What does it mean when the percentage change in quantity demanded the percentage change in price are the same?

When the price of a good or service changes, the quantity demanded changes in the opposite direction. Total revenue will move in the direction of the variable that changes by the larger percentage. If the variables move by the same percentage, total revenue stays the same. If quantity demanded changes by a larger percentage than price (i.e., if demand is price elastic), total revenue will change in the direction of the quantity change. If price changes by a larger percentage than quantity demanded (i.e., if demand is price inelastic), total revenue will move in the direction of the price change. If price and quantity demanded change by the same percentage (i.e., if demand is unit price elastic), then total revenue does not change.

When demand is price inelastic, a given percentage change in price results in a smaller percentage change in quantity demanded. That implies that total revenue will move in the direction of the price change: a reduction in price will reduce total revenue, and an increase in price will increase it.

Consider the price elasticity of demand for gasoline. In the example above, 1,000 gallons of gasoline were purchased each day at a price of $4.00 per gallon; an increase in price to $4.25 per gallon reduced the quantity demanded to 950 gallons per day. We thus had an average quantity of 975 gallons per day and an average price of $4.125. We can thus calculate the arc price elasticity of demand for gasoline:

The demand for gasoline is price inelastic, and total revenue moves in the direction of the price change. When price rises, total revenue rises. Recall that in our example above, total spending on gasoline (which equals total revenues to sellers) rose from $4,000 per day (=1,000 gallons per day times $4.00) to $4037.50 per day (=950 gallons per day times $4.25 per gallon).

When demand is price inelastic, a given percentage change in price results in a smaller percentage change in quantity demanded. That implies that total revenue will move in the direction of the price change: an increase in price will increase total revenue, and a reduction in price will reduce it.

Consider again the example of pizza that we examined above. At a price of $9 per pizza, 1,000 pizzas per week were demanded. Total revenue was $9,000 per week (=1,000 pizzas per week times $9 per pizza). When the price rose to $10, the quantity demanded fell to 900 pizzas per week. Total revenue remained $9,000 per week (=900 pizzas per week times $10 per pizza). Again, we have an average quantity of 950 pizzas per week and an average price of $9.50. Using the arc elasticity method, we can compute:

Demand is unit price elastic, and total revenue remains unchanged. Quantity demanded falls by the same percentage by which price increases.

Consider next the example of diet cola demand. At a price of $0.50 per can, 1,000 cans of diet cola were purchased each day. Total revenue was thus $500 per day (=$0.50 per can times 1,000 cans per day). An increase in price to $0.55 reduced the quantity demanded to 880 cans per day. We thus have an average quantity of 940 cans per day and an average price of $0.525 per can. Computing the price elasticity of demand for diet cola in this example, we have:

The demand for diet cola is price elastic, so total revenue moves in the direction of the quantity change. It falls from $500 per day before the price increase to $484 per day after the price increase.

A demand curve can also be used to show changes in total revenue. Figure 5.3 "Changes in Total Revenue and a Linear Demand Curve" shows the demand curve from Figure 5.1 "Responsiveness and Demand" and Figure 5.2 "Price Elasticities of Demand for a Linear Demand Curve". At point A, total revenue from public transit rides is given by the area of a rectangle drawn with point A in the upper right-hand corner and the origin in the lower left-hand corner. The height of the rectangle is price; its width is quantity. We have already seen that total revenue at point A is $32,000 ($0.80 × 40,000). When we reduce the price and move to point B, the rectangle showing total revenue becomes shorter and wider. Notice that the area gained in moving to the rectangle at B is greater than the area lost; total revenue rises to $42,000 ($0.70 × 60,000). Recall from Figure 5.2 "Price Elasticities of Demand for a Linear Demand Curve" that demand is elastic between points A and B. In general, demand is elastic in the upper half of any linear demand curve, so total revenue moves in the direction of the quantity change.

Figure 5.3 Changes in Total Revenue and a Linear Demand Curve

Moving from point A to point B implies a reduction in price and an increase in the quantity demanded. Demand is elastic between these two points. Total revenue, shown by the areas of the rectangles drawn from points A and B to the origin, rises. When we move from point E to point F, which is in the inelastic region of the demand curve, total revenue falls.

A movement from point E to point F also shows a reduction in price and an increase in quantity demanded. This time, however, we are in an inelastic region of the demand curve. Total revenue now moves in the direction of the price change – it falls. Notice that the rectangle drawn from point F is smaller in area than the rectangle drawn from point E, once again confirming our earlier calculation.

We have noted that a linear demand curve is more elastic where prices are relatively high and quantities relatively low and less elastic where prices are relatively low and quantities relatively high. We can be even more specific. For any linear demand curve, demand will be price elastic in the upper half of the curve and price inelastic in its lower half. At the midpoint of a linear demand curve, demand is unit price elastic.

Learning Objectives

• Mathematically differentiate between elastic, inelastic, and unitary elasticities of demand
• Calculate percentage changes, or growth rates
• Differentiate between the midpoint elasticity approach and the point elasticity approach in calculating elasticity

The formula for calculating elasticity is:

[latex]\displaystyle\text{Price Elasticity of Demand}=\frac{\text{percent change in quantity}}{\text{percent change in price}}[/latex].

Let’s look at the practical example mentioned earlier about cigarettes. Certain groups of cigarette smokers, such as teenage, minority, low-income, and casual smokers, are somewhat sensitive to changes in price: for every 10 percent increase in the price of a pack of cigarettes, the smoking rates drop about 7 percent. Plugging those numbers into the formula, we get

[latex]\displaystyle\text{Price Elasticity of Demand}=\frac{\text{percent change in quantity}}{\text{percent change in price}}=\frac{-7\%}{10\%}=-0.7[/latex]

Inelastic, Elastic, and Unitary Demand

So what does the number -0.7 tell us about the elasticity of demand? The negative sign reflects the law of demand: at a higher price, the quantity demanded for cigarettes declines. All price elasticities of demand have a negative sign, so it’s easiest to think about elasticity in absolute value, ignoring the negative sign. The fact that the result is less than one is more important than the negative sign. It tells us that the size of the quantity change is less than the size of the price change (i.e. the numerator in the elasticity formula is less than the denominator). This tells us that it would take a relatively large price change in order to cause a relatively small change in quantity demanded. In other words, consumer responsiveness to a change in price is relatively small. Therefore, when the elasticity is less than 1, we say that demand is inelastic.

The data above indicate that the demand for cigarettes by teenagers, minority, low income and casual smokers is relatively inelastic. Addicted adult smokers, though, are even less sensitive to changes in the price—most are willing to pay whatever it takes to support their smoking habit. We can say that their demand is even more inelastic than low income or casual smokers.

Different products have different price elasticities of demand. If the absolute value of the elasticity of some product is greater than one, it means that the change in the quantity demanded is greater than the change in price. This indicates a larger reaction to price change, which we describe as elastic. If the elasticity is equal to one, it means that the change in the quantity demanded is exactly equal to the change in price, so the demand response is exactly proportional to the change in price. We call this unitary elasticity, because unitary means one.  

Watch this video carefully to understand how to solve for elasticity and to see what the numerical values for elasticity mean when applied to economic situations.

You can view the transcript for “Episode 16: Elasticity of Demand” here (opens in new window).

Calculating Percentage Changes and Growth Rates

Before we dive deeper into solving for elasticity, let’s first make sure we are comfortable calculating percentage changes, also known as a growth rates. The formula for computing a growth rate is straightforward:

[latex]\text{Percentage change}=\frac{\text{Change in quantity}}{\text{Quantity}}[/latex]

Suppose that a job pays $10 per hour. At some point, the individual doing the job is given a $2-per-hour raise. The percentage change (or growth rate) in pay is

[latex]\frac{\$2}{\$10}=0.20\text{ or }20\%[/latex].

Now to solve for elasticity, we use the growth rate, or percentage change, of the quantity demanded as well as the percentage change in price in order to to examine how these two variables are related. The price elasticity of demand is the ratio between the percentage change in the quantity demanded (Qd) and the corresponding percent change in price:

[latex]\text{Price elasticity of demand}=\frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}}[/latex]

There are two general methods for calculating elasticities: the point elasticity approach and the midpoint (or arc) elasticity approach. Elasticity looks at the percentage change in quantity demanded divided by the percentage change in price, but which quantity and which price should be the denominator in the percentage calculation? The point approach uses the initial price and initial quantity to measure percent change. This makes the math easier, but the more accurate approach is the midpoint approach, which uses the average price and average quantity over the price and quantity change. (These are the price and quantity halfway between the initial point and the final point.) Let’s compare the two approaches. Suppose the quantity demanded of a product was 100 at one point on the demand curve, and then it moved to 103 at another point. The growth rate, or percentage change in quantity demanded, would be the change in quantity demanded [latex]{(103-100)}[/latex] divided by the average of the two quantities demanded:

[latex]\frac{(103+100)}{2}[/latex].

In other words, the growth rate:

[latex]\begin{array}{r}{\frac{103-100}{(103+100)/2}}\\{=\frac{3}{101.5}}\\{=0.0296}\\{=2.96\%\text{ growth}}\end{array}[/latex]

Note that if we used the point approach, the calculation would be:

[latex]\frac{(103–100)}{100}=3\%\text{ growth}[/latex]

This produces nearly the same result as the slightly more complicated midpoint method (3% vs. 2.96%). If you need a rough approximation, use the point method. If you need accuracy, use the midpoint method. Note: as the two points become closer together, the point elasticity becomes a closer approximation to the arc elasticity.

In this module you will often be asked to calculate the percentage change in the quantity. Keep in mind that this is same as the the growth rate of the quantity. As you work through the course and find other applications for calculate growth rates, you will be well prepared.

These next questions allow you to get as much practice as you need, as you can click the link at the top of the questions (“Try another version of these questions”) to get a new version of the questions. Practice until you feel comfortable with this concept.

elastic demand: when the calculated elasticity of demand is greater than one, indicating a high responsiveness of quantity demanded or supplied to changes in price elastic supply: when the calculated elasticity of either supply is greater than one, indicating a high responsiveness of quantity demanded or supplied to changes in price inelastic demand: when the calculated elasticity of demand is less than one, indicating that a 1 percent increase in price paid by the consumer leads to less than a 1 percent change in purchases (and vice versa); this indicates a low responsiveness by consumers to price changes inelastic supply: when the calculated elasticity of supply is less than one, indicating that a 1 percent increase in price paid to the firm will result in a less than 1 percent increase in production by the firm; this indicates a low responsiveness of the firm to price increases (and vice versa if prices drop) midpoint elasticity approach: Most accurate approach to solving for elasticity in which the percent changes in quantity demanded and price are measured relative to the average quantity demanded and price; the initial quantity demand is subtracted from the new quantity demanded; then divided by the average of the two quantities demanded; similarly, the initial price is subtracted from the new price, then divided by the average of the two prices   point elasticity approach: approximate method for solving for elasticity in which the percent changes are measured relative to the initial quantity demanded and price;  the initial quantity demanded is subtracted from the new quantity demanded, then divided by the initial quantity demanded; similarly, the initial price is subtracted from the new price, then divided by the initial price.   unitary elasticity: when the calculated elasticity is equal to one indicating that a change in the price of the good or service results in a proportional change in the quantity demanded or supplied

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