# 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 $c$. Jun 18, 2013 at 22:12

• 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 $C$. Jun 18, 2013 at 22:40
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 $$Q(p)=e^{-k}p^\eta.$$