In economics, elasticity is the ratio of the percent change in one variable to the percent change in another variable. It is a tool for measuring the responsiveness of a function to changes in parameters in a relative way. Commonly analyzed are elasticity of substitution, price and wealth. Elasticity is a popular tool among empiricists because it is independent of units and thus simplifies data analysis.
An "elastic" good is one whose price elasticity of demand has a magnitude greater than one. Similarly, "unit elastic" and "inelastic" describe goods with price elasticity having a magnitude of one and less than one respectively.
The degree to which a demand or supply curve reacts to a change in price is the curve's elasticity. Elasticity varies among products because some products may be more essential to the consumer. Products that are necessities are more insensitive to price changes because consumers would continue buying these products despite price increases. Conversely, a price increase of a good or service that is considered less of a necessity will deter more consumers because the opportunity cost of buying the product will become too high.
A good or service is considered to be highly elastic if a slight change in price leads to a sharp change in the quantity demanded or supplied. Usually these kinds of products are readily available in the market and a person may not necessarily need them in his or her daily life. On the other hand, an inelastic good or service is one in which changes in price witness only modest changes in the quantity demanded or supplied, if any at all. These goods tend to be things that are more of a necessity to the consumer in his or her daily life.
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The definition of elasticity is based on the mathematical notion of point elasticity. For example, it applies to price elasticity of demand and price elasticity of supply, in which case the functions of interest are Q^{d}(P) and Q^{s}(P). When working with graphs, it is common to put Quantity on the xaxis and Price on the yaxis, thus the function of interest is x(y) rather than (as commonly used in mathematics) y(x).
In general, the "yelasticity of x" is:
The "yelasticity of x" is also called "the elasticity of x with respect to y". It is typical to represent elasticity as 'E', 'e' or lowercase epsilon, 'ε'.
Elasticity can be approximated using percent changes:
where , and percentage change , and similarly for .
Another way to approximate elasticity is using the average value (see arc elasticity):
PED can also be expressed as (dQ/dP)/Q/P or the ratio of the marginal function to the average function for a demand curve Q = f(P).^{[1]} This relationship provides an easy way of determining whether a point on a demand curve is elastic or inelastic. The slope of a line tangent to the curve at the point is the marginal function. The slope of a secant drawn from the origin through the point is the average function. If the slope of the tangent is greater than the slope of the secant (M > A) then the function is elastic at the point.^{[1]} If the slope of the secant is greater than the slope of the tangent then the curve is inelastic at the point.^{[1]}
If the tangent line is extended to the horizontal axis the problem is simply a matter of comparing angles formed by the lines and the horizontal axis.^{[2]} If the marginal angle is numerically greater than the average angle then the function is elastic at the point. If the marginal angle is less than the average angle then the function is inelastic at that point. If you follow the convention adopted by economist and plot the independent variable on the vertical axis and the dependent variable on the horizontal axis then the marginal function will be dP/dQ and the average function will be P/Q meaning that you are deriving the reciprocal of elasticity. Therefore opposite rules would apply. The tangency line slope would be dP/dQ and the slope of the secant would be the numerical value P/Q.
This method is not limited to demand functions; it can be used with any functions. For example a linear supply curve drawn through the origin has unitary elasticity (if you use the method the marginal function is identical to the slope). If a linear supply function intersects the y axis then the marginal function will be less than the average and the function is inelastic at any point and becomes increasingly inelastic as one moves up the curve.^{[2]} With a supply curve that intersects the x axis then the slope of the curve will exceed the slope of the secant at all point meaning that the M > A the slope is elastic and will become increasingly elastic as one moves up the slope.^{[1]} Again this assumes that the dependent variable is drawn on the Y axis.
It is important to keep in mind that PED measures the magnitude of the movement along the demand curve while AED, YED and XED measure the magnitude of the shift of the demand curve.
Price elasticity of demand (PED) measures the percentage change in quantity demanded (Q) caused by a one percent change in the price variable (P). A change in price induces a movement along the demand curve that reflects the change in quantity demanded. PED is a measurement of how far along the curve the movement is or how much quantity demanded changes. Mathematically PED = (∂Q/∂P) (P/Q). The partial derivative ∂Q/∂P indicates that all other determinants of demand are being held constant. PEDs are almost always negative. Conventionally, economists use absolute values in discussing elasticities. If the PED is greater than 1 demand is said to be elastic; between zero and one demand is inelastic and if PED equals one demand is unitelastic. A perfectly inelastic demand curve, perpendicular to the Xaxis, has zero elasticity. A perfectly elastic demand curve, horizontal to X axis, is infinitely elastic.
There is a predictable relationship between revenue and elasticity. Depending on PED, one may raise revenue either by increasing prices and sacrificing quantity or by reducing them and outputting more.
Income elasticity of demand (YED) measures the percentage change in demand caused by a one percent change in income. A change in income causes the demand curve to shift reflecting the change in demand. YED is a measurement of how far the curve shifts horizontally along the Xaxis. Mathematically YED = (∆Q/∆Y) (Y/Q). Again the partial derivative indicates that all other determinants of demand including the price of the good are being held constant. When YED is less than one (YED < 1) demand is income inelastic.^{[3]} When YED is greater than one (YED > 1) demand is income elastic.^{[4]} If YED is negative (YED < 0) the good is sometimes referred to as an inferior good as opposed to normal goods ( 0 < YED < 1) and superior goods (1 < YED).
The income elasticity of demand measures the demand response to a change in income all other determinants of demand being held constant. Income is a nonprice determinant of demand. Therefore a change in income will cause a shift in the demand curve. If income increases the demand curve will shift out (for normal goods). Income decreases will induce a decrease in demand  the curve shifts inward. YED measures the magnitude of this shift in relative terms. If the coefficient of elasticity is greater than one the response is elastic. If the coefficient is less than one the response is inelastic.
YED's are used to classify goods as superior, normal or inferior. With a normal good demand varies in the same direction and at approximately the same rate as income.^{[5]} With an inferior good demand and income move in opposite directions. With a superior good demand varies in the same direction but at a greater rate.^{[5]} A superior good has a YED greater than 1; an normal good, from zero to 1 and an inferior good have negative YED's. An example of a superior good is restaurant meals with a YED of 1.40. An example of a normal good is butter with a YED of 0.42. Margarine with a YED of 0.20 is an inferior good. As with all elasticities the coefficients vary with time place circumstance.
Income elasticity gauges the sensitivity of a consumer's demand for a good to changes in the consumers income. If YED is high the percentage increase in expenditures for the good will be greater than the percentage increase in the consumer's income. In plain English if her income goes up the consumer will buy a lot more of the good.^{[6]} If YED is high the percentage decrease in expenditures for the good will be greater than the percentage decrease in the consumer's income. Again if her income goes down she will greatly cut her purchases of the good. Consumers with low YEDs will react opposite reaction. Their response to income changes in terms of purchasing patterns is muted. If their income increases the purchase of a good will go up a little bit. If their income decreases they will cut their purchases by a small amount.
Cross price elasticity of demand (XED) measures the percentage change in demand for the good in question caused by a one percent change in the price of a related good. Related goods are complements and substitutes. A change in the price of a related good causes the demand curve to shift reflecting the change in demand. XED is a measurement of how far the curve shifts horizontally along the Xaxis. Mathematically XED = (∆Q/∆Prg)(Prg/Q) where Prg is the price of the related good. The crossprice elasticity of demand measures the demand response of good x to a change in the price of a related good, y. If the price of a complement (substitute) goes up the demand for the good in question will go down (up) the relationship is inverse (positive) and the sign of the coefficient for the related good will be negative (positive). The coefficient of elasticity show the strength of the relationship how strong of a complement (substitute) the related good is.
Crossprice elasticity of demand was used to study the consumption habits of heavy smokers after a substantial increase in the price of cigarettes.^{[7]} In one study a ten percent rise in cigarettes caused "poor smoking families" to cut back on cigarettes by 9%, food by 17% and health care by 12%.^{[7]} In other words people will reduce purchases of basic goods and services to continue to smoke.^{[8]}
Cross price elasticities of demand measure the strength of the relationship between goods. A positive crossprice elasticity means that the goods are substitute goods. A substitute good is one that a consumer will buy instead of the good in question. if the price of a good goes up, the demand for the substitute good goes up as well. A negative crossprice elasticity means that the goods are complement goods. Complementary goods are bought together, for example a hotdog and mustard. With complements an increase in the price of one good causes a drop in the demand for the complementary good. A low or a zero value mean that the two goods are weakly related or not related. XEDs are used in antitrust litigation as a measure of monopoly power. A monopolized good has no substitutes. Therefore, the XED should be zero between a monopolized good and any other good.
Advertising elasticity of demand (AED) is an elasticity measuring the effect of an increase or decrease in advertising on a market.^{[9]}^{[10]} Mathematically, then, AED measures the percentage change in the quantity of a good demanded induced by a given percentage change in spending on advertising in that sector:^{[11]}
In other words, the percentage by which sales will increase after a 1% increase in advertising expenditure assuming all other factors remain equal (ceteris paribus).^{[10]} AED is usually positive.^{[11]} Negative advertising may, however, result in a negative AED. In general, the greater the AED the more worth is gained through advertising, though this is also dependent on price elasticity of demand.
Cross elasticity of demand for firms is a measure of the interdependence between firms. It is a measure of the extent to which one firm reacts to changes made by other firms. The change could be price, quantity, advertising  any fact or circumstance that is a basis for competition. Monopolistic competition is an example of a market structure with a low cross elasticity of demand between firms while oligopoly is a market structure with a high coefficient of elasticity.^{[12]}
The primary supply elasticity is the price elasticity of supply (PES) which equals %∆Q_{S}/%∆P = (∆Q/∆P) x P/Q = (∆Q/∆P) x P/Q. The price elasticity of supply is (in the vast majority of cases) positive or zero. For example, if the PES for a good is 0.67 a 1% rise in price will induce a twothirds increase in quantity supplied.^{[13]} If the coefficient, η, is less than 1, supply is said to be inelastic. If η>1 supply is said to be elastic. If the supply curve slopes upward elasticity varies along the curve.^{[14]} Linear supply curves which pass through the origin display unitary elasticity because the slope and the ratio of price to quantity remain constant along the curve.^{[14]} The coefficient of elasticity of any linear supply curve that cuts the yaxis is greater than 1 (elastic), and the coefficient of elasticity of any linear supply curve that cuts the xaxis is less than 1 (inelastic).
If the PES is zero the quantity supplied is fixed. The quantity supplied will not change regardless of the price offered. If the PES is infinite the supply curve is perfectly elastic and the buyer can buy all the goods he wants at the existing price.
Elasticity of scale or output elaticities measure the percent increase in output induced by a one percent increase in inputs due to an increase in the scale of the production process.^{[15]} This number provides a means to determine whether a production function is exhibiting increasing, decreasing or constant returns to scale for any value of x where y = f(x) is a production function. In English, this means that a production function while overall exhibiting increasing, decreasing or constant returns to scale may at some specific level of operatations exhibit different returns to scale. The elasticity of scale provides a means of determining the scale for any level of operations. It is a local rather than global measure. The output elasticity for constant returns to scale is 1; for increasing returns to scale, greater than 1 and for decreasing returns to scale, less than 1. For example, an output elasticity of 0.75 would mean that a ten percent increase in all the inputs would result in a 7.5 percent increase in output.^{[16]}
Output elasticities can also be calculated for each input while holding the other inputs constant. For example the output elasticity with respect to labor is the percentage change in output caused by a one percent change in labor or (∆Q/∆L)(L/Q).^{[13]} (∆Q/∆L) is the marginal product of labor and (L/Q) is the reciprocal of the average product of labor. Therefore, (∆Q/∆L)(L/Q) is MPL/APL.^{[13]}
The concept of elasticity applies to any situation where variables are functionally related.^{[17]} For example, one could calculate the "TV football game elasticity of divorce rates".^{[17]} or "swine flu elasticity of gambling junkets". The only limitations are that the variables be quantifiable and that they be causally related.^{[18]}
With certain limited exceptions elasticity is not the same as slope.^{[19]} though they are easily confused.^{[20]} This is best illustrated by calculating elasticity along a linear demand curve. Intuitively with a straight demand curve it would make sense to assume that the slope and elasticity are identical and constant. However, while slope is constant along the curve elasticity varies from ∞ to zero.^{[19]} The reason for the difference between slope and elasticity is the ratio of price to quantity P/Q. This ratio continuously falls as one moves "down" the demand curve. As the ratio falls the coefficient of elasticity increases. There are two linear demand curves for which the slope and elasticity are identical.^{[19]} The slope and coefficient of elasticity of a perfectly inelastic demand curve is zero. The slope and coefficient of elasticity for a perfectly inelastic demand curve equal negative infinity. The elasticity is constant along each curve.^{[19]}
For example, in the figure to the right the slope of S_{1} is clearly different from the slope of S_{2}, but since the rate of change of P relative to Q is always proportionate, both S_{1} and S_{2} are unit elastic (i.e. E = 1).
The demand curve (D_{1}) is perfectly ("infinitely") elastic. 
The demand curve (D_{2}) is perfectly inelastic. 
Unit elasticity for a supply line passing through the origin 
As the price of a good rises, consumers will usually demand a lower quantity of that good; they may consume less of that good, substitute it with another product, etc. The greater the extent to which demand falls as price rises, the greater the price elasticity of demand. Conversely, as the price of a good falls, consumers will usually demand a greater quantity of that good: consuming more, dropping substitutes, etc. However, there may be some goods of which consumers cannot consume less or for which adequate substitutes cannot be found. Prescription drugs, fuel, and food are some examples of these. For such goods, demand does not greatly decrease as the price rises, and elasticity of demand can be considered low.
Further, elasticity will normally be different in the short term and the long term. For example, for many goods the supply can be increased over time by locating alternative sources, investing in an expansion of production capacity, or developing competitive products which can substitute. One might therefore expect that the price elasticity of supply will be greater in the long term than the short term for such a good, that is, that supply can adjust to price changes to a greater degree over a longer time.
This applies to the demand side as well. For example, if the price of petrol rises, consumers will find ways to conserve their use of the resource. However, some of these ways, like finding a more fuelefficient car, take time. So consumers as well may be less able to adapt to price shocks in the short term than in the long term.
The concept of elasticity has an extraordinarily wide range of applications in economics. In particular, an understanding of elasticity is useful to understand the dynamic response of supply and demand in a market, to achieve an intended result or avoid unintended results. Supply and demand does not always guarantee buyers or sellers; this depends on their competitive positions within the market. For example, a business considering a price increase might find that doing so lowers profits if demand is highly elastic, as sales would fall sharply. Similarly, a business considering a price cut might find that it does not increase sales, if demand for the product is price inelastic.
An example of how elasticity can be useful in business situations can be shown by the equation
MR = P * (1+E)/E,
where MR is marginal revenue, P is price of the good, and E is the own price elasticity of demand for the good. Notice that when E is less than negative one, demand is elastic. When E is between negative one and zero, demand is inelastic. And at E=1, demand is unit elastic (or unitary elastic), and thus MC=MB and MNB=0.
Elasticity is also used to analyze social policies. For example, the tobacco settlement in the United States led to significant price and tax increases on cigarettes. PED's could be used to determine the incidence of taxes and the demand response to the increase prices. Further, YED's could show how cigarette smokers responded to the price increases in terms of reallocation of their incomes. The estimated PED for all smokers is 0.7 while the PED for smokers ages 15–18 is 1.4.^{[21]}
In 1954 the price elasticity of demand for local newspapers was 0.1. This highly inelastic demand can be attributed to the lack of substitute news sources. Today there is the internet, local and national news and radio news. Plus there are many more things to do with one's spare time. One would guess that the ready availability would be reflected in a much higher PED. The effect of substitutes would also mean more competition for the local newspaper industry and possibly the consolidation of the industry. According to a recent survey only 55 per cent of regular readers of local newspapers would miss the paper "a lot" if the paper closed while 42% of readers would not miss the paper much or at all.^{[citation needed]} In 2009 13,700 jobs were lost in the local newspaper sector by closings and buyouts.^{[22]}
YED can also give an idea as to the growth potential of an industry. For example assume that the YED for a product is 0.45. This means that if income in the economy increases by 10% this industry will experience a 4.5% Expansion in output. Therefore the industry although growing is losing market share.^{[23]} This fact may make it difficult to attract new capital and to attract top management candidates to an industry with declining power and presence in the aggregate economy. An example might be the electronic typewriter industry after the introduction of dedicated word processors in the late 1970s. The typewriter industry continued to grow for several years after processors came on the market but eventually the growth of the industry declined eventually turning into an absolute decline in output.
Firms determine their profit maximizing output by equating MR to MC. Alternatively the firm can determine its profit maximizing price directly by using the markup rule.^{[24]}
The derivation of the markup rule can be explained in three steps.
Step #1, the price elasticity of demand (PED) and marginal revenue.
The total revenue for a firm equals
Differentiating by Q, this implies that the firm's marginal revenue equals:
The negative sign appears here since (dP/dQ)/(Q/P) is negative (given a downwardsloping demand curve).^{[25]}
Step #2, PED and the markup. The markup rule is equivalent to the standard profit maximizing rule MR = MC. Let e be the absolute value of PED. Then^{[26]}
The markup as a percentage of MC = m =(P  MC) MC
This says that as the (absolute value of the) price elasticity of demand (e) for a firm's product rises due to increased competition for it, then the percentage markup falls. Similarly, with less competition for a firm's output and lower elasticity, the percentage markup rises. This is why the markup has been dubbed the "degree of monopoly," a measure of pricesetting power. If e is infinite, as for a perfectly competitive firm, then m = 0.
Step #3, the markup pricing rule. This gives the standard markup pricing rule, with P as the percentage markup over MC:
With MC constant, a rising price elasticity of demand implies a lower degree of monopoly and lower prices. As e rises toward infinity (perfect competition), m falls toward zero and P approaches MC. Similarly, with falling price elasticity of demand, there is a higher degree of monopoly and higher prices compared to MC. Note that the markup rule applies only to elastic demand (e > 1). However, firms almost never operate in a range of priceinelastic demand because by raising price and cutting production it would increase total revenue, reduce total costs and increase profits.^{[27]}
A firm that is considering building a new plant in a developing country will want to know the income elasticity in that country for the goods the company will be manufacturing. A firm can use weighted income elasticities to estimate the range on YEDs for the good in question. For example an electronic firm is considering building a manufacturing plant in China. The firm knows that China has experienced a remarkable rise in income which is expected to continue. The firm estimates that the YED for electronics in China is 1.5 meaning that a 10% rise in income will induce a 15% increase in expenditures on electronics.^{[13]} The point is that determining how persons spend their income reveals their tastes and preferences. The study has revealed that a significant number of people in China prefer to spend their income on electronics. Further income elasticities tend to be stable over time.
Elasticities can provide governments with valuable information about the efficacy of laws and regulations. For example, federal and state governments have substantially increased the taxes on cigarettes in an effort to raise revenue and to discourage smoking. In one study price elasticities and cross price elasticities were used to study the consumption habits of heavy smokers after a substantial increase in the price of cigarettes.^{[7]} In one study a ten percent rise in cigarettes caused "poor smoking families" to cut back on cigarettes by 9%, food by 17% and health care by 12%.^{[7]} In other words people will reduce purchases of basic goods and services to continue to smoke.^{[8]} Thus a policy of heavily taxing cigarettes to reduce consumption and promote health is counterproductive at least among the segment of the population studied. However, studies have also estimated that the PED for smoker generally was approximately 0.04 and for children 0.065. Thus a tax increase on cigarettes would have a greater impact on children than adults.
If the demand curve is inelastic relative to the supply curve the tax will be disproportionately borne by the buyer rather than the seller. If the demand curve is elastic relative to the supply curve the tax will be born disproportionately by the seller. If PED = PES the tax burden split equally between buyer and seller. If PED < PES, the buyer will bear the burden. If PES < PED the seller disproportionately bears the tax burden. The pass through fraction for buyers is PES/PES  PED.^{[26]} So if PED for cigarettes is 0.4 and PES is 0.5^{[26]} then the pass through fraction to the buyer would be calculated as follow:
Thus 56% of any tax increase would be "paid" by the buyer; 44% would be paid by the seller. From the seller/producer's perspective the formula is PED/(PES PED)
If a purpose of the tax is to reduce smoking its deterrent effect will be undercut by two factors. First the seller can pass through 56% of the tax to the buyer. In other words the buyer has a relatively high tolerance for absorbing the tax and continuing to smoke; they may prioritise smoking over the purchase of other goods.
Elasticities can be used to predict the change in equilibrium price from a change in demand or supply.^{[28]} To calculate the change divide the percentage change in demand (supply) by the sum of the PED (absolute value) and PES. As an example, demand increases by 40%, PED is 1.8 and PES is 2.2. the formula is percentage change in equilibrium price = %∆Demand/PED + PES = 40/(1.8 + 2.2) = 40/4 = 10%.^{[28]}
Elasticity is one of the more important concepts in microeconomic theory. It is useful in understanding the incidence of indirect taxation, marginal concepts as they relate to the theory of the firm, distribution of wealth and different types of goods as they relate to the theory of consumer choice and the Lagrange multiplier. Elasticity is also crucially important in any discussion of welfare distribution, in particular consumer surplus, producer surplus, or government surplus. The concept of elasticity was also an important component of the SingerPrebisch thesis which is a central argument in dependency theory as it relates to development economics. Elasticity measures the intensity of the relationship between to variables. It provides a gauge of not only the nature of the relationship but also the extent of the relationship. It answers the question of how much a change in one variable will affect a change in another variable.
Elasticity as described above is necessarily dimensionless – meaning that it is independent of units of measurement. For example, the value of the price elasticity of demand for gasoline would be the same whether prices were measured in dollars or euros, or quantities in tonnes or gallons. This unitindependence is the main reason why elasticity is so popular a measure of the responsiveness of economic behavior.
A major study of the price elasticity of supply and the price elasticity of demand for US products was undertaken by Hendrik S. Houthakker and Lester D. Taylor.^{[29]}


In economics, people talk about elasticity of supply, or demand, or of price.
If the price for an item with an elastic demand goes up, the demand for it will go down. For example, people will buy fewer DVDs if DVDs get more expensive. DVDs have an elastic demand.
If an item has inelastic demand, the demand for it will not be affected by the price of it. For example, if the price of salt increases, people will not buy less of it. They need their daily salt intake. The demand of salt is therefore inelastic.
Topics in microeconomics 

Scarcity • Opportunity cost • Supply and demand • Elasticity • Economic surplus • Economic shortage • Aggregation of individual demand to total, or market, demand • Consumer theory • Production, costs, and pricing • Market forms • Welfare economics • Market failure 
