1. Chapter Fifteen Market Demand

2. From Individual to Market Demand Functions Thinkof an economy containing n consumers, denoted by i = 1, ,n. Consumer is ordinary demand function for commodity j is x*i (p1 , p 2 , mi ) j 3. From Individual to Market Demand Functions Whenall consumers are price-takers, the market demand function for commodity j isn *i X j ( p1 , p 2 , m ,, m ) = x j (p1 , p 2 , mi ). i= 1 1 Ifnall consumers are identical then X j ( p1 , p 2 , M) = n x* ( p1 , p 2 , m) jwhere M = nm. 4. From Individual to Market Demand Functions Themarket demand curve is the horizontal sum of the individual consumers demand curves. E.g. suppose there are only two consumers; i = A,B. 5. From Individual to Market Demand Functions p1p1p1 p1p1 p1 20 x*A 115x*B 1 6. From Individual to Market Demand Functions p1p1p1 p1p1 p1p120 x*A 115p1x*A + xB 1 1x*B 1 7. From Individual to Market Demand Functions p1p1p1 p1p1 p1p120 x*A 115p1 p1 x*A + xB 1 1x*B 1 8. From Individual to Market Demand Functions p1p1p1 p1p1 p1p120 x*A 115x*B 1The horizontal sum of the demand curves of individuals A and B.p1 p1 35x*A + xB 1 1 9. Elasticities Elasticitymeasures the sensitivity of one variable with respect to another. The elasticity of variable X with respect to variable Y is% x x,y = . % y 10. Economic Applications of Elasticity Economistsuse elasticities to measure the sensitivity of quantity demanded of commodity i with respect to the price of commodity i (own-price elasticity of demand) demand for commodity i with respect to the price of commodity j (cross-price elasticity of demand). 11. Economic Applications of Elasticity demand for commodity i with respect to income (income elasticity of demand) quantity supplied of commodity i with respect to the price of commodity i (own-price elasticity of supply) 12. Economic Applications of Elasticity quantity supplied of commodity i with respect to the wage rate (elasticity of supply with respect to the price of labor) and many, many others. 13. Own-Price Elasticity of Demand Q:Why not use a demand curves slope to measure the sensitivity of quantity demanded to a change in a commoditys own price? 14. Own-Price Elasticity of Demand p1 10slope =-25p1 10X1*slope = - 0.250 X * 1In which case is the quantity demanded X1* more sensitive to changes to p1? 15. Own-Price Elasticity of Demand p1 10slope =-25p1 10X1*slope = - 0.250 X * 1In which case is the quantity demanded X1* more sensitive to changes to p1? 16. Own-Price Elasticity of Demand p1 1010-packs slope =-25p1 10X1*Single Units slope = - 0.250 X * 1In which case is the quantity demanded X1* more sensitive to changes to p1? 17. Own-Price Elasticity of Demand p1 1010-packs slope =-25p1 10X1*Single Units slope = - 0.250 X * 1In which case is the quantity demanded X1* more sensitive to changes to p1? It is the same in both cases. 18. Own-Price Elasticity of Demand Q:Why not just use the slope of a demand curve to measure the sensitivity of quantity demanded to a change in a commoditys own price? A: Because the value of sensitivity then depends upon the (arbitrary) units of measurement used for quantity demanded. 19. Own-Price Elasticity of Demand * % x1 * = x1 ,p1 % p 1 is a ratio of percentages and so has no units of measurement. Hence own-price elasticity of demand is a sensitivity measure that is independent of units of measurement. 20. Arc and Point Elasticities Anaverage own-price elasticity of demand for commodity i over an interval of values for pi is an arcelasticity, usually computed by a mid-point formula. Elasticity computed for a single value of pi is a point elasticity. 21. Arc Own-Price Elasticity pi pi+h p i pi-hWhat is the average own-price elasticity of demand for prices in an interval centered on pi?X i* 22. Arc Own-Price Elasticity pi pi+h p i pi-hWhat is the average own-price elasticity of demand for prices in an interval centered on pi?Xi"Xi '"X i* 23. Arc Own-Price Elasticity pi pi+h p i pi-hWhat is the average own-price elasticity of demand for prices in an interval centered on pi?% X* i * = Xi ,pi % piXi"Xi '"X i* 24. Arc Own-Price Elasticity pi pi+h p i pi-hWhat is the average own-price elasticity of demand for prices in an interval centered on pi?% X* i * = Xi ,pi % piXi" 2h % pi = 100 pi 'Xi '"X i* 25. Arc Own-Price Elasticity pi pi+h p i pi-hWhat is the average own-price elasticity of demand for prices in an interval centered on pi?% X* i * = Xi ,pi % piXi" 2h % pi = 100 pi 'Xi '"X i*% X* = 100 i( Xi" Xi '") ( Xi"+ Xi '") / 2 26. Arc Own-Price Elasticity * % Xi * = Xi ,pi % pi2h % pi = 100 pi ' % X* = 100 i( Xi" Xi '") ( Xi"+ Xi '") / 2 27. Arc Own-Price Elasticity * % Xi * = Xi ,pi % pi2h % pi = 100 pi ' % X* = 100 i( Xi" Xi '") ( Xi"+ Xi '") / 2So %X* pi ' ( Xi " Xi '" ) i = X* ,p = . i i %pi ( Xi "+ Xi '" ) / 2 2his the arc own-price elasticity of demand. 28. Point Own-Price Elasticity What is the own-price elasticity of demand in a very small interval of prices centered on pi?pi pi+h p i pi-hXi"Xi '"X i*%X* pi ' ( Xi " Xi '" ) i = X* ,p = . i i %pi ( Xi "+ Xi '" ) / 2 2h 29. Point Own-Price Elasticity What is the own-price elasticity of demand in a very small interval of prices centered on pi? As h 0,pi pi+h p i pi-hXi"Xi '"X i*%X* pi ' ( Xi " Xi '" ) i = X* ,p = . i i %pi ( Xi "+ Xi '" ) / 2 2h 30. Point Own-Price Elasticity pi pi+h p i pi-hWhat is the own-price elasticity of demand in a very small interval of prices centered on pi? As h 0,Xi" Xi '"X i*%X* pi ' ( Xi " Xi '" ) i = X* ,p = . i i %pi ( Xi "+ Xi '" ) / 2 2h 31. Point Own-Price Elasticity pi pi+h p i pi-hWhat is the own-price elasticity of demand in a very small interval of prices centered on pi? As h 0,Xi 'X i*%X* pi ' ( Xi " Xi '" ) i = X* ,p = . i i %pi ( Xi "+ Xi '" ) / 2 2h 32. Point Own-Price Elasticity pip iWhat is the own-price elasticity of demand in a very small interval of prices centered on pi? As h 0, pi ' dX* i * Xi ,pi Xi ' dpiXi 'X i*%X* pi ' ( Xi " Xi '" ) i = X* ,p = . i i %pi ( Xi "+ Xi '" ) / 2 2h 33. Point Own-Price Elasticity pip iWhat is the own-price elasticity of demand in a very small interval of prices centered on pi? * pi ' dXi * = Xi ,pi Xi ' dpi is the elasticity at the point ( Xi ', pi ' ).Xi 'X i* 34. Point Own-Price Elasticity dX* i * = Xi ,pi X* dpi i piE.g. Suppose pi = a - bXi. Then Xi = (a-pi)/b and * dXi1 = . Therefore, dpi bpi 1 = pi . X* ,p = i i ( a pi ) / b b a pi 35. Point Own-Price Elasticity pipi = a - bXi*aa/bXi* 36. Point Own-Price Elasticity pipi = a - bXi*pi X* ,p = i i a piaa/bXi* 37. Point Own-Price Elasticity pi api = a - bXi*pi X* ,p = i i a pip = 0 = 0a/bXi* 38. Point Own-Price Elasticity pi api X* ,p = i i a pipi = a - bXi* p = 0 = 0 =0a/bXi* 39. Point Own-Price Elasticity pi api X* ,p = i i a pipi = a - bXi*a a/2 p = = = 1 2 a a / 2 =0a/bXi* 40. Point Own-Price Elasticity pi a a/2pi X* ,p = i i a pipi = a - bXi*a a/2 p = = = 1 2 a a / 2 = 1 =0a/2ba/bXi* 41. Point Own-Price Elasticity pi a a/2pi X* ,p = i i a pipi = a - bXi*a p = a = = a a = 1 =0a/2ba/bXi* 42. Point Own-Price Elasticity pi = a - bXi*pi a = a/2pi X* ,p = i i a pia p = a = = a a = 1 =0a/2ba/bXi* 43. Point Own-Price Elasticity pipi X* ,p = i i a pipi = a - bXi*a = own-price elastic a/2 = 1own-price inelastic =0a/2ba/bXi* 44. Point Own-Price Elasticity pipi X* ,p = i i a pipi = a - bXi*a = own-price elastic a/2 = 1 (own-price unit elastic)own-price inelastic =0a/2ba/bXi* 45. Point Own-Price Elasticity dX* i * = Xi ,pi X* dpi i piE.g. soX* = kpia . iThendX* i = apa 1 i dpia pi pi a 1 X* ,p = a kapi =a = a. a i i kpi pi 46. Point Own-Price Elasticity pik * a 2 Xi = kpi = kpi = 2 pi = 2 everywhere along the demand curve.X i* 47. Revenue and Own-Price Elasticity of Demand Ifraising a commoditys price causes little decrease in quantity demanded, then sellers revenues rise. Hence own-price inelastic demand causes sellers revenues to rise as price rises. 48. Revenue and Own-Price Elasticity of Demand Ifraising a commoditys price causes a large decrease in quantity demanded, then sellers revenues fall. Hence own-price elastic demand causes sellers revenues to fall as price rises. 49. Revenue and Own-Price Elasticity of Demand R( p ) = p X* (p ). Sellers revenue is 50. Revenue and Own-Price Elasticity of Demand R( p ) = p X* (p ). Sellers revenue is dR dX* So = X* (p ) + p dp dp 51. Revenue and Own-Price Elasticity of Demand R( p ) = p X* (p ). Sellers revenue is dR dX* So = X* (p ) + p dp dp * p dX * = X (p )1 + * X (p ) dp 52. Revenue and Own-Price Elasticity of Demand R( p ) = p X* (p ). Sellers revenue is dR dX* So = X* (p ) + p dp dp * p dX * = X (p )1 + * X (p ) dp = X* (p )[ 1 + ] . 53. Revenue and Own-Price Elasticity of Demand dR = X* (p )[ 1 + ] dp 54. Revenue and Own-Price Elasticity of Demand dR = X* (p )[ 1 + ] dpso if = 1thendR =0 dpand a change to price does not alter sellers revenue. 55. Revenue and Own-Price Elasticity of Demand dR = X* (p )[ 1 + ] dp dR >0 but if 1 < 0 then dpand a price increase raises sellers revenue. 56. Revenue and Own-Price Elasticity of Demand dR = X* (p )[ 1 + ] dpAnd if < 1dR 0. 63. Marginal Revenue and Own-Price Elasticity of Demand If = 1 then MR( q) = 0. Selling one more unit does not change the sellers revenue. If 1 < 0 then MR( q) < 0. Selling one more unit reduces the sellers revenue. If < 1 then MR( q) > 0. Selling one more unit raises the sellers revenue. 64. Marginal Revenue and Own-Price Elasticity of Demand An example with linear inverse demand. p( q) = a bq.Then R( q) = p( q)q = ( a bq)q andMR( q) = a 2bq. 65. Marginal Revenue and Own-Price Elasticity of Demand p a p( q) = a bqa/2ba/bqMR( q) = a 2bq 66. Marginal Revenue and Own-Price p Elasticity of Demand aMR( q) = a 2bqp( q) = a bq$a/2ba/bqa/bqR(q)a/2b