Price Elasticity of Demand
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Price Elasticity of Demand A Primer on the Price Elasticity of Demand By Mike Moffatt, About.com Guide
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elasticity formula price elasticity of demand elasticity
The Price Elasticity of Demand (commonly known as just price elasticity) measures the rate of response of quantity demanded due to a price change. The formula for the Price Elasticity of Demand (PEoD) is: PEoD = (% Change in Quantity Demanded)/(% Change in Price) Calculating the Price Elasticity of Demand You may be asked the question "Given the following data, calculate the price elasticity of demand when the price changes from $9.00 to $10.00" Using the chart on the bottom of the page, I'll walk you through answering this question. (Your course may use the more complicated Arc Price Elasticity of Demand formula. If so you'll need to see the article on Arc Elasticity) First we'll need to find the data we need. We know that the original price is $9 and the new price is $10, so we have Price(OLD)=$9 and Price(NEW)=$10. From the chart we see that the quantity demanded when the price is $9 is 150 and when the price is $10 is 110. Since we're going from $9 to $10, we have QDemand(OLD)=150 and QDemand(NEW)=110, where "QDemand" is short for "Quantity Demanded". So we have: Price(OLD)=9 Price(NEW)=10 QDemand(OLD)=150 QDemand(NEW)=110 To calculate the price elasticity, we need to know what the percentage change in quantity demand is and what the percentage change in price is. It's best to calculate these one at a time. Calculating the Percentage Change in Quantity Demanded The formula used to calculate the percentage change in quantity demanded is: [QDemand(NEW) - QDemand(OLD)] / QDemand(OLD) By filling in the values we wrote down, we get:
[110 - 150] / 150 = (-40/150) = -0.2667 We note that % Change in Quantity Demanded = -0.2667 (We leave this in decimal terms. In percentage terms this would be -26.67%). Now we need to calculate the percentage change in price. Calculating the Percentage Change in Price Similar to before, the formula used to calculate the percentage change in price is: [Price(NEW) - Price(OLD)] / Price(OLD) By filling in the values we wrote down, we get: [10 - 9] / 9 = (1/9) = 0.1111 We have both the percentage change in quantity demand and the percentage change in price, so we can calculate the price elasticity of demand. Final Step of Calculating the Price Elasticity of Demand We go back to our formula of: PEoD = (% Change in Quantity Demanded)/(% Change in Price) We can now fill in the two percentages in this equation using the figures we calculated earlier. PEoD = (-0.2667)/(0.1111) = -2.4005 When we analyze price elasticities we're concerned with their absolute value, so we ignore the negative value. We conclude that the price elasticity of demand when the price increases from $9 to $10 is 2.4005. How Do We Interpret the Price Elasticity of Demand? A good economist is not just interested in calculating numbers. The number is a means to an end; in the case of price elasticity of demand it is used to see how sensitive the demand for a good is to a price change. The higher the price elasticity, the more sensitive consumers are to price changes. A very high price elasticity suggests that when the price of a good goes up, consumers will buy a great deal less of it and when the price of that good goes down, consumers will buy a great deal more. A very low price elasticity implies just the opposite, that changes in price have little influence on demand. Often an assignment or a test will ask you a follow up question such as "Is the good price elastic or inelastic between $9 and $10". To answer that question, you use the following rule of thumb:
If PEoD > 1 then Demand is Price Elastic (Demand is sensitive to price changes)
If PEoD = 1 then Demand is Unit Elastic If PEoD < 1 then Demand is Price Inelastic (Demand is not sensitive to price changes)
Recall that we always ignore the negative sign when analyzing price elasticity, so PEoD is always positive. In the case of our good, we calculated the price elasticity of demand to be 2.4005, so our good is price elastic and thus demand is very sensitive to price changes. Next: Price Elasticity of Supply Data Price $7 $8 $9 $10 $11
Quantity Demanded 200 180 150 110 60
Quantity Supplied 50 90 150 210 250
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. Contents
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1 Mathematical definition 2 Estimating point elasticities 3 PED, YED and XED o 3.1 Examples 4 Price elasticity of demand (PED) o 4.1 Determinants of PED o 4.2 Factors that make demand for a good elastic o 4.3 Factors that make demand for a good inelastic o 4.4 Interpreting price elasticities of demand o 4.5 Selected price elasticities 5 Relationship between revenue and elasticity o 5.1 Calculating percentage change in total revenue 6 Elasticity and slope 7 Applications o 7.1 An indicator of industry health o 7.2 Product pricing
7.2.1 Derivation of the markup rule 7.3 Investment decisions 7.4 Government policy 7.5 Tax Incidence 7.6 Predicting Changes in Price 8 Examples 9 Other important elasticity measures o 9.1 Income elasticity of demand o 9.2 Interpreting income elasticities of demand o 9.3 Selected income elasticities o 9.4 Cross-price elasticity of demand o 9.5 Interpreting cross-price elasticity of demand o 9.6 Advertising elasticity of demand o 9.7 Calculating AED o 9.8 Applications of AED o 9.9 Selected advertising elasticities of market demand o 9.10 Cross elasticity of demand between firms o 9.11 Supply elasticities o 9.12 Determinants of PES o 9.13 Elasticities of linear supply curves o 9.14 Non-traditional elasticities o 9.15 Elasticity of scale o 9.16 Output Elasticity (Partial) 10 Importance 11 See also 12 References o 12.1 Footnotes o o o o
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13 External links
 Mathematical definition
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 Qd(P) and Qs(P). When working with graphs, it is common to put Quantity on the x-axis and Price on the y-axis, thus the function of interest is x(y) rather than (as commonly used in mathematics) y(x). In general, the "y-elasticity of x" is:
The "y-elasticity 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:
, and percentage change
, and similarly for
Another way to approximate elasticity is using the average value (see arc elasticity):
.  Estimating point elasticities
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). 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. If the slope of the secant is greater than the slope of the tangent then the curve is inelastic at the point. 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. 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. 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. Again this assumes that the dependent variable is drawn on the Y axis.
 PED, YED and XED
Three of the most commonly used demand elasticities are price elasticity of demand, income elasticity of demand and cross price elasticity of demand. Price elasticity of demand measures the percentage change in quantity demanded caused by a one percent change in price. 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. If the PED is between zero and one demand is inelastic and if PED equals one demand is unit-elastic. A perfectly inelastic demand curve, perpendicular to X axis, has zero elasticity. A perfectly elastic demand curve, horizontal to X axis, is infinitely elastic. 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 X-axis. 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. When YED is greater than one (YED > 1) demand is income elastic. 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 X-axis. Mathematically XED = (∂Q/∂Prg)(Prg/Q) where Prg is the price of the related good. •
It is important to keep in mind that PED measures the magnitude of the movement along the demand curve while YED and XED measure the magnitude of the shift of the demand curve.
 Examples Assume that the demand for processed pork in western Canada is Qd = 195 - 20P + 20Pb + 3Pc + 2Y  Where P = price of processed pork in western Canada Pb is the price of beef in terms of dollars per kg. Pc is the price of chicken Y is income •
P = $3 per kg Pb = $4 per kg. Pc = $3 ⅓ per kg. Y = 12.5 thousand dollars •
YED = (∂Q/∂Y) (Y/Q) (∂Q/∂Y) = 2 (Y/Q) = 12.5/250 = 0.05 (∂Q/∂Y) (Y/Q) = (2)(0.05) = 0.1 YED = 0.1
PED = (∂Q/∂P) (P/Q) (∂Q/∂P) = -20 (P/Q) = .012 (∂Q/∂P) (P/Q) = -0.24
XEDPb = (∂Q/∂Pb) (Pb/Q) (∂Q/∂Pb) = 20 (P/Q) = 0.016 (∂Q/∂P) (P/Q) = 0.32
These results and the original function tell us (1) that processed pork is a normal good (2) the demand for processed pork is inelastic and (3) that beef is a moderately weak substitute for processed pork in Western Canada. Note in calculating elasticity coefficients, for example PED, you must hold all other determinants of demand constant to isolate the effect of price changes in the good in question with changes in quantity demanded. For example, if you didn't hold income constant then your data for Q would reflect not only the effects of price changes but also the effects on changes income on the Q variable. Also, the general formula for calculating PED is PED = (∆Q/∆P) x P/Q. (∆Q/∆P) is the slope of the demand equation Q = f(P). However, the demand curve is traditionally depicted using the inverse demand equation and thus the slope of the curve shown is ∆P/∆Q. So to calculate elasticity using a graph make sure that you use the formula PED = (1/∆P/∆Q) x P/Q. It is possible to consider the combined effects of two or more determinant of demand. The steps are as follows: PED = (∆Q/∆P) x P/Q. Convert this to the predictive equation: ∆Q/Q = PED(∆P/P) if you wish to find the combined effect of changes in two or more determinants of demand you simply add the separate effects: ∆Q/Q = PED(∆P/P) + YED(∆Y/Y)
Remember you are still only considering the effect in demand of a change is two of the variables. All other variables must be held constant. Note also that graphically this problem would involve a shift of the curve and a movement along the shifted curve. Finally elasticity is dependent on time, place and circumstance.
 Price elasticity of demand (PED)
PED is a measure of the sensitivity of the quantity variable, Q, to changes in the price variable, P. Elasticity answers the question of how much the quantity will change in percentage terms for a 1% change in the price. As noted previously the formula for calculating PED is :(∂Q/∂P) (P/Q).
 Determinants of PED The overriding factor in determining PED is the willingness and ability of consumers after a price changes to postpone immediate consumption decisions concerning the good and to search for substitutes (wait and look). The greater the incentive the consumer has to delay consumption and search for substitutes and the more readily available substitutes are the more elastic the demand will be. Specific factors are: Availability of substitutes: The more choices that are available, the more elastic is the demand for a good. If the price of Pepsi goes up by 20%, one can always purchase Coke, 7-Up, Dr. Pepper and so forth. One's willingness and ability to postpone the consumption of Pepsi and get by with a "lesser brand" makes the PED of Pepsi relatively elastic. Necessity: With a true necessity a consumer has neither the willingness nor the ability to postpone consumption. There are few or no satisfactory substitutes. Insulin is an example of such a good. Proportion of income spent on a good: Most consumers have both the willingness and ability to postpone the purchase of big ticket items. If an item constitutes a significant portion of one's income, it is worth one's time to search for substitutes. A consumer will give more time and thought to the purchase of a $3000 television than a $1 candy bar. Duration: The more time a consumer has to search for substitute goods, the more elastic the demand. Breadth of definition: How specifically the good is defined. For example, the demand for automobiles is less elastic than the demand for Toyotas which is in turn less elastic than the demand for Red Toyota Priuses. If the price of Toyotas increased, then consumers might purchase a Ford or a Honda. If the price of all cars increased, consumers would react less since the substitutes are not as good.
Availability of Information Concerning Substitute Goods: The easier it is for a consumer to locate the substitute goods, the more willing he will be to undertake the search.
 Factors that make demand for a good elastic 1. There are many substitutes 2. The substitutes are readily obtainable 3. The good is a luxury - it is something you can do without 4. The good is important in terms of proportion of income spent of the goods 5. The consumer had plenty of time to search for the substitutes
 Factors that make demand for a good inelastic 1. There are few substitutes 2. substitutes are difficult to obtain 3. the good is a necessity - it is something you have to have 4. the good is unimportant in terms of proportion of income spent of the goods 5. the consumer has little time or inclination to search for substitutes
 Interpreting price elasticities of demand The coefficient of elasticity indicates how sensitive the demand for a good is to a price change. If the PED is between zero and 1 demand is said to be inelastic, if PED equals 1, the demand is unitary elastic and if the PED is greater than 1 demand is elastic. A low coefficient implies that changes in price have little influence on demand. A high elasticity indicates that consumers will respond to a price rise that buying a lot less of the good and that consumers will respond to a price cut buy a lot more "
 Selected price elasticities •
Cigarettes -0.3 to -0.6 US population -0.6 to -0.7 US children
Airline travel -0.3 First Class US -0.4 Unrestricted Coach US -0.9 Discount
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Local newspaper -0.1 Oil -0.4 World Rice -0.47Austria -0.8 Bangladesh -0.8 China
-0.25 Japan -0.55 US •
Beef -1.6 US
Legal gambling -1.9 US -0.80 to -1.0 Indiana/Kentucky 
Movies -0.87 US -0.2 Teenagers US -2.0 Adults US
Medical insurance -0.31 US
Bus travel  -0.20 US
Insulin -0.01 daily users US
Ford compact automobile 2.8
Mountain Dew 4.4
 Relationship between revenue and elasticity • •
Total revenue equals price P times quantity demanded Q. The relationship between price and quantity is inverse:
P↑ Q↓ P↓ Q↑ •
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If a firm is operating in the elastic range of a linear demand curve the firm can increase profits by reducing prices since the percentage decrease in price will be more than offset by the percentage increase in quantity demanded - Total revenue goes up. So the firm can always increase revenue by reducing price. On the other hand if the firm is operating in an inelastic range of a linear demand curve the firm can increase revenue by upping the price. Increasing the price would cause quantity demanded to fall. However, the percentage decrease in Q would be less than the percentage increase in price. Therefore total revenue would increase. Further it can be unambiguously said that profit will increase as well. So a firm operating in the inelastic range of a linear demand curve can always increase revenue and profits by raising the price. Revenue is maximized where elasticity equals one because when demand is inelastic revenue can be increased by raising prices and when demand is elastic revenue can be increased by lowering prices. If PED 1 P↓ Q↑ TR↑
If PED = 0 P↑ TR↑ If PED = 0 P↓ TR↓ If PED = ∞ P↑ TR⇒0 If PED = ∞ P↓ TR↓ •
% change in revenue = % change in quantity demanded + % change in price.
 Calculating percentage change in total revenue Example: Assume that a 5 % increase in price produces a 4 % decrease in quantity demanded. The PED is % ∆ Q / % ∆ P = 4 %/ 5% = -0.8 Effect on total revenue: if firm is operating in the inelastic range a price increase will cause total revenue to increase - the quesiton is by how much? % Change in TR = % Change in quantity demanded + % Change in price % Change in TR = - 4% + 5% % Change in TR = 1 % Total revenue will increase 1%  Elasticity and slope
With certain limited exceptions elasticity is not the same as slope. 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. 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. 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.  Applications
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 fuel-efficient 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 re allocation of their incomes. The estimated PED for all smokers is -0.7 while the PED for smokers ages 15–18 is -1.4.
 An indicator of industry health 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. In 2009 13,700 jobs were lost in the local newspaper sector by closings and buyouts. 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. 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.
 Product pricing 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.  Derivation of the markup rule 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 TR = P x Q
Differentiating by Q, this implies that the firm's marginal revenue equals: MR = dTR/dQ = P (dQ/dQ) + Q (dP/dQ) = P [1 + (dP/dQ)/(Q/P)], so MR = P [1 – (1/⎮PED⎮)] The negative sign appears here since (dP/dQ)/(Q/P) is negative (given a downwardsloping demand curve).  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  MR = MC, so P(1 – [1/e]) = MC and: P/MC = 1/(1 - [1/e]) The markup as a percentage of MC = m =(P - MC)- MC = P/MC - 1 = 1/(1 - [1/e]) - 1 = [1/e]/(1 - [1/e]), so m = 1/(e - 1) 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 price-setting power. If e is infinite, as for a perfectly competitive firm, then m = 0. Step #3, the markup pricing rule. This gives the standard mark-up pricing rule, with P as the percentage mark-up over MC: P = (1 + m)MC = (1 + (1/[e - 1)])MC or: P = (e/[e - 1])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 price-inelastic demand because by raising price and cutting production it would increase total revenue, reduce total costs and increase profits. ––
 Investment decisions 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. 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.
 Government policy 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. 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%. In other words people will reduce purchases of basic goods and services to continue to smoke. 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.
 Tax Incidence 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 , buyer bears burden. If PES > PED the seller disproportionately bears the tax burden. The pass through fraction for buyers is PES/PES - PED. So if PED for cigarettes is -0.4 and PES is 0.5 then the pass through fraction to the buyer would be calculated as follow: PES/PES - PED 0.5/ 0.5 - (-.0.4) = 0.5/0.9 = 56% •
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)
-(-0.4)/(0.5 -(-0.4) 0.4/).9 44% •
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 tax to buyer. In other words the buyer has a relatively high tolerance for absorbing the tax and continuing to smoke. And to compensate for the increased costs of cigarettes the buyer cuts back on basic food and health services.
 Predicting Changes in Price Elasticities can be used to predict the change in equilibrium price from a change in demand ro supply. 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%.  Examples
A common mistake for students and teachers of economics is to confuse elasticity with slope. (Case & Fair, 1999: 108, 109). Elasticity is the slope of a curve on a loglog graph only, not on a regular graph (taking into account whether the independent variable is on the horizontal or the vertical axis). Consider the information in the figure. This is a special case which illustrates that slope and elasticity are different. In the figure to the right the slope of S1 is clearly different from the slope of S2, but since the rate of change of P relative to Q is always proportionate, both S1 and S2 are unit elastic (i.e. E = 1). Illustrations of perfect elasticity and perfect inelasticity.
The demand curve (D1) is perfectly ("infinitely") elastic.
The demand curve (D2) is perfectly inelastic.
Unit elasticity for a supply line passing through the origin
The above figures show x = Q horizontal and y = P vertical. •
Note: Values given for lines are elasticities and not slope. A horizontal line has a slope of zero (0) and a vertical line has no slope. On the contrary, perfectly
inelastic and perfectly elastic demand curves are the two exceptions to the rule that elasticity is not the same as slope.  Other important elasticity measures
Other important elasticities are income elasticity of demand (YED) = %∆Q/%∆Y; the cross price elasticity of demand (XED) = %∆QY/%∆PXand output elasticity which is %∆Q/%∆F.
 Income elasticity of demand 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. 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. 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.
 Interpreting income elasticities of demand 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. 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.
 Selected income elasticities • •
Automobiles 2.46 Books 1.44
• • •
Restaurant Meals 1.40 Tobacco 0.64 Margarine -0.20
Income elasticities are notably stable over time and across countries.
 Cross-price elasticity of demand The cross-price 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. Cross-price elasticity of demand was used to study the consumption habits of heavy smokers after a substantial increase in the price of cigarettes. 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%. In other words people will reduce purchases of basic goods and services to continue to smoke.
 Interpreting cross-price elasticity of demand Cross price elasticities of demand measure the strength of the relationship between goods. A positive cross-price elasticity means that the goods are substitutes. A substitute good is one that a consumer will buy instead of the good is question. if the price of a good goes up, the demand for the substitute good goes up as well. A negative cross-price elasticity means that the goods are complements. Complementary goods are used together. An example would be 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 Advertising elasticity of demand measures the change demand induced by a change in advertising. Advertising and demand although traditionally considered as being positively related to demand for the good that is subject of the advertising campaign can be inversely related if the advertising is negative. For example, the Coke versus Pepsi blind taste test ads.
 Calculating AED AED is calculated using the same general formula as any elasticity - AED = %∆Q/%∆A = (∆Q/∆A) X (A/Q)
 Applications of AED • • •
AED can be used to make sure advertising expenses are in line. The rule of thumb combines the PED and AED. The rule is A/PQ = -(AED/PD) To quote Pindyck and Rubinfeld, "to maximize profit, the firms advertising to sales ratio should be equal to minus the ratio of the advertising and price elasticities of demand." As noted by Pindyck and Rubinfeld, firms should advertise heavily if there AED is high - they get a lot of bang for their advertising buck or if their PED is low - a low PED implies high markups so for every added sales there is significant profit. 
 Selected advertising elasticities of market demand • • • • •
Beer US 0.0 Wine US 0.08 Cigarettes US 0.04 Recreation US 0.08 The elasticity figures are surprisingly low. For example, a one percent increase in advertising expenditures results in no increase in beer sales. Similarly the response to cigarette advertising is negligible. Both the beer and cigarette industries advertised heavily. Given the anemic response the question is why? The answer is that the coefficients are for industry demand not firm demand. the AED for individual brands would be substantially higher.
 Cross elasticity of demand between firms 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.
 Supply elasticities The primary supply elasticity is the price elasticity of supply PES which equals %∆QS/ %∆P = (∆Q/∆P) x P/Q = (∂Q/∂P) x P/Q. The price elasticity of supply is positive. For example if the PES for a good is 0.67 a 1% rise in price will induce a two-thirds increase in quantity supplied. 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. 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. 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.
 Determinants of PES PES measures the responsiveness of quantity supplied to change in prices. Significant determinants include: Reaction time: The PES coefficient will largely be determined by how quickly producers react to price changes by increasing (decreasing) production and delivering (cutting deliveries of) goods to the market. Complexity of Production: Much depends on the complexity of the production process. Textile production is relatively simple. The labor is largely unskilled and production facilities are little more than buildings - no special structures are needed. Thus the PES for textiles is elastic. On the other hand, the PES for specific types of motor vehicles is relatively inelastic. Auto manufacture is a multi-stage process that requires specialized equipment, skilled labor, a large suppliers network and large R&D costs. Time to respond: The more time a producer has to respond to price changes the more elastic the supply. For example, a cotton farmer cannot immediately respond to an increase in the price of soybeans. Excess capacity: A producer who has unused capacity can quickly respond to price changes in his market assuming that variable factors are readily available. Inventories: A producer who has a supply of goods or available storage capacity can quickly respond to price changes.
 Elasticities of linear supply curves The coefficient of elasticity of any linear supply curve that passes through the origin is 1. The coefficient of elasticity of any linear supply curve that cuts the y-axis is greater than 1. The coefficient of elasticity of any linear supply curve that cuts the x-axis is less than 1.
 Non-traditional elasticities The concept of elasticity applies to any situation where variables are functionally related.  For example, one could calculate the "TV football game elasticity of divorce rates".
or swine flu elasticity of gambling junkets. The only limitations are that the variables be quantifiable and that they be causally related.
 Elasticity of scale 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.  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.
 Output Elasticity (Partial) 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). (∆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.  Importance
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 Singer-Prebisch 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 unit-independence 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.