Price elasticity of demand can be calculated using the arc or price point method. We will walk through an example using both methods. Essentially, both methods are trying to determine what the change in quantity demanded will be if there is a change in price. The more sensitive, i.e. more change, in quantity demanded, the more elastic the demand curve. Think of something that you treat yourself to. If that price increases, you would probably treat yourself less often. If the price goes up 1% and you decrease the amount you buy by more than 1%, it is considered to have an elastic demand.

**Arc Price Elasticity Formula**

- Where P
`1` and P`2` are two price points on the demand curve.
- Q
`d1` and Q`d2` are the quantity demanded given P`1` and P`2`.
- Delta Q
`d` = Qd`2` - Qd`1`
- Delta P = P
`1` - P`2`

This method is used primarily when you don't have a formula for the demand curve, or you aren't familiar with taking derivatives of equations. The arc method essentially takes the average elasticity between the two points chosen. The more curved the demand is between the two points, the more inaccurate the estimate.

Arc Price Elasticity Example:

The following graph shows the demand curve for Pinot Noir after the movie *Sideways*. The demand curve is Q`d` = 600 - 5P.

Using the arc method, determine the price elasticity at $90/bottle.

Given the demand equation above, first determine the quantity demanded at $90/bottle.

Quantity demanded = 600 - (5 x 90) = 600 - 450 = 150

(P`1`,Q`d1`) = (90,150)

Now choose a second price to use as the end point to the arc and determine the quantity demanded at that point. The closer the second point is to the original price point, the less inaccurate the estimate will be. Let's pick $31/bottle as P`2`.

Quantity demanded = 600 - (5 x 91) = 600 - 455 = 145

(P`1`,Q`d1`) = (91,145)

Using the formula from earlier:

E`d` = ((90 + 91) / (150 + 145)) x ((145 - 150) / (91 - 90))

E`d` = (181 / 295) x (-5 / 1) = .61 x -5 = -3.07

At $90/bottle, Pinot Noir with this demand curve is considered elastic. Quantity demanded will decrease by 3.07% with a 1% increase in price.

**Price Point Method Formula**

- Where P is the price at which you are evaluating the elasticity of demand
- Q
`d` is the quantity demanded at the point you are evaluating elasticity of demand
- dQ
`d` / dP is the first derivative of quantity demanded with respect to price

This is the more accurate method since it uses derivatives to determine the price elasticity at a given point on the demand curve. Calculus allows us to minimize the 'arc' used to estimate elasticity in the arc method, such that it becomes a single point on the demand curve.

Price-Point Method Example:

Using the same demand curve and price of $90/bottle, lets evaluate the price elasticity using the price-point elasticity method. The first derivate of the demand curve with respect to price is -5.

SHORTCUT: Even if you do not know calculus, if the demand curve is a straight line, the first derivative with respect to a given variable will always be the value in front of the price variable (P). Again, ONLY if the demand curve is a straight line.

Quantity demanded at $90/bottle = 600 - 5 x $90 = 600 - 450 = 150.

Using the formula from earlier:

E`d` = (90 / 150) x - 5 = -3

At $90/bottle, a 1% increase in price will result in a 3% decrease in quantity demanded. Therefore, this Pinot Noir demand curve at $90\bottle is elastic. The percent change in quantity demanded is greater than the percent change in price.

## Lesson Summary

Let's review. An **elastic demand** is one that is sensitive to price change, such that the percent change in quantity demanded will be greater than the percent change in price. As seen in the **price-point elasticity of demand** example, elastic demand has a price elasticity more negative than negative one. The two examples also illustrate the slight difference in results when using the two methods. The **arc elasticity of demand** is a less precise estimation, especially when the curve is not exactly linear.