Cross Price Elasticity of Demand Formula

Cross Price Elasticity of Demand Formula Complementary vs Substitute

The cross price elasticity of demand formula can determine if goods are complementary vs substitute goods. Complementary goods are goods that are often bought together or require the use of one good to make the other function.

• Some examples of complementary goods are:
  • Tennis balls and rackets;
  • Cars and gasoline;
  • Paint and paintbrushes;
  • Pencils and pencil sharpeners.

If the price of tennis rackets increases, we can assume fewer tennis balls are sold since fewer people will buy and play tennis rackets. Conversely, maybe there is a sale on tennis rackets for 50% off and people take advantage and buy the associated tennis balls as well. Complementary goods have an inverse relationship to each other where, as the price of one good increases the quantity demanded of the other falls. This means the elasticity value for complementary goods is negative because the slope of the curve is negative, as seen in Figure 1.

Fig. 1 - Relationship Between Complementary Goods

Figure 1 shows how the price decrease from P₁ to P₂ of good 1 results in an increase in the quantity demanded from Q₁ to Q₂ of good 2. This is because as good one becomes more affordable, more people will want to buy it. As more people buy good one, more people will buy its complement, good two, as well.

Substitute goods are goods that are not bought together, but rather instead. A good is a substitute when the price of a good rises, we buy more of another good. It is important to note that goods are not always perfect substitutes since their characteristics may differ slightly. To see how much the price has to change to motivate people to buy the substitute instead is why we calculate cross price elasticity in the first place!

• Some examples of substitute goods are:
  • Glasses and contact lenses;
  • Coffee and tea;
  • Paper books and E-books;
  • Text messages and WhatsApp.

Figure 2, below, shows the positive relationship between two substitute goods. When calculating the elasticity between substitute goods, the elasticity value will be positive. Even if we do not have graphs like Figures 1 and 2, to see the slope, performing the calculations using the cross price elasticity of demand formula will tell us if the slope is positive or negative.

Fig. 2 - Relationship Between Substitute Goods

Figure 2 shows us how an increase in the price of a good from P₁ to P₂ will cause the quantity demanded of the other good to increase from Q₁ to Q₂. This is because as a good becomes more expensive people will buy less of it. Then, to replace the good that they no longer want to consume because it became too expensive, they buy the substitute good. The quantity demanded of the substitute good then increases.

Cross Price Elasticity of Demand Midpoint Formula

The cross price elasticity of demand midpoint formula uses the midpoint of the two data points to calculate an elasticity value that is the same, no matter if the price is increasing or decreasing. If we calculate elasticity without using the midpoint, we have to be clear on which value is the initial value and which is the new value, otherwise, we end up with two different elasticity values.

\[\hbox{Cross Price Elasticity of Demand}=\frac{\frac{New\ Quantity-Old\ Quantity}{Old \ Quantity}}{\frac{New\ Price-Old\ Price}{Old\ Price}}\]

For example, if we use the formula above, and the price increases from $5 to $7 the percent change is \(($7-$5)/$5=0.4\) or 40%. But if we are not clear that the price is increasing, the percent change could also be calculated as \(($5-$7)/$7=-0.286\) or -28.6%.

Both values are correct but it depends on if we want to calculate the elasticity for an increase or decrease in price.

Let's investigate the midpoint formula and how it can avoid this issue. The formula for cross price elasticity of demand using the midpoint method is:

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

The midpoint formula is different because of the value it uses as its base. In the first formula, the base is simply the "old value" which means that it changes depending on which number is considered the old value. The midpoint method uses the average between the two values, which means it will be the same base regardless of which value comes first. This can be seen in the \((Q_2+Q_1)/2\) and \((P_2+P_1)/2\) portions of the formula.

Calculate the Cross Price Elasticity of Demand

To calculate the cross price elasticity of demand, we need at least two data points that show the quantity demanded of good one in relation to the price of good two.

If the resulting elasticity value is negative, the slope of the curve will slope down. A downward or negative slope indicates that as the price of one good decreases, the quantity demanded of the other increases making the two goods complementary goods. When the elasticity value is positive, the slope of the curve will slope up. This tells us that as the price of one good increases, the quantity demanded of another also increases making them substitute goods for each other.

Cross Price Elasticity of Demand Formula Example

To calculate, it is best to start with an example using the cross price elasticity of demand formula. Let's work through the calculations of cross price elasticity of demand for complementary goods first.

One brother sells peanut butter for a living while the other sells jelly. The first brother wants to increase the sales of his peanut butter so he wants to lower the price. Before he does this, however, he wants to know how this price change will affect his brother's business. He calculates that if the price of peanut butter falls from $6 per jar to $4.50, the quantity of jelly demanded increases from 7 jars to 10 jars. We will use the midpoint formula.

\(\hbox{Cross Price Elasticity of Demand}\ (Midpoint\ Method)=\frac{\frac{Q_2-Q_1}{(Q_2+Q_1)/2}}{\frac{P_2-P_1}{(P_2+P_1)/2}}\)

\(\hbox{Cross Price Elasticity of Demand}=\frac{\frac{10-7}{(10+7)/2}}{\frac{$4.50-$6}{($4.50+$6)/2}}\)

\(\hbox{Cross Price Elasticity of Demand}=\frac{\frac{3}{8.5}}{\frac{-$1.50}{$5.25}}\)

\(\hbox{Cross Price Elasticity of Demand}=\frac{0.353}{-0.286}\)

\(\hbox{Cross Price Elasticity of Demand}=-1.234\)

The negative elasticity value tells us that the two goods are complementary goods and that they are strong complements since the cross price elasticity is above one. The closer the elasticity value gets to infinity, the stronger the relationship. The first brother decides to lower the price of peanut butter because he hopes for more sales and he hopes that his brother's jelly sales will increase too!

Figure 3 shows what the graph for peanut butter and jelly would look like.

Fig. 3 - Complementary Relationship between Peanut Butter and Jelly

Now, to calculate the cross price elasticity for substitutes we will use the same midpoint formula. When the price of coffee went from $10 per bag to $15, the quantity of tea demanded went from 20 boxes to 25 boxes.

\(\hbox{Cross Price Elasticity of Demand}\ (Midpoint\ Method)=\frac{\frac{Q_2-Q_1}{(Q_2+Q_1)/2}}{\frac{P_2-P_1}{(P_2+P_1)/2}}\)

\(\hbox{Cross Price Elasticity of Demand}=\frac{\frac{25-20}{(25+20)/2}}{\frac{$15-$10}{($15+$10)/2}}\)

\(\hbox{Cross Price Elasticity of Demand}=\frac{\frac{5}{22.5}}{\frac{$5}{$12.5}}\)

\(\hbox{Cross Price Elasticity of Demand}=\frac{0.222}{0.4}\)

\(\hbox{Cross Price Elasticity of Demand}=0.555\)

Coffee and tea are substitute goods since the elasticity value is positive. However, they are not very strong substitutes since the elasticity value is less than one. The increase in the price of coffee only had a small effect on the quantity demanded of tea.

Fig. 4 - Substitute Relationship Between Coffee and Tea

Figure 4 shows us what the graph of coffee and tea looks like when they are weak substitutes for each other. Maybe tea is not a close enough alternative to coffee to motivate coffee drinkers to convert.