# Unit 3. Elasticity Learning objectives Questions for revision: 3.1. Price elasticity of demand

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1 Unit 3. Elasticity Learning objectives To comrehen an aly the concets of elasticity, incluing calculating: rice elasticity of eman; cross-rice elasticity of eman; income elasticity of eman; rice elasticity of suly. Questions for revision: Deman scheule an the law of eman; Factors of eman, comlements an substitutes; Suly scheule an the law of suly; Market equilibrium: welfare asects of government controls rice elasticity of eman rice elasticity of eman is the measure of intencity of reaction of quantity emane in resonse to a change in rice. It shows a ercentage change in quantity emane that results from one ercent change in rice: This is the oint rice elasticity of eman in the form of escrete changes in rice an quantity emane. If the changes in rice an quantity emane are infinitesimal, one can calculate rice elasticity of eman in ifferential form as the limiting case of the escrete ratio: To eliminate the inetrminance of the starting oint of change that gives the reference rice an quantity for the corresoning changes in the variables one can calculate arc rice elasticity of eman where average values of the variables serve as the base for relative changes: Min the ifference between elasticity an the sloe of eman curve. For instance, esite the sloe of the eman curve D a in the figure 3.1 is twice as much as the sloe of the curve D b, taking into consieration 1

2 that for the curve D a an for the curve D b at the rice Р * elasticity of eman is one an the same for both curves: An vice versa, rice elasticity may be ifferent esite equal sloes of linear eman curves. Ror instance, in the figure 3.2 at the given rice Р * quantity of eman for D b is twice as much as for D a :. The lines D a an D b are arallel. It imlies that at the given oints the curve D a is twice as much elastic as D b : n Figure 3.1. Equal elasticities an ifferent sloes of eman curves Figure 3.2. Different elasticities an equal sloes of eman curves * D a D b * D a D b Q a Q b k 2k Q Q a Q b =2Q a Q The reason to use elasticity instea of erivative of a function is that elasticity is a relative ratio that is ineenent of numeraires of the variables y an x: Using the erivative of logarithm of quantity emane, i.e., an the erivative of logarithm of market rice, i.e., one can calculate rice elasticity of eman as the ratio of ifferentials of the corresoning logarithms: Deman is sai to be elastic if. Deman is sai to be inelastic if. Deman is sai to be unit elastic if. We suose here that the law of eman hols. 2

3 3.2. Elasticity of eman an total revenue of roucers Total revenue of roucers as a function of outut can be exresse using rice elasticity of eman: where is marginal revenue of roucer(s), rice elasticity of eman, elasticity of quantity emane. Note that erivatives of inverse functions an are inverse values:. The same alies to elasticities of inverse functions they are inverse values:. In case of elastic eman. It imlies that total revenue is an increasing function of the quantity of the goo rouce: when Q goes u, TR grows as well; an a reuction in Q yiels a ecline in TR. If eman is inelastic, i.e.,, an total revenue is a ecreasing function of outut: when Q goes u, TR eclines; when Q goes own, TR grows. In case of unit elasticity of eman, an total revenue is at the maximum. In the figure below eman is assume to be linear:, where an are constant numbers. It imlies that total revenue is arabola, an marginal revenue is linear an twice as stee as the eman curve with the same intercet at the vertical axes:. A tyical relationshi between total revenue of roucers an rice elasticity of linear eman, or, is eicte below. 3

4 TR Elasticity of linear eman an revenues of roucers as relate to outut TR rice-elastic eman rice-inelastic eman 1 E E E 1 E, MR a Q MR D a/2b Q a/b Use the efinition of elasticity b of quantity emane for linear eman curve to get:. Elasticities of inverse functions are. rice elasticity of inverse linear eman curve is an inverse value of elasticity of quantity emane : For the reasons that will be clarifie later let s rotate the total revenue curve an lot it in the irection to the bottom of the grah. 4

5 a Elasticity of linear eman an revenues E of roucers with resect to outut rice-elastic eman a/2 MR 1 E rice-inelastic D eman E a/2b a/b Q 1 E TR Elasticity of linear eman is equal to the ratio of its sloe (with resect to axis) an the sloe of the ray coming from the origin an going through the given oint (,q ) on the eman curve:. The sloe of linear eman curve is constant, an ratio goes u with Q (see the figure below). So, elasticity of linear eman curve is not constant, its absolute value eclines with Q: q Geometry of elasticity of linear eman β D q q 5

6 Quantity of the goo <Q<a/2b corresons to rice elastic eman, an the range a/2b<q<a/b to rice inelastic eman. In the mioint (Q=a/2b) linear eman is unit-elastic. There is the following relationshi between total revenue of roucers which is treate as a function of market rice an rice elasticity of eman: If eman is rice elastic,. A rice increase will reuce total exeniture, an a rice reuction will increase total exeniture. If eman is rice inelastic,. In this case a rice increase will ush total exeniture u, an a rice reuction will ecrease total exeniture. Total exeniture is highest when eman is unit elastic (see the figure below). Elasticity of linear eman an revenues of E roucers with resect to rice a rice-elastic eman TR 1 Combime the grah with the similar figure above to get a joint illustration of eenence between elasticity of linear eman an total revenue of roucers (see the figure below). E a/2 a/2b 1 E rice-inelastic eman E a/b Q 6

7 Elasticity of linear eman an revenues of roucers a E rice-elastic eman TR 1 E a/2 a/2 b 1 1 E rice-inelastic eman E E a/b Q TR Note that rice elasticity of ower eman curve constant an is equal to the ower : is For instance, in the constant revenue case (TR=Q=const), when eman curve is hyerbola: emane (or rice)., it is unit-elastic regarless of quantity rice elasticity of eman can be calculate as the negative of the ratio of segments of a line tangent to the eman curve from the oint of tangency to the horizontal an vertical axis corresoningly: (see the figure below). roof:, where. It imlies:. Taking into consieration similarity of triangles:, we get:, q.e.. 7

8 B Geometry of elasticity of eman C q In a linear case eman scheule will coincie with its tangent line. Thus, moving along the eman curve we observe a ecline of absolute value of elasticity from to Cross-rice elasticity of eman. Income elasticity of eman Cross-rice elasticity of eman is the ratio that shows a ercentage change of quantity emane of a goo in resonse to one ercent change in the rice of the other: A D q By efinition the goos are comlements if their quantities change in one an the same irection: when quantity of a goo goes u consumers increase the eman for the other one. An vice versa, a ecrease in eman for a goo ushes own consumtion of the other one. Uner the law of eman an uwar movement of the rice of a goo x tens to ecrease its consumtion. As the goo y is comlementary for x, it shifts eman curve for y inwars (towars the origin), an both its quantity an rice go own (see the figure below). It follows that an increase in the rice of a goo ushes own consumtion of its comlement, so cross-rice elasticity of eman for comlementary goos is negative, as a rule 1. 1 The excetion is the case of a Giffen goo. It will be consiere later, in unit 4 Consumer choice. 8

9 x Markets for comlements y D y S y E E 1 Q x Q y If the goos are substitutes, an increase in rice of a goo results in an increase in rice of the other one: x Q x Q y (see the corresoning figure in unit 2). Consequently, cross-rice elasticity of substitutes is ositive 2. Income elasticity of eman is the ratio that shows a ercentage change of quantity emane of a goo in resonse to one ercent change in income of consumers: If the goo is normal, an increase in consumers income yiels an increase in the quantity emane. Consequently, income elasticity of eman for normal goos is ositive:. If the goo is inferior, an increase in consumers incomes yiels a fall in quantity emane. It follows that income elasticity of eman for inferior goos is negative: rice elasticity of suly rice elasticity of suly is the ercentage change in quantity sulie that occurs in resonse to a 1 ercent change in rice: This is the exression of rice elasticity of suly in case of iscrete, finite changes in rice an quantity of suly. If the changes in rice an quantity 2 The more recise treatment is ostone until unit 4 Consumer choice. 9

10 of suly are infinitesimal one can use marginal analysis to calculate rice elasticity of suly: 3.5. Alication of elasticity theory: tax buren of roucers an consumers Incience of a tax escribes who eventually bears the buren of it. Let s calculate tax buren of consumers an roucers to rove that it eens on the relative elasticities of eman an suly:,,, where T c is the tax buren of consumers, T tax buren of roucers, T =T c +T is the total tax revenue of the government (see the two figures below). Use the exressions of elasticities of eman an suly to get: The relative tax buren of consumers an roucers is the inverse ratio of absolute values of corresoning elasticities, i.e. to the negative of the ratio of elasticities of suly an eman:. The more elastic eman an the less elastic suly curves are the greater is the share of the tax levie on roucers as comare to that of consumers (see the two figures below). Elasticity of suly an eman an tax buren of consumers an roucers: relatively elastic suly an inelastic eman t T c T t E t E S t D S Elasticity of suly an eman an tax buren of consumers an roucers: relatively inelastic suly an elastic eman t T T c E t t S t E D S Q Q 1

11 Let s go on with the examles of government fiscal olicies in the unit 2 Suly an eman. Suose at first that market eman an suly curves are given by the following exressions: Q = 1 - ; Qs = 3-2. If the government imoses a tax equal to 8 ollars er unit sol by roucers total tax revenues of the government will be T=8. 64=512; an the shares of tax ai by consumers an roucers are: Tc=(36-3)64=384 (75%), T=(3-28)64=128 (25%) (see the left han sie of the last but one figure in the unit 2 Suly an eman ). If the government imoses a valorem (sales) tax equal to 15% of roucers revenues total tax revenues of the government will be: T=,15. 66,2. 33,8=335,634; an the shares of tax ai by consumers an roucers are: Tc=(33,8-3)66,2=251,56 (75%); T=(3-28,7)66,2=86,6 (25%) (see the right han sie of the last but one figure in the unit 2 Suly an eman ). 11

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