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Source: p 96, Principles of Microeconomics, 7 Ed, 2014, by N Gregory Mankiw

Even though the slope of a linear demand curve is constant, the elasticity is not. This is true because the slope is the ratio of changes in the two variables, whereas the elasticity is the ratio of percentage changes in the two variables.

[Source:] Elasticity along a straight line demand curve varies from zero at the quantity axis to infinity at the price axis. ... At the midpoint ... elasticity is equal to one, or unit elastic.

Please explain the above intuitively? I ask NOT for a formal proof.

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  • $\begingroup$ @FooBar I apologise if you mind the rollback, but I prefer to include 'Intuition' because the excerpt above truly does explain why, but I seek intuition. $\endgroup$ – Greek - Area 51 Proposal Mar 3 '15 at 21:19
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    $\begingroup$ I disagree to the extent that it doesn't add relevant information to the title. "Intuition" actually is the standard type of question/answer for undergraduate questions. We could add that to almost all undergraduate question without adding any information. And the second part of the title makes it clear that the question is an undergraduate one. Also, as you emphasize in the question, you seek intuition. I do not see a reason to add additional "fluff" into the title. See also: meta.stackexchange.com/questions/10647/… $\endgroup$ – FooBar Mar 3 '15 at 21:23
  • $\begingroup$ @FooBar I can't speak for all undergrad questions, but I entitle my question 'Intuition' lest someone reads the title, thinks that my question excludes intuition, but then reads the content and then becomes confused as to whether I wish to focus on intuition? Does this explain my choice? $\endgroup$ – Greek - Area 51 Proposal Mar 3 '15 at 21:35
  • $\begingroup$ Why would someone think your question excludes intuition? Trust me, anyone who reads a title about such a basic thing, will assume that intuition is required. Even if not, "no intuition" is not the default - and your question makes clear what you want. By your logic, you might as well put "No Italian answers please" in your title, just to ensure that you won't get any. $\endgroup$ – FooBar Mar 3 '15 at 22:41
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Think of it this way. A yacht costs $\$0.01$. How much do you demand? Then double the price. Now how many would you demand?

Then consider something closer to a market price. Something in 6 or 7 figures I guess. If you've got some money, you might buy one at the market price. But then if that price doubles, wouldn't that affect your demand a lot more than the 1 to 2 pennies change?

Simply put, as you go up the demand curve, the same percentage change in price is a much bigger absolute movement in price.

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Say elasticity (of demand) gives the percentage change in quantity demanded in response to a one percent change in price. Since the change is porcentual, if you are in a point of the demand where consumption is low, then a one percent decrease in price will result in a relatively big change in consumption, so elasticity is relatively high. As the quantity demanded increases along the demand curve, the percentage increase in quantity resulting of a one percent decrease of price will decrease.

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Mathematically, the intuition is simple. (This is not a formal proof, though it can easily be turned into one. It's just another dimension of intuition -- symbolic intuition.)

The price elasticity of demand at $(P_0, Q_0)$ is the infinitesimal ratio of percentage change in quantity demanded ($dQ/Q_0$) to percentage change in price ($dP/P_0$).

$$\epsilon_d=\left|\frac{dQ/Q}{dP/P}\right|_{(P_0, Q_0)}=\color{red}{\left|\frac{dQ}{dP}\right|_{(P_0,Q_0)}}\cdot\color{blue}{\frac{P_0}{Q_0}}$$

When the demand curve is linear, the red expression is constant: it's just the slope of the demand curve. The blue expression, however, depends on the point at which the elasticity is calculated; even for a linear demand curve, it is not constant.

In fact, if you draw a typical demand diagram, you plot the inverse demand function, with $P$ on the vertical axis and $Q$ on the horizontal axis. In this case,

  • $P_0/Q_0>1$ when the point $(P_0, Q_0)$ lies above the line $P=Q$
  • $P_0/Q_0=1$ when the point $(P_0, Q_0)$ lies on the line $P=Q$
  • $P_0/Q_0<1$ when the point $(P_0, Q_0)$ lies below the line $P=Q$

Thus, it is easy to see why $\epsilon_d$ changes along even a linear demand curve. The inverse demand curve will cut through Quadrant I of the $QP$-plane. The closer the point $(P_0, Q_0)$ gets to the vertical axis, the larger the elasticity.

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