Problem 2. Let\'s take a look at how we might model the effect of increased inco

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Problem 2. Let's take a look at how we might model the effect of increased income (or wealth) on people's preferences for different kinds of goods. We'll start with a consumer named Alice, 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 Alice feels about trade-offs between food and money, and how she feels about trade-offs between travel and money, but not directly at how she feels about trade-offs 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 afford, but this is the kind of simplification that economic analysts make all the time, and in many cases it is "good enough". To remind yourself how we use the composite "all other goods," review section 2.2 in Varian.] (a) Suppose Alice has preferences over travel, T, measured in weekend trips, and all other goods, Y, measured in thousands of dollars, represented by the utility function (i) Compute Alice'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 Alice's life. Suppose that, given her current income, her current plan is to consume six weekend trips and sixty thousand dollars worth of other goods this year. What is her willingness to pay for additional units of travel? [To remind yourself about the concept of willingness to pay in a composite-good setting, review section 3.7 of Varian.] i) Now, suppose we gave Alice 830,000 worth of "all other goods", so that her new consumption bundle is six weekend trips and ninety thousand dollars worth of other goods. This is effectively 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 effect of increased income on Alice's illingness to pay for travel seem re- alistic? Comment on the appropriateness of using this utility function to analyze her travel decisions. (b) Next, suppose Alice has quasilinear preferences over food, F, measured in pounds of food, and all other goods, Y, measured in thousands of dollars, represented by the utility function U(F, Y) Y. To remind yourself what you know about quasilinear preferences, review section 4.3 of Varian.] (i) Compute Alice'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 Alice's life. Suppose that, given her current income, she is currently planning to consume three thousand pounds of food and sixty thousand dollars worth of other goods. What is her willingness to pay for additional units of food? (ii) Once again, suppose we gave Alice $30,000 worth of non-food goods, so that her new consumption bundle is three thousand pounds of food and ninety thousand dollars worth of other goods. Again, we have effectively 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 effect of increased income on Alice's willingness to pay for food seem realistic? Comment on the appropriateness of using a quasilinear utility function to analyze her food decisions

Explanation / Answer

(a)(i) U(T,Y)=T1/5 +Y4/5

MRS=MUT/MUY

MRS=Y1/5/4T4/5

(ii) MRS gives the marginal willingness to pay.

T=6 and Y=$60, OOO

MRS=(60,000)1/5 /4(6)4/5 =0.538

(iii) T=6 and Y=$90,000

MRS=(90,000)1/5 /4(6)4/5 =0.584

As compared to before the marginal willingness to pay has gone up.

(iv) Yes, with increase in income the increased marginal willingness to pay seems realistic as the more we have of one good the more we are willing to give some of it for the other good.

In this utility function when we increase number of weekend trips (i.e.T ) the MRS falls as denominator increases. So this implies that the utility function exhibit diminishing marginal rate of substitution. We have strictly convex IC for this utility function as MRS decreases when T increases.

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