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Finance 30210: Managerial Economics Consumer Demand Analysis

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

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

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

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

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

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

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For example…. Quantity Price $ $80

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Lets try something a little more complicated…a non-linear demand relationship Quantity Price $400 $

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

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Quantity Price $12 40

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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

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

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Quantity Price $10 3 Quantity LN(Price) 2.3 3

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

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

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

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

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Quantity Price $ $80,396 If you wanted to increase revenues, should you increase or decrease your price? You should decrease price. A $1 price decrease will raise revenues by $400 $ Why didnt revenue go up by $400?

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Quantity Price $ $90,000 Suppose that you wanted to maximize revenues? $

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Now, lets go about this a little differently… Quantity Price $ $80,000 Every 1% drop in price will raise revenues by 1%

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Using the point elasticity gives you the same answer… Quantity Price $ $86,400 $ Why didnt revenues go up by 10%?

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

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

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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 Quantity Price $ $75,000 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?

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Quantity Price $ $72,000 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 $120

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Now, again with elasticity… Suppose that median income is equal to $50,000. Quantity Price $ $75,000 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?

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Quantity Price $ $123 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? $73,800

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

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

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Recall the aggregate demand curve we had previously….what we had was actually a piece of what aggregate demand really looked like Quantity Price $ $400 $90,000 At a price above $200, recreational travelers dont fly. The only demand is coming from business travelers. At a price below $200, we now have two types of demanders flying. + $ Could we do better?

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Suppose that we decided to ignore recreational travelers and cater to business clients… Quantity Price $ $400 $80, No…thats not a good idea!

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Why dont we just charge them different prices? Quantity Price $ Quantity Price $ Business Recreational $400 $200 $80,000 $20,000

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Quantity Price $200 $400 The real question is: 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.

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

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

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Now, lets change it up a little…. Quantity Price $ Quantity Price $ BusinessRecreational $600 $200 $67,500 $22,500

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

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Quantity Price $200 $600 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. $300 +

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

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

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

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

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So, here is how we attack the problem… We see if we can come up with a numerical representation of preferences… P Q D P Q D P Q D P D We then derive the resulting demand curves that come from the consumer choice problem… P Q D 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

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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 cant afford Television B Suppose that you choose Television B You must prefer Television B

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

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

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

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Is strictly preferred to Choice BChoice A Is strictly preferred to Choice CChoice B Is strictly preferred to Choice CChoice A Rational preferences exhibit transitivity C > B > A

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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

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

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We require that utility functions satisfy a few basic properties A B C There is a definite ranking of all choices (i.e. transitivity) This tells us that indifference curves cant cross on another

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

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

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

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A B C People prefer extremes! What if we didnt have decreasing marginal utility? Increasing marginal utility produces some weird decisions!!

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We can characterize preferences with a few statistics. First, how does a consumer prefer one good relative to another. Marginal Utility of Y Marginal Utility of X 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

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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!

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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

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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

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X Y X Y 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 Elasticity is 0Elasticity is Infinite

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

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First Order Necessary Conditions

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

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

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The elasticity of substitution will effect the price elasticity of the demand curve D

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Elasticity of Substitution vs. Price Elasticity

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Perfect Complements vs. Perfect Substitutes

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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

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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

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An Example: Cobb-Douglas Utility Marginal Rate of Substitution

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An Example: Cobb-Douglas Utility

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Constant price elasticity

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Zero cross price elasticity

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Constant income elasticity

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Log-linear demand curves…

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

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

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

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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 alpha will govern the marginal rate of substitution which influences the intercept of the demand curve The parameter rho will govern the elasticity of substitution which influences the slope of the demand curve

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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 goods prices

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

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

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

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