# Price elasticity of demand always increases with price?

Is there a closed-form continuous demand function whose price elasticity of demand decreases with the price?

Taking hint from 1muflon1's answer, consider the following demand function:

\begin{align} Q &= e^{1/p} \\ \frac{dQ}{dp} &= -\frac{e^{1/p}}{p^2} \\ \frac{dQ}{dp}\frac{p}{Q} &= -1/p \end{align}

So absolute elasticity, $$1/p$$ is decreasing.

There are many functions where absolute value of elasticity is decreasing and which are continuous closed form.

One example of such function would be:

$$Q= a-\ln(p), p\geq1 \implies EL = - \frac{1}{p}\frac{p}{a-\ln p} = -\frac{1}{a - \ln p},$$

which is continuous closed form and decreasing in price in its absolute value.

Originally I interpreted the request too mathematically I am leaving it here since other answer references it.

Yes for example consider the following demand function:

$$Q = e^{-p}$$

price elasticity of demand is given by:

$$EL = \frac{dQ}{dp}\frac{p}{Q} = -e^{-p} \frac{p}{e^{-p}} = -p$$

So the demand function has elasticity that is always decreasing in price $$p$$. The function has closed form, and it is continuous.

• I should have made it clearer. I was talking about the absolute value of elasticity.
Nov 30 '20 at 1:44
– 1muflon1
Nov 30 '20 at 8:23
• @1muflon1: I think you don't need the negative sign. $Q=\ln p$ is what you perhaps meant. Nov 30 '20 at 9:51
• @Dayne we need negative sign just so the demand function makes sense $Q=\ln p$ would make sense only for giffen goods and even there with a caveat as giffen goods are often giffen goods only for a portion of demand curve. (you can try to plot $\ln p$ and $-\ln p$, you will see what I mean. I mean we should also probably add some constant to raise the demand above zero I will add that
– 1muflon1
Nov 30 '20 at 9:56
• The absolute value of the elasticity would be 1/Q, increasing in p. Dayne's works.