Finance 30210: Managerial Economics Consumer Demand Analysis

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?

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!

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

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

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”

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

We require that utility functions satisfy a few basic properties There is a definite ranking of all choices A C B

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

We require that utility functions satisfy a few basic properties People Prefer Moderation! 15 A C 10 5 B 5 10 15

Suppose you are given a little extra of good X Suppose you are given a little extra of good X. How much Y is needed to return to the original indifference curve? Marginal Utility of X Marginal Utility of Y The marginal rate of substitution (MRS) measures the amount of Y you are willing to give up in order to acquire a little more of X

The marginal rate of substitution (MRS) measures the amount of Y you are willing to give up in order to acquire a little more of X If you have a lot of X relative to Y, then X is much less valuable than Y MRS is low!

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

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

We often assume that the marginal rate of substitution is dependant only on the ratio of X and Y – i.e. preferences are homogeneous

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

First Order Necessary Conditions

Demand Curves present the same information in a different format – therefore, all the properties of preferences are present in the demand curve

Demand relationships are based off of the theory of consumer choice Demand relationships are based off of the theory of consumer choice. We can characterize the average consumer by their utility function. “Utility” is a function of lemonade and hot dogs Consumers make choices on what to buy that satisfy two criteria: Their decision on what to buy generates maximum utility Their decision on what to buy generates is affordable These decisions can be represented by a demand curve

Example: Suppose that you have $10 to spend Example: Suppose that you have $10 to spend. Hot Dogs cost $4 apiece and glasses of lemonade cost $2 apiece. # Hot Dogs MU (Hot Dogs) # Lemonade MU (Lemonade) 1 9 4 2 8 3 7 1.5 6 5 .5 This point satisfies both conditions and, hence, is one point of the demand curve

The marginal rate of substitution controls the height of the demand curve Willingness to pay is low $2 $10 Willingness to pay is high

Now, suppose that the price of hot dogs rises to $6 (Lemonade still costs $2 and you still have $10 to spend) # Hot Dogs MU (Hot Dogs) # Lemonade MU (Lemonade) 1 9 4 2 8 3 7 1.5 6 5 .5 You can’t afford what you used to be able to afford – you need to buy less of something! (Income effect) Your decision at the margin has been affected. You need to buy less hot dogs and more lemonade (Substitution effect)

Now, suppose that the price of hot dogs rises to $6 (Lemonade still costs $2 and you still have $10 to spend) # Hot Dogs MU (Hot Dogs) # Lemonade MU (Lemonade) 1 9 4 2 8 3 7 1.5 6 5 .5 This point satisfies both conditions and, hence, is one point of the demand curve

Demand curves slope downwards – this reflects the negative relationship between price and quantity. Elasticity of Demand measures this effect quantitatively Price $6.00 $4.00 Quantity 1 2

The elasticity of substitution will control the slope of the demand curve

Elasticity of Substitution vs. Price Elasticity

Perfect Complements vs. Perfect Substitutes (Almost)

Now, suppose that the price of a hot dog is $4, Lemonade costs $2, but you have $20 to spend. # Hot Dogs MU (Hot Dogs) # Lemonade MU (Lemonade) 1 9 4 2 8 3 7 1.5 6 5 .5 Your decision at the margin is unaffected, but you have some income left over (this is a pure income effect)

Now, suppose that the price of a hot dog is $4, Lemonade costs $2, but you have $20 to spend. # Hot Dogs MU (Hot Dogs) # Lemonade MU (Lemonade) 1 9 4 2 8 3 7 1.5 6 5 .5 This point satisfies both conditions and, hence, is one point of the demand curve

For any fixed price, demand (typically) responds positively to increases in income. Income Elasticity measures this effect quantitatively Price $4.00 Quantity 2 4

Income elasticity measures the response of consumers to changes in income holding prices constant – the homogeneity of preferences will effect this

Cross price elasticity refers to the impact on demand of another price changing Note: These numbers aren’t coming from the previous example!! Price $4.00 Quantity 2 6 A positive cross price elasticity refers to a substitute while a negative cross price elasticity refers to a compliment

Cross price elasticity measures consumer response to changes in other prices – this is influenced by both homogeneity and elasticity of substitution

An Example: Cobb-Douglas Utility

An Example: Cobb-Douglas Utility

An Example: Cobb-Douglas Utility With Cobb-Douglas Utility functions, your MRS is directly proportional to your relative consumption of the two goods.

An Example: Cobb-Douglas Utility Cobb-Douglas Utility functions have constant elasticity of substitution

Cobb-Douglas demands are independent of other prices!