Jacob: What does it mean that the elasticity varies over a linear curve but is constant over a logarithmic curve?
Rachel: The price elasticity of demand (η) = MQ/MP × P/Q.
For a linear demand curve, Q = α – ßP, so the elasticity (η) = MQ/MP × (P/Q) = –ßP / (α – ßP).
~ If P is near zero, the elasticity is close to zero.
~ If Q is near zero, α . ßP, so the elasticity is close to –4.
If the relation between two variables is multiplicative, or Y = α Zβ, we take logarithms of both sides to get ln(Y) = ln(α) + β ln(Z). This is a logarithmic curve.
β is the derivative of ln(Y) with respect to ln(Z).
Mln(Y) = MY/Y and Mln(Z) = MZ/Z.
Mln(Y) / Mln(Z) is the elasticity of Y with respect to Z.
The elasticity is constant over the curve.