Percentage Change and Price Elasticity of Demand: Videos & Practice Problems

Elasticity measures the sensitivity between two variables, such as quantity demanded and price. The price elasticity of demand is calculated using the formula: $\frac{\mathrm{\%\; change\; in\; quantity\; demanded}}{\mathrm{\%\; change\; in\; price}}$. Demand is classified as elastic (E > 1), inelastic (E < 1), or unit elastic (E = 1). Understanding these concepts helps analyze consumer behavior in response to price changes, revealing how demand can be affected by factors like substitutes and complements.

Percentage Change and Price Elasticity of Demand

Now let's move our discussion to elasticity, which is basically a measure of sensitivity between two variables such as quantity and price. So when we're calculating elasticities, we're going to use percentage changes just like you see in this box here, and we're going to do that because it just makes it easier to compare across products. We get rid of units altogether; we're not talking about dollars or cents, so different prices of products, different sizes of markets can be compared using elasticity because it gets rid of those units.

So let's go ahead and start here with our percentage change formula. In the numerator, we have the change in x, which is basically the new value minus the old value. Say the price was $5, now it's $6; the change in x would be the 6 minus the 5, a $1 change, right, over the original value of x. In the denominator, we've got the original value, which would have been, say in that case, the $5, right? Whatever we started with. So we're going to be using this formula when we're calculating elasticities quite a bit.

So let's go ahead and define what is elasticity. It is a ratio between two variables. It's relating the changes between two calculating elasticities, most commonly we are going to use these variables, right? The one we've been using. We'll use quantity demanded, we'll use quantity supplied, we'll use price, and we're also going to use income in this chapter when we're calculating these.

So let's start with our first one, price elasticity of demand. This is the one we're going to spend the most time with. You get a lot of information out of it and it helps us answer the question: How does quantity demanded respond to a change in price? So when the price goes up, does quantity demanded go down a little bit? Does quantity demanded go down a lot, right? It's how much is it going to be changing? That's the question we're answering here.

So look at our formula. In the numerator, we've got percentage change in quantity demanded, and in the denominator, we have another percentage change. So we've got two percentage changes in one formula. We've got that percentage change in price and notice this real shorthand that I've got on the right here. Right? We've got percentage change in quantity demanded over percentage change in price, so that's a really shorthand way we can write that out.

So let's go ahead and see what this means. Let's do an example with it. When the price of dog bills rises by 20%, you buy 10% fewer dog bills. What is your price elasticity of demand for dog bills? So, we've got the price of dog bills is going up 20% and the percentage change in price is that 20% since it tells us the price rises by 20%. Then, we're told that we buy 10% fewer dog bills, so we're going to have a percentage change in quantity demanded of -10. Let's go ahead and do this in our formula. The calculation would look like this:

- 10 % 20 %

We put this in our calculator and we're going to get an answer of -0.5. Since we always get a negative answer when calculating price elasticity of demand due to the inverse relationship between price and quantity demanded, we use the absolute value which makes it a positive 0.5.

Let's define some ranges where we're going to call demand elastic, inelastic, or unit elastic. Demand is elastic when the price elasticity of demand is greater than 1. It's inelastic when the price elasticity of demand is less than 1 — which is always the absolute value; this situation makes our demand inelastic in this case. That means that you're not so sensitive to price changes. The price went up 20%, but you only bought 10% less stuff. So, you changed your spending habits less than the price changed. On the other hand, elastic means the quantity demanded will change more than the price.

And the last one, unit elastic, means that the changes are going to be the same. For us to get an answer of 1, the percentage change in quantity demanded would have had to be equal to the percentage change in price.

I want to go ahead and show you guys that there's a problem with our price elasticity of demand formula, and we're going to go ahead and do some examples using our percentage change, and you'll see how we can actually get different answers using the same data.

Let's do that on the next page.

Different Answers when Increasing or Decreasing Prices! (Part 1)

So now let's see how we can actually get different elasticities using the same data. The problem here ends up being with our percentage change formula. Remember that when we were using percentage change, by the way, this is the shorthand, percentage and the delta, the triangle here is change. We would do the change over the original which was basically the new minus the original, right? That's the numerator divided by the original right. So the change divided by the original gives us the percentage change and it's actually in this denominator that we end up having our problem. So let's go ahead and see through these examples. Let's see it in action. We're going to get a different elasticity when we're raising the price and when we're decreasing the price. So let's see this example. A pizza company's lunch special currently costs $5. At this price, the weekly demand is 2,000 lunch specials. If they raise their price to $6, the weekly demand will drop to 1,400 lunch specials. What is the price elasticity of demand? Alright, so let's go ahead and start. Remember our formula for elasticity of demand was our percentage change in quantity demanded over our percentage change in price, right?

So let's go ahead and start with the quantity demanded and I'm gonna go here and we're gonna use our percentage change formula just as I've written it above, new minus original divided by original. The percentage change for quantity demanded, let's see. First I'm going to circle all our data. We've got in blue, I'll circle our quantity demanded: 2,000, and it went down to 1,400, and in red I'll do the prices. Well, I've been using red. I'll use green for the prices here, the color of money. 56, right? So let's start with our quantity demanded and we had a demand of 2,000 and it dropped to 1,400, right? So our new is 1,400 minus the 2,000 divided by the original of 2,000, right? So our original demand was 2,000. Our new demand was 1,400. What is going to be the difference here? We're gonna get negative 600 over 2,000. Right? We put that in our calculator and we're gonna get -0.3. Right. But remember we're gonna drop all the negatives and positives because we're always gonna get one of them negative so we're just gonna say 0.3 absolute value

Let's do the same thing for the price, right? For the price, we had a price of $5 and it went up to $6 so the new was 6 5 минус 5 5 equals 1 divided by 5. Put that in our calculator and we're gonna get 0.2. Right, that oops. Can you see that there? Alright. 6 minus 5 divided by 5. So it gives us 1 / 5 and we're gonna get 0.2 here for our percentage change in price and we had 0.3 here for our percentage change in quantity demanded, right?

Okay. So let's go ahead and solve for elasticity in this case and I'll do it in red here. So elasticity of demand is going to equal that percentage change in quantity demanded, 0.3 divided by our percentage change in price which was 0.2. Right and what does that give us? It's going to give us 1.5. So our elasticity in demand in this case was 1.5, right, and when we get an elasticity of demand greater than 1, right, that means that it's elastic. So in this case, we got an elasticity of demand greater than 1 and elastic, 1.5. So let's go ahead in the next video. We're going to do a similar example with similar data. Check it out.

Different Answers when Increasing or Decreasing Prices! (Part 2)

Alright. So now let's check out this problem. A pizza company's lunch special currently costs $6. At this price, the weekly demand is 1400 lunch specials. If they lower their price to $5, the weekly demand will increase to 2,000 lunch specials. What is the price elasticity of demand? And you should be able to notice that we've got the same numbers here, right, except in this case we started at a price of $6 and we're decreasing to $5. In the previous problem up here, we started at a price of $5 and increased to $6, but the quantity demanded are all the same, right. It's just which direction we're moving. So let's go ahead and solve our price elasticity of demand here. Right. We're going to use still our percentage change formula. Remember that was new minus original divided by original, and just like I said, this is where we're going to end up seeing the problem is going to be in this denominator is the problem with this formula where we're going to get different answers. So let's go ahead and do our price elasticity. So elasticity of demand is going to equal percentage change in quantity demanded over percentage change in price, right? So let's go ahead and do like we did before. We'll get our quantities here and our prices. We'll use green.

Alright, so let's go ahead and start with quantity demanded. Let's get our percentage change in quantity demanded. Alright, so in this case, they started at 1400 lunch specials and demand will increase to 2,000. So the new in this case is 2,000 and the original was 1400, right? They were at 1,400. They're going to decrease the price which will increase quantity demanded. So new minus original divided by the original which in this case was 1400, and we are going to get 600 in the numerator divided by 1400. We put that in our calculator and we're going to get 0.429, I'm going to say. Okay, I'm going to cut it off at 3 decimals there. You can stop at 2 or 3. We're just rounding, and for price, let's go ahead and do the same thing. So price, we had a new price of $5 right? In this case, they're lowering their price to $5. They had an original price of $6 and we're going to divide by $6 there and we're going to get $5 minus $6 is negative one over $6, and remember we just get rid of the negative, right? We're going to be dealing with absolute values here. So I'm just going to get rid of that negative there. We have 1/6 which we put that in and we're going to get 0.167. Right. So there we go. That is going to be a percentage change in price. Over here, we have percentage change in quantity demanded right. Oops. Right there and right there.

Let's go ahead and put this into our elasticity of demand formula and see what answer we get. So we get 0.429 is going to be in our numerator. I'll do it in blue just to keep it even. And in our denominator, we will have 0.167. So let's go ahead and do that math. 0.4290.167 gives us an elasticity of demand equal to approximately 2.569. So look at this. In this case, we've got 2.569. Before we got 1.5, right. We've gotten different answers in both cases and we used the same numbers. We used a price of $5 and a price of $6 and a quantity demanded of 2,000 and 1,400, right? The numbers stayed the same, but we got a different answer whether we were going up in price or down in price and we don't want that. We want a consistent answer, right? This problem came from what was in our denominators, the original values. Notice in this first problem for the quantity demanded, our denominator was 2,000 right? Our numerator was 600 still, but our denominator was 1,400 and that's the problem. We're seeing the same thing is going to happen with price. We have different denominators in either case, so what we're going to end up doing is taking an average and putting that in the denominator instead. So let's go ahead to the next video where I'm going to show you a step-by-step method to do the averaging and get a consistent answer when we solve these problems. So let's go ahead and do that now.

Here’s what students ask on this topic:

What is the formula for calculating price elasticity of demand?

The formula for calculating price elasticity of demand is the percentage change in quantity demanded divided by the percentage change in price. Mathematically, it is expressed as:

$\frac{\mathrm{\Delta Q}/Q}{\mathrm{\Delta P}/P}$

where $\mathrm{\Delta Q}$ is the change in quantity demanded, $Q$ is the original quantity demanded, $\mathrm{\Delta P}$ is the change in price, and $P$ is the original price. This formula helps determine how sensitive the quantity demanded is to a change in price.

How do you interpret the elasticity of demand values?

Elasticity of demand values can be interpreted as follows:

• Elastic Demand: If the elasticity is greater than 1, demand is elastic, meaning consumers are sensitive to price changes. A small change in price leads to a larger change in quantity demanded.
• Inelastic Demand: If the elasticity is less than 1, demand is inelastic, indicating consumers are less sensitive to price changes. A change in price results in a smaller change in quantity demanded.
• Unit Elastic Demand: If the elasticity equals 1, demand is unit elastic, meaning the percentage change in quantity demanded is equal to the percentage change in price.

Understanding these values helps businesses and economists predict consumer behavior in response to price changes.

Why do we use percentage changes in calculating elasticity?

Percentage changes are used in calculating elasticity to standardize the measure across different products and markets. By using percentage changes, we eliminate the units of measurement, allowing for a direct comparison of elasticity across various goods and services. This approach provides a more accurate reflection of consumer sensitivity to price changes, regardless of the initial price or quantity levels. It also helps in understanding the relative change rather than the absolute change, which is crucial for analyzing economic behavior and making informed business decisions.

What is the problem with using original values in the percentage change formula for elasticity?

The problem with using original values in the percentage change formula for elasticity is that it can lead to inconsistent results depending on the direction of the price change. When calculating elasticity, using the original values as denominators can yield different elasticity values for the same numerical changes in price and quantity, depending on whether the price is increasing or decreasing. This inconsistency arises because the original values differ between scenarios, even though the changes are the same. To address this issue, it is recommended to use an average of the original and new values as the denominator, providing a more consistent and reliable calculation of elasticity.

How can we ensure consistent elasticity calculations?

To ensure consistent elasticity calculations, it is advisable to use the midpoint or arc elasticity method. This approach involves using the average of the original and new values as the denominator in the percentage change formula. By doing so, we calculate elasticity as:

$\frac{\mathrm{\Delta Q}/\frac{Q}{}}{}$₁+Q₂2ΔP/P₁+P₂2

where $Q$₁ and $Q$₂ are the original and new quantities, and $P$₁ and $P$₂ are the original and new prices. This method provides a more accurate and consistent measure of elasticity, regardless of the direction of the price change.