Constant unit elasticity

we've already talked about linear demand functions that actually have changing price elasticity as we go 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 in this video and it'll 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 one so it has constant unit elasticity of demand so let's think about how we can create that and hopefully it'll give us a little bit more intuition on how this elasticity business even works so let's draw our axes so I can see there you go that is price price and that right there is quantity quantity and let me put Kwan T now that I put one extra t-there quantity now let's put some numbers there that'll just help us draw this this demand curve that has unit elasticity at every point at every price in every quantity so I'm just going to put some numbers here so let's say that this that right over there is 10 so that's $10 or whatever we're doing and then this is 10 units per time period 10 units per week or 10 units per month or whatever else now we want we want the absolute value of the elasticity of demand to be equal to 1 at all points and we're going to assume that this curves is meets the law of demand which means this price goes down quantity demanded goes up so let's think of so it's going to be a downward sloping so really we're going to say that the elasticity of demand is going to be equal to negative 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 if we're up here where the price is near $10 and maybe where the quantity is closer to $1 let's think about what a 10 percent movement in price would look like a 10 percent movement down it would be roughly about this size a 10 percent movement would be roughly there and I'm just trying to get the general shape of this curve I'm not going to go into the deep mathematics or the calculus of it so that is a 10% price movement down and we also want a 10% quantity movement up but remember our quantity is only at 1 so 10% quantity movement up would only be would only be 10% of 1 so if we're moving 10% in price downwards this is a 10% upwards in quantity so our curve up here would look something like this it would actually have to be quite steep now let's think about what the curve would look over here once again we want 10% for both of them because we want this we want the price elasticity of demand to be 1 throughout the curve so if we go over here 10% a 10% movement in price so let's say we're down here where prices where price is close to 1 a 10% movement in price is going to be very small so 10% movement in price is going to be like that it's going to be like what's going to be rough roughly 1/10 of a movement so that's a 10% movement in price but a 10% movement in quantity demanded over here it's going to be much larger it's going to look something like that because where the quantity is approaching 10 so 10% of that is about one 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 then when the price and the quantity is about the same so let's say this point right over here where the price and the quantity is about the same so let's say that this is so that is 2 this is 3 this is 2 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 something like that it should have a slope of 1 and so if you connect the dots you get a the general shape of a demand curve that has a price elasticity of demand at negative 1 throughout the curve or whose absolute value of the price elasticity of demand is 1 so let's just do that so the curve would look something like I'll just draw a dotted line it's easier to do a do so it'll look something like that it'll keep getting steeper as we get the quantity closer to zero and we'll keep flattening out as the quantity grows and grows and grows anyway hopefully you found that interesting