### Compute the formula for bobs indi erence curves

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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) = CαP(1- α ). However, their preferences are not identical, because they have di erent values of . Bob's preferences are de ned by αBob = 2/3 , while Bette has α Bette = 1/3 .

(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 indifference curves are just sets of all bundles of cheese and peanut butter that give the same|i.e.constant|utility. If you're not sure how to do this, reread section 4.3 of Varian.]

(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.

(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.

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