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Constant Unit Elasticity
We've aready talked about linear demand functions that actually have changing price elasticity As we go down the curve and we've shown the extremes we've shown things that are perfectly inelastic and things that are perfectly elastic What I want to do with this video and it will be a quick little video is think about, can we construct a demand curve -or at least understand what it looks like- that has a constant elasticity across the curve and just for fun let's make it a constant elasticity of 1 so it has constant unit elasticity of demand So let's think about how we can create that and hopefully it will give us a little more intuition on how this elasticity business even works so let's draw our axes that is price, and that right there is quantity Now let's put some numbers there, that will just help us draw this demand curve that has unit elasticity at every point, every price, and every quantity I'm just going to put some numbers here, that over there is 10, so that's $10 or whatever we're doing, and that this is 10 units per time period, 10 units per week, 10 units per month, or whatever else. Now, we want the absolute value of the elasticity of the demand to be equal to 1 at all points. We're going to assume that this curve meeets the law of demand which means that as price goes down, quantity demanded goes up. So let's think, this is going to be a downward slope, and really, we're going to say so the elasticity of demand is going to be equal to -1 If we have a 1% decrease in price, we're going to have a 1% increase in quantity and vice versa so let's think about it. If we're up here, where the price is nearly $10, and where maybe the quantity is closer to 1, let's think about what a 10% movement in price would look like, a 10% movement down. It would be about roughly about this size A 10% movement would be roughly there. I'm just trying to get the general shape of this curve, I'm not going into the deeper mathematics behind it or the calculus of it. So that is a 10% price movement down and we also want a 10% in quantity movement up but remember, our quantity is only 1 so 10% quantity movement up, would only be 10% of 1 so if we're moving 10% in price downwards, this is a 10% move upwards in quantity so our curve up here, would look something like this, which would have to be quite steep. Now, let's think about what the curve would look like over here. Now, once again, we want 10% for both of them. Because we want the price elasticity of demand to be 1 throughout the curve. So, if we go over here for a 10% movement in price, so let's say we're down here where price is close to 1. A 10% movement in price is going to be very small. A 10% movement in price is going to be like that, It's going to be roughly a tenth of a unit so that's a 10% movement in price, but a 10% movement in quantity demanded is going to be much larger, it's going to look like that because the quantity is approaching 10, so 10% of that is about 1 unit, just like that, so at this point in the graph it would look something like this it would flatten out a good bit, just like that, and when the price and quantity are the same so let's say that this is 2, and this is 3, right over here, your percent movements are going to be the same, but since the price and quantity are the same the absolute movements are also going to be the same so at that point our curve should look somthing like that, it should have a slope of one so if you connect the dots, you would get the general shape of a demand curve that has a price elasticity of demand of -1 throughout the curve, whose absolute value of the price elasticity of demand is 1, so let's just do that, so the curve will look something like that it will keep getting steeper, as we get the quantity closer to zero and it would keep flattening out as the quantity grows and grows and grows.