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Sensitivity of quantity to price

A good's price elasticity of demand (, 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). 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. 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.

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:

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.

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:

In other words, it is equal to the absolute value of the first derivative of quantity with respect to price multiplied by the point's price (P) divided by its quantity (Qd). However, the point elasticity can be computed only if the formula for the demand function, , is known so its derivative with respect to price, , can be determined. In terms of partial differential calculus, point elasticity of demand can be defined as follows: let be the demand of goods as a function of parameters price and wealth, and let be the demand for good The elasticity of demand for good with respect to price is

Together with the concept of an economic "elasticity" coefficient, Alfred Marshall is credited with defining "elasticity of demand" in Principles of Economics, published in 1890. Alfred Marshall invented price elasticity of demand only four years after he had invented the concept of elasticity. He used Cournot's basic creating of the demand curve to get the equation for price elasticity of demand. He described price elasticity of demand as thus: "And we may say generally:— the elasticity (or responsiveness) of demand in a market is great or small according as the amount demanded increases much or little for a given fall in price, and diminishes much or little for a given rise in price". He reasons this since "the only universal law as to a person's desire for a commodity is that it diminishes but this diminution may be slow or rapid. If it is slow a small fall in price will cause a comparatively large increase in his purchases. But if it is rapid, a small fall in price will cause only a very small increase in his purchases. In the former case the elasticity of his wants, we may say, is great. In the latter case the elasticity of his demand is small." Mathematically, the Marshallian PED was based on a point price definition, using differential calculus to calculate elasticities.

In most situations, such as those with nonzero variable costs, revenue maximizing prices are not profit maximizing prices. For these situations, using a technique for Profit maximization is more appropriate.