ELASTICITY ON THE LINEAR DEMAND CURVE P Consider a linear demand function denoted by Qd = a − bP I a b We find the endpoint co-ordinates on each axis: at P = 0 ⇒ Qd = a … horizontal intercept € a at Qd = 0 ⇒ P = … vertical intercept b II € III a 2b € IV € V € a 2 Q a € Also, find the midpoint co-ordinate of P and Qd: a a at P = … midpoint ⇒ Qd = 2b 2 Recall the own-price elasticity of demand: %ΔQd dQd P ηQ d ,P = = %ΔP dP Qd d dQd € the demand function: dQ = −b . Hence, we can simply find the ownWe need to find from dP € €dP P ηQ d ,P = −b d . price elasticity of demand by substituting P and Qd into: Q € € Observation I – At the vertical-intercept endpoint: ηQ d ,P = −∞ (perfectly elastic) Proof Substitute Qd = 0 and P = ( ) a € a a ; therefore, ηQ d ,P = −b b = − = −∞ . b 0 0 € Observation II – At the quantity demanded lower than midpoint: ηQ d ,P < −1 (elastic) € €a " 1 % 2" a % "P% "1% a €1 2 Proof If Qd < ⇒ d > and P > ; hence, P$ d ' > $ ' ∴ ηQ d ,P = −b$ d ' < −b$ ' < −1. #b& 2 Q a 2b # Q & a # 2b & #Q & € = −1 (unitary elastic) Observation III – At the midpoint: ηQ d ,P € € € a a a a d Proof Substitute Q = and P = ; therefore, ηQ d ,P = −b 2b = − 2 = −1. a a 2 2b 2 2 € ( ) Observation IV – At the quantity demanded higher than midpoint: −1 < ηQ d ,P < 0 (inelastic) € € " 1 % 2" a % "P% "1% a 1 2 a€ d Proof If Q > ⇒ d < and P < ; hence, P$ d ' < $ ' ∴ ηQ d ,P = −b$ d ' > −b$ ' > −1. #b& 2 Q a 2b # Q & a # 2b & #Q & € Observation V – At the horizontal-intercept endpoint: ηQ d ,P = 0 (perfectly inelastic) € € € 0 d Proof Substitute Q = a and P = 0; therefore, ηQ d ,P = −b = 0 . a € € € €