Intermediate Microeconomics Module 2 Utility

UTILITY 19 CHAPTER Utility Er Up by is licensed under THE POLICY QUESTION IS A TAX CREDIT ON HYBRID CAR PURCHASES THE GOVERNMENT BEST CHOICE TO REDUCE FUEL CONSUMPTION AND CARBON EMISSIONS ?

US residents and the government are concerned about the dependence on imported foreign oil and the release of carbon into the atmosphere . In 2005 , Congress passed a law to provide consumers with tax credits toward the purchase of electric and hybrid cars , This tax credit may seem like a good policy choice , but it is costly because it directly lowers the amount of revenue the US government collects , Are there more effective approaches to reducing dependency on fossil fuels and carbon emissions ?

How do we decide which policy is best ?

To answer this question , need to predict with some accuracy how consumers will respond to this tax policy before these spend millions of federal dollars , 19 20 PATRICK EMERSON We can apply the concept of this policy question . In this chapter , we will study utility and utility functions . We will then be able to use an appropriate utility function to derive indifference curves that describe our policy question . EXPLORING THE POLICY QUESTION . Suppose that the tax credit to subsidize hybrid car purchases is wildly successful and the average fuel economy of all cars on US result that is clearly not realistic but useful for our subsequent discussions . What do you think would happen to the fuel consumption of all US motorists ?

Should the government expect fuel consumption and bon emissions from cars to decrease by half in response ?

Why or why not ?

LEARNING OBJECTIVES Utility Functions Learning Objective Describe a utility function . Utility Functions and Typical Preferences Learning Objective Identify utility functions based on the typical preferences they represent . Relating Utility Functions and Indifference Curve Maps Learning Objective Explain how to derive an indifference curve from a utility function . Finding Marginal Utility and Marginal Rate Learning Objective Derive marginal utility and for typical utility functions . Policy Example Is a Tax Credit on Hybrid Car Purchases the Government Best Choice ?

Learning Objective The Hybrid Tax Credit and customer utility UTILITY FUNCTIONS Learning Objective Describe a utility function . Our preferences allow us to make comparisons between different consumption bundles and choose the preferred bundles . We could , for example , determine the rank ordering of a whole set of bundles based on our preferences . A utility function is a mathematical function that ranks bundles of tion goods by assigning a number to each , where larger numbers indicate preferred bundles . Utility have the properties we identified in chapter regarding preferences . That is , they are able to order bundles , they are complete and transitive , more is preferred to less , and in relevant cases , mixed bundles are better .

UTILITY 21 The number that the utility function assigns to a specific bundle is known as utility , the satisfaction a consumer gets from a specific bundle . The utility number for each bundle does not mean anything in absolute terms there is no uniform scale against which we measure satisfaction . Its only purpose is in relative terms we can use utility to determine which bundles are preferred to others . If the utility from bundle A is higher than the utility from bundle , it is equivalent to saying that a consumer prefers bundle A to bundle . Utility functions , therefore , rank consumer preferences by assigning a number to each bundle . We can use a utility function to draw the indifference curve maps described in chapter . Since all bundles on the same indifference curve provide the same satisfaction and therefore none is preferred , each bundle has the same utility . We can therefore draw an indifference curve by determining all the bundles that return the same number from the utility function . Economists say that utility functions are ordinal rather than cardinal . Ordinal means that utility only rank only indicate which one is better , not how much better it is than another bundle . Suppose , for example , that one utility function indicates that returns ten and . We do not say that bundle is twice as good , or ten better , only that the prefers bundle . For example , suppose a friend entered a race and told you she came in third . This information is ordinal You know she was faster than the finisher and slower than the finisher . You only know the order in which runners finished . The individual times are nal if the finisher ran the race in exactly one hour and your friend finished in one hour and six minutes , you know your friend was exactly 10 percent slower than the fastest runner . Because utility functions are ordinal , many different utility functions can represent the same preferences . This is true as long as the ordering is preserved . Take , for example , the utility function that describes preferences over bundles of goods A and ( A , We can apply any positive monotonic transformation to this function ( which means , that we do not change the ordering ) and the new function we have created will represent the same preferences . For example , we could multiply a positive constant , 04 , or add a positive or a negative constant , So ( A , represents exactly the same preferences as ( A , because it will order the bundles in exactly the same way . This fact is quite useful because sometimes applying a tive monotonic transformation ofa utility function makes it easier to solve problems . UTILITY FUNCTIONS AND TYPICAL PREFERENCES Learning Objective Identify utility functions based on the typical preferences they represent . Consider bundles of apples , A , and bananas , A utility function that describes Isaac preferences for bundles of apples and bananas is the function ( A , But what are Isaac particular preferences for bundles of apples and bananas ?

Suppose that Isaac has fairly standard preferences for apples and bananas that lead to our typical indifference curves he prefers more to less , and he likes variety . A utility function that represents these preferences might be ( A , AB and bananas are perfect complements in Isaac preferences , the utility function would look something like this ( A , MIN A , where the MIN function simply assigns the smaller of the two numbers as the function value . and bananas are perfect substitutes , the utility function is an additive and would look thing like this ( A , A

22 PATRICK EMERSON utility functions are commonly used in economics for two reasons . They represent standard preferences , such as more is better and preference for variety . They are very flexible and can be adjusted to fit data very easily . utility functions have this form ( A , A Because positive monotonic transformations represent the same preferences , one such tion can be used to set or , which later we will see is a convenient condition that simplifies some math in the consumer choice problem . Another way to transform the utility function in a useful way is to take the natural log of the function , which creates a new function that looks like this ( A , aln ( A ) To derive this equation , simply apply the rules of natural logs . It is important to keep in mind the level of abstraction here . We typically can not make specific utility functions that precisely describe individual preferences . Probably none of us could describe our own preferences with a single equation . But as long as consumers in general have preferences that follow our basic assumptions , we can do a pretty good job finding utility functions that match consumption data . We will see evidence of this later in the course . Table es of references and the functions that them Preferences Utility function Type of utility function Love or well behaved Love or well behaved Love or well behaved Natural log Perfect complements Min function Perfect substitutes A Additive RELATING UTILITY FUNCTIONS AND INDIFFERENCE CURVE MAPS Learning Objective Explain how to derive an indifference curve from a utility function . Indifference curves and utility functions are directly related . In fact , since indifference curves represent preferences graphically and utility functions represent preferences mathematically , it follows that curves can be derived from utility functions . In functions , the dependent variable is plotted on the vertical axis , and the independent variable is plotted on the horizontal axis , like the graph of In contrast , graphs of bivariate functions are , like ( A , Figure shows a graph of A . graphs are useful for understanding how utility increases with the increased sumption of both A and .

UTILITY 23 Figure . latex Figure clearly shows the assumption that consumers have a preference for variety . Each bundle , which contains a specific amount of A and , represents a point on the surface . The vertical height of the surface represents the level of utility . By increasing both A and , a consumer can reach higher points on the surface . So where do indifference curves come from ?

Recall that an indifference curve is a collection of all bundles that a consumer is indifferent about with respect to which one to consume . Mathematically , this is equivalent to saying all bundles , when put into the utility function , return the same functional value . So if we set a value for utility , and find all the bundles of A and that generate that value , we will define an indifference curve . Notice that this is equivalent to finding all the bundles that get the consumer to the same height on the surface in . Indifference curves are a representation of elevation ( utility level ) on a flat surface . In this way , they are analogous to a contour line on a topographical map . By taking the graph back to A , can show the contour lines indifference curves that resent different elevations or utility levels . From the graph in figure , you can already see how this utility function yields indifference curves that are or concave to the origin .

24 PATRICK EMERSON So indifference curves follow directly from utility functions and are a useful way to represent utility functions in a graph . 24 FINDING MARGINAL UTILITY AND MARGINAL RATE OF SUBSTITUTION Learning Objective Derive marginal utility and for typical utility functions . Marginal utility ( MU ) is the additional utility a consumer receives from consuming one additional unit of a good . Mathematically , we express this as AU or the change in utility from a change in the amount of A consumed , where A represents a change in the value of the item , so AU ( A AA , A , AA AA Note that when we are examining the marginal utility ofthe consumption of A , we hold constant . Using calculus , the marginal utility is the same as the partial derivative of the utility function with respect to A ( A , BA Consider a consumer who sits down to eat a meal of salad and pizza . Suppose that we hold the amount of salad side salad with a dinner , for example . Now let increase the slices of pizza pose with one slice , utility is ten with two , it is eighteen with three , it is and with four , it is . Let plot these numbers on a graph that has utility on the vertical axis and pizza on the horizontal axis ( table and ) Marginal utility

UTILITY 25 Utility 10 I I I I I I Slices of Pizza Figure Diminishing From the positive slope of the graph , we can see the increase in utility from additional slices of pizza . From the concave shape of the graph , we can see another common phenomenon the additional utility the consumer receives from each additional slice of pizza decreases with the number of slices consumed . The fact that the additional utility gets smaller with each additional slice of pizza is called the principle of diminishing marginal utility . This principle applies to preferences where mixed are preferred . Marginal rate of substitution ( is the amount of one good a consumer is willing to give up to get one more unit of another good . This is why it is the same thing as the slope ofthe indifference we keep satisfaction level constant , we stay on the same indifference curve , just moving along it as we trade one good for another . How much ofone you are willing to trade for one more of another depends on the marginal utility of each . Using our previous example , if by consuming one more side salad , your utility goes up by ten , then at a current consumption of four slices of pizza , you could give up two slices of pizza and go from eight to eighteen . Ten more from salad and ten fewer by giving up two slices of pizza leave overall utility we must still be on the same indifference curve . As you move along the indifference curve , you must be riding the is , you must be giving up the good on the vertical axis for more of the good on the horizontal axis , which yields a negative rise over a positive run . We can go directly from marginal utility by recognizing the connection between the two . In our case , for the utility function ( A , is represented as MU A Note that when we substitute , we can simplify the equation AU AB AA AB

26 PATRICK EMERSON Inserting the calculus it equates to ( A , A , BEE THE POLICY EXAMPLE IS A TAX CREDIT ON HYBRID CAR PURCHASES THE GOVERNMENT BEST CHOICE TO REDUCE FUEL CONSUMPTION AND CARBON EMISSIONS ?

We determined in chapter that the relevant consumer decision between more miles driven and other consumption probably conforms to the standard assumptions about consumer choice . Therefore , using the utility function to represent a consumer who likes to drive a car as well as consume other goods and who sees them as a ( money spent on gas is money not spent on other goods ) is a good choice . It also has the benefits of both conforming to the assumptions and being flexible ( where miles driven and other consumption . In fact , the function itself can be taken to data , where the parameters can be estimated for this market , the market for miles driven in the consumer car . Policy Other Consumption Miles Driven Figure Graph curves for the policy example

UTILITY 27 EXPLORING THE POLICY QUESTION would other preference types be more appropriate in this example ?

What would have to be true for perfect complements to be the appropriate preference type to use to analyze this policy ?

What would have to be true for perfect substitutes ?

Given that we are considering a cal consumer who drives , is it appropriate to choose a typical utility function ?

Are wejust guessing , or do we have some basis in theory to support our choice of behaved preferences or a utility function ?

REVIEW TOPICS AND RELATED LEARNING OUTCOMES Utility Functions Learning Objective Describe a utility function . Utility Functions and Typical Preferences Learning Objective Identify utility functions based on the typical preferences they represent , Relating Utility Functions and Indifference Curve Maps Learning Objective Explain how to derive an indifference curve from a utility function . Finding Marginal Utility and Marginal Rate of Substitution Learning Objective Derive marginal utility and for typical utility functions . Policy Example Is a Tax Credit on Hybrid Car Purchases the Government Best Choice ?

Learning Objective The Hybrid Tax Credit and customer utility LEARN KEY TOPICS Terms Utility The number that a utility function assigns to a bundle . functions The dependent variable is plotted on the vertical axis and the independent variable is plotted on the horizontal axis , ie , the graph of ( a ) mac be

28 PATRICK EMERSON Bivariate functions Functions containing two potentially independent variables . If one variable is another variable , then you will have bivariate data that has an independent and dependent variable . Useful for understanding how utility increases with the increased consumption of both A and , the graph of . Ordinal A function that deals in ranking as opposed to pure they only indicate which bundle is better , more preferred , but not how much better it is than another bundle . After application of positive monotonic transformation to an ordinal function , meaning the ordering is not changed , the new function will represent the same preferences . If an ordinal function is reported in , regardless of the amounts used there is no relationship implied , bundle A having 10 and bundle having 20 does not imply bundle is two times as preferred as bundle A , only that it is more preferred . Cardinal The utility of the product measured with the help of the product weight , length , temperature , etc . A unit used to transform the logical utility of a product to empirical . The ordinal utility might say that the consumer prefers the apple to the orange . Cardinal utility might say that the apple provides 80 while the orange only provides 40 . Contour line A way to represent a topographical map in a space . Diminishing marginal utility The additional utility a customer receives from a product diminishes with each additional iteration of the product the customer receives . Example Pizza slices Marginal ut ty 24 28 As a customer consumes more and more slices of pizza , the additional utility having one more slice of pizza excess hunger , producing dopamine from a delicious , with each additional slice . Graph utility function and contour line

UTILITY Figure A graph showing a graph of equals AA times to the half power Equations 29 utility function A utility function describing goods which are neither complements nor substitutes . A , A Perfect complement utility function A utility function that describes goods that are perfect complements . See table for more details . A , A , Perfect substitute utility function A utility function that describes goods that are perfect substitutes . See table for more details . A , A Marginal rate of substitution

30 PATRICK EMERSON as the amount of one good a consumer is willing to give up to get one more unit of another good . From There is a direct connection between marginal utility and the marginal rate of substitution . This connection varies somewhat by utility function . From our basic utility function , is represented as Substitution results in the following TA AB AA AB Utilizing calculus ( A , A , Marginal utility AA The additional utility a consumer receives from consuming one additional unit of a good . Utility function Functions that rank bundles of consumption goods by assigning a number to each , where larger numbers indicate preferred bundles . Table Summarizing Utility Functions Table es of references and the functions that them Preferences Utility function Type of utility function Love of variety or well behaved Love or well behaved Love or well behaved Natural log function A Additive