Editing Price elasticity of demand (section)

===Point elasticity===

In order to avoid the accuracy problem described above, the difference between the starting and ending prices and quantities should be minimised. This is the approach taken in the definition of ''point'' elasticity, which uses [[differential calculus]] to calculate the elasticity for an infinitesimal change in price and quantity at any given point on the demand curve:<ref name="sloman">Sloman, John (2006). p.&nbsp;55.</ref>
:<math>E_d = \frac{\mathrm{d}Q_d}{\mathrm{d}P} \times \frac{P}{Q_d}</math>

In other words, it is equal to the absolute value of the first derivative of quantity with respect to price <math>\frac{\mathrm{d}Q_d}{\mathrm{d}P}</math> multiplied by the point's price (''P'') divided by its quantity (''Q''<sub>d</sub>).<ref name="Wessels2000">Wessels, Walter J. (2000). p.&nbsp;296.</ref> However, the point elasticity can be computed only if the formula for the [[Demand schedule|demand function]], <math>Q_d = f(P)</math>, is known so its derivative with respect to price, <math>{dQ_d/dP}</math>, can be determined.

In terms of partial-differential calculus, point elasticity of demand can be defined as follows:<ref>Mas-Colell; Winston; Green (1995).</ref> let <math>\displaystyle x(p,w)</math> be the demand of goods <math>x_1,x_2,\dots,x_L</math> as a function of parameters price and wealth, and let <math>\displaystyle x_\ell(p,w)</math> be the demand for good <math>\displaystyle\ell</math>. The elasticity of demand for good <math>\displaystyle x_\ell(p,w)</math> with respect to price <math>p_k</math> is

:<math>E_{x_\ell,p_k} = \frac{\partial x_\ell(p,w)}{\partial p_k}\cdot\frac{p_k}{x_\ell(p,w)}  = \frac{\partial \log x_\ell(p,w)}{\partial \log p_k}</math>