Problem 1. Bob and Bette (rhymes with \jetty") each have Cobb-Douglas preferences for cheese, C, and peanut butter, P (each of which will be measured in ounces1), de ned by the utility function U(C; P) = C P(1 ). However, their preferences are not identical, because
they have di erent values of . Bob's preferences are de ned by Bob = 23 , while Bette has Bette = 13 .
(a) Let's begin by getting a graphical picture of their preferences by plotting their indifference curves.
(i) Compute the formula for Bob's indi erence curves by setting U(C; P) = k for some constant, k, and solving for P as a function of C. [Remember that indi erence curves are just sets of all bundles of cheese and peanut butter that give the same constant|utility.
(ii) Plot two or three of Bob's indi erence curves for di erent values of k on a neat and clear graph. [Your graph does not need to be perfectly precise, but it should be as neat as possible, and it should give a reasonably accurate representation of the shapes of the indi erence curves. For a free, online graphing tool
(iii) Repeat steps (i) and (ii) for Bette's indi erence curves, plotting them on a separate graph.
(iv) Who likes cheese better, and who likes peanut butter better? Explain how you can determine this by looking at the indi erence curves. Also explain how you could have determined it just by looking at their utility functions.
(b) Now let's get more precise about Bob and Bette's likes and dislikes by looking at their MRS's.
(i) Compute Bob's MRS as a function of C and P. Explain what this term means economically, and what it tells us about Bob's preferences, and about his willingness to trade peanut butter for cheese and vice versa.
(ii) Do the same for Bette.
(iii) For any given bundle of cheese and peanut butter, how much more willing is Bob to trade peanut butter for cheese than Bette? How much more willing is Bette to trade cheese for peanut butter than Bob?
(c) Now suppose that Bob has three ounces of cheese and three ounces of peanut butter, and Bette has exactly the same bundle. (We'll assume they both have plenty of crackers.)
(i) How much peanut butter would Bob be willing to give up for one more ounce of cheese, starting from this bundle? Would he be willing to make a one-for-one trade where he gave up one ounce of peanut butter for one ounce of cheese?
(ii) How much cheese would Bette be willing to give up for one more ounce of peanut butter? Would she be willing to make a one-for-one trade where she gave up one ounce of cheese for one ounce of peanut butter?
(iii) Show that both Bob and Bette would be made better o in terms of utility if they made a one-for-one trade in which Bob gave Bette one ounce of peanut butter and Bette gave Bob one ounce of cheese. [Hint: you will need to compute their actual utility.]
(iv) Compute Bob and Bette's MRS's at their new bundles. Could both of them be made any better o if they continued to trade? Explain what economists mean when they say that free trade can make both parties better.
Problem 2. Let's take a look at how we might model the e ect of increased income (or wealth) on people's preferences for di erent kinds of goods. We'll start with a consumer named Florence, and we'll investigate her preferences for food versus all other goods, and her preferences for vacation travel versus all other goods. In other words, we are going to separately look at how Florence feels about trade-o s between food and money, and how she feels about trade-o s between travel and money, but not directly at how she feels about trade-o s between food and travel. This is obviously somewhat unrealistic, because most people probably take their food expenditures into account when deciding how much travel they can a ord, but this is the kind of simpli cation that economic analysts make all the time, and in many cases it is \good enough".
(a) Suppose Florence has Cobb-Douglas preferences over travel, T, measured in weekend trips, and all other goods, Y , measured in thousands of dollars, represented by the utility function U(T; Y ) = T
(i) Compute Florence's MRS of all other goods for travel. (In other words, compute her MRS with travel on the horizontal axis.
(ii) Let's look at a single year in Florence's life. Suppose that her current plan is to consume ve weekend trips and fty thousand dollars worth of other goods this year. What is her willingness to pay for additional units of travel?
(iii) Now, suppose we gave Florence $20,000 worth of non-travel goods, so that her new consumption bundle is ve weekend trips and seventy thousand dollars worth of other goods. This is e ectively the same thing as increasing her income. What is her willingness to pay for additional units of travel now? Compare it to her original willingness to pay. Has it gone up, gone down, or stayed the same?
(iv) Does the e ect of increased income on Florence's willingness to pay for travel seem realistic? Comment on the appropriateness of using a Cobb-Douglas utility function to analyze her travel decisions.
(b) Next, suppose Florence has quasilinear preferences over food, F, measured in pounds of food2, and all other goods, Y , measured in thousands of dollars, represented by the utility function U(F; Y ) = ln F + Y .
(i) Compute Florence's MRS of money for food. (In other words, compute her MRS with food on the horizontal axis.)
(ii) Once again, let's look at a single year in Florence's life. Suppose that she is currently planning to consume two thousand pounds of food and fty thousand dollars worth of other goods. What is her willingness to pay for additional units of food?
(iii) Once again, suppose we gave Florence $20,000 worth of non-food goods, so that her new consumption bundle is two thousand pounds of food and seventy thousand dollars worth of other goods. Again, we have e ectively increased her income. What is her willingness to pay for additional units of food now? Compare it to her original willingness to pay. Has it gone up, gone down, or stayed the same?
(iv) Does the e ect of increased income on Florence's willingness to pay for food seem realistic? Comment on the appropriateness of using a quasilinear utility function to analyze her food decisions.
(c) To convince yourself you fully understand what is going on here, a graph may be useful.
(i) Graph two or more of Florence's indi erence curves for travel and money. Draw a dashed vertical line at ve weekend trips. Graphically show her MRS at ve weekend trips on the two indi erence curves. What has happened to her MRS as she has gotten richer? [Your graph should be neat and clear, but does not need to be precise. There's a handy graph of Cobb-Douglas indi erence curves on page 64 of Varian.]
(ii) Graph two or more of Florence's indi erence curves for food and money. Draw a dashed vertical line at two thousand pounds of food. Graphically show her MRS at two thousand pounds of food on the two indi erence curves. What has happened to her MRS as she has gotten richer?
(iii) If you were a consultant, and a client asked you to analyze consumer decisions about di erent kinds of goods, for what kinds of goods might you choose to use a Cobb-Douglas model of preferences, and for what other kinds of goods might you choose to use a quasilinear model of preferences?