### The Economic Concept of Elasticity

Economists use the concept of elasticity to describe quantitatively the impact on one economic variable (such as supply or demand) caused by a change in another economic variable (such as price or income). This concept of elasticity has two formulas that one could use to calculate it, on called point elasticity and the other called arc elasticity. Let's describe these formulas and examine the difference between the two.

As a representative example, we will talk about price elasticity of demand, but the distinction between point elasticity and arc elasticity holds in an analogous fashion for other elasticities, such as price elasticity of supply, income elasticity of demand, cross-price elasticity, and so on.

### The Basic Elasticity Formula

The basic formula for price elasticity of demand is the percent change in quantity demanded divided by the percent change in price. (Some economists, by convention, take the absolute value when calculating price elasticity of demand, but others leave it as a generally negative number.) This formula is technically referred to as "point elasticity." in fact, the most mathematically precise version of this formula involves derivatives and really does only look at one point on the demand curve, so the name makes sense!

When calculating point elasticity based on two distinct points on the demand curve, however, we come across an important downside of the point elasticity formula. To see this, consider the following two points on a demand curve:

- Point A: Price = 100, Quantity Demanded = 60
- Point B: Price = 75, Quantity Demanded = 90

If we were to calculate point elasticity when moving along the demand curve from point A to point B, we would get an elasticity value of 50%/-25%=-2. If we were to calculate point elasticity when moving along the demand curve from point B to point A, however, we would get an elasticity value of -33%/33%=-1. The fact that we get two different numbers for elasticity when comparing the same two points on the same demand curve is not an appealing feature of point elasticity since it's at odds with intuition.

### The "Midpoint Method," or Arc Elasticity

To correct for the inconsistency that occurs when calculating point elasticity, economists have developed the concept of arc elasticity, often referred to in introductory textbooks as the "midpoint method," In many instances, the formula presented for arc elasticity looks very confusing and intimidating, but it actually just uses a slight variation on the definition of percent change.

Normally, the formula for percent change is given by (final - initial)/initial * 100%. We can see how this formula causes the discrepancy in point elasticity because the value of the initial price and quantity is different depending on what direction you are moving along the demand curve. To correct for the discrepancy, arc elasticity uses a proxy for percent change that, rather than dividing by the initial value, divides by the average of the final and the initial values. Other than that, arc elasticity is calculated exactly the same as point elasticity!

### An Arc Elasticity Example

To illustrate the definition of arc elasticity, let's consider the following points on a demand curve:

- Point A: Price = 100, Quantity Demanded = 60
- Point B: Price = 75, Quantity Demanded = 90

(Note that these are the same numbers we used in our earlier point elasticity example. This is helpful so that we can compare the two approaches.) If we calculate elasticity by moving from point A to point B, our proxy formula for percent change in quantity demanded is going to give us (90 - 60)/((90 + 60)/2) * 100% = 40%. Our proxy formula for percent change in price is going to give us (75 - 100)/((75 + 100)/2) * 100% = -29%. Out value for arc elasticity is then 40%/-29% = -1.4.

If we calculate elasticity by moving from point B to point A, our proxy formula for percent change in quantity demanded is going to give us (60 - 90)/((60 + 90)/2) * 100% = -40%. Our proxy formula for percent change in price is going to give us (100 - 75)/((100 + 75)/2) * 100% = 29%. Out value for arc elasticity is then -40%/29% = -1.4, so we can see that the arc elasticity formula fixes the inconsistency present in the point elasticity formula.

### Comparing Point Elasticity and Arc Elasticity

Let's compare the numbers that we calculated for point elasticity and for arc elasticity:

- Point elasticity A to B: -2
- Point elasticity B to A: -1
- Arc elasticity A to B: -1.4
- Arc elasticity B to A: -1.4

In general, it will be true that the value for arc elasticity between two points on a demand curve will be somewhere in between the two values that can be calculated for point elasticity. Intuitively, it is helpful to think about arc elasticity as a sort of average elasticity over the region between points A and B.

### When to Use Arc Elasticity

A common question that students ask when they are studying elasticity is, when asked on a problem set or exam, whether they should calculate elasticity using the point elasticity formula or the arc elasticity formula.

The easy answer here, of course, is to do what the problem says if it specifies which formula to use and to ask if possible if such a distinction is not made! In a more general sense, however, it's helpful to note that the directional discrepancy present with point elasticity gets larger when the two points used to calculate elasticity get further apart, so the case for using the arc formula gets stronger when the points being used are not that close to one another.

If the before and after points are close together, on the other hand, it matters less which formula is used and, in fact, the two formulas converge to the same value as the distance between the points used becomes infinitely small.