Midpoint Method

When we calculate the elasticity of demand, we usually calculate it as the percent change in quantity demanded by the percent change in price. However, this method will give you different values depending on if you calculate the elasticity from point A to B or from B to A. But what if there was a way to calculate the elasticity of demand and avoid this frustrating issue? Well, good news for us, there is! If you want to learn about the midpoint method, you've come to the right spot! Let's get started!

Midpoint Method Economics

The midpoint method in economics is used to find the price elasticity of supply and demand. Elasticity is used to gauge how responsive the quantity supplied or quantity demanded is when one of the determinants of supply and demand changes.

To calculate the elasticity, there are two methods: the point elasticity method and the midpoint method. The midpoint method, also referred to as arc elasticity, is a method to calculate the elasticity of supply and demand using the average percent change in price or quantity.

The midpoint method avoids any confusion or mix-ups that result from using other methods of calculating elasticity. The midpoint method does this by giving us the same percent change in value regardless of if we calculate the elasticity from point A to point B or from point B to point A.

As a reference, if point A is 100 and point B is 125, the answer changes depending on which point is the numerator and which one is the denominator.

\[ \frac {100}{125}=0.8 \ \ \ \hbox{versus} \ \ \ \frac{125}{100}=1.25\]

Using the midpoint method eliminates the scenario above by using the midpoint between the two values: 112.5.

Midpoint Method vs Point Elasticity

Let's have a look at the midpoint method vs the point elasticity method. Both are perfectly acceptable ways of calculating the elasticity of supply and demand, and they both require mostly the same information to perform. The difference in the information required comes from needing to know which value is the initial value for the point elasticity method since this will tell us if the price rose or fell.

Midpoint Method vs Point Elasticity: Point Elasticity Formula

The point elasticity formula is used to calculate the elasticity of a demand or supply curve from one point to another by dividing the change in value by the starting value. This gives us the percent change in value. Then, to calculate the elasticity, the percent change in quantity is divided by the percent change in price. The formula looks like this:

\[\hbox{Point Elasticity of Demand}=\frac{\frac{Q_2-Q_1}{Q_1}}{\frac{P_2-P_1}{P_1}}\]

Let's set this into practice by looking at an example.

If we graph the points from the example on a chart, it will look something like Figure 1 below.

Fig. 1 - Elastic Demand Curve for Bread

To briefly illustrate the problem with the point elasticity method, we will use Figure 1 again, only this time calculating an increase in the price of bread.

See how using the point elasticity method can give us two different impressions of the market even though it is the same curve? Let's look at how the midpoint method can avoid this situation.

Midpoint Method vs Point Elasticity: Midpoint Method Formula

The midpoint method formula has the same purpose of calculating the elasticity of supply and demand, but it uses the average percent change in value to do so. The formula for calculating elasticity using the midpoint method is:

\[\hbox{Elasticity of Demand}=\frac{\frac{(Q_2-Q_1)}{(Q_2+Q_1)/2}}{\frac{(P_2-P_1)}{(P_2+P_1)/2}}\]

If we examine this formula closely, we see that rather than dividing the change in value by the initial value, it is divided by the average of the two values.

This average is calculated in the \((Q_2+Q_1)/2\) and the \((P_2+P_1)/2\) portions of the elasticity formula. This is where the midpoint method gets its name. The average is the midpoint between the old value and the new value.

Rather than using two points to calculate the elasticity, we will use the midpoint because the midpoint between two points is the same no matter the direction of the calculation. We will use the values in Figure 2 below to prove this.

As demonstrated in the example above, when the midpoint method is used, the percentage change in price and quantity is the same in either direction.

Purpose of the Midpoint Method

The main purpose of the midpoint method is that it gives us the same elasticity value from one price point to another, and it does not matter if the price decreases or increases. But how? It gives us the same value because the two equations use the same denominator when dividing the change in value to calculate the percent change.

The change in value is always the same, regardless of an increase or decrease, since it is simply the difference between the two values. However, if the denominators change depending on if the price increases or decreases when we are calculating the percent change in value, we will not get the same value. The midpoint method is more useful when the values or data points provided are further apart, such as if there is a significant price change.

The disadvantage of the midpoint method is that it is not as precise as the point elasticity method. This is because as the two points get farther apart, the elasticity value becomes more general for the whole curve than just a portion of the curve. Think of it this way. High-income people are going to be insensitive or inelastic to a price increase because they have the disposable income to be more flexible. Low-income people are going to be highly elastic to increases in price because they are on a set budget. Mid-income people are going to be more elastic than high-income people and less elastic than low-income people. If we lump them all together, we get the elasticity of demand for the entire population, but this is not always useful. Sometimes it is important to understand the elasticity of individual groups. This is when using the point elasticity method is superior.

Midpoint Method Example

To finish off, we will look at a midpoint method example. If we pretend that the price of pick-up trucks jumped from $37,000 to $45,000 because the world ran out of steel, the number of trucks demanded would fall from 15,000 to just 8,000. Figure 3 shows us what it would look like on a graph.

Fig. 3 - Elastic Demand Curve for Pick-up Trucks

Figure 3 shows us how consumers would react if the price suddenly increased from $37,000 to $45,000. Using the midpoint method, we will calculate the elasticity of demand for pick-up trucks.

\(\hbox{Elasticity of Demand}=\frac{\frac{(8,000-15,000)}{(8,000+15,000)/2}}{\frac{($45,000-$37,000)}{($45,000+$37,000)/2}}\)

\(\hbox{Elasticity of Demand}=\frac{\frac{-7,000}{11,500}}{\frac{$8,000}{$41,000}}\)

\(\hbox{Elasticity of Demand}=\frac{-0.61}{0.2}\)

\(\hbox{Elasticity of Demand}=-3.05\)

The elasticity of demand for pick-up trucks is 3.05. That tells us that people are very elastic to the price of trucks. Since we used the midpoint method, we know that the elasticity would be the same even if the price of trucks decreased from $45,000 to $37,000.