Price elasticity

Question

If $P = 90 – 0.05Q$ is the demand function for calculators in an engineering college.

(a) Derive expression for $\epsilon$ in terms of (i) $P$ only, (ii) $Q$ only.

(b) Calculate the value $\epsilon$ when the calculators are priced at $P = £20; £30; £70$.

(c) Determine the number of calculators demanded when $\epsilon = – 1$ and $\epsilon = 0$. of demand.

(d) Use $\epsilon$ to calculate the response ($\%$ change in $Q$) to a $10\%$ increase in the price of calculators at each of the following prices: $P = £20; £30; £45; £70; £90$.

Answer 1

The price elasticity of demand is defined by $$ \epsilon_{Q,P}=\epsilon = \frac{dQ/Q}{dP/P}=\frac{d\log(Q)}{d\log(P)}, $$ i.e. percentage change of $Q$ in response to a one percent change in price. So if $P=90-0.05Q$, then we have $$ \epsilon = \frac{dQ}{dP}\frac{P}{Q}=\frac{-20P}{Q}=\frac{-1800+Q}{Q}=\frac{P}{P-90.} $$ For (b) and (c), just plug the value in the above expression and solve the equation. For (d), calculate $10\epsilon$ when $P=20,30,\ldots, 90$.