1. Finance 30210: Managerial Economics
Consumer Demand Analysis

2. We can begin our representation of the consumer with a demand function…
Is a function of…
Quantity demanded
Price
Income
Prices of Compliments
Prices of Substitutes
Price
$20
Quantity 40

3. Price elasticity will always be a negative number
Given a demand function we can characterize the behavior of demand with elasticity..
Price
Price elasticity will always be a negative number
$20
$15
Quantity 40 60

4. Income elasticity will generally be a positive number
Given a demand function we can characterize the behavior of demand with elasticity..
Income elasticity will generally be a positive number
Price
$20
Quantity 40 56

5. Cross price elasticity will be a positive number for substitutes
Given a demand function we can characterize the behavior of demand with elasticity..
Cross price elasticity will be a positive number for substitutes
Price
$20
Quantity 40 44

6. Cross price elasticity will be a negative number for compliments
Given a demand function we can characterize the behavior of demand with elasticity..
Cross price elasticity will be a negative number for compliments
Price
$20
Quantity 32 40

7. Note that if the demand relationship is linear, elasticity is not constant

8. For example….
Price
$80
$30
Quantity
100

9. Let’s try something a little more complicated…a non-linear demand relationship
Price
$2916
$400
Quantity 68

10. Sometimes we use demand functions that are linear in logs…
Price
Price
$12
$12
Quantity
LN(Quantity) 40
3.7

11. Price
$12
Quantity 40

12. A little math trick…recall the derivative of the natural log
This just says that the difference in logs is a percentage change
Therefore, if we start with elasticity

13. Sometimes we use demand functions that are linear in logs…
Price
$12
LN(Quantity)
3.7

14. Sometimes we use demand functions that are linear in logs…
Price
LN(Price)
$10
2.3
Quantity
Quantity 3 3

15. Sometimes we use demand functions that are linear in logs…
LN(Price)
2.3
Quantity 3

16. Sometimes we use demand functions that are linear in logs…
Price
LN(Price) $5
1.6
Quantity
LN(Quantity)
8.8
2.2

17. Sometimes we use demand functions that are linear in logs…
LN(Price)
1.6
Log linear demand curves have constant elasticities!
LN(Quantity)
2.2

18. Suppose that you setting prices for US Air
Suppose that you setting prices for US Air. You know that you face the following demand curve for the New York/Chicago Shuttle…
Price
$200
$80,000
If you wanted to increase revenues, should you increase or decrease your price?
Quantity
400

19. If you wanted to increase revenues, should you increase or decrease your price?
Why didn’t revenue go up by $400?
$200
$199
$80,396
Quantity
400
404
You should decrease price. A $1 price decrease will raise revenues by $400

20. Suppose that you wanted to maximize revenues?
Price
$200
$150
$90,000
Quantity
400
600

21. Now, let’s go about this a little differently…
Price
$200
$80,000
Every 1% drop in price will raise revenues by 1%
Quantity
400

22. Using the point elasticity gives you the same answer…
Why didn’t revenues go up by 10%?
Price
$200
$180
$86,400
Quantity
400
480

23. We could also maximize revenues using elasticity…
When the elasticity hits -1, revenues stop growing when you lower your price.
Price
$150
$90,000
Quantity
600

24. However, knowing a few elasticities is quite helpful as well!
Obviously, the more information you have, the better decisions you will make…
However, knowing a few elasticities is quite helpful as well!
The best pricing decisions would come from a demand curve
Price
Price
Revenue maximizing price
Revenue maximizing price
$250
$200
$150
$140
$90,000
$50
Quantity
Quantity
600
600

25. Suppose that median income is equal to $50,000.
Suppose that you setting prices for US Air. You know that you face the following demand curve for the New York/Chicago Shuttle…
Suppose that median income is equal to $50,000.
Income in thousands
Price
Suppose that a recession causes a 20% drop in income. How much would we have to lower our price if we wanted to keep sales constant?
$125
$75,000
Quantity
600

26. Suppose that a recession causes a 20% drop in income
Suppose that a recession causes a 20% drop in income. How much would we have to lower our price if we wanted to keep sales constant?
Now income is $40,000
Price
$125
$120
$72,000
Quantity
600

27. Now, again with elasticity…
Suppose that median income is equal to $50,000.
Price
$125
Suppose that a recession causes a 20% drop in income. How much would we have to lower our price if we wanted to keep sales constant?
$75,000
Quantity
600

28. Suppose that a recession causes a 20% drop in income
Suppose that a recession causes a 20% drop in income. How much would we have to lower our price if we wanted to keep sales constant?
Price
$125
$123
$73,800
Quantity
600

29. However, knowing a few elasticities is quite helpful as well!
Again, the more information you have, the better decisions you will make…
However, knowing a few elasticities is quite helpful as well!
The best pricing decisions would come from a demand curve
Price
Price
$125
$125
$75,000
$75,000
Quantity
Quantity
600
600

30. Now, suppose that you realized that you actually faced two types of customers : Recreational and business travelers. Could you do better?
Business
Recreational
Price
Price
$400
$200
$150
$150
$75,000
$15,000
Quantity
Quantity
500
100

31. Recall the aggregate demand curve we had previously…
Recall the aggregate demand curve we had previously….what we had was actually a piece of what aggregate demand really looked like
Price
At a price above $200, recreational travelers don’t fly. The only demand is coming from business travelers.
$400
At a price below $200, we now have two types of demanders flying.
$200 +
$150
$90,000
Could we do better?
Quantity
400
600

32. No…that’s not a good idea!
Suppose that we decided to ignore recreational travelers and cater to business clients…
Price
$400
$200
$80,000
No…that’s not a good idea!
Quantity
400
600

33. Why don’t we just charge them different prices?
Recreational
Business
Price
Price
$400
$200
$200
$100
$80,000
$20,000
Quantity
Quantity
400
200

34. The real question is: Is this feasible to do empirically?
Price
$400
This is “the Truth”. Two types of individuals making purchasing decisions. Individual decisions added up across all individuals create an aggregate demand curve.
$200
Quantity

35. If you do a linear estimation, what you end up with something like this…not really what we want
Price
$400
$200
Quantity

36. However, if you put a dummy variable in the regression for business/recreational traveler, you could get at the truth…
Price
$400
$200
YES! We can do this!
Quantity

37. Now, lets change it up a little….
Business
Recreational
Price
Price
$600
$150
$200
$67,500
$150
$22,500
Quantity
Quantity
450
150

38. Now, if we charge different prices….
Recreational
Business
Price
Price
$600
$300
$200
$90,000
$100
$30,000
Quantity
Quantity
300
300

39. Again, we have to ask: Is this feasible to do empirically?
This is “the Truth”. Two types of individuals making purchasing decisions. Individual decisions added up across all individuals create an aggregate demand curve.
Price
$600
$300
$200 +
Quantity

40. If you do a linear estimation, what you end up with something like this…
Price
$600
$200
Quantity

41. What if we tried the dummy variable trick…
Price
$600
$200
We’re Screwed! Time to try something else.
Quantity

42. Here’s the process that takes place in the economy…D Q D Q P P
Individual consumers have preferences over a variety of goods…they have limited incomes and face market prices. Consumers make choices on what to buy D Q D Q
Individual decisions can be represented by individual demand curves P
The market aggregates those decisions into a market demand curve D Q

43. Here is the problem we face…
We can estimate a market demand curve… D Q
The problem is that the market demand often tells us very little about what is happening in the background… P P D Q D Q P P D Q D Q

44. So, here is how we attack the problem…D Q D Q P P D Q
We see if we can come up with a numerical representation of preferences… D
We then derive the resulting demand curves that come from the consumer choice problem… P
We then aggregate the individual demand curves to get a market demand. This we can compare to aggregate data to see if we are correct D Q

45. How do we get an understanding of consumer preferences
How do we get an understanding of consumer preferences? We observe what they do!
Television A
Cost = $500
Television B
Cost = $2500
Suppose you walk into the store with a choice between two TVs.
Suppose that you choose Television A
Either you prefer Television A or you can’t afford Television B
Suppose that you choose Television B
You must prefer Television B

46. Suppose that you observed the following consumer behavior
P(Bananas) = $4/lb.
P(Apples) = $2/Lb.
Q(Bananas) = 10lbs
Q(Apples) = 20lbs
Choice A
P(Bananas) = $3/lb.
P(Apples) = $3/Lb.
Q(Bananas) = 15lbs
Q(Apples) = 15lbs
Choice B
What can you say about this consumer?
Is strictly preferred to
Choice B
Choice A
How do we know this?

47. Consumers reveal their preferences through their observed choices!
Choice A
Choice B
Q(Bananas) = 10lbs
Q(Apples) = 20lbs
Q(Bananas) = 15lbs
Q(Apples) = 15lbs
P(Bananas) = $4/lb.
P(Apples) = $2/Lb.
Cost = $80
Cost = $90
P(Bananas) = $3/lb.
P(Apples) = $3/Lb.
Cost = $90
Cost = $90
B Was chosen even though A was the same price!

48. What about this choice?
Choice C
Cost = $90
P(Bananas) = $2/lb.
P(Apples) = $4/Lb.
Q(Bananas) = 25lbs
Q(Apples) = 10lbs
Q(Bananas) = 15lbs
Q(Apples) = 15lbs
Cost = $90
Choice B
Q(Bananas) = 10lbs
Q(Apples) = 20lbs
Cost = $100
Choice A
Is strictly preferred to
Is choice C preferred to choice A?
Choice C
Choice B

49. Is strictly preferred to
Choice B
Choice A
Is strictly preferred to
Choice C
Choice B
C > B > A
Is strictly preferred to
Choice C
Choice A
Rational preferences exhibit transitivity

50. Consumer theory begins with the assumption that every consumer has preferences over various combinations of consumer goods. Its usually convenient to represent these preferences with a utility function
Set of possible consumption choices
“Utility Value”

51. Using the previous example (Recall, C > B > A)
Choice A
Q(Bananas) = 10lbs
Q(Apples) = 20lbs
Choice B
Q(Bananas) = 15lbs
Q(Apples) = 15lbs
Choice C
Q(Bananas) = 25lbs
Q(Apples) = 10lbs

52. We require that utility functions satisfy a few basic properties
There is a definite ranking of all choices (i.e. transitivity) A C B
This tells us that indifference curves can’t cross on another

53. Suppose that indifference curves did cross…
We run into a problem! C A B

54. More is always better! C A B
We require that utility functions satisfy a few basic properties
More is always better! C A B

55. People Prefer Moderation!
We require that utility functions satisfy a few basic properties
People Prefer Moderation! 15 A C 10 5 B 5 10 15
Note: This is a result of diminishing marginal utility…this guarantees that demand curves slope down!

56. People prefer extremes!
What if we didn’t have decreasing marginal utility?
People prefer extremes! A C B
Increasing marginal utility produces some weird decisions!!

57. We can characterize preferences with a few statistics
We can characterize preferences with a few statistics. First, how does a consumer prefer one good relative to another.
Marginal Utility of X
Marginal Utility of Y
The marginal rate of substitution (MRS) measures the amount of Y you need to be get in order to give up a little of X

58. The marginal rate of substitution (MRS) measures the amount of Y you require to give up a little of X
If you have a lot of X relative to Y, then X is much less valuable than Y MRS is low!

59. The elasticity of substitution measures the curvature of the indifference curve
Elasticity of substitution measures the degree to which your valuation of X depends on your holdings of X

60. The elasticity of substitution measures the curvature of the indifference curve
If the elasticity of substitution is small, then small changes in x and y cause large changes in the MRS
If the elasticity of substitution is large, then large changes in x and y cause small changes in the MRS

61. Elasticity is Infinite Elasticity is 0
Elasticity of Substitution measures the degree in which you can alter the mix of goods. Consider a couple extreme cases:
Perfect substitutes can always be can always be traded off in a constant ratio
Perfect compliments have no substitutability and must me used in fixed ratios Y Y
Elasticity is Infinite
Elasticity is 0 X X

62. Consumers solve a constrained maximization – maximize utility subject to an income constraint.
As before, set up the lagrangian…

63. First Order Necessary Conditions

65. Individual demand Curves represent a summary of the individual consumer choice problem

66. The marginal rate of substitution controls the height of the demand curve
Willingness to pay is low
Willingness to pay is high

67. The elasticity of substitution will effect the price elasticity of the demand curve

68. Elasticity of Substitution vs. Price Elasticity

69. Perfect Complements vs. Perfect Substitutes

70. An Example: Cobb-Douglas Utility
Recall, Marginal Rate of Substitution measures the relative value of x. This will determine the intercept of the demand curve

71. Recall, elasticity of substitution is measuring the degree of flexibility in the consumption of X and Y and will determine the slope of the demand curve

72. An Example: Cobb-Douglas Utility
Marginal Rate of Substitution

73. An Example: Cobb-Douglas Utility

74. Constant price elasticity

75. Zero cross price elasticity

76. Constant income elasticity

77. Log-linear demand curves…

78. Different intercepts but the same slope…
Returning to our airline example…suppose we used a customer survey or some other method and came up with preferences of airline travelers
Recreational
Business
Airline Travel
Everything Else
Different intercepts but the same slope…

79. Returning to our airline example…suppose we used a customer survey or some other method and came up with preferences of airline travelers
Recreational
Business
Airline Travel
Everything Else

80. We end up with something like this…
LN(Price)
Business +
Recreation
Aggregate
LN(Quantity)
We could estimate a log linear demand curve with a dummy variable to check this

81. Suppose that we wanted different slopes…
The Cobb-Douglas utility function is a special case of the CES (Constant elasticity of substitution) utility functions…
The parameter rho will govern the elasticity of substitution which influences the slope of the demand curve
The parameter alpha will govern the marginal rate of substitution which influences the intercept of the demand curve

82. Suppose that we wanted different slopes…
The Cobb-Douglas utility function is a special case of the CES (Constant elasticity of substitution) utility functions…
The resulting demand curve looks like this…
A price index comprised of both good’s prices

83. Returning to our airline example…suppose we used a customer survey or some other method and came up with preferences of airline travelers
Recreational
Business
Different intercepts and different slopes…

84. Returning to our airline example…suppose we used a customer survey or some other method and came up with preferences of airline travelers
Recreational
Business

85. Now, we have something like this…
Business +
Recreation
Price
Aggregate
We could estimate a log linear demand curve to check this
Quantity