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 you’ll either consume all ice cream or all olives. Thus the two choices for the optimal consumption bundles will be x1 = m/p??, x2 = 0, or x1 = 0, x2 = m/p??.
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 doesn’t induce any change in demand.
6 Demand
1. 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.
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2. Show that perfect substitutes are an example of homothetic preferences.
6.2. The utility function for perfect substitutes is u( x??, x??)= x?? + x??. Thus if u( x??, x??) >u ( y??, y??), we have x?? + x??> y?? + y??. It follows that t x?? + t x?? > t y?? + ty??, so that u(t x??,t x??) >u (t y??, ty??).
3. Show that Cobb-Douglas preferences are homothetic preferences. 6.3. The Cobb-Douglas utility function has the property that u(t x??,t x??)=( t x??)a( t x??)1?a = tat1?a x??a x??1?a 2 = t x??a x??1?a2 = t*u(x1, x??). Thus if u( x??, x??) >u ( y??, y??), we know that u(t x??,t x??) >u (t y??,t y??), so that Cobb-Douglas preferences are indeed homothetic.
4. The income o?er curve is to the Engel curve as the price o?er 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?
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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 Preference
1. 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 =1×1+2×2=5> 4=1×2+2×1=p1y1 + p2y2 and
q1y1 + q2y2 =2×2+1×1=5> 4=2×1+1×2=q1x1 + q2x2.
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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 a?ordable 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
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base year, for all price changes? 7.5. Perfect complements.
8 Slutsky Equation
1. 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 e?ect?
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 e?ect.
2. Suppose that preferences are concave. Is it still the case that the substitution e?ect 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 e?ect would cancel out. All that would be left would be the pure substitution e?ect, which would automatically be
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