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.
Worthwhile Canadian Initiative 
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.
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.
PS: Sorry, obviously you are thinking:
(q1(p)+q2(p)+…+ qn(p))=Q(P)=a-bP
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?
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.
Nick,
Q = a – bP
dQ/dP = -b
P = a/b – Q/b
dP/dQ = -1/b : Slope for market
Q = Sum (q) = nq
nq = a – bP
q = a/n – bP/n
dq/dP = -b/n
P = a/b – qn/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
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.
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?
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 – 0.9Q/b
% Change in Price = [ P(New) – P(Old) ] / P(Old) = [ -0.9Q/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.
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: 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.
“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. 🙂
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.
@ 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.
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.
Jacques René Giguère: “We are on the same side on this one.”
I am always heartened when I find myself agreeing with you. 🙂
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.
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.
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.