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 ?


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.


Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.