Income Elasticity (Normal Goods) Elasticity

Income Elasticity (Normal Goods) Elasticity Elasticity is a measure of how responsive a dependent variable is to a small change in an independent variable(s) Elasticity is defined as a ratio of the percentage change in the dependent variable to the percentage change in the independent variable Elasticity can be computed for any two related variables

Income Elasticity (Normal Goods) Elasticity can be computed to show the effects of: a change in price on the quantity demanded [ “a change in quantity demanded” is a movement on a demand function] a change in income on the demand function for a good a change in the price of a related good on the demand function for a good a change in the price on the quantity supplied a change of any independent variable on a dependent variable

Income Elasticity (Normal Goods) “Own” Price Elasticity

Income Elasticity (Normal Goods) Sometimes called “price elasticity” can be computed at a point on a demand function or as an average [arc] between two points on a demand function e p,  are common symbols used to represent price elasticity Price elasticity [e p ] is related to revenue “How will a change in price effect the total revenue?” is an important question.

Income Elasticity (Normal Goods) Elasticity as a measure of responsiveness The “law of demand” tells us that as the price of a good increases the quantity that will be bought decreases but does not tell us by how much. ep [“own”price elasticity] is a measure of that information] “If you change price by 5%, by what percent will the quantity purchased change?

or, e p  %  Q %  P At a point on a demand function this can be calculated by: e p = Q 2 - Q 1 Q1Q1 P 2 - P 1 P1P1 Q 2 - Q 1 =  Q P 2 - P 1 =  P =  Q Q1Q1  P P1P1

 Q Q1Q1  P P1P1 ep =ep = Price decreases from $7 to $5 3 PxPx Q x / ut D $5 B 5 $7 A P 1 = P 2 = P 2 - P 1 = = PP = -2  PP = -2 Q 1 = Q 2 = Q 2 - Q 1 = = QQ = +2  QQ = [2/3 =.66667] [-2/7= ] = %  Q = 67% %  P = -28.5% = -2.3 [rounded] The “own” price elasticity of demand at a price of $7 is -2.3 This is “point” price elasticity. It is calculated at a point on a demand function. It is not influenced by the direction or magnitude of the price change.. There is a problem! If the price changes from $5 to $7 the coefficient of elasticity is different! -2

3 PxPx Q x / ut D $5 B 5 $7 A  Q Q1Q1  P P1P1 ep =ep = When the price increases from $5 to $7, P 1 = P2 =P2 =  PP = Q1=Q1= Q2=Q2=  QQ = [-2/5 = -.4] [+2/5 =.4] = %  Q = -40% %  P = 40% = -1 [this is called “unitary elasticity] the ep ep = [“unitary”] e p = -1 In the previous slide, when the price decreased from $7 to $5, e p = -2.3 e p = -2.3 The point price elasticity is different at every point! There is an easier way!

An easier way! Q1Q1  P P1P1 ep =ep =  Q Q1Q1 = Q1Q1 P 1  P * By rearranging terms = P1P1 Q1Q1 *  Q  P this is the slope of the demand function this is a point on the demand function  Q P1P1 Q1Q1 = *  P epep Given that when: P 1 = $7, Q 1 = 3 P 2 = $5, Q 2 = 5 P 2 - P 1 = =  P = -2 Q 2 - Q 1 = =  Q = +2 Then,  Q  P = = - 1 This is the slope of the demand Q = f(P) P 1 = $7, Q 1 = = On linear demand functions the slope remains constant so you just put in P and Q

P 1 = $7, Q 1 = 3 P 2 = $5, Q 2 = 5 P 2 - P 1 = =  P = -2 Q 2 - Q 1 = =  Q = +2 3 PxPx Q x / ut D $5 B 5 $7 A The following information was given Q = f (P) The slope of the demand function [ Q = f(P) ] is  Q  P = = -1 The slope-intercept form Q = a + m P - 1 What is the Q intercept? P x must decrease by 5. The slope [-1] indicates that for every 1 unit increase in Q, P x will decrease by 1. Since P x must decrease by 5, Q must increase by 5 Q increases by 5 Q = 10 Q = 10 when P x = 0 10 The equation for the demand function we have been using is Q = P. A table can be constructed.

For a simple demand function: Q = P pricequantityepTotal Revenue $010 $19 $28 $37 $46 $55 $64 $73 $82 $91 $100 The slope is -1 The intercept is 10 using our formula, epep =  Q P 1 Q1Q1 *  P epep =  Q P1P1 Q1Q1  P * the slope is -1, (-1) price is 7 7 at a price of $7, Q = 3 3 = Calculate e p at P = $9 Q = 1 e p = (-1) 9 1 = -9 Calculate e p for all other price and quantity combinations undefined

For a simple demand function: Q = P pricequantityepTotal Revenue $010 $19 $28 $37 $46 $55 $64 $73 $82 $91 $ undefined Notice that at higher prices the absolute value of the price elasticity of demand,  e p  is greater. Total revenue is price times quantity; TR = PQ Where the total revenue [TR] is a maximum,  e p  is equal to 1 In the range where  e p  < 1, [less than 1 or “inelastic”], TR increases as price increases, TR decreases as P decreases. In the range where  e p  > 1, [greater than 1 or “elastic”], TR decreases as price increases, TR increases as P decreases.

3 PxPx Q x / ut D $5 B 5 $7 A To solve the problem of a point elasticity that is different for every price quantity combination on a demand function, an arc price elasticity can be used. This arc price elasticity is an average or midpoint elasticity between any two prices. Typically, the two points selected would be representative of the usual range of prices in the time frame under consideration. The formula to calculate the average or arc price elasticity is: epep =  Q P 1 + P 2 Q 1 + Q 2 *  P The average or arc e p between $5 and $7 is calculated, epep =  Q P 1 + P 2 Q 1 + Q 2 *  P Slope of demand  Q  P = - 1 P 1 = $7, Q 1 = 3 P 2 = $5, Q 2 = 5 P 2 - P 1 = =  P = -2 Q 2 - Q 1 = =  Q = +2 P 1 + P 2 = 12 Q 1 + Q 2 = 8 8 = The average ep ep between $5 and $7 is -1.5

Calculate the point e p at each price on the table. Calculate the TR at each price on the table. Calculate arc e p at between $10 and $20. Calculate arc e p at between $25 and $28. Calculate arc e p at between $20 and $28. Graph the demand function [labeling all axis and functions], identify which ranges on the demand function are price elastic and which are price inelastic.

Calculate the point e p at each price on the table Calculate the TR at each price on the table. TR = PQ $800 $500 $224 Calculate arc e p at between $10 and $20. e p = -1 Calculate arc e p at between $25 and $28. e p = -7.6 Calculate arc e p at between $20 and $28. e p = -4 Graph the demand function [labeling all axis and functions], identify which ranges on the demand function are price elastic and which are price inelastic. At what price will TR by maximized? P = $15

Q/ut Price e p = |  e p | > 1 [elastic] The top “half” of the demand function is elastic. |  e p | < 1 inelastic The bottom “half” of the demand function is inelastic. Graphing Q = P, TR TR is a maximum where e p is -1 or TR’s slope = 0 When e p is -1 TR is a maximum. When |  e p | > 1 [elastic], TR and P move in opposite directions. (P has a negative slope, TR a positive slope.) When |  e p | < 1 [inelastic], TR and P move in the same direction. (P and TR both have a negative slope.) Arc or average e p is the average elasticity between two point [or prices] point  e p is the elasticity at a point or price. Price elasticity of demand describes how responsive buyers are to change in the price of the good. The more “elastic,” the more responsive to  P.

Income Elasticity (Normal Goods) Determinants of Price Elasticity Availability of substitutes [greater availability of substitutes makes a good relatively more elastic] Portion of the expenditures on the good to the total budget [lower portion tends to increase relative elasticity] Time to adjust to the price changes [longer time period means there are more adjustment possible and increases relative elasticity Price elasticity for “brands” is tends to be more elastic than for the category of goods

Income Elasticity (Normal Goods) An application of price elasticity. The price elasticity of demand for milk is estimated between -.35 and -.5. Using -.5 as a reasonable figure, there are several important observations that can be made. What effect does a 10% increase in the P milk have on the quantity that individuals are willing to buy ? e p  %  Q %  P e p  %  Q %  P Since e p = = +10% To solve for %  Q Multiply both sides by +10% (+10%)x ( ) x (+10%) -5% = %  Q A 10% increase in the price of milk would reduce the quantity demanded by about 5%. P milk Q milk D milk P1P1 Q1Q1 P2P2 A 10% increase in P Q2Q2 reduces Q by 5% +10% -5% If price were decreased by 5%, what would be the effect on quantity demanded?

e p  %  Q %  P The price elasticity of demand is a measure of the %  Q that will be “caused” by a %  P. If the price elasticity of demand for air travel was estimated at -2.5, what effect would a 5% decrease in price have on quantity demanded ? -2.5 = %  Q %  P - 5% = +12.5% change in quantity demanded If the price elasticity of demand for softdrink was estimated at -.8, what effect would a 6% increase in price have on quantity demanded ? -.8 = %  Q %  P +6% = -4.8% decrease in quantity demanded

If the price elasticity of demand for milk were -.5, the effects of a price change on total revenue [TR] can also be estimated. Since, e p  %  Q %  P When  e p  < 1, demand is “inelastic. “ This means that the  %  Q  <  %  P . Since the % price decrease is greater than the % increase in Q, TR [TR = PQ] will decrease. When  e p  < 1, a price decrease will decrease TR; a price increase will increase TR, Price and TR “move in the same direction.” [inelastic demand with respect to price] When  e p  > 1, demand is “elastic.” This means that the  %  Q  >  %  P . When the % price decrease is less than the % increase in Q, TR [TR = PQ] will increase. When  e p  > 1, a price decrease will increase TR; a price increase will decrease TR, price and TR “move in opposite directions.” [elastic demand wrt price]

Graphically this can be shown P Q/ut D at the midpoint, e p = -1 P1P1 Q1Q1 TR TR is a maximum TR TR = PQ, so the maximum TR is the rectangle 0Q 1 EP 1 0 E elastic price rises P 2 Q 2 (P 2 Q 2 ) is less than (P 1 Q 1 ) Loss in TR when  P +TR As price rises into the elastic range the TR will decrease. Notice that in this range the slope of demand is negative, the slope of TR is positive Price and TR move in opposite directions

P Q/ut D inelastic TR at the midpoint, e p = -1 0 E P1P1 Q1Q1 TR is a maximum TR = P 1 Q 1 [Maximum] TR When price elasticity of demand is inelastic A price decrease P0P0 Q 0 results in a smaller PQ [TR] will result in a decrease in TR [PQ]. notice that both TR and Demand have a negative slope in the inelastic range of the demand function. Price and TR “move in the same direction.” A price decrease will reduce TR; a price increase will increase TR. Note that this information is useful but does not provide information about profits!

Fall '97Economics 205Principles of MicroeconomicsSlide 24 “Own” Price Elasticity of Demand e p is a measure of the responsiveness of buyers to changes in the price of the good. e p will be negative because the demand function is negatively sloped. A linear demand function will have unitary elasticity at its “midpoint.” AT THIS POINT TR IS A MAXIMUM! A linear demand function will be more “elastic” at higher prices and tends to be more “inelastic” in the lower price ranges

Elastic e p When  ep  > 1 [greater than 1], the demand is “elastic”  %  Q  >  %  P  this shows buyers are responsive to changes in price An increase in the price of the good results in a decrease in total revenue [TR], a decrease in the price increases TR. Price and TR move in opposite directions. The demand for a good tends to become more elastic as the number of substitutes increases “luxury” good more elastic than “necessities” % of price [or expenditure on the good] of the budget as the amount of time for adjustments increases elasticity

Fall '97Economics 205Principles of MicroeconomicsSlide 26 Examples Goods that are relatively price elastic lamb, restaurant meals, china/glassware, jewelry, air travel [LR], new cars, Fords in the long run,  e p  tends to be greater Goods that are relatively price inelastic electricity, gasoline, eggs, medical care, shoes, milk in the short run,  e p  tends to be less

Fall '97Economics 205Principles of MicroeconomicsSlide 27 Reference: Principles of Economics, 6/e by Karl Cas, Ray Fair Slides prepared by: Fernando Quijano and Yvonn Quijano