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© 2010 W. W. Norton & Company, Inc. 15 Market Demand.

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1 © 2010 W. W. Norton & Company, Inc. 15 Market Demand

2 © 2010 W. W. Norton & Company, Inc. 2 From Individual to Market Demand Functions u Think of an economy containing n consumers, denoted by i = 1, …,n. u Consumer i’s ordinary demand function for commodity j is

3 © 2010 W. W. Norton & Company, Inc. 3 From Individual to Market Demand Functions u When all consumers are price-takers, the market demand function for commodity j is u If all consumers are identical then where M = nm.

4 © 2010 W. W. Norton & Company, Inc. 4 From Individual to Market Demand Functions u The market demand curve is the “horizontal sum” of the individual consumers’ demand curves. u E.g. suppose there are only two consumers; i = A,B.

5 © 2010 W. W. Norton & Company, Inc. 5 From Individual to Market Demand Functions p1p1 p1p p1’p1’ p1”p1” p1’p1’ p1”p1”

6 © 2010 W. W. Norton & Company, Inc. 6 From Individual to Market Demand Functions p1p1 p1p1 p1p p1’p1’ p1”p1” p1’p1’ p1”p1” p1’p1’

7 © 2010 W. W. Norton & Company, Inc. 7 From Individual to Market Demand Functions p1p1 p1p1 p1p p1’p1’ p1”p1” p1’p1’ p1”p1” p1’p1’ p1”p1”

8 © 2010 W. W. Norton & Company, Inc. 8 From Individual to Market Demand Functions p1p1 p1p1 p1p p1’p1’ p1”p1” p1’p1’ p1”p1” p1’p1’ p1”p1” The “horizontal sum” of the demand curves of individuals A and B.

9 © 2010 W. W. Norton & Company, Inc. 9 Elasticities u Elasticity measures the “sensitivity” of one variable with respect to another. u The elasticity of variable X with respect to variable Y is

10 © 2010 W. W. Norton & Company, Inc. 10 Economic Applications of Elasticity u Economists use 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 © 2010 W. W. Norton & Company, Inc. 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 © 2010 W. W. Norton & Company, Inc. 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 © 2010 W. W. Norton & Company, Inc. 13 Own-Price Elasticity of Demand u Q: Why not use a demand curve’s slope to measure the sensitivity of quantity demanded to a change in a commodity’s own price?

14 © 2010 W. W. Norton & Company, Inc. 14 Own-Price Elasticity of Demand X1*X1* slope = - 2 slope = p1p1 p1p1 In which case is the quantity demanded X 1 * more sensitive to changes to p 1 ? X1*X1*

15 © 2010 W. W. Norton & Company, Inc. 15 Own-Price Elasticity of Demand slope = - 2 slope = p1p1 p1p1 X1*X1* X1*X1* In which case is the quantity demanded X 1 * more sensitive to changes to p 1 ?

16 © 2010 W. W. Norton & Company, Inc. 16 Own-Price Elasticity of Demand slope = - 2 slope = p1p1 p1p1 10-packsSingle Units X1*X1* X1*X1* In which case is the quantity demanded X 1 * more sensitive to changes to p 1 ?

17 © 2010 W. W. Norton & Company, Inc. 17 Own-Price Elasticity of Demand slope = - 2 slope = p1p1 p1p1 10-packsSingle Units X1*X1* X1*X1* In which case is the quantity demanded X 1 * more sensitive to changes to p 1 ? It is the same in both cases.

18 © 2010 W. W. Norton & Company, Inc. 18 Own-Price Elasticity of Demand u Q: Why not just use the slope of a demand curve to measure the sensitivity of quantity demanded to a change in a commodity’s own price? u A: Because the value of sensitivity then depends upon the (arbitrary) units of measurement used for quantity demanded.

19 © 2010 W. W. Norton & Company, Inc. 19 Own-Price Elasticity of Demand 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 © 2010 W. W. Norton & Company, Inc. 20 Arc and Point Elasticities u An “average” own-price elasticity of demand for commodity i over an interval of values for p i is an arc- elasticity, usually computed by a mid-point formula. u Elasticity computed for a single value of p i is a point elasticity.

21 © 2010 W. W. Norton & Company, Inc. 21 Arc Own-Price Elasticity pipi Xi*Xi* pi’pi’ p i ’+h p i ’-h What is the “average” own-price elasticity of demand for prices in an interval centered on p i ’?

22 © 2010 W. W. Norton & Company, Inc. 22 Arc Own-Price Elasticity pipi Xi*Xi* pi’pi’ p i ’+h p i ’-h What is the “average” own-price elasticity of demand for prices in an interval centered on p i ’?

23 © 2010 W. W. Norton & Company, Inc. 23 Arc Own-Price Elasticity pipi Xi*Xi* pi’pi’ p i ’+h p i ’-h What is the “average” own-price elasticity of demand for prices in an interval centered on p i ’?

24 © 2010 W. W. Norton & Company, Inc. 24 Arc Own-Price Elasticity pipi Xi*Xi* pi’pi’ p i ’+h p i ’-h What is the “average” own-price elasticity of demand for prices in an interval centered on p i ’?

25 © 2010 W. W. Norton & Company, Inc. 25 Arc Own-Price Elasticity pipi Xi*Xi* pi’pi’ p i ’+h p i ’-h What is the “average” own-price elasticity of demand for prices in an interval centered on p i ’?

26 © 2010 W. W. Norton & Company, Inc. 26 Arc Own-Price Elasticity

27 © 2010 W. W. Norton & Company, Inc. 27 Arc Own-Price Elasticity So is the arc own-price elasticity of demand.

28 © 2010 W. W. Norton & Company, Inc. 28 Point Own-Price Elasticity pipi Xi*Xi* pi’pi’ p i ’+h p i ’-h What is the own-price elasticity of demand in a very small interval of prices centered on p i ’?

29 © 2010 W. W. Norton & Company, Inc. 29 Point Own-Price Elasticity pipi Xi*Xi* pi’pi’ p i ’+h p i ’-h What is the own-price elasticity of demand in a very small interval of prices centered on p i ’? As h  0,

30 © 2010 W. W. Norton & Company, Inc. 30 Point Own-Price Elasticity pipi Xi*Xi* pi’pi’ p i ’+h p i ’-h What is the own-price elasticity of demand in a very small interval of prices centered on p i ’? As h  0,

31 © 2010 W. W. Norton & Company, Inc. 31 Point Own-Price Elasticity pipi Xi*Xi* pi’pi’ p i ’+h p i ’-h What is the own-price elasticity of demand in a very small interval of prices centered on p i ’? As h  0,

32 © 2010 W. W. Norton & Company, Inc. 32 Point Own-Price Elasticity pipi Xi*Xi* pi’pi’ What is the own-price elasticity of demand in a very small interval of prices centered on p i ’? As h  0,

33 © 2010 W. W. Norton & Company, Inc. 33 Point Own-Price Elasticity pipi Xi*Xi* pi’pi’ What is the own-price elasticity of demand in a very small interval of prices centered on p i ’? is the elasticity at the point

34 © 2010 W. W. Norton & Company, Inc. 34 Point Own-Price Elasticity E.g. Suppose p i = a - bX i. Then X i = (a-p i )/b and Therefore,

35 © 2010 W. W. Norton & Company, Inc. 35 Point Own-Price Elasticity pipi Xi*Xi* p i = a - bX i * a a/b

36 © 2010 W. W. Norton & Company, Inc. 36 Point Own-Price Elasticity pipi Xi*Xi* p i = a - bX i * a a/b

37 © 2010 W. W. Norton & Company, Inc. 37 Point Own-Price Elasticity pipi Xi*Xi* p i = a - bX i * a a/b

38 © 2010 W. W. Norton & Company, Inc. 38 Point Own-Price Elasticity pipi Xi*Xi* p i = a - bX i * a a/b

39 © 2010 W. W. Norton & Company, Inc. 39 Point Own-Price Elasticity pipi Xi*Xi* a p i = a - bX i * a/b

40 © 2010 W. W. Norton & Company, Inc. 40 Point Own-Price Elasticity pipi Xi*Xi* a p i = a - bX i * a/b a/2 a/2b

41 © 2010 W. W. Norton & Company, Inc. 41 Point Own-Price Elasticity pipi Xi*Xi* a p i = a - bX i * a/b a/2 a/2b

42 © 2010 W. W. Norton & Company, Inc. 42 Point Own-Price Elasticity pipi Xi*Xi* a p i = a - bX i * a/b a/2 a/2b

43 © 2010 W. W. Norton & Company, Inc. 43 Point Own-Price Elasticity pipi Xi*Xi* a p i = a - bX i * a/b a/2 a/2b own-price elastic own-price inelastic

44 © 2010 W. W. Norton & Company, Inc. 44 Point Own-Price Elasticity pipi Xi*Xi* a p i = a - bX i * a/b a/2 a/2b own-price elastic own-price inelastic (own-price unit elastic)

45 © 2010 W. W. Norton & Company, Inc. 45 Point Own-Price Elasticity E.g.Then so

46 © 2010 W. W. Norton & Company, Inc. 46 Point Own-Price Elasticity pipi Xi*Xi* everywhere along the demand curve.

47 © 2010 W. W. Norton & Company, Inc. 47 Revenue and Own-Price Elasticity of Demand u If raising a commodity’s price causes little decrease in quantity demanded, then sellers’ revenues rise. u Hence own-price inelastic demand causes sellers’ revenues to rise as price rises.

48 © 2010 W. W. Norton & Company, Inc. 48 Revenue and Own-Price Elasticity of Demand u If raising a commodity’s price causes a large decrease in quantity demanded, then sellers’ revenues fall. u Hence own-price elastic demand causes sellers’ revenues to fall as price rises.

49 © 2010 W. W. Norton & Company, Inc. 49 Revenue and Own-Price Elasticity of Demand Sellers’ revenue is

50 © 2010 W. W. Norton & Company, Inc. 50 Revenue and Own-Price Elasticity of Demand Sellers’ revenue is So

51 © 2010 W. W. Norton & Company, Inc. 51 Revenue and Own-Price Elasticity of Demand Sellers’ revenue is So

52 © 2010 W. W. Norton & Company, Inc. 52 Revenue and Own-Price Elasticity of Demand Sellers’ revenue is So

53 © 2010 W. W. Norton & Company, Inc. 53 Revenue and Own-Price Elasticity of Demand

54 © 2010 W. W. Norton & Company, Inc. 54 Revenue and Own-Price Elasticity of Demand so ifthen and a change to price does not alter sellers’ revenue.

55 © 2010 W. W. Norton & Company, Inc. 55 Revenue and Own-Price Elasticity of Demand but ifthen and a price increase raises sellers’ revenue.

56 © 2010 W. W. Norton & Company, Inc. 56 Revenue and Own-Price Elasticity of Demand And ifthen and a price increase reduces sellers’ revenue.

57 © 2010 W. W. Norton & Company, Inc. 57 Revenue and Own-Price Elasticity of Demand In summary: Own-price inelastic demand; price rise causes rise in sellers’ revenue. Own-price unit elastic demand; price rise causes no change in sellers’ revenue. Own-price elastic demand; price rise causes fall in sellers’ revenue.

58 © 2010 W. W. Norton & Company, Inc. 58 Marginal Revenue and Own- Price Elasticity of Demand u A seller’s marginal revenue is the rate at which revenue changes with the number of units sold by the seller.

59 © 2010 W. W. Norton & Company, Inc. 59 Marginal Revenue and Own- Price Elasticity of Demand p(q) denotes the seller’s inverse demand function; i.e. the price at which the seller can sell q units. Then so

60 © 2010 W. W. Norton & Company, Inc. 60 Marginal Revenue and Own- Price Elasticity of Demand and so

61 © 2010 W. W. Norton & Company, Inc. 61 Marginal Revenue and Own- Price Elasticity of Demand says that the rate at which a seller’s revenue changes with the number of units it sells depends on the sensitivity of quantity demanded to price; i.e., upon the of the own-price elasticity of demand.

62 © 2010 W. W. Norton & Company, Inc. 62 Marginal Revenue and Own- Price Elasticity of Demand Ifthen Ifthen Ifthen

63 © 2010 W. W. Norton & Company, Inc. 63 Selling one more unit raises the seller’s revenue. Selling one more unit reduces the seller’s revenue. Selling one more unit does not change the seller’s revenue. Marginal Revenue and Own- Price Elasticity of Demand Ifthen Ifthen Ifthen

64 © 2010 W. W. Norton & Company, Inc. 64 Marginal Revenue and Own- Price Elasticity of Demand An example with linear inverse demand. Then and

65 © 2010 W. W. Norton & Company, Inc. 65 Marginal Revenue and Own- Price Elasticity of Demand a a/b p qa/2b

66 © 2010 W. W. Norton & Company, Inc. 66 Marginal Revenue and Own- Price Elasticity of Demand a a/b p qa/2b q $ a/ba/2b R(q)


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