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College Macroeconomics: Tutoring Solution15 chapters | 148 lessons

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Lesson Transcript

Instructor:
*Kallie Wells*

Do people buy more when prices drop? How much more do they buy? These questions can be answered by evaluating a good's elasticity of demand, which is defined and illustrated in this lesson with a few examples.

The **elasticity of demand (Ed)**, also referred to as the price elasticity of demand, measures how responsive demand is to changes in a price of a given good. More precisely, it is the percent change in quantity demanded relative to a one percent change in price, holding all else constant (*ceteris paribus*). Demand of goods can be classified as either perfectly elastic, elastic, unitary elastic, inelastic, or perfectly inelastic based on the elasticity of demand. This table shows the values of elasticity of demand that correspond to the different categories.

The graph illustrates the demand curves and places along the demand curve that correspond to the table. The elasticity of demand changes as one moves along the demand curve. This is an important concept - the elasticity of demand for a good changes as you evaluate it at different price points. These charts are for illustration only. To determine if a demand is elastic at a given price, you have to evaluate it with the methods we're about to discuss.

The formula for elasticity of demand can be formulated two different ways depending on what is available to you at the time. To calculate the elasticity of demand in either case, you will need a demand curve for a good. This can be in graphical or equation format. Essentially, when determining the elasticity of demand, you are trying to determine the slope of the demand curve at a given point on the curve. The first method is called **arc elasticity of demand**. This method is used when either:

- You don't have the exact equation for demand
- You aren't familiar with taking derivatives

The second method is called **point-price elasticity of demand**. This method is used when you:

- Have the mathematical equation for demand
- Are familiar with taking derivatives of equations

As we've already said, the elasticity of demand is evaluating the slope of the demanded curve at a given point. The arc elasticity of demand takes the difference between two points along the curve. Therefore, there can be slight inaccuracies when using this method since it uses an arc on the curve rather than a single point, which is how the point-price elasticity of demand calculates the elasticity of demand.

The arc elasticity of demand formula is:

*E*sub*d*= (*P*sub 1 +*P*sub 2)/(*Q*sub*d*1 +*Q*sub*d*2) * change in*Q*sub*d*/change in*P*, where:*P*sub 1 is the original price point,*P*sub 2 is the new price point,*Q*sub*d*1 is the quantity demanded at the original price point*Q*sub*d*2 is the quantity demanded at the new price point- Change in
*Q*sub*d*is the change in quantity demanded:*Q*sub*d*2 -*Q*sub*d*1 - Change in
*P*is the change in price:*P*sub 2 -*P*sub 1

The price-point elasticity of demand formula is:

- Ed =
*P*/*Q*sub*d***dQ*/*Dp*, where:*P*is the price at which you are evaluating the elasticity of demand*Q*sub*d*is the quantity demanded at the point you are evaluating elasticity of demand*dQ*/*dP*is the first derivative of quantity demanded with respect to price

When calculating the elasticity of demand for all goods with a downward sloping demand curve, you should get a negative value. Remember this as a good reality check on your work. However, in some reports and texts, people will leave off the negative sign when reporting elasticity of demand because it is almost always negative.

Follow these steps to determine the elasticity of demand via arc elasticity:

- Determine an original and new price point - for this example:
*P*sub 1 and*P*sub 2. - Evaluate the quantity demanded using the demand curve at points:
*P*sub 1 and*P*sub 2;*Q*sub*d*1 and*Q*sub d2, respectively. - Use this formula above and plug in the values for
*P*sub 1 and*P*sub 2;*Q*sub*d*1 and*Q*sub*d*2. - Reality check and interpret your results - you should get a negative value assuming the good is a downward sloping line!

So, assume this is the demand curve for lattes and you have to determine the elasticity of demand if prices increased from $4 per latte to $5 per latte. Since you do not have the exact formula, you have to use the arc elasticity of demand method. Following the four steps we just covered, you would get:

*P*sub 1 = $4;*P*sub 2 = $5- At
*P*sub 1,*Q*sub*d*1 = 20 lattes (this is the quantity demanded at $4); at*P*sub 2,*Q*sub*d*2 = 10 lattes (quantity demanded at $5) - Using the formula provided: Ed = (
*P*sub 1 +*P*sub 2)/(*Q*sub*d*1 +*Q*sub*d*2) * change in*Q*sub*d*/change in*P*; and the values in steps one and two, you would get Ed = ($4 + $5)/(10 + 20) * (10 - 20)/($5 - $4) = $9/30 * (-10)/$1 = 0.3 * -10 = -3 - It's a negative value. For every percent increase in price, quantity demanded will decrease by 3 lattes. Also, please note that the units (dollars and lattes) cancel out; therefore, the elasticity of demand is unit-less.

If you were given the formula rather than a chart, you would have to first chose two price points if they were not given to you, and then solve the formula for quantity demanded for each price point. This would then give you *P* sub 1, *P* sub 2, *Q* sub *d*1, and *Q* sub *d*2. Then, follow steps three and four from earlier.

Follow these steps to determine the elasticity of demand via price-point elasticity:

- Arrange the demand curve, such that it is in
*Q*sub*d*and*f*(*P*) format. - Take the derivative of the demand curve with respect to
*P*. If the demand curve is linear, then you do not necessarily have to take the derivate. The derivate of a linear demand curve with respect to*P*will always be the coefficient (number in front of)*P*. - Multiply the derivative by
*P*/*Q*sub*d*. - If you are given a point at which to evaluate the elasticity of demand,
*P*;*Q*sub*d*, insert those values into the formula after step three and solve. If you were only given price, you can put the given variable into the original demand curve to solve for quantity demanded. Then, insert the two variables into the formula generated from step three. - Reality check and interpret your results.

Given the demand curve, *Q* sub *d* = 500 - 5*P*, we want to evaluate the elasticity of demand for wine when the price is $20 and $50. Following the five steps from earlier:

- We do not need to arrange since it is already in the desired format.
- The derivative of
*Q*sub*d*is*dQ*/*dP*= -5. For linear demand curves, the derivative is always the coefficient of*P*:*P*/*Q*sub*d*- 5 = -5*P*/*Q*sub*d*. - We were only provided two price points at which to evaluate. Therefore, we need to determine the quantity demanded associated with a $20 and $50 price: when the price is $20,
*Q*sub*d*= 500 - 5(20) = 500 - 100 = 400; when the price is $50,*Q*sub*d*= 500 - 5(50) = 500 - 250 = 250. - Now we have to insert ($20; 400) and ($50; 250) into the formula from step three to determine the elasticity of demand at those two prices: Ed($20) = -5(20)/400 = -100/400 = -1/4; Ed($50) = -5(50)/250 = -250/250 = -1.
- Reality check - both results are negative, so that is good!

- When the price is $20, the elasticity of demand is -.25. Therefore, a one percent increase in price will result in a .25 percent decrease in quantity demanded. This wine is relatively inelastic when the price is $20.
- When the price is $50, the elasticity of demand is -1. Therefore, a one percent increase in price will result in a 1 percent decrease in quantity demanded. At $50, the wine is at the price point where it is unitary elastic.

**Elasticity of demand** describes the responsiveness of quantity demanded of a good relative to a small change in price. The more elastic a good is, the more quantity demanded will increase relative to a change in price; quantity demanded of inelastic goods will not be as responsive. There are two methods for determining elasticity. The first method is called **arc elasticity of demand**. This method is used when either:

- You don't have the exact equation for demand
- You aren't familiar with taking derivatives

The second method is called **point-price elasticity of demand**. This method is used when you:

- Have the mathematical equation for demand
- Are familiar with taking derivatives of equations

After this lesson, you should be ready to do the following tasks:

- Describe elasticity of demand
- Recall the equations to calculate arc-elasticity and point-price elasticity of demand
- Explain how to calculate elasticity of demand through the equations for arc-elasticity or point-price elasticity

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College Macroeconomics: Tutoring Solution15 chapters | 148 lessons

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