Price Elasticity of Demand: Definition and Calculation

1. What is Price Elasticity of Demand?

At its core, price elasticity of demand measures the sensitivity of the quantity demanded of a good to a change in its own price. In simpler terms, it answers the question: "If the price of a product changes by 1%, by what percentage will the quantity demanded change?"

To start, let's watch a brief segment from a Khan Academy video that explains why economists use percentage changes to measure this responsiveness, rather than absolute changes in quantity and price.

As the video explained, using percentages allows us to have a universal measure that isn't dependent on the units used (e.g., gallons vs. liters, or dollars vs. yen).

The formal definition is:

$E_d=\frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}}=\frac{\%\Delta Q_d}{\%\Delta P}$

Because the demand curve is downward sloping (an increase in price leads to a decrease in quantity), the price elasticity of demand is technically a negative number. However, for interpretation, economists often discuss its absolute value.

2. Calculating Price Elasticity of Demand

For the CFA exam, you need to be proficient in two methods for calculating price elasticity:

1. Point Elasticity: Calculating elasticity at a single point on the demand curve, typically used when you have a demand function.
2. Arc Elasticity: Calculating elasticity over a range or between two points on the demand curve.

2.1 Point Elasticity

When we have the demand function, we can calculate the elasticity at a specific price. The formula for point elasticity is:

$E_d=\left(\frac{\Delta Q_d}{\Delta P}\right)\times \left(\frac{P}{Q_d}\right)$

Here:

• $\frac{\Delta Q_d}{\Delta P}$ is the change in quantity for a given change in price. If you have a demand function like $Q_d=a-bP$, this term is simply the coefficient $-b$.
• $\frac{P}{Q_d}$ is the ratio of price to quantity at the specific point on the curve you are analyzing.

Let's watch a video that demonstrates this calculation.

As the video showed, to calculate point elasticity, you need three pieces of information:

1. The price ($P$).
2. The quantity demanded at that price ($Q_d$).
3. The coefficient on price from the demand function ($\Delta Q_d/\Delta P$).

2.2 Arc Elasticity (The Midpoint Method)

What if you don't have a demand function, but instead have two price-quantity combinations? For example, you know that at a price of $10, 50 units were sold, and when the price rose to $12, 40 units were sold.

If you use the standard percentage change formula, you'll get a different answer depending on whether you calculate the elasticity for a price increase from $10 to $12 or a price decrease from $12 to $10. To solve this problem, we use the arc elasticity or midpoint method, which uses the average price and average quantity as the denominator.

The formula for arc elasticity is:

$E_d=\frac{(Q_2-Q_1)/((Q_1+Q_2)/2)}{(P_2-P_1)/((P_1+P_2)/2)}$

This formula looks more complex, but it ensures you get a consistent result regardless of the direction of the price change. The following video provides an excellent, step-by-step walkthrough of this method.

3. Interpreting the Elasticity Value

Now that we can calculate the number, we need to interpret it. We use the absolute value of the elasticity, $|E_d|$, to classify demand.

• Elastic Demand: $|E_d|>1$

  • The percentage change in quantity demanded is greater than the percentage change in price.
  • This means demand is highly responsive or sensitive to price changes.
  • Example: A specific brand of soda. If the price goes up, people can easily switch to another brand.

• Inelastic Demand: $|E_d|<1$

  • The percentage change in quantity demanded is less than the percentage change in price.
  • This means demand is not very responsive to price changes.
  • Example: Gasoline. Even if the price increases, most people still need to drive, so their consumption doesn't change much.

• Unit Elastic Demand: $|E_d|=1$

  • The percentage change in quantity demanded is exactly equal to the percentage change in price.

The PrepNuggets video you watched earlier also provides good examples for these classifications.

There are also two theoretical extremes:

• Perfectly Inelastic: $|E_d|=0$. Quantity demanded does not change at all when price changes (a vertical demand curve).
• Perfectly Elastic: $|E_d|=\infty$. Any price increase causes quantity demanded to drop to zero (a horizontal demand curve).

4. Elasticity Varies Along a Linear Demand Curve

A critical concept to master is that elasticity is not the same as slope. A common mistake is to assume that because a straight-line demand curve has a constant slope, it also has constant elasticity. This is incorrect.

As we move down a linear demand curve, price elasticity of demand changes continuously.

Let's break down why this happens by looking at the point elasticity formula again: $E_d=(\text{slope})\times (P/Q)$.

• At the top of the demand curve (high P, low Q), the ratio $P/Q$ is very large, making demand elastic.
• At the bottom of the demand curve (low P, high Q), the ratio $P/Q$ is very small, making demand inelastic.

5. Check Your Understanding

You've learned the definitions and the calculation methods. Let's put your skills to the test with a CFA-style problem.

Question:

The demand function for a product is given by $Q_d=100-4P$.

Calculate the price elasticity of demand when the price of the product is $15.

Take a few minutes to solve this on your own before reading the solution below.

Solution:

1. Find the quantity demanded (Q) at the given price (P).
   Plug $P=15$ into the demand function:
   $Q_d=100-4(15)=100-60=40$

2. Find the slope coefficient ($\Delta Q_d/\Delta P$).
   From the demand function $Q_d=100-4P$, the coefficient on $P$ is -4. So, $\Delta Q_d/\Delta P=-4$.

3. Plug these values into the point elasticity formula.
   $E_d=\left(\frac{\Delta Q_d}{\Delta P}\right)\times \left(\frac{P}{Q_d}\right)$
   $E_d=(-4)\times \left(\frac{15}{40}\right)$
   $E_d=-4\times 0.375=-1.5$

4. Interpret the result.
   The price elasticity of demand is -1.5. The absolute value is $|-1.5|=1.5$. Since this is greater than 1, we can conclude that demand is elastic at a price of $15.

This type of calculation is very common on the exam. You can find another similar practice problem in the AnalystPrep resource below if you'd like more practice.

Conclusion

In this lesson, we introduced the fundamental concept of price elasticity of demand. You now have a solid grasp of what it represents and how to measure it.

Key Takeaways:

• Price Elasticity of Demand measures the percentage change in quantity demanded resulting from a 1% change in price.
• We use the point elasticity formula when we have a demand function and want to find the elasticity at a specific price.
• We use the arc elasticity (midpoint) formula to find the average elasticity between two distinct price-quantity points.
• Based on the absolute value of elasticity, demand is classified as elastic ($>1$), inelastic ($<1$), or unit elastic ($=1$).
• Crucially, elasticity is not the same as slope. For a linear demand curve, demand is elastic at high prices and inelastic at low prices.

Preview of the Next Lesson:

Now that we can define and calculate elasticity, a natural question arises: Why is the demand for some goods elastic while for others it's inelastic? In our next lesson, we will explore the determinants of price elasticity of demand, such as the availability of substitutes, the proportion of income spent on the good, and the time horizon. Understanding these factors will deepen your analytical skills and allow you to predict how responsive demand is likely to be in different market scenarios.