Point Elasticity PDF

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Key points about point elasticity. Prepared for Mackay's ECON 1011...

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Point Elasticity vs. Midpoint/Arc Elasticity Price elasticity of demand is a measurement of the responsiveness of a change in the quantity demanded for a change in the price of a good. Because some goods are much more expensive in absolute terms – houses vs. loaves of bread – the changes in price of a good are measured in percent changes. For example, a $2 price change in a loaf of bread would represent a substantial change in price. While a $2 change in the price of a house would be exceedingly small. This idea of relative change is also used for quantity. Again, buying a few more loaves of bread in a year means something very different from buying a few more houses a year. Measuring elasticity is to measure the percent change in quantity demanded for -- or over -- a percent change in price. Elasticity in its most generalized form is Elasticity =

𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑃𝑟𝑖𝑐𝑒

Notice that there are no units of measure to elastic; the number is a scalar. There are several methods to measure the ratio of percent changes. The texts utilizes a formula known as the midpoint formula for finding elasticities. The midpoint formula is mostly used in empirical projects. While the midpoint formula is correct, most of our applications in economics will utilizes an alternative formula known as the point elasticity formula. The difference in the calculation is quite straightforward; moreover, the interpretation of any calculated number is identical: ɛ < 1 is inelastic, ɛ > 1 is elastic, and ɛ = 1 is unit elastic. Should you be researching this in alternative textbooks or online, the midpoint formula is sometimes known as arc elasticity. Both formulas can look complex when first inspected. Indeed, the formula used in most introductory classes – the midpoint formula – can be quite intimidating. As given in our text, on page 186, the midpoint elasticity formula is: Price Elasticity of Demand =

(𝑄2 − 𝑄1 ) (𝑃2 − 𝑃1 ) ÷ 𝑃 +𝑃 𝑄 +𝑄 ( 1 2 2) ( 1 2 2)

When written in the somewhat more cumbersome format:

(𝑄2 − 𝑄1 ) 𝑄 +𝑄 ( 1 2 2) Price Elsticity of Demand = (𝑃2 − 𝑃1 ) 𝑃 +𝑃 ( 1 2 2) One can begin to see this is a ratio of percent changes. Price elasticity is measuring the percent change in quantity for a percent change in the price. The (𝑄 −𝑄 ) numerator above, 𝑄22 +𝑄11 , is just a measurement of the percentage change from the midpoint of (

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the change. For example, we might be measuring the percentage change in quantity of loaves of bread for someone who went from buying 30 (Q1=30) loaves a year when the price was $3 (P1=$3) to 20 loaves (Q2=20) a year when the price increased to $4 (P2=$4). The numerator above would be

(20−30) 30+20 ) 2

(

, or -10/25 , or -.4, or a 40% decrease. This suggests a 12

unit decrease in the quantity of loaves: (30 × .4 = 12). Confusingly, we know that the actual from the previous paragraph. The actual change is known to be 10 (Q1 –Q2); the mid-point elasticity method sacrifices some accuracy for great convenience. On the price side, the (4−3) (𝑃 −𝑃 ) (1) denominator above 𝑃12+𝑃21 would be 3+4 , or 7 , or 2/7, or about .29, or 29%. This suggests that (

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(

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( 2)

the change was: ($3 × .29) = $. 87). Again, we know the actual change in price was $1 (P2-P1). Again, there are reasons to tolerate this somewhat dramatic-seeming loss in accuracy. While accuracy is sacrificed with the book’s mid-point method, only the observed – or reported – percent changes in price and quantity are needed to make the calculation. Moreover, we don’t have to worry about whether it’s an increase in quality or a decrease in quantity in order to find (30−20) (20−30) the absolute percent change: 30+20 = −.4, and 30+20 = 4. It’s either a 40% increase or a 40% (

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(

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decrease. We need know just two prices and two quantities. In addition, with the book method there can be any change in the price – an increase or decrease – for the good. That is, we could go from a loaf of bread costing $2 to it costing $10. When utilizing the alternative type of elasticity measurement – point elasticity – there can be only a one unit – in this case $1 – change in the denominator. While you should be familiar with mid-point elasticity, we will almost always use point elasticity in this class. Point elasticity finds the elasticity at a specific point. The formula for point elasticity can also look a bit intimidating; however, if you follow it step by step it’s straightforward. ∆𝑄 𝑄2 − 𝑄1 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 %∆𝑄 𝑄 𝑄 = = = 𝑃 −1 𝑃 %∆𝑃 ∆𝑃 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑃𝑟𝑖𝑐𝑒 2 1 𝑃1 𝑃 Using this method resolves the accuracy problem we demonstrated above. If we look at the 20−30 1 change in the loaves of bread purchased we can calculate = − 3 = −. 33 ≈ −33%. This 30 gives us an accurate answer. We can multiply 30 by -.33 to get -10: 20 is 10 units less than 30. 4−3 1 Doing the same with price we get an equally accurate answer: 3 = =. 33 = 33%. If we take 3 1

$3 and multiply it by , or 33%, it gives us $1: the exact difference in price. 3

The shortcoming of the point elasticity is that the percent change depends on whether it a positive or negative change. If we go from 30 loaves to 20 loaves we’ve gone down by a third;

30−20

1 however, it we go from 20 loaves to 30 loaves, = 2 = .5, we’ve gone up by 50%. It’s the 20 same change, but its meaning changes with perspective.

For this, and some other, reasons one should always calculate the point elasticity such that the denominator is positive and, therefore, the numerator is negative. (Because demand curves slope down and increase in price will always lower the quantity demanded.) Another constraint is that the elasticity is measured at a particular point and not over a range as with the mid-point elasticity. We shouldn’t use point elastic to find the elastic for a multi-dollar change in price. Again, if we had a price change that spanned more than a $1 change, point elasticity would not be appropriate. In that case, we would instead want to use mid-point, or arc, elasticity mentioned in the text. Let’s do an example. Let’s take the (inverse) demand curve of P=12-Q and calculate the elasticity at a price of $4 (P=4). ∆𝑄 𝑄2 − 𝑄1 −1 7−8 %∆𝑄 −1 4 1 𝑄1 𝑄 = = 𝜀= ∙ =− = 8 = 8 = 5 − 4 1 𝑃 − 𝑃 ∆𝑃 2 %∆𝑃 8 1 2 1 𝑃 𝑃1 4 4 As I mentioned in class, it is customary to take the absolute value of the calculated elasticity. The above calculation tells us that the point elasticity is ½ which is, naturally, less than 1. Therefore this is an inelastic point. 1 1 𝜀 = |− | = < 1 ⟹ 𝑖𝑛𝑒𝑙𝑎𝑠𝑡𝑖𝑐 2 2 This point elasticity approach can be used with any of the other types of elasticities that we will discuss in class: supply elasticity, cross-price elasticity, and income elasticity. While we might technically consider the point elasticity approach only appropriate for a one-unit change in the denominator, we’ll be using this approach for all our applications for simplicity....