There are 9 questions on this exam. These 9 questions are independent of each other.

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1 Economics 21: Microeconomics (Summer 2001) Midterm Exam 1 Professor Andreas Bentz instructions You can obtain a total of 100 points on this exam. Read each question carefully before answering it. Do not use any books or notes. The use of non-programmable calculators, or the use of only the non-programming and non-graphing functions of a programmable calculator, is permitted. You have 65 minutes to answer all questions. Please use the space provided to answer questions; if you need more space, use the back of the page. There are 9 questions on this exam. These 9 questions are independent of each other. Good luck. Name: questions (5 points) There are only two goods a student can consume: beer (b) and textbooks (t). Econometric estimates have shown that a typical student spends 10% of his income on beer. Write down this student's utility function. The Cobb-Douglas utility function has the property that the demand function exhibits "constant expenditure shares." That is, for a Cobb-Douglas utility function of the form x 1 a x 2 1-a, the demand function for good 1 is x 1 = am/p 1, that is the individual spends a fraction a of her income on good 1. So a good model of the student's utility function would be: u(b, t) = b 0.1 t 0.9, or any positive monotonic transformation thereof. 2. (7 points) A consumer has the quasilinear utility function (defined over good 1, x 1, and good 2, x2) of the following form: u(x 1, x 2 ) = x 2 + x 1. The consumer has wealth of $100,000. Suppose the price of good 2 is $1, and the price of good 1 initially is $1 also. Now the price of good 1 increases to $2. How big is the income effect (on your consumption of good 1) of that price increase? This is a quasilinear utility function. As you know from the workout, quasilinear utility functions (quasilinear in good 1) have a vertical Engel curve for good 1. There is therefore no change in consumption of good 1 as income changes: the income effect is zero. page 1 of 5

2 3. The following utility function represents Mona Tonic's preferences over good 1 (x 1 ) and good 2 (x 2 ): u(x 1, x 2 ) = ln[5(x 1 1/2 x 2 1/2 + 1)]. Good 1 costs p 1 per unit and good 2 costs p 2 per unit. Mona has wealth m. (a) (11 points) Write down the Lagrangean for her constrained utility maximization problem, and derive the first-order necessary conditions for maximization. Then derive her demand functions for good 1 and for good 2. Your answer will be much simpler if you realize that the utility function given in the question is just a positive monotonic transformation of a utility function x 1 1/2 x 2 1/2 (this is also where Mona's name comes from). Since any positive monotonic transformation of a utility function represents the same preferences, you can just work the problem using the utility function x 1 1/2 x 2 1/2. This is just the Cobb-Douglas utility function we used in class, and the Lagrangean and first-order conditions are on the class handouts (topic 1, slide 56). The demand function for good 1 is, of course, x 1 = (1/2)(m/p 1 ), and the demand function for good 2 is x 2 = (1/2)(m/p 2 ). (b) (6 points) Suppose Mona's demand function for good 1 is x 1 = (1/2)(m/p 1 ). Is this good an inferior good? Why or why not? Good 1 is not an inferior good since the derivative of demand with respect to income is 1/(2p 1 ), which is positive. (c) (6 points) Suppose Mona's demand function for good 1 is x 1 = (1/2)(m/p 1 ). Further suppose, for this part of the question only, that Mona's wealth is $100, and the current price of good 1 is $5. What is Mona's price elasticity of demand? Price elasticity of demand is (dx 1 /dp 1 )(p/x) = -(1/2)(m/p 1 2 ) {p 1 /((1/2)(m/p 1 ))]} = -1. page 2 of 5

3 4. Suppose a consumer with wealth m and facing prices p 1 for good 1 and p 2 for good 2, has the following demand function for good 1: x 1 = m a p 1 + bp 2. (Of course p 1, p 2, and m are all positive.) For which values of the parameters a and b is good 1 (if you think any value of a parameter is admissible, write "this parameter can be anything"): (a) (6 points) a normal good? dx 1 /dm = am a-1 p 1 we need this > 0 for it to be normal, so a > 0, and b can be anything (b) (6 points) a gross substitute for good 2? dx 1 /dp 2 = b so we need this > 0 for substitutes, so b > 0, and a can be anything 5. (11 points) Anne Themme can consume two goods: c g and c b (you can think of the good c g as consumption when Anne is healthy, that is the "g"ood thing happens; and think of the good c b as consumption when Anne is sick, that is the "b"ad thing happens). Her utility function is u(c g, c b ) = (1-π) ln(c g ) + π ln(c b ), where π is a number between 0 and 1 (you can think of π as the probability with which Anne gets sick). She has the following budget constraint: π m b + (1-π) m g = π c b + (1-π) c g. (You may think of m g as Anne's income when she is healthy, and of m b as her income when she is sick. Again, both m g and m b are just numbers.) How much of good c g and good c b will Anne choose to consume? Write the Lagrangean: L(c g, c b, λ) = (1-π) ln(c g ) + π ln(c b ) - λ[π c b + (1-π) c g - π m b - (1-π) m g ] The first-order conditions are: (i) (ii) (iii) (1-π)/c g = λ(1-π) π/c b = λπ π m b + (1-π) m g = π c b + (1-π) c g Combining (i) and (ii) yields c g = c b. Putting this into (iii) gives us: c g = c b = π m b + (1-π) m g. (Anne Themme is called that because Dartmouth's employee health plan is with Anthem BC/BS.) page 3 of 5

4 6. (15 points) "For an inferior (but not a Giffen) good, and a price fall, the Slutsky income effect is greater than the Hicks income effect." True or false? Draw a diagram to give your reasoning. The Hicks income effect is greater (in absolute terms) than the Slutsky income effect. See below: original consumption final cons. Slutsky Hicks page 4 of 5

5 7. (11 points) A monopolist knows that the inverse market demand curve for its product (the quantity of the product is x) is p = 100-2x. Currently, the monopolist charges a price of $50. The monopolist thinks about increasing output slightly. Would this increase or decrease the monopolist's revenue? At what rate would revenue change? Revenue is (100-2x)x = 100x - 2x 2. Marginal revenue is the derivative of this w.r.t. x, that is MR = 100-4x. If she charges $50 right now, she sells quantity x = 50 - (1/2)p = 25. Putting this into MR, you get MR = 0. It would neither increase or decrease. 8. (7 points) A bond with a face value of $100, a coupon of $10 and a maturity date of July 11, 2002 (i.e. you do not get this year's (July 11, 2001) coupon payment) currently trades for $105. What is therefore the interest rate that the buyers and sellers of this bond use to discount future payments? (This is sometimes called the "yield" of a bond.) The bond's present value is $110/(1+r). This is the price at which it should trade. It trades at $105, so we know $110/(1+r) = $105, or (110/105) - 1 = r. That is, r = 0.048, or 4.8%. 9. (7 points) You have a piece of real estate that can be used in only one of two possible ways. You can either harvest the timber that grows on the land. This would give you a profit of $10,000 per year (from next year on), for ever. Or, you could build a shopping mall on the land. The mall costs $180,000 to build (today), and yields $24,000 per year (from next year on), for ever. The interest rate is 10%. What will you do? If you were to sell the land, how much would you be able to sell it for? PV logging = $10,000/0.1 = $100,000 PV mall = -$180,000 + $24,000/0.1 = -$180,000 + $240,000 = $60,000 Logging has the higher present value. So the land is worth $100,000 - which is just what you should be able to sell it for. page 5 of 5