# What is the equation representing a constant elasticity of 1?

I'm reading the chapter in my textbook about the price elasticity of demand, and it was pointed out that most demand curves do not represent a constant elasticity of demand - even linear curves like $f(x)=x$ is not constant elasticity although it has constant slope. It then points out three curves that have constant elasticity all throughout the curve.

I'm just curious, from a mathematical standpoint, what function represents the rounded curve? It's definitely something akin to an exponential function, but I'm not quite sure how to calculate it. I played around on google and found that $f(x)=e^{-x}$ looks something like it, but it's not quite the right shape. The straight lines are simple and straightforward - they have a slope of 0 and a slope of infinity, respectively. But what about the round curve?

FWIW, the equation I have to calculate elasticity is:

$$\eta=\frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}}$$

Would I have to use integration to find this out?

-
Rectangular Hyperbola - given by $pQ = c^2$ for some $c$. – ronno Jun 18 '13 at 22:12

To expand on ronno's comment, recall that you can use the price elasticity of demand to determine whether you should raise or lower your prices (If elasticity of demand is below 0, higher prices yield greater profit and vice versa) until the elasticity is exactly 1, and then you are maximizing revenue. Thus, if the elasticity is 1 throughout the entire curve, then you should be making the same revenue regardless of what point on the curve you "choose" by setting your price. Revenue being price times quantity, it follows that...

pQ = c for some constant c.

I've only actually taken one semester of microeconomics, so I'm not sure why you would write c^2 instead of just c. Perhaps someone else can explain that.

-
The use of $c^2$ here would be somewhat justified if $p$ and $Q$ had similar units, e.g. if both were in dollars then using $c^2$ would make $c$ come out also in dollars. But in this case, $p$ is in a money unit while $Q$ is only a quantity unit, so likely no need for the $c^2$ rather than the $c$. One thing it does do is to make sure the value is positive -- you want a positive constant on the right of $pQ=c$ provided you want the usual inverse relation between $p$ and $C$. – coffeemath Jun 18 '13 at 22:40
Ah, thank you!! – Upside Jun 18 '13 at 23:13
That's a pretty good explanation. Thanks for the insight. – agent154 Jun 18 '13 at 23:18

The general isoelastic demand function is given by $$Q(p)=Ap^\eta$$ where $\eta$ is the constant elasticity of demand.

To derive this function, note that $$\text{constant}=\eta=\frac{dQ/Q}{dp/p}$$ or $$\frac{Q'(p)}{Q(p)}=\frac{\eta}{p}.$$ Integrate to get $$\eta \ln p = \ln Q(p) +k$$ where $k$ is the constant of integration. Then $$Q(p)=e^{-k}p^\eta.$$

-