3.4. It is transitive, but not complete. What if A were bigger but slower than B? Which one would he prefer?
5. Can an indi?erence curve cross itself? For example, could Figure 3.2 depict a single indi?erence curve?
3.5. Yes. An indi?erence curve can cross itself, it just can’t cross another distinct indi?erence curve.
6. Could Figure 3.2 be a single indi?erence curve if preferences are monotonic?
3.6. No, because there are bundles on the indi?erence curve that have strictly more of both goods than other bundles on the (alleged) indi?erence curve.
7. If both pepperoni and anchovies are bads, will the indi?erence curve have a positive or a negative slope?
3.7. A negative slope. If you give the consumer more anchovies, you’ve made him worse o?, so you have to take away some pepperoni to get him back on his indi?erence curve. In this case the direction of increasing utility is toward the origin.
8. Explain why convex preferences means that “averages are preferred
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to extremes.”
3.8. Because the consumer weakly prefers the weighted average of two bundles to either bundle.
9. What is your marginal rate of substitution of $1 bills for $5 bills? 3.9. If you give up one $5 bill, how many $1 bills do you need to compensate you? Five $1 bills will do nicely. Hence the answer is ?5 or?1/5, depending on which good you put on the horizontal axis.
10. If good 1 is a “neutral,” what is its marginal rate of substitution for good 2?
3.10. Zero—if you take away some of good 1, the consumer needs zero units of good 2 to compensate him for his loss. ANSWERS A13
11. Think of some other goods for which your preferences might be concave.
3.11. Anchovies and peanut butter, scotch and Kool Aid, and other similar repulsive combinations.
4 Utility
1. The text said that raising a number to an odd power was a monotonic transformation. What about raising a number to an even
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power? Is this a monotonic transformation? (Hint: consider the case f(u)=u^2.)
4.1. The function f(u)=u^2 is a monotonic transformation for positive u, but not for negative u.
2. Which of the following are monotonic transformations?
(1) u =2 v?13; (2) u = ?1/v^2; (3)u =1/v^2; (4)u = ln v; (5)u = ?e^?v; (6)u = v^2; (7) u = v^2 for v>0; (8) u = v^2 for v<0.
4.2. (1) Yes. (2) No (works for v positive). (3) No (works for v negative). (4) Yes (only de?ned for v positive). (5) Yes. (6) No. (7) Yes. (8) No.
3. We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indi?erence curve exactly once. Can you prove this rigorously? (Hint: what would happen if it intersected some indi?erence curve twice?)
4.3. Suppose that the diagonal intersected a given indi?erence curve at two points, say (x,x) and (y,y). Then either x>y or y>x, which means that one of the bundles has more of both goods. But if preferences are monotonic, then one of the bundles would have to be preferred to the other.
4. What kind of preferences are represented by a utility function of the
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form u(x1,x2)= ???? + ????? What about the utility function v(x1,x2)= 13x1 + 13x2?
4.4. Both represent perfect substitutes.
5. What kind of preferences are represented by a utility function of the form u(x1,x2)=x1 + ????? Is the utility function v(x1,x2)=x2 1 +2x1 ???? +x2 a monotonic transformation of u(x1,x2)? 4.5. Quasilinear preferences. Yes.
6. Consider the utility function u(x1,x2)= ???? ???? . What kind of pref- erences does it represent? Is the function v(????, ????) = ????????????a monotonic transformation of u(????, ????)? Is the function w(????, ????) = ???????????? a monotonic transformation of u(????, ????)?
4.6. The utility function represents Cobb-Douglas preferences. No. Yes.
7. Can you explain why taking a monotonic transformation of a utility function doesn’t change the marginal rate of substitution?
4.7. Because the MRS is measured along an indi?erence curve, and utility remains constant along an indi?erence curve.
5 Choice
1. If two goods are perfect substitutes, what is the demand function for good 2?
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5.1. ????=0 when p2>p??, ???? = m/p?? when p?? 2. Suppose that indi?erence curves are described by straight lines with a slope of ?b. Given arbitrary prices and money income p1, p2, and m, what will the consumer’s optimal choices look like? 5.2. The optimal choices will be x1 = m/p1 and x2 = 0 ifp1/p2 b, and any amount on the budget line if p1/p2 = b. 3. Suppose that a consumer always consumes 2 spoons of sugar with each cup of co?ee. If the price of sugar is p1 per spoonful and the price of co?ee is p2 per cup and the consumer has m dollars to spend on co?ee and sugar, how much will he or she want to purchase? 5.3. Let z be the number of cups of co?ee the consumer buys. Then we know that 2z is the number of teaspoons of sugar he or she buys. We must satisfy the budget constraint 2p??z + p??z = m. Solving for z we have z =. 4. Suppose that you have highly nonconvex preferences for ice cream 10 m 2p??+ p??