Editing Price elasticity of demand (section)

==Definition==

The variation in demand in response to a variation in price is called price elasticity of demand. It may also be defined as the [[ratio]] of the percentage change in quantity demanded to the percentage change in price of particular commodity.<ref name="Png57">Png, Ivan (1989). p.&nbsp;57.</ref> The formula for the coefficient of price elasticity of demand for a good is:<ref>Parkin; Powell; Matthews (2002). pp.&nbsp;74–5.</ref><ref name="Gillespie43">Gillespie, Andrew (2007). p.&nbsp;43.</ref><ref name="Gwartney425">Gwartney, Yaw Bugyei-Kyei.James D.; Stroup, Richard L.; Sobel, Russell S. (2008). p.&nbsp;425.</ref> 

:<math>E_{\langle P \rangle} = \frac{\Delta Q/Q}{\Delta P/P}</math>

where  <math>P</math> is the initial price of the good demanded, <math>\Delta P</math> is how much it changed,  <math>Q</math> is the initial quantity of the good demanded, and <math>\Delta Q</math> is how much it changed. In other words, we can say that the price elasticity of demand is the percentage change in demand for a commodity due to a given percentage change in the price. If the quantity demanded falls 20 tons from an initial 200 tons after the price rises $5 from an initial price of $100, then the quantity demanded has fallen 10% and the price has risen 5%, so the elasticity is (−10%)/(+5%) = −2.

The price elasticity of demand is ordinarily negative because quantity demanded falls when price rises, as described by the "law of demand".<ref name="Gillespie43"/>  Two rare classes of goods which have elasticity greater than 0 (consumers buy more if the price is ''higher'') are [[Veblen good|Veblen]] and [[Giffen good|Giffen]] goods.<ref name="Gillespie2007">Gillespie, Andrew (2007). p.&nbsp;57.</ref> Since the price elasticity of demand is negative for the vast majority of goods and services (unlike most other elasticities, which take both positive and negative values depending on the good), economists often leave off the word "negative" or the minus sign and refer to the price elasticity of demand as a positive value (i.e., in [[absolute value]] terms).<ref name="Gwartney425"/> They will say "Yachts have an elasticity of two" meaning the elasticity is −2. This is a common source of confusion for students. 

Depending on its elasticity, a good is said to have 
elastic demand (>&nbsp;1),
inelastic demand (<&nbsp;1), or 
unitary elastic demand (=&nbsp;1). If demand is elastic, the quantity demanded is very sensitive to price, e.g. when a 1% rise in price generates a 10% decrease in quantity. If demand is inelastic, the good's demand is relatively insensitive to price, with quantity changing less than price. If demand is unitary elastic, the quantity falls by exactly the percentage  that the price rises. Two important special cases are perfectly elastic demand (=&nbsp;∞), where even a small rise in price reduces the quantity demanded to zero; and perfectly inelastic demand (= 0), where a rise in price leaves the quantity unchanged. 
The above measure of elasticity is sometimes referred to as the ''own-price'' elasticity of demand for a good, i.e., the elasticity of demand with respect to the good's own price, in order to distinguish it from the elasticity of demand for that good with respect to the change in the price of some other good, i.e., an independent, [[complementary good|complementary]], or [[substitute good]].<ref name="Png57"/> That two-good type of elasticity is called a [[Cross-price elasticity of demand|''cross''-price elasticity of demand]].<ref>Ruffin; Gregory (1988). p.&nbsp;524.</ref><ref>Ferguson, C.E. (1972). p.&nbsp;106.</ref> If a 1% rise in the price of gasoline causes a 0.5% fall in the quantity of cars demanded, the cross-price elasticity is <math>E^d_{cg} = (-0.5%)/(+1%) = -0.5.</math> 

As the size of the price change gets bigger, the elasticity definition becomes less reliable for a combination of two reasons. First, a good's elasticity is not necessarily constant; it varies at different points along the [[demand curve]] because a 1% change in price has a quantity effect that may depend on whether the initial price is high or low.<ref>Ruffin; Gregory (1988). p.&nbsp;520</ref><ref>McConnell; Brue (1990). p.&nbsp;436.</ref> Contrary to [[List of common misconceptions|common misconception]], the price elasticity is not constant even along a linear demand curve, but rather varies along the curve.<ref>Economics, Tenth edition, John Sloman</ref>  A linear demand curve's slope is constant, to be sure, but the elasticity can change even if  <math>\Delta P/\Delta Q</math> is constant.<ref name="parkin75">Parkin; Powell; Matthews (2002). p&nbsp;.75.</ref><ref>McConnell; Brue (1990). p.&nbsp;437</ref> There does exist a nonlinear shape of demand curve along which the elasticity is constant: <math>P = aQ^{1/E}</math>, where <math>a</math> is a shift constant and <math>E</math> is the elasticity. 

Second, percentage changes are not symmetric; instead, the [[Percentage#Percentage increase and decrease|percentage change]] between any two values depends on which one is chosen as the starting value and which as the ending value. For example, suppose that when the price rises from $10 to $16, the quantity falls from 100 units to 80. This is a price increase of 60% and a quantity decline of 20%, an elasticity of <math>(-20%)/(+60%) \approx -0.33 </math> for that part of the demand curve. If the price falls from $16 to $10 and the quantity rises from 80 units to 100, however, the price decline is 37.5% and the quantity gain is 25%, an elasticity of <math>(+25%)/(-37.5%) = -0.67 </math> for the same part of the curve. This is an example of the [[Index_(economics)#Index_number_problem|index number problem]].<ref name="Ruffin">Ruffin; Gregory (1988). pp.&nbsp;518–519.</ref><ref name="Ferguson">Ferguson, C.E. (1972). pp.&nbsp;100–101.</ref>

Two refinements of the definition of elasticity are used to deal with these shortcomings of the basic elasticity formula: ''arc elasticity'' and ''point elasticity''.

===Arc elasticity===

{{main|arc elasticity}}
Arc elasticity was introduced very early on by Hugh Dalton. It is very similar to an ordinary elasticity problem, but it adds in the index number problem.  A second solution to the asymmetry problem of having an elasticity dependent on which of the two given points on a demand curve is chosen as the "original" point and which as the "new" one is Arc Elasticity, which is to compute the percentage change in P and Q relative to the ''average'' of the two prices and the ''average'' of the two quantities, rather than just the change relative to one point or the other.  Loosely speaking, this gives an "average" elasticity for the section of the actual demand curve—i.e., the ''arc'' of the curve—between the two points. As a result, this measure is known as the ''[[arc elasticity]]'', in this case with respect to the price of the good. The arc elasticity is defined mathematically as:<ref name="Ferguson"/><ref name="wall">Wall, Stuart; Griffiths, Alan (2008). pp.&nbsp;53–54.</ref><ref name="McConnell; Brue">McConnell;Brue (1990). pp.&nbsp;434–435.</ref>

:<math>E_d = \frac{ \left(\frac{P_1 + P_2}2\right) }{\left( \frac{Q_{d_1} + Q_{d_2}} 2 \right)}\times\frac{\Delta Q_d}{\Delta P} = \frac{P_1 + P_2}{Q_{d_1} + Q_{d_2}}\times\frac{\Delta Q_d}{\Delta P}</math>

This method for computing the price elasticity is also known as the "midpoints formula", because the average price and average quantity are the coordinates of the midpoint of the straight line between the two given points.<ref name="Ruffin"/><ref name="McConnell; Brue"/> This formula is an application of the [[midpoint method]]. However, because this formula implicitly assumes the section of the demand curve between those points is linear, the greater the curvature of the actual demand curve is over that range, the worse this approximation of its elasticity will be.<ref name="wall" /><ref>Ferguson, C.E. (1972). p.&nbsp;101n.</ref>

===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>