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ANSWERS1 The Market1. Suppose that there were 25 people who had a reservation price of $500, and the 26th person had a reservation price of $200. What would the demand curve look like?1.1. It would be constant at $500 for 25 apartments and then drop to $200.2. In the above example, what would the equilibrium price be if there were 24 apartments to rent? What if there were 26 apartments to rent? What if there were 25 apartments to rent?1.2. In the rst case, $500, and in the second case, $200. In the third case, the equilibrium price would be any price between $200 and $500.3. If people have dierent reservation prices, why does the market demand curve slope down?1.3. Because if we want to rent one more apartment, we have to oer a lower price. The number of people who have reservation prices greater than p must always increase as p decreases.4. In the text we assumed that the condominium purchasers came from the inner-ring peoplepeople who were already renting apartments. What would happen to the price of inner-ring apartments if all of the condominium purchasers were outer-ring peoplethe people who were not currently renting apartments in the inner ring?1.4. The price of apartments in the inner ring would go up since demand for apartments would not change but supply would decrease.5. Suppose now that the condominium purchasers were all inner-ring people, but that each condominium was constructed from two apartments. What would happen to the price of apartments?1.5. The price of apartments in the inner ring would rise.6. What do you suppose the eect of a tax would be on the number of apartments that would be built in the long run?1.6. A tax would undoubtedly reduce the number of apartments supplied in the long run.7. Suppose the demand curve is D(p) = 1002p. What price would the monopolist set if he had 60 apartments? How many would he rent? What price would he set if he had 40 apartments? How many would he rent?1.7. He would set a price of 25 and rent 50 apartments. In the second case he would rent all 40 apartments at the maximum price the market would bear. This would be given by the solution to D(p) = 1002p = 40, which is p = 30.8. If our model of rent control allowed for unrestricted subletting, who would end up getting apartments in the inner circle? Would the outcome be Pareto ecient?1.8. Everyone who had a reservation price higher than the equilibrium price in the competitive market, so that the nal outcome would be Pareto ecient. (Of course in the long run there would probably be fewer new apartments built, which would lead to another kind of ineciency.)2 Budget Constraint1. Originally the consumer faces the budget line p1x1 + p2x2 = m. Then the price of good 1 doubles, the price of good 2 becomes 8 times larger, and income becomes 4 times larger. Write down an equation for the new budget line in terms of the original prices and income.2.1. The new budget line is given by 2p1x1 +8p2x2 =4m.2. What happens to the budget line if the price of good 2 increases, but the price of good 1 and income remain constant?2.2. The vertical intercept (x2 axis) decreases and the horizontal intercept (x1 axis) stays the same. Thus the budget line becomes atter.3. If the price of good 1 doubles and the price of good 2 triples, does the budget line become atter or steeper?2.3. Flatter. The slope is 2p1/3p2. 4. What is the denition of a numeraire good?2.4. A good whose price has been set to 1; all other goods prices are measured relative to the numeraire goods price.5. Suppose that the government puts a tax of 15 cents a gallon on gasoline and then later decides to put a subsidy on gasoline at a rate of 7 cents a gallon. What net tax is this combination equivalent to?2.5. A tax of 8 cents a gallon. 6. Suppose that a budget equation is given by p1x1 +p2x2 = m. The government decides to impose a lump-sum tax of u, a quantity tax on good 1 of t, and a quantity subsidy on good 2 of s. What is the formula for the new budget line?2.6. (p1+ t) x1 +(p2s) x2 = mu. 7. If the income of the consumer increases and one of the prices decreases at the same time, will the consumer necessarily be at least as well-o?2.7. Yes, since all of the bundles the consumer could aord before are aordable at the new prices and income.3 Preferences1. If we observe a consumer choosing (x1, x2) when (y1, y2) is available one time, are we justied in concluding that (x1, x2) (y1, y2)?3.1. No. It might be that the consumer was indierent between the two bundles. All we are justied in concluding is that (x1, x2) (y1, y2). 2. Consider a group of people A, B, C and the relation “at least as tall as,” as in “A is at least as tall as B.” Is this relation transitive? Is it complete?3.2. Yes to both.3. Take the same group of people and consider the relation “strictly taller than.” Is this relation transitive? Is it reexive? Is it complete?3.3. It is transitive, but it is not completetwo people might be the same height. It is not reexive since it is false that a person is strictly taller than himself.4. A college football coach says that given any two linemen A and B, he always prefers the one who is bigger and faster. Is this preference relation transitive? Is it complete?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 indierence curve cross itself? For example, could Figure 3.2 depict a single indierence curve?3.5. Yes. An indierence curve can cross itself, it just cant cross another distinct indierence curve.6. Could Figure 3.2 be a single indierence curve if preferences are monotonic?3.6. No, because there are bundles on the indierence curve that have strictly more of both goods than other bundles on the (alleged) indierence curve.7. If both pepperoni and anchovies are bads, will the indierence curve have a positive or a negative slope?3.7. A negative slope. If you give the consumer more anchovies, youve made him worse o, so you have to take away some pepperoni to get him back on his indierence curve. In this case the direction of increasing utility is toward the origin.8. Explain why convex preferences means that “averages are preferred 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 or1/5, 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. Zeroif you take away some of good 1, the consumer needs zero units of good 2 to compensate him for his loss.ANSWERS A1311. 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 Utility1. The text said that raising a number to an odd power was a monotonic transformation. What about raising a number to an even power? Is this a monotonic transformation? (Hint: consider the case f(u)=u2.)4.1. The function f(u)=u2 is a monotonic transformation for positive u, but not for negative u.2. Which of the following are monotonic transformations? (1) u =2 v13; (2) u = 1/v2; (3)u =1/v2; (4)u = ln v; (5)u = ev; (6)u = v2; (7) u = v2 for v0; (8) u = v2 for vy or yx, 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 form u(x1,x2)=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 +x2? Is the utility function v(x1,x2)=x2 1 +2x1x2 +x2 a monotonic transformation of u(x1,x2)?4.5. Quasilinear preferences. Yes.6. Consider the utility function u(x1,x2)= x1 x2 . What kind of pref- erences does it represent? Is the function v(x1, x2) = x12x22a monotonic transformation of u(x1, x2)? Is the function w(x1, x2) = x12x22 a monotonic transformation of u(x1, x2)?4.6. The utility function represents Cobb-Douglas preferences. No. Yes.7. Can you explain why taking a monotonic transformation of a utility function doesnt change the marginal rate of substitution?4.7. Because the MRS is measured along an indierence curve, and utility remains constant along an indierence curve.5 Choice1. If two goods are perfect substitutes, what is the demand function for good 2?5.1. x2=0 when p2p1, x2 = m/p2 when p2p1, and anything between 0 and m/p2 when p1 = p2.2. Suppose that indierence curves are described by straight lines with a slope of b. Given arbitrary prices and money income p1, p2, and m, what will the consumers 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 coee. If the price of sugar is p1 per spoonful and the price of coee is p2 per cup and the consumer has m dollars to spend on coee and sugar, how much will he or she want to purchase?5.3. Let z be the number of cups of coee the consumer buys. Then we know that 2z is the number of teaspoons of sugar he or she buys. We must satisfy the budget constraint2p1z + p2z = m.Solving for z we havez =m2p1+ p2 .4. Suppose that you have highly nonconvex preferences for ice cream and olives, like those given in the text, and that you face prices p1, p2 and have m dollars to spend. List the choices for the optimal consumption bundles.5.4. We know that youll either consume all ice cream or all olives. Thus the two choices for the optimal consumption bundles will be x1 = m/p1, x2 = 0, or x1 = 0, x2 = m/p2.5. If a consumer has a utility function u(x1,x2)=x1x4 2, what fraction of her income will she spend on good 2?5.5. This is a Cobb-Douglas utility function, so she will spend 4/(1 + 4) = 4/5 of her income on good 2.6. For what kind of preferences will the consumer be just as well-o facing a quantity tax as an income tax?5.6. For kinked preferences, such as perfect complements, where the change in price doesnt induce any change in demand.6 Demand1. If the consumer is consuming exactly two goods, and she is always spending all of her money, can both of them be inferior goods?6.1. No. If her income increases, and she spends it all, she must be purchasing more of at least one good.2. Show that perfect substitutes are an example of homothetic preferences.6.2. The utility function for perfect substitutes is u( x1, x2)= x1 + x2. Thus if u( x1, x2) u ( y1, y2), we have x1 + x2 y1 + y2. It follows that t x1 + t x2 t y1 + ty2, so that u(t x1,t x2) u (t y1, ty2).3. Show that Cobb-Douglas preferences are homothetic preferences.6.3. The Cobb-Douglas utility function has the property that u(t x1,t x2)=( t x1)a( t x2)1-a = tat1-a x1a x21-a 2 = t x1a x21-a2 = t*u(x1, x2). Thus if u( x1, x2) u ( y1, y2), we know that u(t x1,t x2) u (t y1,t y2), so that Cobb-Douglas preferences are indeed homothetic.4. The income oer curve is to the Engel curve as the price oer curve is to .?6.4. The demand curve.5. If the preferences are concave will the consumer ever consume both of the goods together?6.5. No. Concave preferences can only give rise to optimal consumption bundles that involve zero consumption of one of the goods.6. Are hamburgers and buns complements or substitutes?6.6. Normally they would be complements, at least for non-vegetarians.7. What is the form of the inverse demand function for good 1 in the case of perfect complements?6.7. We know that x1 = m/(p1 + p2). Solving for p1 as a function of the other variables, we have p1 = m x1 p2.8. True or false? If the demand function is x1 = p1, then the inverse demand function is x = 1/p1.6.8. False.7 Revealed Preference1. When prices are (p1,p2) = (1 ,2) a consumer demands (x1,x2) = (1 ,2), and when prices are ( q1,q2) = (2 ,1) the consumer demands (y1,y2) = (2 ,1). Is this behavior consistent with the model of maximizing behavior?7.1. No. This consumer violates the Weak Axiom of Revealed Preference since when he bought (x1,x2) he could have bought (y1,y2) and vice versa. In symbols: p1x1 + p2x2 =11+22=5 4=12+21=p1y1 + p2y2andq1y1 + q2y2 =22+11=5 4=21+12=q1x1 + q2x2.2. When prices are (p1,p2) = (2 ,1) a consumer demands (x1,x2) = (1 ,2), and when prices are ( q1,q2) = (1 ,2) the consumer demands (y1,y2) = (2 ,1). Is this behavior consistent with the model of maximizing behavior?7.2. Yes. No violations of WARP are present, since the y-bundle is not aordable when the x-bundle was purchased and vice versa.3. In the preceding exercise, which bundle is preferred by the consumer, the x-bundle or the y-bundle?7.3. Since the y-bundle was more expensive than the x-bundle when the x-bundle was purchased and vice versa, there is no way to tell which bundle is preferred.4. We saw that the Social Security adjustment for changing prices would typically make recipients at least as well-o as they were at the base year. What kind of price changes would leave them just as well-o, no matter what kind of preferences they had?7.4. If both prices changed by the same amount. Then the base-year bundle would still be optimal.5. In the same framework as the above question, what kind of preferences would leave the consumer just as well-o as he was in the base year, for all price changes?7.5. Perfect complements.8 Slutsky Equation1. Suppose a consumer has preferences between two goods that are perfect substitutes. Can you change prices in such a way that the entire demand response is due to the income eect?8.1. Yes. To see this, use our favorite example of red pencils and blue pencils. Suppose red pencils cost 10 cents a piece, and blue pencils cost 5 cents a piece, and the consumer spends $1 on pencils. She would then consume 20 blue pencils. If the price of blue pencils falls to 4 cents a piece, she would consume 25 blue pencils, a change which is entirely due to the income eect.2. Suppose that preferences are concave. Is it still the case that the substitution eect is negative?8.2. Yes.3. In the case of the gasoline tax, what would happen if the rebate to the consumers were based on their original consumption of gasoline, x, rather than on their nal consumption of gasoline, x?8.3. Then the income eect would cancel out. All that would be left would be the pure substitution eect, which would automatically be negative.4. In the case described in the preceding question, would the government be paying out more or less than it received in tax revenues?8.4. They are receiving tx in revenues and paying out tx, so they are losing money.5. In this case would the consumers be better o or worse o if the tax with rebate based on original consumption were in eect?8.5. Since their old consumption is aordable, the consumers would have to be at least as well-o. This happens because the government is giving them back more money than they are losing due to the higher price of gasoline.9 Buying and Selling1. If a consumers net demands are (5,3) and her endowment is (4,4), what are her gross demands?9.1. Her gross demands are (9,1).2. The prices are (p1,p2) = (2 ,3), and the consumer is currently consuming (x1,x2) = (4 ,4). There is a perfect market for the two goods in which they can be bought and sold costlessly. Will the consumer necessarily prefer consuming the bundle (y1,y2) = (3 ,5)? Will she necessarily prefer having the bundle (y1,y2)?9.2. The bundle (y1,y2) = (3 ,5) costs more than the bundle (4,4) at the current prices. The consumer will not necessarily prefer consuming this bundle, but would certainly prefer to own it, since she could sell it and purchase a bundle that she would prefer.3. The prices are (p1,p2) = (2 ,3), and the consumer is currently consuming (x1,x2) = (4 ,4). Now the prices change to (q1,q2) = (2 ,4). Could the consumer be better o under these new prices?9.3. Sure. It depends on whether she was a net buyer or a net seller of the good that became more expensive.4. The U.S. currently imports about half of the petroleum that it uses. The rest of its needs are met by domestic production. Could the price of oil rise so much that the U.S. would be made better o?9.4. Yes, but only if the U.S. switched to being a net exporter of oil.5. Suppose that by some miracle the number of hours in the day increased from 24 to 30 hours (with luck this would happen shortly before exam week). How would this aect the budget constraint?9.5. The new budget line would shift outward and remain parallel to the old one, since the increase in the number of hours in the day is a pure endowment eect.6. If leisure is an inferior good, what can you say about the slope of the labor supply curve?9.6. The slope will be positive.10 Intertemporal Choice1. How much is $1 million to be delivered 20 years in the future worth today if the interest rate is 20 percent?10.1. According to Table 10.1, $1 20 years from now is worth 3 cents today at a 20 percent interest rate. Thus $1 million is worth .031,000,000 = $30,000 today.2. As the interest rate rises, does the intertemporal budget constraint be- come steeper or atter?10.2. The slope of the intertemporal budget constraint is equal to (1+r). Thus as r increases the slope becomes more negative (steeper).3. Would the assumption that goods are perfect substitutes be valid in a study of intertemporal food purchases?10.3. If goods are perfect substitutes, then consumers will only purchase the cheaper good. In the case of intertemporal food purchases, this implies that consumers only buy food in one period, which may not be very realistic.4. A consumer, who is initially a lender, remains a lender even after a decline in interest rates. Is this consumer better o or worse o after the change in interest rates? If the consumer becomes a borrower after the change is he better o or worse o?10.4. In order to remain a lender after the change in interest rates, the consumer must be choosing a point that he could have chosen under the old interest rates, but decided not to. Thus the consumer must be worse o. If the consumer becomes a borrower after the change, then he is choosing a previously unavailable point that cannot be compared to the initial point (since the initial point is no longer available under the new budget constraint), and therefore the change in the consumers welfare is unknown.5. What is the present value of $100 one year from now if the interest rate is 10%? What is the present value if the interest rate is 5%?10.5. At an interest rate of 10%, the present value of $100 is $90.91. At a rate of 5% the present value is $95.24.11 Asset Markets1. Suppose asset A can be sold for $11 ne
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