Please do not leave the exam room within the final 15 minutes of the exam, except in an emergency.

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1 Economics 21: Microeconomics (Spring 2000) Midterm Exam 1 - Answers 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 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. Please do not leave the exam room within the final 15 minutes of the exam, except in an emergency. At the end of the exam, you must turn in your answers promptly: if you continue writing I will take off points. Good luck. Name: Class time (circle one): 9 (section 1) 10 (section 2) questions You have estimated that the market demand curve for gas looks like this: X(p) = 1,000,000 p -2, where X is the quantity of gas (in gallons), and p is the price per gallon (in dollars). a. Currently, gas costs $1.49. What is the price elasticity of demand? (5 points) answer: elasticity = [dx(p)/dp][p/x(p)] = -2,000,000 p -3 p / (1,000,000 p -2 ) = - 2 b. OPEC controls the quantity of gas on the market. If it were to increase output slightly, would OPEC's revenue increase or decrease? At what rate? (5 points) answer: MR = p[1+1/elasticity] = p 1/2 > 0. Therefore revenue would increase, at a rate of 1/2 p (here: 0.745) page 1 of 1

2 2. You are advising the government on its labor policy. From econometric estimates you know that a representative worker's utility function for the two goods general consumption (c) and leisure (l) looks like this: u(l, c) = ln(l) + c, where consumption (c) is measured in units per day and leisure (l) in hours per day, and ln is the natural logarithm. a. Every unit of consumption costs p (the price of consumption), and for every hour a worker works she receives a gross wage of w (the hourly wage). There is an income tax at the uniform rate t (that is, for every dollar she earns, she takes (1 - t) dollars home). Write down the budget constraint for this problem. (3 points) answer: pc = (24 - l) w (1-t) b. Write down the Lagrangean for the constrained maximization problem (maximize utility subject to the budget constraint). Then solve for the optimal choices of l (leisure) and c (consumption). (10 points) answer: L(l, c, λ) = ln(l) + c - λ(pc - 24 w (1-t) + l w (1-t)) FOC: (i) (ii) (iii) 1/l = λ w (1-t) 1 = λp pc = 24 w (1-t) - l w (1-t) from (ii): λ = 1/p put this into (i): 1/l = (1-t) w/p, or: l = p/[w(1-t)] work out consumption from (iii): pc = 24 w (1-t) - p/[w(1-t)] w (1-t) or: pc = 24 w (1-t) - p, or: c = 24 [w(1-t)]/p - 1 page 2 of 2

3 c. Denote by n the number of hours this worker will work (generally, it will depend on the wage, prices, and the income tax rate). Write down the labor supply function n(w, p, t). The government thinks about increasing income tax (to pay for better healthcare provision). How does this worker's labor supply respond (i.e. does it increase or decrease)? At what rate does this worker's labor supply change as income tax is increased slightly? (4 points) answer: n(w, p, t) = 24 - l = 24 - p/[w(1-t)] since dn(w, p, t)/dt = - p/w (1-t) -2, which is < 0, labor supply decreases as the tax increases, and it does so at a rate of - p/w (1-t) -2. d. Suppose the tax is 20%, the price of consumption (p) is $80 per unit, and the (gross) wage rate (w) is $10 per hour. How much will the worker work per day? What is the tax revenue to the government therefore (per day)? (2 points) answer: n(10, 80, 0.2) = 24-80/8 = 14 government makes $2 in taxes for every hour this worker works, and since she works for 14 hours, govt. makes $28. e. If the government redistributed some of the tax revenue back to the worker (as a straightforward money payment to the worker), would this worker's labor supply increase, decrease, or remain the same? Why? (4 points) answer: The worker's labor supply is independent of wealth you can see this from drawing the (quasilinear) indifference curves: you will see that there is no income effect. 3. You have the following preferences over goods 1 and 2: u(x 1, x 2 ) = c(x 1/2 1 )(x 1/2 2 ), where x 1, x 2 are the quantities of good 1 and 2, and c is a positive parameter. a. What is the marginal rate of substitution (MRS)? (4 points) answer: MRS = - MU1 / MU2 = - [c (1/2) (x -1/2 1 )(x 1/2 2 )] / [c (1/2) (x 1/2 1 ) (x -1/2 2 )] = - x 2 / x 1. b. If good 1 is twice as expensive as good 2 (its price is twice that of the price of good 2), in what ratio will you (optimally) consume goods 1 and 2? (2 points) answer: Optimality implies that MRS = p 1 / p 2. Therefore, if p 1 / p 2 = 2 (because good 1 is twice as expensive as good 2), at the optimum, x 2 / x 1 = 2: the consumer will consume twice as much of good 2 as of good 1. page 3 of 3

4 4. Your preferences over timed consumption can be represented by the following utility function: u(c 1, c 2 ) = c 1 c 2, where c 1 is consumption this period, and c 2 is consumption next period. (This consumer lives only for two periods.) In period 1, this consumer's endowment is m 1, and in period 2, m 2. There is an interest rate r at which you can borrow and save, and inflation is π. a. Write down the budget constraint for this problem. (2 points) answer: c 2 = m 2 + [(1+r)/(1+π)](m 1 - c 1 ) b. Write down (without solving it) the Lagrangean for this problem. (2 points) answer: L(c 1, c 2, λ) = c 1 c 2 - λ{c 2 - m 2 - [(1+r)/(1+π)](m 1 - c 1 )} c. The optimal demands for consumption this period and for consumption next period for the above preferences are given by: 1 + π c 1 = 0.5 m1 + m r Suppose that the interest rate is 10% and there is no inflation. Your endowment in this period (m 1 ) is 40 and your endowment (m 2 ) next period is 22. Are you a lender or a borrower? How much do you lend/save (or borrow)? (4 points) 1 answer: c 1 = = since you consume less this period than your endowment, you are a lender. You lend (save) 10. d. Suppose the inflation rate increases to 10%. How much will you lend (or borrow) now? Is this more or less than in part (c)? Is this result generally predictable, or could it theoretically have been different (if you didn't know precisely the utility function)? Explain. In your explanation, make sure you give a good intuition for wealth ("income") and substitution effects. (No calculation of these effects is required.) (8 points) answer: Put in the numbers again to obtain c 1 = 31. You now lend less (although you are still a lender). Here is the explanation: as the inflation rate increases, the real rate of interest decreases. This means that there is a substitution effect that tells you to consume more of period-1 consumption. If you are a lender, you are now less wealthy, so you will consume less of all normal goods: the income effect tells you to consume less of period-1 consumption. So the overall effect is uncertain. page 4 of 4

5 5. Your preferences over two goods can be represented by the following utility function: u(x 1, x 2 ) = x 1 x 2 + x 2, and good 1 costs p 1 per unit and good 2 costs p 2 per unit. Your wealth is m. a. Write down the Lagrangean and solve for the demand function for goods 1 and 2. (10 points) answer: L(x 1, x 2, λ) = x 1 x 2 + x 2 - λ( p 1 x 1 + p 2 x 2 - m) FOC: (i) x 2 = λp 1 (ii) x = λp 2 (iii) p 1 x 1 + p 2 x 2 = m from (i): λ = x 2 / p 1 putting this into (ii): x = x 2 (p 2 /p 1 ), and multiplying through by p 1 we get: p 1 x 1 + p 1 = p 2 x 2. Adding p 1 x 1 on both sides, we get 2p 1 x 1 + p 1 = p 2 x 2 + p 1 x 1. But from (iii), p 2 x 2 + p 1 x 1 = m, so that 2p 1 x 1 + p 1 = m. Solve for x 1 to get 0.5 (m/p 1 ) Putting this into (iii) and solving for x 2 you get x 2 = 0.5(m/p 2 ) + 0.5(p 1 /p 2 ). b. Is good 1 a normal good or an inferior good? Is it a Giffen good or a non-giffen good? For each of these, explain very briefly why. (4 points) answer: good 1 is a normal good: the partial derivative with respect to m is 0.5/p 1 which is positive. good 1 is also non-giffen: the partial derivative of demand with respect to p 1 is (m/p 1 2 ) page 5 of 5

6 c. Suppose your demand functions for goods 1 and 2 look like this: x 1 = 0.5(m/p 1 ) - 1, and x 2 = 0.5(m/p 2 ) + (p 1 /p 2 ). Further, suppose that p 1 = 11 and p 2 = 11 and your money income is m = 110. Now the price of good 1 falls to p 1 ' = 10. What is the total effect on demand of good 1 of this price fall? What size (and direction) is the substitution effect for good 1, what is the income effect for good 1 (use the Slutsky decomposition)? (10 points) Instructions: You can either use the rate-of-change [derivative] version of the Slutsky equation, or you can calculate the effects more directly. answer: First, you want to consume 4 units of good 1 and 6 units of good 2. (Just put the numbers p 1 = 11, p 2 = 11 and m = 110 into the demand functions for good 1 and good 2.) After the price change, you want to consume 4.5 units of good 1. (Just put the numbers p 1 ' = 10, p 2 = 11 and m = 110 into the demand function for good 1.) Now, in order for you still to be able to buy the bundle (4, 6) (the original bundle) after the price change, you need 10x4 + 11x6 = 106 income. If you had that income (and faced the new prices, p 1 ' = 10, p 2 = 11), you would want to consume 4.3 units of good 1. So the substitution effect tells you to consume 0.3 units more (from 4 to 4.3), and then the income effect tells you to consume 0.2 units more (from 4.3 to 4.5). 6. You are hired by Barings Bank to value financial products. Suppose you think that the interest rate over the lifetime of this bond will be constant at 10%. a. How much is a $22 coupon bond with a face value of $110 and a maturity date of April 19, 2001, worth to you today? (Assume you will get today's coupon.) (4 points) answer: PV = /(1.1) + 110/(1.1) = = 142. b. Suppose in the bond market this bond currently trades at $120. This gets you worried that you have estimated the interest rate higher, or lower, than everyone else. Which? (2 points) answer: You have clearly underestimated the interest rate. The market values the bond at a lower price than you, so the market (all the other individuals out there who buy and sell this bond) must have used a higher interest rate to calculate how much the bond is worth to them. page 6 of 6

7 7. General Electric has recently invested heavily in projects that have low immediate returns, but promise high returns in the future - in fact, GE has always pursued its investment strategy so as to maximize GE's present value. However, management fears that myopic (and economically badly informed) shareholders may dislike this investment strategy. Therefore GE hires you to write a short justification, based on the Fisher Separation Theorem, for why GE has acted in the interest of all shareholders. Use diagrams as appropriate. (15 points) answer: In your answer, you should point out that firms can structure the payouts to shareholders differently, according to the investment strategy that they pursue. If they invest in projects that have no immediate payout but a future payout, that would mean that a shareholder's income this period is low, and income next period is high. If shareholders are myopic they will not like this investment strategy if they cannot lend or borrow. If, however, they can lend and borrow at the same rate of interest, then they should be interested in the firm pursuing the investment strategy that gives the shareholders the highest present value of their payouts (i.e. the investment that maximizes the present value of the firm). Since this is what GE has just done, shareholders should not object to this strategy. Here is the diagram that you might have wanted to draw: Maximizing present value means shifting the shareholders' (black) budget constraint out as far as possible. All shareholders are as well off as possible (on the highest indifference curve). PV PV page 7 of 7