Unitary Elastic Demand

Question

My teacher defined Unitary elastic demand as demand which has $PED = (-)1$ . The formula I have been told for $PED$ is $\%\Delta Q_d /\% \Delta P$. But how can total revenue $( = P\times Q)$ remain constant for such a demand? Suppose $P$ goes up $10\%$, so $P' = 1.1P$. For unit elasticity, $Q$ must go down $10\%$ so $Q' = 0.9Q$. So $T' = Q' \times P' = 0.9Q \times 1.1P = 0.99 \times PQ = 99\%$ of $T$. So when we say $T$ stays constant, do we mean that as an approximation ?

Answer 1

The way to examine such issues is by general mathematical treatment, not arithmetic examples.

We have total revenue

$$TR = Q(P)\cdot P$$

When will Total Revenue remain constant as price changes?

We want

$$\frac{\partial TR}{\partial P} =0 \implies \frac{\partial Q(P)}{\partial P}\cdot P + Q(P) =0$$

$$\implies \frac{\partial Q(P)}{\partial P} = -\frac {Q(P)}{P}$$

which is to say that the price elasticity of demand is equal to minus unity.

Is it possible for this to hold over the whole range of $P$ and not just momentarily, at a specific value of $P$?

This entails to find a solution to a simple differential equation (using generic notation)

$$y' + \frac{y}{x} = 0$$

which gives

$$Q(P) = \frac {A}{P},\;\;\; A>0$$

So if the demand function is as above, it has unitary price elasticity everywhere.