Kallie has a B.S. in Agribusiness and minor in Statistics from California Polytechnic State University, San Luis Obispo and M.S. in Agricultural and Resource Economics from University of California, Davis. She has extensive experience designing and performing economic analysis of wholesale energy markets and investigations of market participant behavior within these markets.
A demand curve is considered inelastic when it is not very sensitive to price changes. More specifically, a one percent change in price will result in less than a one percent change in quantity demanded, that is, price elasticity between zero and negative one (not inclusive of zero and negative one).
Since demand for most goods is downward sloping, the elasticity of demand will always be negative. Sometimes the negative sign is ignored by taking an absolute value. In that event, an elastic demand will have price elasticity greater than one. In this lesson, the negative sign will be represented. Elasticity of demand is evaluated in terms of percent change in quantity relative to a percent change in price.
Elasticity of demand = % change in quantity demanded / % change in price
It is not to be confused with the slope of the demand curve. The slope of the demand curve remains constant; however, the elasticity of the demand curve changes as you move along. The only curve that has a constant inelastic demand is a vertical demand curve. Any change in price will result in no change in quantity demanded.
This graph illustrates elasticity changes along the demand curve. This is for illustration purposes only. To accurately determine at what point a demand curve is inelastic, you will have to calculate the elasticity at varying points on the demand curve.
Arc Price Elasticity Formula
Price elasticity of demand can be calculated using the arc or price point method. The arc price formula is Ed equals P1 plus P2 over Qd1 plus Qd2, multiplied by the change in Qd over the change in P, where P1 and P2 are two price points on the demand curve and Qd1 and Qd2 are the quantity demanded given P1 and P2.
Therefore, delta Qd = Qd2 - Qd1
delta P = P1 - P2
This method is used primarily when you either don't have a mathematical formula for the demand curve, or you aren't familiar with taking derivatives of equations. The arc method essentially assumes a linear demand curve between the two points when estimating the price elasticity. Therefore, the more curved the demand is between the two points, the more inaccurate the estimate.
Price Point Elasticity Formula
The price point elasticity formula is where elasticity of demand equals P over Qd multiplied by d Qd over d P.
P is the price at which you are evaluating the elasticity of demand.
Qd is the quantity demanded at the point you are evaluating elasticity of demand.
dQd/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.
Example: Arc Method
It's summer - let's go to Jamba Juice! This graph shows the demand curve for Jamba Juice in the summer. The demand curve is:
Qd = 100 - 2P
Using the arc method, determine the price elasticity at $4/smoothie. Given the demand equation, first determine the quantity demanded at $4/bottle.
Quantity demanded = 100 - 2(4) = 100 - 8 = 92
(P1, Qd1) = (4, 92)
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 $4.50/smoothie as P2.
Quantity demanded = 100 - 2(4.5) = 100 - 9 = 91
(P1, Qd1) = (4.5, 91)
Using this formula:
Ed = ((4 + 4.5 ) / (92 + 91)) * ((91 - 92) / ( 4.5 - 4))
Ed = (8.5 / 183) * (-1 / 0.5) = .046 * (-2) = -0.092
At $4/bottle, Jamba Juice in summer, given this demand curve, is considered inelastic. Quantity demanded will decrease only by 0.092 percent - that's less than a tenth of a percent - with a one percent increase in price.
Using the same demand curve and price of $4/smoothie, lets evaluate the price elasticity at using price-point elasticity method.
The first derivative of the demand curve with respect to price is -2.
NOTE: Even if you do not know calculus, so long as the demand curve is linear, the first derivative with respect to a given variable will always be the coefficient of that variable. Again, this is only if the demand curve is linear.
Quantity demanded at $4/smoothie = 100 - 2 * $4 = 100 - 8 = 92
Using this formula:
Elasticity of demand = (4 / 92) * (-2) = -0.087
At $4/smoothie, a one percent increase in price will result in only a 0.087 percent decrease in quantity demanded. Again, that is less than a tenth of a percent decrease in quantity demanded. Therefore, this Jamba Juice demand curve at $4/smoothie is inelastic. The percent change in quantity demanded is less than the percent change in price.
An inelastic demand is one that is not very sensitive to price change, such that the percent change in quantity demanded will be less than the percent change in price. As seen in the price point elasticity of demand example, inelastic demand has a price elasticity between zero and negative one, not inclusive. 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.
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