A quantitative finance problem by Steven Yuan

Clearly, a perfectly elastic demand curve would maximise the quantity

The elasticity of demand is given by

$\left | \frac{dQ}{dP} \times \frac{P}{Q} \right |.$

The total revenue a company can make is equal to $PQ$ . Since $P = \frac{a - Q}{b}$ , we have

$PQ = \frac{Q(a - Q)}{b} = \frac{aQ - Q^2}{b}.$

The quantity that maximizes revenue can be found by taking the derivative of the above expression:

$\frac{d}{dQ} \; \frac{aQ - Q^2}{b} = \frac{a - 2Q}{b}.$

Setting this equal to 0, we find $Q = \frac{a}{2}$ . Since the second derivative is always negative, the revenue is maximized at this value.

We have $\frac{dQ}{dP} = -b$ from our function. Thus, the elasticity equals

$\left | - \frac{bP}{Q} \right | = \left | - \frac{b \left (\frac{a - Q}{b} \right)}{Q} \right | = \left | - \frac{a - Q}{Q} \right |.$

Plugging in $Q = \frac{a}{2}$ , we find that the elasticity equals $\boxed{1}$ .