing and pricing decisions without thinking about what they'd expect revenues and profits to look like under different choices.

Obviously the quantity and price decisions of very few real world firms resemble those in (almost) perfectly competitive markets, and I'm sure there are lots of impeccably mainstream (industrial organization) economists who specialise in the details of firm pricing behaviour that could add a lot of detail here. Plus, I'm sure there's always going to be a gap between theory and practise, and a lot of rule-of-thumb behaviour goes on. None of this suggests that concepts such as elasticity of demand or marginal cost are of no use.

Posted by: Luis Enrique | December 04, 2012 at 06:19 AM

one more thought ... if I get the gist of Keen's agent based simulations correctly, then it is a nice illustration of the point that ABM is no panacea, because it is not free of eminently questionable simplifying assumptions. The agents are following a simple decision rule, which is just as unrealistic as anything mainstream economics ever uses. Do these agents even consider the strategy of undercutting each other and winning market share? Does the model include frictions such as habit formation amongst consumers that might make winning market share more valuable than a simpler model might predict? etc.

Posted by: Luis Enrique | December 04, 2012 at 06:26 AM

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)

Posted by: Robert | December 04, 2012 at 06:58 AM

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.

Posted by: Nick Rowe | December 04, 2012 at 07:01 AM

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.

Posted by: Nick Rowe | December 04, 2012 at 07:20 AM

And Robert: don't miss my short comment about Sraffa in that other post.

Posted by: Nick Rowe | December 04, 2012 at 07:25 AM

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.

Posted by: Luis Enrique | December 04, 2012 at 07:27 AM

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.

Posted by: Luis Enrique | December 04, 2012 at 07:30 AM

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.

Posted by: Nick Rowe | December 04, 2012 at 07:35 AM

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?

Posted by: Luis Enrique | December 04, 2012 at 07:56 AM

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.

Posted by: Nick Rowe | December 04, 2012 at 08:03 AM

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.

Posted by: Nick Rowe | December 04, 2012 at 08:12 AM

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.

Posted by: Nick Rowe | December 04, 2012 at 08:27 AM

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.

Posted by: Patrick | December 04, 2012 at 12:24 PM

Patrick: Work through the standard Cornot Nash model first. Here's an example with 2 firms by Martin Osborne at U of Toronto.

Posted by: Nick Rowe | December 04, 2012 at 12:52 PM

Nick: OK. Thanks.

Posted by: Patrick | December 04, 2012 at 04:04 PM

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

Posted by: Frank Restly | December 04, 2012 at 05:31 PM

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.

Posted by: Robert | December 04, 2012 at 07:40 PM

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.

Posted by: Nick Rowe | December 04, 2012 at 08:05 PM

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!

Posted by: Rumplestatskin | December 04, 2012 at 08:43 PM

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*.

Posted by: Evan | December 04, 2012 at 08:58 PM

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.

Posted by: Nick Rowe | December 04, 2012 at 09:39 PM

Ooops. I think that should be:

100 firms will set P=1.005

(Yep, I'm cr*p at math).

Posted by: Nick Rowe | December 04, 2012 at 09:43 PM

But it isn't about the math, Nick. It's about the assumptions. The math is just arm-waving.

Posted by: Sandwichman | December 04, 2012 at 11:10 PM

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".

Posted by: Evan | December 05, 2012 at 12:27 AM

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 - b*Q

With Cournot competition (assuming as you did MC = 1), the profit maximizing equation will be:

for two firms: 3 - 2*q*b - q*b = 1 => q = 2/(3*b)

for three firms: 3 - 2*q*b - q*b - q*b = 1 => q = 2/(4*b)

...

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 :)

Posted by: bankster | December 05, 2012 at 07:33 AM

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.)

Posted by: Nick Rowe | December 05, 2012 at 08:03 AM

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.

Posted by: Chris Auld | December 05, 2012 at 11:41 AM

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.

Posted by: bankster | December 05, 2012 at 02:04 PM

"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.

Posted by: UnlearningEcon | December 05, 2012 at 02:48 PM

Chris: that is very helpful. Thanks.

"Conjectural variations". That lovely old phrase takes me back a bit!

Back soon.

Posted by: Nick Rowe | December 05, 2012 at 02:59 PM

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?

Posted by: Min | December 05, 2012 at 09:19 PM

I've put up a follow-up blog post:

link here NR

Posted by: Chris Auld | December 06, 2012 at 04:41 PM

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.Posted by: Nick Rowe | December 06, 2012 at 04:55 PM

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).

Posted by: ??? | December 06, 2012 at 05:29 PM

??? 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.

Posted by: Nick Rowe | December 06, 2012 at 05:51 PM

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.

Posted by: Chris Auld | December 06, 2012 at 07:46 PM

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.

Posted by: Nathanael | December 10, 2012 at 03:06 AM

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.

Posted by: Nathanael | December 10, 2012 at 03:09 AM

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!

Posted by: Nathanael | December 10, 2012 at 03:11 AM

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.

Posted by: Nick Rowe | December 10, 2012 at 05:40 AM

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.

Posted by: UnlearningEcon | December 12, 2012 at 07:20 AM

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.

Posted by: Evan | December 12, 2012 at 12:39 PM

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.

Posted by: Nick Rowe | December 12, 2012 at 05:18 PM

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.

Posted by: Nathanael | December 13, 2012 at 01:54 AM

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?

Posted by: UnlearningEcon | December 19, 2012 at 02:26 PM

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.

Posted by: Nick Rowe | December 19, 2012 at 02:51 PM

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.

Posted by: Evan | December 19, 2012 at 05:40 PM