Recall the “Law of Demand”: Other things equal, when the price of a good rises, the quantity demanded of the good falls. An example of qualitative information about the relationship between two variables (... relates directions of changes). Often we need quantitative information (relating magnitudes of changes). When price rises, does quantity demanded fall a “little”?... a “lot”?

How can we measure the degree of responsiveness of quantity demanded to price changes? D1D1 D2D2 “big”  Q It seems to be related to slope: Relatively “flat” demand -- big response of Q to  p Relatively “steep” demand -- small response. “small”  Q  p p Q Q p

This suggests using either the slope or the reciprocal of the slope of demand as a measure of the responsiveness of Q to changes in p. Reciprocal of the slope = “change in Q / change in p” Greater value for reciprocal of slope means quantity demanded is “more responsive” to price changes?

Problems with the use of demand curve slope (or its reciprocal) as a measure of responsiveness: 1. Numerical value has “units” attached. -- can’t compare across goods measured in different units. -- can get any numerical value you want just by changing units. 2. The slope (or its reciprocal) involves an implicit comparison (ratio) of absolute changes. -- may not be very informative.

Here’s an example to show why: At first, the demand for good A seems more responsive: a smaller price change produces a bigger quantity response. But when we think in terms of relative (rather than absolute) changes, the demand for good B is clearly more responsive than the demand for good A.

The lesson: A comparison of relative (or percentage) changes gives a more informative measure of responsiveness. (own price) elasticity of demand =

One remaining question: When quantity changes (from Q 1 to Q 2, say), what quantity should be used as the “base Q” in a percentage change calculation? The usual choice: the initial quantity (Q 1 ) (A change from Q = 100 to Q = 110 is an increase of 10%, calculated by this method.)

Problem: This method gives a different (absolute) value for the percentage change going in the other direction. (A change from Q = 110 to Q = 100 is a decrease of 9.09%) Standard convention: The “Midpoint Method.” For the “base Q” use the simple average of the initial and final quantities: (Q 1 + Q 2 ) ÷ 2 (Same for percentage changes in price)

Earlier, we had the (own price) elasticity of demand expressed as: = (Q 2 - Q 1 )/ [ (Q 2 + Q 1 ) ÷ 2] (p 2 - p 1 )/ [ (p 2 + p 1 ) ÷ 2 ] or, more simply = (Q 2 - Q 1 ) / (Q 2 + Q 1 ) (p 2 - p 1 ) / (p 2 + p 1 )

An example of an elasticity calculation:  = (Q 2 - Q 1 ) / (Q 2 + Q 1 ) (p 2 - p 1 ) / (p 2 + p 1 ) ( ) / ( ) ( ) / ( ) = = Pt. 1 Pt ($/lb.) Demand (lbs./day)

A brief digression for those who know some calculus: The formula on previous slide gives “arc elasticity,” because it assigns an elasticity value to any “arc,” or segment, of a demand curve. We can write arc elasticity as:.. or, simplifying..... where “midpoint” p and Q are the average values of p and Q over the arc.

Now let the arc gradually shrink to a single point. “Midpoint” p and Q become the unique price and quantity coordinates of the point and the ratio of discrete changes in Q and p converges to: -- the derivative of Q w.r.t. p. “Point elasticity”: (For the purposes of Econ 101, arc elasticity is good enough.)

Now returning to the example of an elasticity calculation:  = What do we make of this number? First note negative sign. Along a demand curve, price and quantity changes are always of opposite signs so (own price) elasticities of demand are always negative. Sometimes we automatically think in terms of absolute values (... but this can be a dangerous habit).

What information is conveyed by the numerical value ( |  | = )? Let’s look at some cases, remembering With vertical demand... %  Q = 0 so  = 0 “perfectly” or (“completely”) inelastic demand Exactly the same quantity demanded -- no matter what the price. Pt. 1 Pt. 2 p Q p p

With “relatively steep” demand... p Q pp  Q | %  p | > | %  Q | so... 0 < |  | < 1 “inelastic demand”

For certain special demand curves, the relative (%) changes in price and quantity are the same (in absolute value). p Q p p  Q | %  p | = | %  Q | so... |  | = 1 “unit elastic demand”

For “relatively flat” demand curves... p Q p p Q Q | %  p | < | %  Q | so... 1 < |  | < “elastic demand”

For a horizontal demand curve... p Q p 1  Q %  p = 0 so... At any price equal to or below p 1, quantity demanded is unlimited (for practical purposes). At any price even slightly above p 1, quantity demanded is zero. |  | = “perfectly” (or “completely”) elastic demand