To solve our different answer dilemma, we use the midpoint method.

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concept

The Midpoint Method

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now we're gonna use what's called the midpoint method to calculate elasticity, which is gonna give us a consistent answer whether we're raising the price or decreasing the price. So, I've got our updated formula for our price elasticity of demand when we're using the midpoint formula, which is what we're gonna use from now on. Um But don't let this trip you up, There's a lot going on there, but this is still just percentage change in quantity demanded, divided by percentage change in price. Okay, we're still just dealing with that same formula. We're just changing how we calculate the percentage change in each situation. So, if you remember with percentage change before we had the original value in the denominator, right? It was the change divided by the original value. Well, instead of the original value. Now we're gonna use uh this notice some of quantities divided by two. This is the average quantity instead of the original quantity. And over here we're gonna use the average price instead of the original price for the denominator. Right? So notice we've got a lot of divisions here, A lot of, you know, divide over, divide by two. There's a division in the middle of the two. There's there's just division everywhere. This can turn into a huge orders of operations nightmare. Let me tell you in what order you do. This could really screw you up. So I've included what we're gonna use this step by step method. This is how I always calculate it. And it just keeps it simple and it makes you can't make those mistakes. So one more thing I want to note before we go into that step by step is that this whole first uh formula right here, This whole first piece of it, this is our percentage change in quantity demanded all of that, that's going on there. And we're gonna divide notice, we've got that division in the middle and this second piece which I'll circle in green like we've been using that is our percentage change in price, right? So we've got the percentage change in quantity demanded divided by the percentage change in price. We just have everything in one place there. Alright so I've got our step by steps here. It's gonna be some subtracting than summing. It's just easier to do it with an example. So let's go ahead and use the example that we were using before with the pizza company lunch special. Right? So we saw in the previous videos that when we raised the price from $5 to $6 we got a different answer than when we decreased the price from $6 to $5. We were getting different elasticity ease. So with this midpoint method we're actually gonna get the same answer whether we raise the price or decrease the price in our example. We're just gonna do the raise the price you can on your own do to decrease the price and just confirm that it's the same. But I guarantee you you're gonna get the same answer right? So let's go ahead and use this step by step um to calculate our elasticity of demand. So just like before we've got a pizza companies lunch special currently costs $5 at this price. The weekly demand is 2000 lunch specials. If they raise their price to $6, the weekly demand will drop to 1400 lunch specials. What is price elasticity of demand is elastic or in elastic? So let's go ahead. Remember we're still going to be dealing with the elasticity of demand formula where percentage change in quantity demanded, divided by percentage change in price. Right? So we're gonna use these step by steps and it's gonna calculate everything like we did above. So I'm gonna have two columns here. This is how I like to do it. I'm gonna have a column for quantity demanded and a column here for price. I'll have to get out of the way in a second. I'll just do it now. Alright so let's go ahead with step one. Alright. Step one here. So step one says subtract the two quantities and subtract the two prices. That's easy enough. Well let's go ahead and circle right are quantities here just so we can see them all at once. Little circle of price is $5.06 dollars. Remember in this case it doesn't matter which one was first which one was second, we're gonna get the same answer. So what I like to do is I like to keep the math easy and I just do the bigger one minus the smaller one every time just to get a positive number. Since we always use positive numbers anyways. So the difference, right, subtract the two quantities, we're gonna get 600 here and I'd like to box off my answers so that I keep track of them. And for price, let's just do it the easy way six minus five equals one. Right? So there we go. That's step one. Easy enough. Let's go on to step two here. So step two, we are going to some of the two quantities and some of the two prices. Hey, I can do that. So 2000 plus 1400 that's gonna be the sum of the two quantities. That's gonna give me 3400 right there and I'll box it off again. Let's do the same thing with price six plus five equals 11. Right? So there we go. Step two. Done very easy stuff so far. Let's go on to step three, divide your quantity sum by two, divide your price sum by two. So this is just what we got in step two, we're gonna divide those by two. So for for um Quantity. So just just so you see it's clear we're doing the 3400 divided by two and we're gonna get 1700 here. Um and for price we've got 11 divided by two, which is going to give us 5.5 right A price of 5 50 is the average price. Right? So notice what we've done so far, I just want to reiterate how we're using these steps in the formula step one. The subtraction step that is our numerator for both of these, it's gonna be our numerator for quantity demanded and our numerator for price. We calculated the change in quantity and the change in price. That's the difference of the two in step two. We sum the quantities and some of the prices which you can see is there in the denominator. Right? And then we also in step three divided by two for each of them. So you can see we've kind of done all the math inside each of them and now we're gonna go ahead and finish up the problem. So let's do step four here where we're gonna divide our answers from step one and step three. Right, so let's do that for quality demanded. First divide your answer from step one and step three which is gonna be our 600 from step one divided by r 1700 from step three. And let's see what that gives us. Um I'm gonna round off to three decimal places here. 0.353. Right 0.353. And this is just so you understand this that we just calculated 0.353. That's our percentage change in quantity demanded right there. Okay, so we've just calculated the percentage change in quantity demanded. Which as a percentage would have been? 35.3%. Right Um the quantity demanded changed by 35.3%. Let's see how much the price changed. So step four we're going to divide steps one and three. We're gonna do one divided by 5.5 which was our answer to step three. And that's going to give us, I'm gonna also round 23 decimals. 0.182. Okay. And just as you might have guessed this is our percentage change in price is 0.182 which is 18.2%. Price changed 18.2% quantity demanded change 35.3%. Just thinking of that right there without even going to our elasticity formula. Let's think about that real quick quantity demanded changed a lot compared to price. Right? Price only changed 18% in this case where quantity demanded. When the price changes 18% quantity demanded shifted 35%. Alright so we're seeing a big shift in quantity demanded for a small change in price. What does that tell us? It tells us that it's gonna be elastic right? Demand is very sensitive to the price. But let's just confirm that with our formula. So elasticity of demand is going to equal the percentage change in quantity demanded. And I think I can come back in now. Hey guys um so our percentage change in quantity demanded was .353 Divided by our percentage change in price .182. Right? And what's that gonna give us? It's gonna give us an elasticity of demand, whoops. Let me do that d small Equal to 1.934. I'm gonna round it off there. So there it is. And just like we just discussed, right, that quantity demanded changed a lot compared to the price change. So what we have here is an elastic demand, right? And we can confirm it because we got in elasticity of demand greater than one. So we have elastic here. Cool. So that's how we're gonna use our step by step to do the midpoint formula. I know there's a lot there, but once you do it a few times it's gonna be a no brainer. So let's go ahead and do a practice problem where you can try using the midpoint formula and the steps involved. Cool, let's do that. Now

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Problem

Problem

The price of widgets is currently $44 with a quantity demanded of 200,000 units. If the price decreases to $36, the quantity demanded increases 280,000. Using the midpoint method, what is the price elasticity of demand? Is demand elastic or inelastic?

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Problem

Problem

Assume that the price elasticity of demand for cigarettes is 0.4. If a pack of cigarettes currently costs $6 and the government aims to decrease smoking by 20 percent, by how much should it increase the price?