Intermediate Microeconomics Module 4 Consumer Choice

CONSUMER CHOICE 47 CHAPTER Consumer Choice Er Up by is licensed under THE POLICY QUESTION HYBRID CAR PURCHASE TAX IT THE BEST CHOICE TO REDUCE FUEL CONSUMPTION AND CARBON EMISSIONS ?

The US government offered a tax credit toward the purchase of hybrid cars with the goal of reducing the output of carbon emissions produced by cars annually . We are using the tools of theory to study this policy and assess its effectiveness in reducing emissions . We are now very close to being able to predict how consumers will change their driving and gasoline purchases in response to a government tax credit on hybrid cars . As we will see , this is simply a specific example of the general question we first raised in chapter of how to predict consumer behavior when prices or incomes change . For our policy example and in general , we address this question by combining the budget constraint with the concept of preferences and utility maximization . Consumer theory in is based on the premise that each person will try to do their best given the money they have and 47

48 PATRICK EMERSON the prices of the goods and services they like . This is what we mean by utility the affordable bundle of goods and services that returns the highest utility . Think about a consumer who goes grocery shopping . There are many ways to fill a shopping up with many different possible bundles . This chapter is concerned with how each consumer picks the best affordable good . As we will see shortly , consumers think about the income they have and the relative prices of all the possible goods they could buy and then choose among all the bundle combinations that their budget can support . EXPLORING THE POLICY QUESTION . What is your prediction about how consumers driving and behavior will change when their income increases or decreases ?

When the price of gasoline increases or decreases ?

What implications will these behavior changes have for the hybrid car tax credit policy ?

LEARNING OBJECTIVES The Consumer Choice Problem Maximizing Utility Learning Objective Define the consumer choice problem . Solving the Consumer Choice Problem Learning Objective Solve a consumer choice problem with the typical utility function . Corner Solutions and Kinked Indifference Curves Learning Objective Solve a consumer choice problem with utility functions for perfect complements and perfect substitutes . Policy Example Hybrid Car Purchase Tax It the Best Choice to Reduce Fuel Consumption and Carbon Emissions ?

Learning Objective The Hybrid Car Tax Credit and Consumers Budgets THE CONSUMER CHOICE PROBLEM MAXIMIZING UTILITY Learning Objective Define the consumer choice problem . What is the consumer optimal choice among competing bundles ?

This question summarizes the choice problem . To resolve this problem , we can combine our understanding of the budget constraint and preferences as represented by utility functions . The budget constraint describes all the bundles the consumer could possibly choose . The utility function describes the consumers preferences and relative level of satisfaction from the consumption of bundles . We put these two pieces together by answering the question , Among all the bundles the consumer could possibly choose , which one returns the highest level of utility ?

CONSUMER CHOICE 49 Conceptually , we are overlaying the indifference curve map on the budget constraint and looking for the point or points of intersection , as in figure . Recall that there can be more than one ence curve for bundles of goods A and . The goal of solving the consumer choice problem is to get on the highest indifference curve that is the farthest to the upper also satisfying the budget constraint . The highest indifference one that represents the highest level of utility or the one thatjust touches the budget line at a single point . It is not possible to get on a higher indifference curve given the budget constraint , and though it is possible to get on a lower one , doing so necessarily means a lower level of utility or satisfaction . MU A A Bundle Slope of the Slope of indifference the budget curve line Figure curve on the budget constraint Figure summarizes the solution to the consumer choice problem The consumer should pick the one bundle that returns the highest level of utility while also satisfying the budget constraint . This graph also shows us the two fundamental conditions that represent the solution to the consumer choice problem . The consumer optimal choice is on the budget line itself , not inside the budget constraint . This is why we can focus on the line rather than the whole set of affordable bundles . At the optimal choice , the indifference touches the budget line , and so at this one point , they have exactly the same slope . This second condition has a significant intuitive interpretation . Recall that the slope of the indifference curve is the marginal rate of substitution ( rate at which the consumer is ing to trade off one good for the other and remain just as well off . Recall also that the slope of the budget

50 PATRICK EMERSON line is the economic rate of substitution ( ERS ) rate at which the consumer is able to trade one good for the other at current prices . What figure shows is that at the optimal choice , these two things must be equal . But why ?

Consider a point , like point in figure , where the indifference curve intersects the budget line . Here , the is greater than the ERS absolute value . The tells us the amount of good the consumer is willing to give up to get one more unit of good A and remain just as satisfied . In this case , the ERS tells us how much of a consumer must give up in exchange for one unit of A in the marketplace . Here , the consumer is willing to give up a lot of to get a little more A and remain just as satisfied , but the ERS indicates that the consumer does not have to give up that much of to get the same amount of A . So it must be the case that if the consumer trades some of for A in the market , they will be better . In other words , a consumer who moves down along the budget line will be on a higher indifference curve . Point Can do better by I trading some for some A Point Can do better by trading some A for some Point Optimal bundle Point Unaffordable Figure Budget with three curves So the graph of the budget line and indifference curves illustrates the two conditions that define the optimal choice latex A and PA ERS Note that the negative signs cancel . With these two conditions , we can solve the consumer choice problem mathematically . Note that the ERS condition can be rearranged to . PA

CONSUMER CHOICE 51 Economists sometimes call this the equal bang for the buck condition because the equation cates that at the optimal choice , the consumer must be getting exactly the same marginal utility per lar from the consumption of goods A and . Why ?

Think of a case where these two goods are not equal . If you at a pizza parlor eating slices of pizza and bowls of salad , suppose that This equation indicates that the marginal utility per dollar of pizza is less than the marginal utility per dollar of salad at your current level of consumption . You can not optimize your utility . Why not ?

For simplicity , let assume the price of both a slice of pizza and a small salad is exactly , so Now let take awayfrom pizza and spend it instead on salad . You lose the marginal utility ofthat last slice of pizza , but you gain the marginal utility of one more serving of salad . From the condition above , we know that total utility must have increased , since Salad We also know that the marginal utility for a normal preference diminishes when you consume more of a good and therefore increases when you consume less . Therefore , the marginal utility of pizza will increase , and the marginal utility of salad will decrease , getting you closer to equality . As long as this is not met ( as long as there is an inequality ) a consumer can always make these types of trades to become better off . SOLVING THE CONSUMER CHOICE PROBLEM Learning Objective Solve a consumer choice problem with the typical utility function . Formally , the consumer optimal choice problem looks like this mail ( A , A , subject to I As we discussed in section , the second line of this problem ( the subject to part ) is the budget constraint . Since we have the assumption , the consumer will always spend all of their budget on goods A and . So we can rewrite the consumer choice problem as mow ( A , A , subject to I Let use the utility function and solve this problem analytically . Our problem is now mass A ( A , 43 ) subject to I To solve this problem , we will apply what we know from section at the optimal solution , we have two conditions , ERS , and we are on the budget line . ERS in this case requires that we find the marginal utility of bundles A and . This requires determining the partial derivative of the utility function for good A and then for good .

52 PATRICK EMERSON ( A ) MU A ( 33 The is the ratio of the two marginal utilities MU ) A ( The ERS is the ratio of the prices ( Putting these two equations together gives us A ! PA ( A The second part of the consumer choice problem , the budget constraint , as we are on the budget line or the subject to part , is straightforward I ( At this point , solving the problem is a matter of simple algebra . We have two equations with two unknowns , good A and good . We can solve these equations by repeated substitution solve tion for good A or , and substitute the result into Equation . If we solve equation for , we get Substituting into equation gives us I ( Simplifying this equation gives us a ) FAA I Solving for A , we get I oz A ( PA ( 01 ) Plugging this equation into equation gives us , I ( 04 ) Simplifying this equation gives us the solution for good I ) Remember that the prices , the income , and the parameter values for a and are all just numbers that are given . Therefore , these solutions for goods A and are simply a specific amount of

CONSUMER CHOICE 53 specific bundle of A and consumption that is the very best choice a consumer has among all the possible choices . Equations and ) are the demand functions for goods A and , respectively . Demand are mathematical functions that describe the relationship between quantity demanded and prices , income , and other things that affect purchase decisions . We can use these demand functions to predict what will happen to the consumption of both goods when prices and incomes change . From the demand functions , it is easy to predict that increases in the price of the good will lead to lower consumption and increases in income will lead to greater consumption . We will return to the examination of these demand functions in the next chapter . CORNER SOLUTIONS AND KINKED INDIFFERENCE CURVES Learning Objective Solve a consumer choice problem with utility function for perfect complements and perfect substitutes . So far , we have considered the optimal consumption bundle for a consumer who has preferences , meaning that they have indifference curves that are smooth , curved in , and not touching the vertical or horizontal axes . The optimal choice for these preferences is characterized by the ERS . The solution to the consumer choice problem with these preferences is always an interior solution a utility maximizing bundle that has a positive amount of both goods . But we know that there are other relatively common preference as perfect and perfect have indifference curves that are shaped differently . The tion to the consumer choice problem for these preference types can not be characterized by the ERS condition . As we saw in chapter , perfect complements have indifference curves that are kinked at degree angles , and perfect substitutes have indifference curves that are straight lines that begin and end on the axes . For both perfect complements and perfect substitutes , the solution to the consumer choice problem is the one consumption bundle that puts the consumer on the highest indifference curve possible . But in both cases , this does not have the tangency condition of the equaling the ERS . Since perfect complements have indifference curves that are kinked , they have abrupt changes in slope at a single point . At this kink , the is not defined because there is no slope . The solution to the consumer choice problem with perfect complement preferences is definitely need some of both goods to get any at the kink , there is no to equate to ERS . The solutions to consumer choice problems with perfect complement preferences are usually corner solutions a utility maximizing bundle that consists of only one of the two other words , a consumption bundle that is located at one corner of the budget constraint . Corner solutions are typical when preferences are perfect substitutes but can occur for many other preference types that we will not study . Let see how to find the utility maximizing bundle for these two preference types , starting with perfect complements . Perfect Complements In the case of perfect complements with strictly positive prices , the optimal bundle is always the one at the kink in the indifference curve , as shown in figure . We can use this observation similarly to how we used the fact that ERS in the previous section . If the optimal bundle is

54 PATRICK EMERSON always at the kink and is always on the budget line , we can once again find two equations with two unknowns to solve . Optimal Bundle Indifference curve for perfect complements Figure Solution to the consumer choice problem for perfect complements Recall that the utility function for perfect complements looks like this ( A , aA , A parable is a fixed value given outside the model , one that never changes . is a value that can change , such as prices , income , and so on . Consuming at the kink implies that ( This condition makes intuitive sense since the utility function takes on only the value ofthe smaller ofthe two choices , spending money on some of good A without the corresponding increase in good not raise utility at all but will cost the consumer money . So this choice can not possibly be optimal . The only optimal decision is to spend money on goods A and in the precise proportions that they are consumed . With the condition aA and the budget constraint I , we can easily solve through repeated substitution . Note that A substitutes into the budget constraint as follows or PA I or Solving for gives us 051 Solving for A using A gives us or

CONSUMER CHOICE 55 A aI oz ( Again , we see that the quantity demanded of each item decreases with an increase in its price and increases with increases in income . But now we also have the interesting result that the quantity demanded ofone good decreases as the price of the other good increases . This makes intuitive sense because perfect complements are goods that are only consumed together , such as hot dogs and hot dog buns . Perfect Substitutes With perfect substitutes , the optimal bundle is generally at either one corner of the budget constraint or the other . Ifyou stop and think about it for a moment , the intuition behind this observation becomes clear . Ifyou like Coke and Pepsi equally well and think of them as perfect substitutes for one another , you will logically buy only the one that has the lower price . If Coke costs and Pepsi , and you like them equally well , why would you ever buy any Pepsi at There is one exception to this rule . When Coke and Pepsi have the same prices , you can get all of one , all of the other , or any combination of both . In this case , there is notjust one optimal bundle . In this chapter , we will on the case where there is one optimal bundle , and so you , the consumer , have to purchase all of one or the other . Consider figure , which illustrates a generic budget line ( in black ) for goods A and and a series of indifference curves ( in blue ) representing preferences for the perfect substitutes of goods A and . The slope of the indifference curves ( the ) is always greater than the slope of the budget line ( the ERS ) As you can see from figure , this means that consumers always do better when they more A and less move to a higher indifference curve . Specifically , consider bundle in the graph . This bundle contains both A and Ifwe compare die to the bundle in the corner ofthe budget constraint labeled optimal bundle , you can see that the optimal bundle is on a higher indifference curve , which means that this bundle is better . In fact , this is true of any other bundle in the budget other bundles place consumers on a lower ence curve .

56 PATRICK EMERSON Optimal Bundle Figure Solution to the consumer choice problem for perfect substitutes Finding this optimal bundle is relatively easy mathematically . Since we know it has to be at a corner , we just need to check the two corners ofthe budget constraint and see which one yields the higher utility . In general , utility functions that represent perfect complements look like this ( A , Note that the additive form of the utility function is the can always get to the same utility by taking away some of one good and adding some ofthe other ( even if there is none of the other good in the bundle ) Note also that the is Since 04 and are parameters , this means the slope of the indifference curve is a constant , and so the indifference curves for perfect substitutes are straight lines that intersect the axes . Solving for the optimal consumption bundle for perfect substitutes starts with checking the corners , which means we ask what utility the consumer gets from spending all oftheir income on just one good . So if I and the consumer decides to consume only A , then the total amount consumed of A is I PA Similarly , the total amount consumed of if all of the income is spent on is I

CONSUMER CHOICE 57 So all that is left to do is to check if I I ( A , a PA ( ifthe opposite is true , or ifthey are equal . ifthe above is true , then we know immediately that consuming only A is the optimal choice . Similarly , if the opposite is true , we know consuming is optimal . Finally , if they happen to be equal , any combination of A and consumption along the budget line is optimal . Figure illustrates these three possibilities . A Optimal Bundle All Bundles Equally Good Optimal Bundle A A A Figure Solutions to the consumer choice problem for perfect substitutes . Solutions ( a ) and ( show the corner solutions . POLICY EXAMPLE HYBRID CAR PURCHASE TAX IT THE BEST CHOICE TO REDUCE FUEL CONSUMPTION AND CARBON EMISSIONS ?

Learning Objective The Hybrid Car Tax Credit and Consumer Choice Suppose you are a considering proposals to reduce the consumption fuels and carbon emissions . With the tools from chapters through , you can now conduct an economic analysis to analyze the likely outcome ofa tax credit for electric and hybrid cars . Begin by considering the choice problem prior to the introduction of the tax credit . First , combine the indifference curve mapping from chapters and with the budget constraint from chapter , as shown in figure . Remember , these are simply graphical representations of the equations and .

58 PATRICK EMERSON Miles Driven Optimal Bundle Income Figure Optimal miles driven without car tax credit We are looking for the point on the budget line that puts the consumer on the highest indifference curve possible . This is of course the tangency point between the indifference curve and the budget line . This tangency is characterized by the equality of the marginal rate of substitution between miles driven and other goods consumed and the price ratio of the same two goods , the economic rate of tion . Figure shows that given our assumption of convex indifference curves , there is a unique point that defines the number of miles driven and the amount of money for other goods consumed that make up the optimal bundle for this typical consumer . Mathematically , we can solve for the tangent point by remembering that it is characterized by two things . The equality of the marginal rate of substitution to the price ratio . The budget constraint holding with equality The first equation ensures that the bundle is at a point of tangency between the indifference curve and lines with the same slope as the budget line . But remember that the price ratio describes the slope , not the specific budget line , which is why we need the second equation to make sure we are on the specific line that describes the consumer budget . Specifically , if the utility function that represents the typical consumer preferences over miles ven ( and all other consumption ( is ( then we can solve for the

CONSUMER CHOICE 59 MUG As noted above , this is one condition that characterized the optimal consumption bundle . The other is the budget line Income ( Price , Miles Driven ) Dollars for Other Consumption We can solve this system of equations by the process of repeated substitution to come up with the description of the precise optimal bundle for this consumer . Now let consider what happens after pass the tax credit into law . From chapter , we know that the effect ofthe tax credit on the budget constraint is to make the cost of miles consumers drive less expensive , which represents a drop in the price of miles driven . This will cause the budget line to expand along the vertical axis , but it will not change the horizontal intercept . The consumer must now choose a new optimal bundle , and we must compare the new bundle to the old one . Let first examine this graphically in figure . New Optimal Bundle I Figure Optimal miles consumers drive with car tax credit In this graph , it is clear that the new bundle represents more of both goods . The consumer will choose to consume more of other goods with the extra buying power that the more car affords as well as to consume more miles driven due to the lower cost of the activity . So the policy turns out to have an unintended consequence in an effort to decrease fuel consumption , the policy actually lowers the marginal cost of driving , inducing consumers to drive more .

60 PATRICK EMERSON By making some fairly basic assumptions about typical consumer preferences and modeling the choice problem , economic theory suggests that we should expect an increase in miles driven as a result . This application illustrates the power of models . By simplifying reality into a model framework , we can discover something about the world and human behavior that was not obvious . But this isjust a theory , so it is suggestive rather than definitive . It is important to note that the assumptions we made may not be completely accurate , and so our prediction may be inaccurate . For example , depending on the precise indifference curve mapping , we could actually see decreased miles driven after the hybrid car tax credit . We now need to test the theory by evaluating data . So what does the data suggest ?

Studies have shown that . An increase in increases how much consumers drive and . This effect is in the range of 10 to 30 percent , meaning that a 10 percent increase in would increase how many miles consumers drive by between and percent . In the case consumption , economists generally prefer that increase the gas subsidy other indirect approaches , like the corporate fuel economy standard known as CAFE . The tax on fuel raises the cost of consumption and , as we have now seen , decreases fuel tion . A number of studies have shown that an increase in the gas tax is a more way to decrease gasoline usage . With our analysis of the policy based on both theory and evidence , we are now in a position to answer the original question posed at the beginning of section . Suppose that a hybrid car tax credit was wildly successful and succeeded in doubling the average fuel economy ofall cars on US roads . What do you think would happen to the fuel consumption of all US motorists ?

Should the government expect the fuel consumption and carbon emissions of cars driven in the United States to decrease by half in response ?

The answer is clearly no . Even the most conservative estimates suggest that a 100 percent increase in average fuel efficiency will result in an increase of miles driven by 10 percent , meaning that the decrease in fuel consumption and carbon emissions will fall but by less than half . EXPLORING THE POLICY QUESTION . Suppose the hybrid car tax credit was accompanied by a tax increase on gasoline . How would the analysis of the policy change ?

Could you make a prediction about the change in carbon emissions now ?

How would the analysis change if consumers preferences change and they begin to lower their gas consumption by carpooling , walking , and biking more instead of driving ?

How could you show this in figure ?

The phenomenon of increased energy efficiency leading to increased consumption is known among economists as the rebound effect , and it is common in other practical texts besides vehicles and driving . What other examples of rebound effect would you expect to see in the real world ?

CONSUMER CHOICE 61 REVIEW TOPICS AND RELATED LEARNING OUTCOMES The Consumer Choice Problem Maximizing Utility Learning Objective Define the consumer choice problem . Solving the Consumer Choice Problem Learning Objective Solve a consumer choice problem with the typical utility function . Corner Solutions and Kinked Indifference Curves Learning Objective Solve a consumer choice problem with utility functions for perfect complements and perfect substitutes . Policy Example Hybrid Car Purchase Tax It the Best Choice to Reduce Fuel Consumption and Carbon Emissions ?

Learning Objective The Hybrid Car Tax Credit and Consumers Budgets LEARN KEY TOPICS Terms Consumer choice problem What is the consumers optimal choice among competing bundles ?

Among all the bundles the consumer could possibly choose , which one returns the highest level of utility ?

Corner solutions A utility maximizing bundle that consists of only one of the two other words , a consumption bundle that is located at one corner of the budget constraint . Corner solutions are typical when preferences are perfect substitutes . Demand functions Mathematical functions that describe the relationship between quantity demanded and prices , income , and other things that affect purchase decisions . Interior solutions A utility maximizing bundle that has a positive amount of both goods . Kinked curves Curves that have abrupt changes in slope at a single point . Graphs Solution to the general consumer choice problem

62 PATRICK EMERSON Optimal Bundle Slope ofthe Slope of indifference the budget curve line Figure ence on the budget Point Can do better by I trading some some A Point Can do better by trading some some Point Optimal bundle Point Unaffordable Figure Budget three Solution to the consumer choice problem for perfect complements

CONSUMER CHOICE I , Bundle Indifference curve for perfect complements A IA Figure to me customer ( Solution to the consumer choice problem for perfect substitutes Optimal Bundle Figure to me consumer ( substitutes Equation Demand functions The Demand Function for Good A 63 A iT PA (

64 PATRICK EMERSON The Demand Function for Good , 01 ) The Consumer Optimal Choice Problem mow ( A , A , subject to I The second line of the consumer optimal choice problem ( the subject to part ) is the budget constraint . Using the assumption , the equation is rewritten to indicate the consumer will always spend all of their budget on goods A and , as below max ( A , A , subject to I Applying the utility function max A ( A , subject to I To solve this requires taking the partial derivative . This equation looks as follows a ( MU ( A ( A ) MU A ( 63 The is the ratio of the two marginal utilities MU ( A ( A ) The ERS is the ratio of the prices PA When combined , they take the following form aB PA or PA A

CONSUMER CHOICE The second part of consumer choice problem , the budget constraint , reads like this I Solving the Equation 65 Solving the problem involves repeated substitution . The ratio initially solved for Bundle is as follows 01 The next step is substituting the above result into the ERS ratio equation , providing the formula below a ( I Solved for A PA ( The result of the above equation substituted into the combined equation is the demand function for good A ( Solving again for provides the demand function for good I