Price elasticity of demand using the midpoint method

what we're going to think about in this video is elasticity of demand to sit TT elasticity of demand and what this is is a measure of how does the quantity demanded change given a change in price or how does the change in price impact the quantity demanded so change in price change change in price change in price change in price impact impact quantity want to be careful your quantity demanded and you talk about demand you're talking about the whole curve quantity demanded is a specific quantity quantity demanded quantity demanded and the way that we as economists I'm not really an economist but since we're doing economics we can pretend to be economists the way that economists measure this is they measure it as a percent change percent change in percent change in quantity over a percent over the percent change change in price and the reason why they do this as opposed to just say change in quantity over change in price is because if you did change in quantity over change in price you would have a number that's specific to the unit's you're using so it would depend on whether you're doing quantity in terms of per hour or per week or per year and so you would have different numbers based on the time frame or the units that you might use but when you use a percentage when you use a percentage it is a unitless number because a percentage you're taking a change in some quantity divided by that quantity so the unit's themselves actually cancel out and the reason why it's called elasticity this might make some sense to you or the reason why I like to think it's called elasticity as I imagine something that's elastic like an elastic band or a rubber band and in the rubber band if you pull it depending if something so let's say this one is let's say this one is inelastic in elastic inelastic so if you pull you're not going to able to pull it much it's going to be fairly stiff it's not going to stretch a lot while if something is elastic if something is elastic for a given amount of force so this is for a given amount of force you're not able to pull it much and if something is elastic maybe for the same amount of force you're going to able to you're going to be able to pull it a lot so this right over here is elastic and so the analogy maybe might make a little bit sense relative to apply price and demand something is elastic so let me write this down elastic elastic so let me write very elastic very elastic if a given change in price given price change you have and especially we will talk about percentages a little bit but the given change in price you have a large change in demand so large percentage change and let me just speak in terms of percentage given a percentage change change in P you end up having a large percentage change in Q that would be very elastic so you could imagine the P is like the force and the Q the quantity demanded is how is how far the thing can get stretched through a part and that's why we would call this very elastic just like a very elastic rubber band and it's something is very inelastic very in elastic if given a percent change in P given a percent change in P you have a small small percent change in Q so if just like a rubber band for a given amount of force if you if you're not able to pull it much at all then it's inelastic if you're able to pull it a lot its elastic same thing with price and quantity if you forgiving change in price if a change if the quantity demanded if the percent quantity demanded changes a lot very elastic if it doesn't change a lot very inelastic now with that out of the way let's actually calculate the out elasticity for multiple points along this demand curve right over here and I think that'll give us a better grounding especially because there are a little slightly slightly I would call unusual ways of calculating the percent change in quantity percent change of price just so that we get the same number when we have a positive change in price and the same as we get the negative change in price or a negative and a positive or a drop in price and an increase in price so let me give myself some real estate over here because I want to do some actual mathematics and actually all of this we will be reviewing and what I'm about to do and it'll give me some real estate to work with so let me clear all of that and let me clear that right over here and what I'm going to do is I'm going to calculate the elasticity of demand at several points along this demand curve right over here and so the first one I will do it at point A to point B so let me let me make another column right over here elasticity of demand and actually we're going to have one column that's elasticity of demand so it's a biggie with a little subscript D and the other one I'll just take its absolute value because depending on sometimes people like to just think the think of the number which will tend to be a negative number and sometimes people like to look at the absolute value of it so we'll look at both and see what it actually means so let's say our price drops from point A to point B so from point A to point B from point A to point B we have a $1 a negative $1 change in price and we have a positive so this is a negative $1 change in price and we have a positive $2 start a positive to burger per our change in quantity demanded so what is the elasticity of demand there so let's write it over here I'll do it in I'll do it in a is color so the elasticity of demand remember it's the percent change in quantity so percent change in quality rewrite it as percent change in quantity over percent change in in price and so we have what's our percent change in quantity so that it's going to be the change in quantity over some base quantity so our change in quantity is two so it's going to be equal to two over now in traditional terms and this is what I want to kind of clarify is a little bit unusual in how we do it but we do it so that we get the same to see the domain of demand whether we go from A to B or B to a or essentially we get the same elasticity of demand along this whole part of the curve instead of just dividing the change in quantity dividing divided by our starting point what I want to do is I'm going to divide the change in quantity divided by the average of our starting and our ending points so that's going to be two over and I'll actually do the math explicitly actually let's just think about what's the average between two and four well the average is just going to be three that's the average of two and four I could write let me write it down just so it's clear this right over here that right over here is two plus four over two that's how you get three that's how you work calculate the average all right so that is so that is our proportionate change and then you want to multiply by 100 times 100 to actually get a percentage and then what is our change in price but we're going to do the same thing the act or the percent change in price our change in price is negative one it is negative one over and once again we don't just do negative 1/9 we do it over the average of eight and nine the average of eight and nine is eight point five eight point five and then multiply by 100 to get your percentage now these hundreds obviously cancel out these hundreds cancel out and so we are going to be left with when you divide by a fraction it's the same thing as multiplying by its inverse so we're going to get 2/3 2/3 times negative eight point five over one or times negative eight point five times negative eight point five you get out our calculator and it is so two well multiply two times negative eight point five and then divided by three which gives us negative five point six six six seven it's really negative five and two thirds so I'll just write a negative I'll round it it's negative five point six seven so this is approximately approximately equal to negative five point six seven so right over here it's negative five point six seven and this absolute value is obviously just five point six seven and I'll leave it to you to verify for yourself that you'll get the same elasticity of demand using this technique where you use the average as your base in the percentage going from nine to eight as going for going from nine to eight in price as going from eight to nine in price which is different than if you use the nine as the base or the eight as a base so this right here is the elasticity of demand not just at Point a you can kind of view it as the average elasticity of demand over this little part of the curve which is really aligned in this example over this part of the of the arc so we'll write that part right over here I'll write the absolute value the absolute value of our elasticity of demand is five point six seven now let's do the other two the other two sections right over here so let's think about what happens let's think about what happens when we go from C to D so let's see the D so our elasticity of demand there so from C to D we have a change in quantity once again if plus two and our change in price once again is negative one but we'll see even though that the change in quantity over the change in quantity is the same as the change in price is the same we're going to have a different elasticity of demand because we're start where I have different starting points our starting points for in our ending points for price or lower and our starting and ending points for quantity or higher so it'll actually change the percentage so let's see what we get so our percent change in quantity we have a change in quantity of two and then our average quantity is nine plus 11 which is 20 divided by two is ten ten all of that over change percent change in price so we have negative one negative one let me scroll down a little bit negative one divided by the change in negative one divided by the average price so negative one is the change in price and we want to divide that by the average price well 550 plus 450 is $10 divided by two is five so the average the average price is five and we can multiply the numerator by 100 and then by 100 but that won't that won't change anything because we could just divide both by 100 and so this is equal to 2 over 10 times dividing by a fraction same thing as multiplying by its inverse times negative 5 over 1 and this is just because 2 over 10 is the same thing as 1/5 1/5 times negative 5 over 1 it is negative 1 so this right over here so our elasticity of demand right over here is negative 1 or its absolute value is 1 so the absolute value of the elasticity of demand right over here is equal to 1 and let's just do one more section and maybe in the next video we can think a little bit about what it's telling us so let's do this last section over here just for some practice I'd encourage you to pause it and try it yourself and so we're going to think about this section right over here so once again our change in quantity is +2 and our change in price is negative 1 and our elasticity of demand change in quantity 2 over average quantity which is 17 change in price is negative 1 over average price 1 plus 2 divided by 2 is a dollar 50 or a dollar 50s right in between these two divided by dollar 50 we don't have to multiply the numerator in the denominator by 100 because those just cancel out so we get 2 over 17 times negative well we could just write this as negative dollar 50 over 1 and this is equal to getting our getting our calculator back out so this is equal to I'll just write well it's really just going to be negative 3 over 17 right 2 times negative 51 50 is negative 3 over 17 so negative 3 divided by 17 is equal to negative 0.1 I'll just say negative 0.18 so here it is negative 0.18 and its absolute value is 0.18 so the elasticity of demand over here is 0.18 and I'll leave you there and in the next video we'll we'll think about these results a little bit