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 ) = T1/5 Y4/5.
(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 . [To remind yourself what you know about quasilinear preferences, review section 4.3 of Varian.]
(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?