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Worked Examples · Example 6

Q.Show, using the geometric method, why the price elasticity of demand equals 1 exactly at the midpoint of a straight-line demand curve, is greater than 1 above it, and less than 1 below it.

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The formula. For a point on a straight-line demand curve AB (A on the price axis, B on the quantity axis), Ep=lower segmentupper segmentE_p = \dfrac{\text{lower segment}}{\text{upper segment}}.

At the midpoint M. The point exactly halves the curve, so lower segment == upper segment. Hence

Ep=MBMA=11=1(unitary elastic).E_p = \frac{\text{MB}}{\text{MA}} = \frac{1}{1} = 1 \quad(\text{unitary elastic}).

Above the midpoint (towards A, the price axis). The point is nearer the top, so the lower segment (down to B) is longer than the upper segment (up to A). The ratio exceeds 1:

Ep=longer lowershorter upper>1(elastic),E_p = \frac{\text{longer lower}}{\text{shorter upper}} > 1 \quad(\text{elastic}),

rising towards ∞\infty as the point approaches A.

Below the midpoint (towards B, the quantity axis). Now the lower segment is shorter than the upper, so

Ep=shorter lowerlonger upper<1(inelastic),E_p = \frac{\text{shorter lower}}{\text{longer upper}} < 1 \quad(\text{inelastic}), …

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