Elasticity, slope, scale, and collusion

Every year I teach "elasticity". And every year the students ask "Why not talk about "slope" instead?". They are familiar with "slope", but "elasticity" is a new concept. Why do we teach a new concept, if an old familiar concept would do just as well?

[For non-economists: the slope of a demand curve is the change in price divided by the change in quantity demanded; the elasticity of a demand curve is the percentage change in quantity demanded divided by the percentage change in price. Elasticity = (1/slope)x(Price/Quantity demanded).]

My normal answer is that elasticity is unit-free, while slope has the units dollars.years/tonnes squared (if price is measured in dollars per tonne and quantity in tonnes per year). So you can compare the elasticities of demand for wheat and electricity, but you can't compare the slopes, because you can't compare tonnes with kilowatts-hours.

But I think there's a better answer. "Elasticity" helps us distinguish the individual experiment from the market experiment. "Slope" doesn't. Things that look flat on one scale don't look flat on another scale (e.g. Earth).

Draw a downward-sloping demand curve for wheat. Now expand the scale on the horizontal axis. The demand curve looks flatter and flatter. To the individual farmer, who sees only a tiny slice of the whole demand curve, because even a 100% change in his output will cause only a tiny percentage change in total output, it will look perfectly flat. But what do we mean by "flat"?

The slope of the individual farmer's demand curve is exactly the same as the slope of the market demand curve. One extra tonne of wheat causes the price to drop by exactly the same amount whether he produces that extra tonne, or his neighbour produces that extra tonne, or a million farmers produce one extra gram each.

But the elasticity of the individual farmer's demand curve is very different from the elasticity of the market demand curve. If there are one million farmers, the individual farmer's demand curve will be one million times more elastic than the market demand curve. If one million farmers grow 1% more wheat the drop in price will be a million times bigger than if the individual farmer alone grows 1% more wheat.

When an individual farmer decides whether to grow more wheat, he treats "Price" and "Marginal Revenue" as the same thing. (Yes, I have talked to farmers.) This makes sense. There is a simple relationship between Marginal Revenue, Price, and Elasticity. MR=[1-(1/E)]P. So as E approaches infinity, MR approaches P. So MR and P are (almost) the same thing, if elasticity is (almost) infinite. He sees a flat demand curve for his wheat.

But when that same individual farmer speaks about production quotas and imports at the National Farmer's Union meeting, he doesn't see a flat demand curve for wheat. He knows that restricting production and restricting imports will increase the price.

We want to say, and we need to say, that the individual farmer's demand curve is "flatter" than the market demand curve. When we talk about "flatness" we are talking about elasticity. If we were talking about slope we would be talking nonsense.

If the farmers all colluded, and asked what would happen to their profits if they all increased or decreased output together, the elasticity would be much lower, and so MR would be much lower too, than if just one farmer increased his output. But the slope would be the same. The NFU has a lower elasticity than its members, but the same slope.

Addendum: The same is true when we move away from perfect competition.

Cournot duopolists selling identical products will each have a demand curve twice as elastic as their market demand curve, but with exactly the same slope. With differentiated products it won't be twice as elastic, but it will still be more elastic than if they colluded.

[In the Cournot-Nash equilibrium, each firm chooses output to maximise profits given other firms' outputs. I was implicitly assuming that farmers play Cournot because, well, that's mostly what they do. They decide how much wheat to plant, and then sell it at what they can get. They don't announce a price and then plant the quantity of wheat demanded at that price.]

Bertrand duopolists selling differentiated products will each have a demand curve that is more elastic than if they colluded.

[In the Bertrand-Nash equilibrium, each firm chooses price to maximise profits given other firms' prices.]

You can think of perfect competition as the limiting case of Bertrand-Nash, as the firms' products become perfect substitutes. Or as the limiting case of Cournot-Nash, as the firm's products become perfect substitues and the number of firms gets very large.

Comments

[Frances Woolley]

Nick - yes, spot on.

One other reason that it's potentially misleading to think of an elasticity as a slope: with a linear demand curve, the elasticity goes from being very small when price is close to zero, to one at the middle of the curve, to very large when quantity is closer to zero - even though the slope is the same at every point on the curve.

[Jim Sentance]

Or more generally, slope is only one component of the equation, the other being the relative price and quantity. So for a given slope, elasticity will vary as you move around in price/quantity space, either up and down a given line or from line to line.

[Nick Rowe]

Frances and Jim: when students ask you "Why don't we just use 'slope' instead?", what do you tell them?

[Frances Woolley]

Nick, this is something I struggle to explain - I free ride on your good efforts in ECON 1000.

I talk about elasticity most often in the context of calculating tax incidence. With linear demand and supply curves, tax incidence - i.e. the amount that the consumer pays increases or the amount that the firm receives decreases - depends entirely upon the relative slopes of the curves. So I'm definitely guilty of slipping between elasticity and slope and being overly casual about the difference between the two.

In my last ECON 2001 class I tried to emphasize the difference between the demand curve faced by the firm and the market demand curve - but the students could probably benefit from hearing it again!

[Nick Rowe]

Frances: With tax incidence, I don't *think* it matters whether we talk about relative slopes or relative elasticities. The P/Q terms just cancel out. But when we are comparing the individual demand curve with the market demand curve, we have P/q in the first case and P/Q in the second, and Q is bigger than q. Mankiw's first two applications of elasticity are farmers (comparing individual to market experiment for a new hybrid), and OPEC (again individual vs market experiment).

[Frank Restly]

Nick,

"My normal answer is that elasticity is unit-free, while slope has the units dollars.years/tonnes squared (if price is measured in dollars per tonne and quantity in tonnes per year). So you can compare the elasticities of demand for wheat and electricity, but you can't compare the slopes, because you can't compare tonnes with kilowatts-hours."

To even try to compare the elasticities for two different products, don't they have to be constant, price indifferent elasticities?

Product a: Wheat
Eda = Pa / Qd(Pa) * dQd / dPa = Ka : Demand elasticity for wheat
Ka * dPa / Pa = dQd / Qd(Pa)
Ka * ln (Pa) = ln ( Qd(Pa) )
Qd(Pa) = Pa ^ Ka

Product b: Energy
Edb = Pb / Qd(Pb) * dQd / dPb = Kb : Demand elasticity for kilowatt hours of energy
Kb * dPb / Pb = dQd / Qd(Pb)
Kb * ln (Pb) = ln ( Qd(Pb) )
Qd(Pb) = Pb ^ Kb

If instead Eda is a function of Pa and Edb is a function of Pb, then comparison is impossible.

[Sandwichman]

"One extra tonne of wheat causes the price to drop..."

Are you sure about this, Nick? Is it the extra tonne that causes the price to drop or the expectations of buyers and sellers (Keynes's 'beauty contest') that cause the price to drop -- or maybe not drop?

[Nick Rowe]

Frank: strictly speaking yes. In general, the elasticity will usually change as we move along a demand curve. But the same can be said of slope too. In practice we don't worry about this much. We talk about small changes relative to the original equilibrium point, and hope elasticity (or slope) doesn't change too much. Except for a few cases where we think a curve is sharply kinked. Like Marginal Cost where a firm hits capacity. Or Demand where rival firms match a price cut but don't match a price increase. Or Demand where existing customers hear about a price increase but potential new customers don't hear about a price cut.

(The bigger problem in teaching it is that we aren't allowed to use calculus in first year, so we have to fudge and talk about "arc elasticity" between two points A and B. And then we run into problems because the percentage change from A to B isn't the same as the percentage change from B to A. So we fudge again and calculate percentage change relative to the midpoint between A and B.

[Nick Rowe]

Sandwichman: Depends on whether you are talking about the spot price or the futures price, and on how easily wheat can be taken in or out of storage. Generally, it's both current supply and expected future supply too. At first year, we normally teach that quantity demanded depends both on current price and on expected future price. Sometimes we say that expected future price depends on expected future demand and supply. But we don't try to solve for the intertemporal equilibrium path.

[Jim Sentance]

I basically tell them what I posted, that we're measuring responsiveness and while slope tells us part of the story (what the change is), the other part is that it's relative. Where you are in price/quantity space tells you your reference point. Not hard to show with a few examples - moving down a single demand curve or shifting one out to double the quantity.

[Jim Sentance]

And with respect to using calculus, I think the problem is not so much that we're not allowed to use it in first year as that most economists are an awful lot better at calculus than geometry.

[Frank Restly]

Nick,

Having never actually constructed a supply or demand curve from market data (non-economist), I have a question - what do typical real world supply and demand curves look like? Are they linear (constant slope), are they inverse exponential (constant elasticity), are they something else, or are they an entirely theoretical construct?

"But the elasticity of the individual farmer's demand curve is very different from the elasticity of the market demand curve."

If the elasticity of the market demand curve is a constant, then the elasticity of the individual farmer's demand curve should match that of the market - yes?

[Nick Rowe]

Frank; "If the elasticity of the market demand curve is a constant, then the elasticity of the individual farmer's demand curve should match that of the market - yes?"

Emphatically NO. That's what this post is about.

Suppose they both have the same slope (which they will, at least locally, under the Cournot assumption). Let q be an individual farmer's output of wheat. let Q be total output of wheat. The elasticity of the individual farmer's demand curve is Ei = (1/slope)*(P/q). The elasticity of the market demand curve is Em = (1/slope)*(P/Q). Since Q is much bigger than q, Ei is much bigger than Em.

On your other question: we figure we are doing well empirically if we can get an estimate of the first derivative in the right ballpark, or at least the right sign. Trying to estimate the sign let alone magnitude of the second derivative is usually asking more of the data than it can tell us. There are some exceptions. But I'm not a good person to answer that question well.

[Frank Restly]

Nick,

Let me try to rephrase my question -

If the elasticity of the market demand curve is a constant:
Em = P / Q * ( dQ / dP ) = K

Q is simply the sum of all farmers q's at price P
Q = Sum ( qi )
dQ/dP = Sum ( dqi/dP )

Em = [ P / Sum ( qi ) ] * [ Sum ( dqi/dP ) ]

For the individual farmer
Ei = P / qi * dqi/dP
dqi/dP = Ei * qi / P

Em = Sum ( Ei * qi )/ Sum ( qi ) for all i from 1 to N (N being the number of farmers that make up the market)

If Em is equal to a constant K, can we say that Ei is equal to the same constant or is there some demand function qi (P) and some elasticity function Ei (P) such that the equation above is satisfied and Ei (P) is not a constant? I am making the assumption that all farmer demand curves are equivalent meaning that qi (P) and Ei (P) are not qi (P,i) and Ei (P,i). What if instead of constant slope demand curve we have a constant elasticity demand curve? Which are more prevalent in the real world - constant slope or constant elasticity?

[Nick Rowe]

Frank: start with the market demand curve. I'm going to write it as an inverse function: P = D(Q). Since Q = sum(qi), we can rewrite it as P = D(sum(qi)).

It must be true that dP/dQ = D' = dP/dq. The market demand curve and the individual demand curve must have the same slope (evaluated at a given Q).

Whether D( ) is a linear function (constant slope), or non-linear, is a separate question.

[bankster]

For a linear demand function, arguably, the most intuitive way of looking at elasticity of demand would be to associate elasticity with a position on the curve: the upper part of the curve corresponds to inelastic demand, the lower to elastic with the middle of the curve having the elasticity of one. It is easy to show that for any linear curve elasticity does not depend on the slope but only on the given price (and the intercept): e = P/(P0 - P) where P0 is the intercept.

Of course, for some non-linear demand curves, e.g. Q = P**2, elasticity may be constant at any point of the curve :)

[bankster]

The power of minus two of course in the above: Q=P**-2

[bankster]

Nick:

I am not sure that the individual farmer can discern "his" piece of the market demand curve. Assuming perfect competition, the farmer sees only horizontal demand curve, not some delta pertaining to his specific output, no ?

[Jacques René Giguère]

Nick: computing arc-elasticity with smaller intervals is a good way of showing the distant valley of calculus where flow milk and honey.
And drawing two parallel curves at different distance from the axis should be sufficient to show that slope and elasticity are different, no?

[Nick Rowe]

bankster: " It is easy to show that for any linear curve elasticity does not depend on the slope but only on the given price (and the intercept): e = P/(P0 - P) where P0 is the intercept."

Cool. I never knew that. You must be able to use that to "see" the elasticity of a non-linear demand curve too, by drawing a tangent to the curve, yes? I'm going to mull that one over.

bankster @7.05. Yep. That's how he sees it.

Jacques Rene: Yep on smaller intervals. I don't see how the two parallel curve things works. If you take two parallel linear demand curves, and consider points on them that are on a ray from the origin, I think those two points would have the same elasticity.

[Frank Restly]

Nick,

Em = P / Qm * ( dQm / dP )
Em * dP / P = dQm / Qm
Qm (P) = P ^ Em : Qm is dependent, P is independent
dQm / dP = Em * P ^ (Em - 1)
Pm (Q) = Q ^ ( 1/Em ) : Pm is dependent, Q is independent
dPm / dQ = 1/Em * Q ^ ( (1-Em) / Em )

Ei = P / Qi * ( dQi / dP )
Ei * dP / P = dQi / qi
Qi (P) = P ^ Ei : Qi is dependent, P is independent
dQi / dP = Ei * P ^ (Ei - 1)
Pi (Q) = Q ^ ( 1/Ei ) : Pi is dependent, Q is independent
dPi / dQ = 1/Ei * Q ^ ( ( 1 - Ei) / Ei )

"It must be true that dP/dQ = D' = dP/dq. The market demand curve and the individual demand curve must have the same slope (evaluated at a given Q)."

If dPm / dQ is equal to dPi / dQ then:
1/Em * Q ^ ( (1-Em) / Em ) = 1/Ei * Q ^ ( (1 - Ei) / Ei )
Em / Ei = Q ^ ( (1-Em) / Em - ( 1 - Ei) / Ei )
Q = ( Em / Ei ) ^ [ 1 / ( (1-Em) / Em - (1 - Ei) / Ei ) ]

Given a market demand curve that has a constant elasticity Em and an individual demand curve that has a constant elasticity Ei, there exists only one quantity Q at which the slopes of the curves will be equal. And so I am not sure what you mean when you say that the market and individual demand curves "must" have the same slope (evaluated at a given Q). In this case there is only one quantity ( Q ) where the slopes of both curves are equal.

Even if you set the inverse slopes ( dQm / dP and dQi / dP ) to be equal you get the following:

Em * P ^ (Em - 1) = Ei * P ^ (Ei - 1)
Em / Ei = P ^ ( Ei - Em )
P = ( Em / Ei ) ^ ( 1 / ( Em - Ei ) )

Meaning there is one price ( P ) at which the slopes are equal. At that price P we can rewrite the quantities:

Qi (P) = P ^ Ei = ( Em / Ei ) ^ ( Ei / ( Em - Ei ) )
Qm (P) = P ^ Em = ( Em / Ei ) ^ ( Em / ( Em - Ei ) )

Qm (P) = Sum ( Qi(P) )
Assuming there are N farmers and each farmer has the same demand at a given price P:

Qm(P) = N * Qi(P)

( Em / Ei ) ^ ( Em / ( Em - Ei ) ) = N * ( Em / Ei ) ^ ( Ei / ( Em - Ei ) )
( Em / Ei ) ^ ( ( Em - Ei ) / ( Em - Ei ) ) = N
( Em / Ei ) ^ 1 = N
Em = N * Ei

Now you have an equation that describes the elasticity of the market (Em) relative to the number of market participants (N) and the elasticity of the individual participants (Ei) at the price (P) where the inverse slopes of the individual and market demand curves are equal ( dQi / dP = dQm / dP ).

[Nick Rowe]

Frank: I didn't follow your math. But your last line looks almost right. Given N farms, all the same size, it should be:

Ei = N*Em. (Not "Em = N * Ei")

So I think you made a tiny math slip somewhere.

[Frank Restly]

Thanks Nick,

I think I found the math slip up after I posted. About half way down:

Em / Ei = P ^ ( Ei - Em )
P = ( Em / Ei ) ^ ( 1 / ( Em - Ei ) )

Here I transposed Em - Ei for Ei - Em and carried it through.

[JW Mason]

I usually come here to argue. But, yup, this is all right, and a good point too. I've done far less teaching than you have, but I also have struggled to explain the concept of elasticity -- not just what it means, but why it's needed. The farmer example is good.

[Ritwik]

I've always found the concept of elasticity to be much more economically intuitive than slope in a P,Q graph.

In general, since both P and Q are lower-bounded at 0, all logarithmic, geometric, percentage forms are more intuitive than arithmetic forms.

[Jacques René Giguère]

Nick: on the ray thing: of course they would. It is easy to show that it is a special case. And by gosh, just the formula show that there are two part to computing e, slope and ratio P/Q.
My class is not generally composed of future Nobel ( one is a highly-paid investment banker in Chicago and a couple are econometricians, one of which a student of Stephen) but I think the get it.

[bankster]

Jacques René Giguère:

For a linear demand curve, the most often used in various micro expositions, the slope is irrelevant. At the same time, almost all people I know who went through a typical micro (e.g. Pindyck) tend to equate the slope with elasticity having acquired a wrong intuition of two products linear demand curves having different "elasticities" due to them having different slopes due to the demanded products' nature.

The slope plays a role with non-linear curves (e.g. e(A*Q**e)) = e), but I am not sure how good/better of an approximation to "real" demand such forms might be.

[tomslee]

when students ask you "Why don't we just use 'slope' instead?", what do you tell them

I must be missing something here. Isn't "elasticity" the actual thing, and "slope" just an artefact of one presentation? No one would say you shouldn't use "velocity" instead of "slope of a graph of position vs time".

[bankster]

tomslee:

Instant velocity *is* "slope of a graph of position vs time".

Elasticity is not.

Perhaps, the notion of elasticity is not particularly useful at all since it causes so much confusion.

[Jacques René GIguère]

bankster: of course,for a linear demand curve, the slope is irrelevant along the same demand curve, in the sense that what varies is P/Q. But that is the point: same slope, different elasticity. Same thing with two parallel linear curves: same slope, different elasticity.
I just went through my copy of Pyndick, as well as other texts, and I don't see in them reasion why one would mistake slope for elasticity. Confusion more due to bad, too fast and sloppy exposition plus lack of real interest by the learner than confusing presentation.

[Min]

Just to check:

slope = ΔP/ΔQ (No calculus!)
elasticity = δQ/δP

Right?

Nick Rowe:"The bigger problem in teaching it is that we aren't allowed to use calculus in first year, so we have to fudge and talk about "arc elasticity" between two points A and B. And then we run into problems because the percentage change from A to B isn't the same as the percentage change from B to A. So we fudge again and calculate percentage change relative to the midpoint between A and B."

Are you allowed to use logarithms in the first year? Plotting elasticities on log/log paper would make sense and emphasize the difference between the demand and elasticity. :)

Also, I am not sure that this matters, but humans are often sensitive to logarithms. Examples: the volume of sound, the Richter scale for earthquakes, Bernoulli's moral value of money.

Ritwik: "I've always found the concept of elasticity to be much more economically intuitive than slope in a P,Q graph.

"In general, since both P and Q are lower-bounded at 0, all logarithmic, geometric, percentage forms are more intuitive than arithmetic forms."

:)

Jim Sentance: "And with respect to using calculus, I think the problem is not so much that we're not allowed to use it in first year as that most economists are an awful lot better at calculus than geometry."

Good point. Newton used geometry in his Principia rather than calculus, right? His readers did not know calculus. ;)

Nick Rowe: "It must be true that dP/dQ = D' = dP/dq. The market demand curve and the individual demand curve must have the same slope (evaluated at a given Q)."

Then dq/dQ = 1.

Nick Rowe: "In the Cournot-Nash equilibrium, each firm chooses output to maximise profits given other firms' outputs. I was implicitly assuming that farmers play Cournot because, well, that's mostly what they do."

Then dq/dQ is not constant. Doesn't that contradict the previous statement? ;)

If a particular farmer is contemplating changing his output while assuming that other farmers will not do the same, he cannot be assuming that his output will keep the same proportion to the whole output.

[Min]

This may be slightly off topic, but not much. I would appreciate your comments. :)

I watched Ken Burns's documentary on the Dustbowl the other night. I found the following narrative interesting from an economic point of view.

At first, the market crash and Great Depression were hardly felt in the Great Plains. Then one year (I don't remember exactly when) wheat prices dropped significantly, about 30% (δP !) as I recall. In response the farmers planted more wheat and had a bumper crop. Predictably, the bottom dropped out of the market and wheat was piled up, unsold. In the documentary one person commented that the answer was always to produce more. If the price went up, produce more. If the price went down, produce more.

Thanks. :)

[bankster]

Jacques René GIguère:

"the slope is irrelevant along the same demand curve"

The slope is irrelevant amongst *multiple* non-parallel linear demand curves as long as they have the same origin: for a given price, elasticity would be the same for all the non-parallel linear demand curves, no matter what the slope is.

The statement is mathematically trivial, but completely surprising and non-intuitive to all the folks who took macro at some point that I had an opportunity to discuss E with...

[Jacques René GIguère]

bankster: in introductory, very basic micro, we don't go there. Just don't have the time, and most students have other interests.
Just like those weird curves we see in Grad Micro: downward-sloping supply and upward-sloping demand curves, both curves in the same directions and so on... Wonderful mental practice. In real life, you barely get stranger things than Giffen goos and backward labor supply curves.

[Michael]

Nick wrote: "The slope of the individual farmer's demand curve is exactly the same as the slope of the market demand curve"

I think the "true" market demand curve is a discontinuous step function and the continuous market demand curve is an estimate of that "true" curve. The individual farmer faces the discontinuous step function not the continuous demand function, and so the slop of the farmer's demand curve and the slop of the (continuous) market demand curve are not exactly the same. (As a loyal Marshallian, that's my 2 cent view anyway)

[Michael]

bankster: " It is easy to show that for any linear curve elasticity does not depend on the slope but only on the given price (and the intercept): e = P/(P0 - P) where P0 is the intercept."

Lynne Pepall, Dan Richards, and George Norman, Industrial Organization: Contemporary Theory and Empirical Applications uses this measure of elasticity in a number of models (Chapters 5 & 6 for example).

[Frank Restly]

Min,

"It must be true that dP/dQ = D' = dP/dq. The market demand curve and the individual demand curve must have the same slope (evaluated at a given Q)."
"Then dq/dQ = 1."

Not necessarily, this is only the case at one price P AND only if Ei is equal to Em.

dQm / dP = Em * P ^ (Em - 1)
dQi / dP = Ei * P ^ (Ei - 1)

If we set the two slopes equal to each other then we know that the price P is:

Em / Ei = P ^ ( Ei - 1 - Em + 1 )
P = ( Em / Ei ) ^ ( 1 / ( Ei - Em ) )

Back solving for the derivatives:

dQm / dP = Em * [ ( Em / Ei ) ^ ( 1 / ( Ei - Em ) ) ] ^ ( Em - 1 )
dQm / dP = Em * [ ( Em / Ei ) ^ ( Em - 1 ) / ( Ei - Em ) ]
dQm / dP = [ Em ^ ( ( Ei - 1 ) / ( Ei - Em ) ) ] * [ Ei ^ ( ( Em - 1 ) / ( Em - Ei ) ) ]

For dQm / dP to be equal to 1:

Em ^ ( ( Ei - 1 ) / ( Em - Em ) ) = Ei ^ ( ( Em - 1 ) / ( Em - Ei ) )
Em ^ ( Ei -1 ) = Ei ^ ( Em - 1 )
( Ei - 1 ) * ln ( Em ) = ( Em - 1 ) * ln ( Ei )
ln ( Ei ) / ( Ei - 1 ) = ln ( Em ) / ( Em - 1 )

One solution to this equation is to set Ei equal to Em. That will mean both of the elasticities will be equal and both the slopes will be equal at price P. There are other solutions but they involve numbers in the complex plane.

[Michael]

Regarding Ken Burns's documentary on the Dustbowl: "wheat prices dropped significantly, about 30% .... In response the farmers planted more wheat and had a bumper crop."

In a normal textbook explanation of supply curves, when the price falls some of the resources used in wheat production will be reallocated to the production of other crops. But during the dustbowl era wheat was the only crop planted, regardless of price. Resources were overly specialized and substitution in production was too limited.

[bankster]

Michael:

Great minds think alike:)

Was it in the context of monopolistic behavior ? That's seems to be the only place the whole notion may provide some insight, imho.

I cannot recall specific books and spots in the books, but I do remember some talk of elastic vs inelastic demand *curves* in those books that leads one to believe that elasticity is an attribute of the entire curve rather than the portion of such a curve (in general, excepting border cases).

[Nick Rowe]

tomslee, and Min "slope = ΔP/ΔQ (No calculus!)
elasticity = δQ/δP

Right?"

Wrong. It's got nothing to do with calculus vs discrete changes. (Point elasticity uses calculus, arc elasticity uses discrete changes).

Slope is change in P divided by change in Q. Elasticity is percentage change in Q divided by percentage change in P.

Elasticity = (1/slope)(P/Q)

[Min]

Nick Rowe:

"tomslee, and Min "slope = ΔP/ΔQ (No calculus!)
elasticity = δQ/δP

Right?"

Wrong. It's got nothing to do with calculus vs discrete changes. (Point elasticity uses calculus, arc elasticity uses discrete changes).

Slope is change in P divided by change in Q. Elasticity is percentage change in Q divided by percentage change in P."

Dear Nick,

Those are not calculus symbols. You said you are not using calculus. :)

ΔP = P1 - P0 (or whatever subscript you want to use)
δP = ΔP/P

OK? :)

[Min]

@Michael

Thanks. :)

@Frank Restly

I think that you are arguing with Nick, as I am.

[Michael]

Bankster: Yes, monopolistic behavior. Specifically, 3rd and 2nd degree price discrimination. In the case of 3rd degree price discrimination, a firm will maximize profit by setting a higher price in markets with more inelastic demand. One of the illustrative models used by Pepall, Richards and Norman have linear demand functions with the same slopes but different intercepts:
Market A: P = 36 – 4Q
Market B: P = 24 – 4Q
So, as you said: "elasticity does not depend on the slope"

For these demand functions, if mc=4, then the profit maximizing single price is 17; e=-.89 in market A; and e=-2.4 in market B. With price discrimination, therefore, price in market A should be 20 and in market B should be 14. That is, price should be higher in more inelastic market.

[Nick Rowe]

Min: "δP = ΔP/P"

Wow! Sorry. I had never seen that squiggly d thingy before. What do you call it (how to you say it)?

[Peter]

Frank: in my experience aggregate demand is approximately linear for small price changes. I have not come across large price changes independent of a major structural change as well.

I agree with previous comments that the most important difference between slope and elasticity is that even with a constant slope elasticity changes depending on where on the curve you are. In my experience people understand the change in Q (due to a change in P) far more easily than a change in MR due to a change in Q.

[Min]

Nick Rowe: "I had never seen that squiggly d thingy before. What do you call it (how to you say it)?"

It's a lower case Δ, so I guess you just call it delta. I have never heard it pronounced. I just say, "relative difference." :)

[Min]

Nick Rowe: "But I think there's a better answer. "Elasticity" helps us distinguish the individual experiment from the market experiment. "Slope" doesn't. Things that look flat on one scale don't look flat on another scale (e.g. Earth)."

For elasticity try plotting the demand curve on log/log paper (and switch the axes). I expect that that is the different look that you want. :)

Frances Woolley: "One other reason that it's potentially misleading to think of an elasticity as a slope: with a linear demand curve, the elasticity goes from being very small when price is close to zero, to one at the middle of the curve, to very large when quantity is closer to zero - even though the slope is the same at every point on the curve."

Plot the curve on log/log paper. :)

Your students can then **see** the difference.

[Min]

There are some assumptions floating around here. :) Let Q denote the total quantity demanded and q denote the quantity for the individual farmer.

First, let the farmer assume that the quantities for all other farmers is a constant, C0. Then Q = q + C0. Whether that is a rational assumption is another matter. It violates the Copernican Principle, that you are not special. (OC, in Lake Woebegone, every farmer is special. ;))

Nice Rowe: "The slope of the individual farmer's demand curve is exactly the same as the slope of the market demand curve."

That follows from the assumption. :) If Q = q + C0 then ΔQ = Δq. And ΔP/ΔQ = ΔP/Δq.(Look, Ma! No calculus! ;))

Nick Rowe: "But the elasticity of the individual farmer's demand curve is very different from the elasticity of the market demand curve."

Well, lessee. We already know (by assumption) that ΔQ = Δq. Then the elasticity of the market demand curve is (ΔQ/Q)(P/ΔP) = (Δq/Q)(P/ΔP). I suppose that the individual farmer's elasticity is (Δq/q)(P/ΔP). Then it would be greater than the market elasticity by a factor of Q/q. That's what you mean, right? :)

Nick Rowe: "We want to say, and we need to say, that the individual farmer's demand curve is "flatter" than the market demand curve."

Well, by assumption, you just said that it isn't. If you say that at the same time I think you will confuse your students. Use log/log paper. :)

[Min]

Nick Rowe: "When an individual farmer decides whether to grow more wheat, he treats "Price" and "Marginal Revenue" as the same thing. (Yes, I have talked to farmers.) This makes sense. There is a simple relationship between Marginal Revenue, Price, and Elasticity. MR=[1-(1/E)]P. So as E approaches infinity, MR approaches P. So MR and P are (almost) the same thing, if elasticity is (almost) infinite. He sees a flat demand curve for his wheat."

I think that the individual farmer is simply assuming that ΔP = 0. :) That gives you the flat demand curve. You don't even have to talk about elasticity. ;) (The farmer is not assuming that MR = P(1 - q/Q), he is assuming that MR = P.)

[Min]

Correction:

The farmer is not assuming that MR = P - ΔP/δq, he is assuming that MR = P.

[Frank Restly]

Min,

Starting with these identities:
Qm (P) = P ^ Em
dQm / dP = Em * P ^ (Em - 1)
Qi (P) = P ^ Ei
dQi / dP = Ei * P ^ (Ei - 1)

If Qm = Qi + C0:
P ^ Em = P ^ Ei + C0
dQm / dP = Ei * P ^ (Ei - 1) = dQi / dP
Meaning ΔQm/ΔP = ΔQi/ΔP, so far so good. But you list the inverse derivatives ΔP/ΔQm = ΔP/ΔQi.

For that we need to express P as a function of Qm and then P as a function of Qi.

P = Qm ^ (1/Em) = Qi ^ (1/Ei)
dP / dQm = 1/Em * Qm ^ (1/Em - 1)
dP / dQi = 1/Ei * Qi ^ (1/Ei - 1)

We know that Qm = Qi + C0 and so:
dP / dQm = 1/Em * (Qi + C0) ^ (1/Em - 1)
dP / dQi = 1/Ei * Qi ^ (1/Ei - 1)

Setting these two equal to each other you get:
1/Em * (Qi + C0) ^ (1/Em - 1) = 1/Ei * Qi ^ (1/Ei - 1)
(Qi + C0) ^ (1/Em - 1) = Em/Ei * Qi ^ (1/Ei - 1)
Qi + C0 = [ Em/Ei * Qi ^ (1/Ei - 1) ] ^ [ Em / (Em - 1) ]

This is a rational power polynomial equation. There will be a spectrum of both real and complex solutions for Q1 when Em, Ei, and C0 are constants. What should be obvious though is that dP / dQi is not equal to dP / dQm for all Q.

[nemi]

I do not understand this post.

Assume the demand function:

(q1+q2+q3+…+qn)=Q=a-b*P

The slope and the elasticity of the demand curve is the same for the individual company and the market.

If we are talking about what amount of stuff an individual company can sell at different prices it depends on the demand as well as the other companies cost curves (and behavior). If they have constant marginal cost, and act competitively, the individual company´s elasticity of demand is infinitely elastic (and not a million times more elastic).

Maybe you reason something like this:

If every plant has some high enough fixed cost (so that it is not more efficient to build new plants) and the same increasing marginal cost they will all react the same way to a price change (e.g. increasing their output by one unit each). If you aggregate all these responses you can calculate the elasticity of supply for the market. Given that the firms are alike, the elasticity of supply will be exactly the same for the company (with respect to its supplied quantity) and the market (with respect to the overall supplied quantity), but each company will only make up a small part of the market response (a millionth part of the market response if there are a million firms).

I, however, fail to see what the last steps are in order to get the result that the individual firm face a demand that is a million times more elastic than the markets.

[nemi]

PS: Sorry, obviously you are thinking:

(q1(p)+q2(p)+...+ qn(p))=Q(P)=a-bP

[Gene Callahan]

Michael at 6:11: "and so the slop of the farmer's demand curve and the slop of the (continuous) market demand curve are not exactly the same."

So this must be a pig farmer, not wheat, you are talking about?

[Nick Rowe]

nemi: "Assume the demand function: (q1+q2+q3+…+qn)=Q=a-b*P"

That is what I was thinking (in the case where all firms' outputs are perfect substitutes).

Slope is -1/b, and that's the same for individual and market.

Market elasticity is bP/Q. Individual firm elasticity is bP/q1.

[Frank Restly]

Nick,

Q = a - b*P
dQ/dP = -b
P = a/b - Q/b
dP/dQ = -1/b : Slope for market

Q = Sum (q) = n*q
n*q = a - b*P
q = a/n - b*P/n
dq/dP = -b/n
P = a/b - q*n/b
dP/dq = -n/b : Slope for individual

Market elasticity = -b * P/Q
Individual elasticity = -b/n * P/q = -b * P/Q
Elasticities are equal

[Nick Rowe]

Frank: you just assumed that all firms collude. Yep, if they collude, the elasticities are equal. Your farmers are thinking: "Hmm, if I just my q by 1%, all other farmers will cut their q by 1% too, so P will rise by a lot, and we will all be richer."

They do think like that at the NFU meeting. They don't think like that when they are back at their farms. Which is why the NFU knows it needs legal quotas to stop their members overproducing.

[nemi]

NICK: You are right. That is the elasticity of the demand function with respect to the comany´s output (i do not know why I thought they would be equal). But that is not the demand function that the individual company is facing unless the other companys dont react to price (and/or quantity) changes, right?

[Frank Restly]

Nick,

"Your farmers are thinking: Hmm, if I just my q by 1%, all other farmers will cut their q by 1% too, so P will rise by a lot, and we will all be richer."

Why would prices rise by a lot if the market curve is linear as suggested by:

Q = a - bP
P(Old) = a/b - Q/b

If we cut market Q by 10% then:
P(New) = a/b - 0.9*Q/b

% Change in Price = [ P(New) - P(Old) ] / P(Old) = [ -0.9*Q/b + Q/b ] / [ a/b - Q/b ] = 0.1*Q / ( a - Q )
If 'a' is much, much bigger than Q then the change in price will be the same as the change in quantity produced (about 10%). Meaning collusion gains the farmers nothing.

----------------------------------------------------------------------------------------------------------

However if the market curve has constant elasticity then

Q = P ^ (1/c)
P(Old) = Q ^ c
If we cut market Q by 10% then
P(New) = (0.9 * Q) ^ c

% Change in price = [ Q^c - (0.9*Q)^c ] / Q^c = 1 - (0.9)^c
If the market elasticity (c) is a constant greater than 1, then the % Change in price would be > 10%. In that case it would behoove the farmers to collude.

Isn't this just a case of herd behavior against an unknown curve. No individual farmer is willing to wager his livelihood on a bet that the market curve has a constant elasticity greater than 1. And so all farmers behave as though the market is linear even if it isn't.

[Nick Rowe]

nemi: "But that is not the demand function that the individual company is facing unless the other companys dont react to price (and/or quantity) changes, right?"

Right. Strictly, I was assuming Cournot. Each firm chooses q taking other firms' q's as given. Bertrand, where they choose prices taking others' prices as given, is a little bit different.

Frank: If the market demand curve is very/infinitely elastic, then collusion gains the farmers little/nothing. If the market demand curve is inelastic, then collusion gains the farmers a lot.

[Frank Restly]

"Frank: you just assumed that all firms collude. Yep, if they collude, the elasticities are equal."

But if the individual and market demand curves are linear, and even if firms don't collude, then the elasticities of both curves are still equal (assuming that the market is made up of n farms of equal production q).

Individual elasticity for a single farm = -b/n * P/q = -b * P/Q (same as for the market).

I think where the disagreement occurs is what happens when a single firm tries to increase or decrease production. If all farms collude, then all farms increase or decrease production by the same amount (elasticities of all firms stay the same). If only one farm changes production, then obviously its demand elasticity will change while all others remain fixed if its demand curve is linear. The assumption of n farms of equal production q is no longer valid if just one firm changes production.

[Min]

"collusion: secret agreement or cooperation especially for an illegal or deceitful purpose"

We do not have to assume collusion for the farmers to consider the actions of other farmers. In particular, people in small towns talk, and it is not too hard to find out what your neighbor is up to. If a farmer who had heard about the depression back East and who discovered that the price of wheat had dropped significantly thought at first, "Well, if the price has dropped, I better grow more next year," it would not have taken much imagination to realize that a lot of other farmers were thinking the same thing. Now, of course, the fact that the next year they went ahead and produced a bumper crop the next year betrays a lack of sophistication which, for some reason, passes for rationality in some circles. It also shows that individual elasticity of demand was actually close to market elasticity, in that instance. :)

[Jacques René Giguère]

Min: "lack of sophistication"? A wheat grower is a wheat grower. What is he supposed to do? Build jet airplanes? He knows that the price has dropped. He has no idea if people will stop eating wheat and change their diet to whatever he will try to grow.

[Min]

@ Jacques René Giguère

I meant no disrespect to the farmers of the early 1930s. The overproduction, which happened before the Dustbowl and during the Great Depression was itself tragic, but understandable. Today's small farmers are much more financially sophisticated. :)

But note that the challenge facing farmers when their customers were in a depression was not something that individual farmers could solve. Not that there was any full solution, but it would have been good for them to get together. To collude, as Nick incorrectly puts it.

[Jacques René Giguère]

Min: never intended to say you dissed the farmers. My apologies if I wasn't clear.
In fact, farmers colluded. The whole system of price support, production quotas and various agricultural policies that are currently under attack as inefficient didn't arise in some vacuum peopled only by conniving dishonest peoples. They are solutions to real problems. Imperfect second or third order maybe , to be improved if possible, but still solutions to real and pressing problems. We are on the same side on this one.

[Min]

Jacques René Giguère: "We are on the same side on this one."

I am always heartened when I find myself agreeing with you. :)

[Min]

On the potential deceptiveness of calculus

Consider this equation in three variables:

1) z = x + y

It is true of these partial derivatives that

2) ∂z/∂x = 1 and
3) ∂z/∂y = 1

However, it is also true that

4) Δz/Δx = 1 + Δy/Δx and
5) Δz/Δy = 1 + Δx/Δy

when the denominators are not zero.

Furthermore, in equations 4) and 5) the difference quotients on the right side cannot be expected to go to 0 as the differences go to 0. That means that equations 2) and 3) cannot be considered as the limits of equations 4) and 5).

It also means that equations 2) and 3) together cannot represent a "collusion-free" state of affairs in which equation 1) is true and x and y are otherwise independent. To get equation 2) you hold y constant and to get equation 3) you hold x constant. While it may be true that equation 2) may be taken to represent a state of affairs for a person who controls the value of x and equation 3) may be taken to represent a state of affairs for a person who controls the value of y, the two together cannot be taken to represent a state of affairs in which both people control the values of their respective variables. The conditions of the two equations are incompatible. For equation 2) y is held constant while x varies and for equation 3) x is held constant while varies.

So when Nick says that the slope of the demand curve for an individual farmer is the same as the slope of the demand curve for the market (the sum of all farmers), that can only be true of one farmer at a time, pick a farmer. It cannot be true for all the farmers at the same time.

[Nick Rowe]

Min: "So when Nick says that the slope of the demand curve for an individual farmer is the same as the slope of the demand curve for the market (the sum of all farmers), that can only be true of one farmer at a time, pick a farmer. It cannot be true for all the farmers at the same time."

I'm not quite sure how to interpret that. Not sure if you are agreeing with me or not. Just in case you are disagreeing with me (and I apologise if what I am about to say is already obvious to you):

Assume Q=q1+q2+q3+...+qn and that P=a-bQ

The slope of the market demand curve is dP/dQ = -b. Elasticity = (1/b)P/Q

Consider two cases:

1. dP/dq1, where dq2/dq1=dq3/dq1=....=dqn/dq1=0. That is dP/dq1=-b. Elasticity = (1/b)(P/q) = (Q/q1)(1/b)(P/Q)
The slope is the same as the market demand curve, the elasticity is much smaller.

2. dP/dq1, where dq2/dq1=dq3/dq1=....=dqn/dq1=1. That is dP/dq1=-nb.
Elasticity = (1/nb)(P/q) = (1/n)(Q/q1)(1/b)(P/Q).

The slope is much steeper than the market demand curve, but the elasticity is about the same (exactly the same if the farmer is average).

When the farmer is alone on his farm, deciding if his growing 1 more or less tonne of wheat would increase or reduce his profits, he thinks like 1.

When the farmers is at the NFU meeting, deciding whether to vote on a quota requiring every one of them to grow one more or less tonne of wheat, and whether this would increase or reduce his profits, he thinks like 2.

It's Prisoners' Dilemma.

[Min]

Nick Rowe: "Not sure if you are agreeing with me or not."

I'm not sure, either. That's why I wrote that. :)

Nick Rowe: "When the farmer is alone on his farm, deciding if his growing 1 more or less tonne of wheat would increase or reduce his profits, he thinks like 1."

Not if I am the farmer. I have heard of the Copernican Principle, and I know that I am not special. ;) (Actually, I knew that before I had heard of the Copernican Principle, but it sounds cool. ;)) i usually come out somewhere between 1 and 2. :) OTOH, I may have reason to think that I am special. For instance, I am often an early adopter of change, and I know that most people are not. :)

If I had been a farmer in Oklahoma and had heard about the depression back East, and then had seen the price of my wheat drop significantly, I would have been very afraid.