A good's price elasticity of demand (${\displaystyle E_{d}}$, PED) is a measure of how sensitive the quantity demanded is to its price. When the price rises, quantity demanded falls for almost any good (law of demand), but it falls more for some than for others. The price elasticity gives the percentage change in quantity demanded when there is a one percent increase in price, holding everything else constant. If the elasticity is −2, that means a one percent price rise leads to a two percent decline in quantity demanded. Other elasticities measure how the quantity demanded changes with other variables (e.g. the income elasticity of demand for consumer income changes).^[1]

Price elasticities are negative except in special cases. If a good is said to have an elasticity of 2, it almost always means that the good has an elasticity of −2 according to the formal definition. The phrase "more elastic" means that a good's elasticity has greater magnitude, ignoring the sign. Veblen and Giffen goods are two classes of goods which have positive elasticity, rare exceptions to the law of demand. Demand for a good is said to be inelastic when the elasticity is less than one in absolute value: that is, changes in price have a relatively small effect on the quantity demanded. Demand for a good is said to be elastic when the elasticity is greater than one. A good with an elasticity of −2 has elastic demand because quantity demanded falls twice as much as the price increase; an elasticity of −0.5 has inelastic demand because the change in quantity demanded change is half of the price increase.^[2]

At an elasticity of 0 consumption would not change at all, in spite of any price increases.

Revenue is maximized when price is set so that the elasticity is exactly one. The good's elasticity can be used to predict the incidence (or "burden") of a tax on that good. Various research methods are used to determine price elasticity, including test markets, analysis of historical sales data and conjoint analysis.

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.^[3] The formula for the coefficient of price elasticity of demand for a good is:^[4][5][6]

${\displaystyle E_{\langle P\rangle }={\frac {\Delta Q/Q}{\Delta P/P}}}$

where ${\displaystyle P}$ is the initial price of the good demanded, ${\displaystyle \Delta P}$ is how much it changed, ${\displaystyle Q}$ is the initial quantity of the good demanded, and ${\displaystyle \Delta Q}$ 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".^[5] Two rare classes of goods which have elasticity greater than 0 (consumers buy more if the price is higher) are Veblen and Giffen goods.^[7] 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).^[6] 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 (> 1), inelastic demand (< 1), or unitary elastic demand (= 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 (= ∞), 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, or substitute good.^[3] That two-good type of elasticity is called a cross-price elasticity of demand.^[8][9] 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 ${\displaystyle E_{cg}^{d}=(-0.5\%)/(+1\%)=-0.5.}$

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.^[10][11] Contrary to common misconception, the price elasticity is not constant even along a linear demand curve, but rather varies along the curve.^[12] A linear demand curve's slope is constant, to be sure, but the elasticity can change even if ${\displaystyle \Delta P/\Delta Q}$ is constant.^[13][14] There does exist a nonlinear shape of demand curve along which the elasticity is constant: ${\displaystyle P=aQ^{1/E}}$, where ${\displaystyle a}$ is a shift constant and ${\displaystyle E}$ is the elasticity.

Second, percentage changes are not symmetric; instead, the 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 ${\displaystyle (-20\%)/(+60\%)\approx -0.33}$ 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 ${\displaystyle (+25\%)/(-37.5\%)=-0.67}$ for the same part of the curve. This is an example of the index number problem.^[15][16]

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

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. Arc Elasticity is 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 will and which as the "new" one 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:^[16][17][18]

${\displaystyle E_{d}={\frac {\left({\frac {P_{1}+P_{2}}{2}}\right)}{\left({\frac {Q_{d_{1}}+Q_{d_{2}}}{2}}\right)}}\times {\frac {\Delta Q_{d}}{\Delta P}}={\frac {P_{1}+P_{2}}{Q_{d_{1}}+Q_{d_{2}}}}\times {\frac {\Delta Q_{d}}{\Delta P}}}$

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.^[15][18] 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.^[17][19]

Point elasticity

The point elasticity of demand method is used to determine change in demand within the same demand curve, basically a very small amount of change in demand is measured through point elasticity. One way to avoid the accuracy problem described above is to minimize the difference between the starting and ending prices and quantities. 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:^[20]

${\displaystyle E_{d}={\frac {\mathrm {d} Q_{d}}{\mathrm {d} P}}\times {\frac {P}{Q_{d}}}}$

In other words, it is equal to the absolute value of the first derivative of quantity with respect to price ${\displaystyle {\frac {\mathrm {d} Q_{d}}{\mathrm {d} P}}}$ multiplied by the point's price (P) divided by its quantity (Q_d).^[21] However, the point elasticity can be computed only if the formula for the demand function, ${\displaystyle Q_{d}=f(P)}$, is known so its derivative with respect to price, ${\displaystyle {dQ_{d}/dP}}$, can be determined.

In terms of partial-differential calculus, point elasticity of demand can be defined as follows:^[22] let ${\displaystyle \displaystyle x(p,w)}$ be the demand of goods ${\displaystyle x_{1},x_{2},\dots ,x_{L}}$ as a function of parameters price and wealth, and let ${\displaystyle \displaystyle x_{\ell }(p,w)}$ be the demand for good ${\displaystyle \displaystyle \ell }$. The elasticity of demand for good ${\displaystyle \displaystyle x_{\ell }(p,w)}$ with respect to price ${\displaystyle p_{k}}$ is

${\displaystyle E_{}}$Zdroj:https://en.wikipedia.org?pojem=Price_elasticity_of_demand
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