Economics 21: Microeconomics (Summer 2000) 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 7 questions on this exam. These 7 questions are independent of each other. 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 1-7 1. Adam has preferences over two goods, apples (a) and fig-leaves (f), represented by the following utility function (subscript A denotes Adam's utility function): u A (a, f) = a 1/2 f 1/2. Eve's preferences over the same two goods can be represented by the following utility function (subscript E denotes Eve's utility function): u E (a, f) = (1/2) ln(a) + (1/2) ln(f) + 20. Suppose that Adam's and Eve's consumption of both apples and fig-leaves is positive. (a) (5 points) Do Adam and Eve have the same preferences? Why or why not? Explain very briefly. Eve's utility function is a positive monotonic transformation of Adam's utility function (it is the monotonic transformation u E (a, f) = ln[u A (a, f)] + 20. Since utility is uniquely defined only up to positive monotonic transformations (i.e. a positive monotonic transformation represents the same preferences as the original utility function), the both have identical preferences. cont'd page 1 of 7
(b) If you were to solve Adam's constrained maximization problem, you would find that his demand function for apples is: a = (1/2)m/p a, where p a is the price of apples, and m is Adam's wealth. (i) (4 points) Are apples a Giffen good for Adam? An inferior good? Why or why not? Apples are not a Giffen good: the partial derivative of demand w.r.t. price is negative. Apples are not an inferior good: the partial derivative of demand w.r.t. income is positive. (ii) (2 points) Are apples a gross substitute for fig-leaves for Adam or a gross complement? Explain your answer briefly. Apples are neither a gross substitute nor a gross complement for fig-leaves: the partial derivative of the demand for apples w.r.t. the price of fig-leaves is zero (i.e. neither positive nor negative). 2. (6 points) Sophie O. Moore has the following utility function over beer (b) and pizza (p): u(p, b) = ln(p) + 2b. She currently consumes 2 bottles of beer and 1 pizza. At what rate is she just willing (while remaining at the same level of utility) to exchange a little less beer for a little more pizza? This questions asks for the MRS, where beer is what we usually called good 2, and pizza is good 1. The MRS, therefore is just: minus the partial derivative of utility w.r.t. pizza divided by the partial derivative of utility w.r.t. beer. Here, MRS = - (1/p)/2 = -1/(2p). Since she is currently consuming 1 pizza, her MRS is -1/2. page 2 of 7
3. (10 points) The market for widgets consists of 1,000 identical consumers. Each consumer has an individual inverse demand curve for widgets (w) as follows: p(w) = 100-2w. where p(w) is the price that can be achieved when w widgets are sold. Widgets are only produced by one firm, and that firm currently charges $50 per widget. The widget producer asks you to calculate its marginal revenue. You calculate marginal revenue to be as follows: In order to find the market demand curve, you first need to find the individual demand curve: rewrite inverse demand to find demand: w(p) = 50 - (1/2)p. Now you can add the 1,000 consumers: W(p) = 50,000-500p. To find marginal revenue you can either calculate revenue and then take the derivative, or find elasticity and then use the formula for marginal revenue (p(1+1/elasticity)). This is the direct (first) way. For this, you need the inverse market demand curve, because marginal revenue is "how much does revenue increase as output is increased", so you need everything as a function of output. The inverse market demand curve is: p(w) = 100 - (1/500)W. Multiplying this price by quantity to get revenue we get: R(W) = 100W - (1/500)W 2. And taking the derivative w.r.t. W (in order to get marginal revenue), you have MR(W) = 100 - (2/500)W. The firm currently charges $50 per widget, so they must be selling (read it off the market demand curve) 25,000 widgets. Put this into the expression for MR and you get MR(25,000) = 100-100 = 0. Alternatively, you could calculate the elasticity from W(p) = 50,000-500p. Elasticity is (dw(p)/dp) x (p/w(p)). Here, elasticity is -500 (p/w(p)). Now put in p and W(p) (p = 50, as given, and therefore W(p) = 25,000 as previously calculated), to get an elasticity of - 1. Using the formula MR = p(1+1/elasticity), you see that MR = 0. page 3 of 7
4. Lionel has preferences over consumption this period (c 1 ) and consumption next period (c 2 ) that can be represented by the following utility function: u(c 1, c 2 ) = ln(c 1 ) + c 2. Lionel has income m 1 this period, and m 2 next period. The real rate of interest is 10%. (a) (10 points) Write down, and solve, the Lagrangean for Lionel's choice problem: that is, derive Lionel's optimal demands for c 1 and c 2. The budget constraint is c 2 = m 2 + (1+ρ)(m 1 - c 1 ), as usual, except that now ρ denotes the real interest rate. The Lagrangean therefore is: L = ln(c 1 ) + c 2 - λ(c 2 - m 2 - (1+ρ)(m 1 - c 1 )) The first-order conditions are: (i) 1/c 1 - λ(1+ρ) = 0 (ii) 1 - λ = 0 (iii) c 2 = m 2 + (1+ρ)(m 1 - c 1 ) Using (ii) which solves as λ = 1 in (i), you get 1/c 1 = (1+ρ), or c 1 = 1/(1+ρ). Putting this into (iii) you get c 2 = m 2 + (1+ρ)(m 1-1/(1+ρ)), or c 2 = m 2 + (1+ρ)m 1-1. (b) (4 points) The real rate of interest falls. Which two factors could have lowered the real rate of interest? If the real rate of interest falls, either the (nominal) interest rate has fallen or the inflation rate has risen. Recall that (1+ρ) is defined to be (1+r)/(1+π), where ρ is the real rate of interest. cont'd page 4 of 7
(c) (10 points) Suppose Lionel is a lender. When the interest rate falls, there are two effects on Lionel's consumption choices. Describe, in words (no calculation is required), the wealth and substitution effects on Lionel's consumption of c 1. Be as specific as possible, including all the information you have about Lionel's preferences. Can you predict whether Lionel will lend more, or less? An interest rate fall is a fall in the relative price of period 1 consumption. So the substitution effect is going to make Lionel consume more of period 1 consumption (c 1 ). Since Lionel is a lender, the interest rate fall, however, has made him "poorer" (since he gets some of his income from interest income, that is now lower). As a result, he will want to consume less of all normal goods. In this particular case of a quasilinear utility function, however, there is no wealth effect on c 1. You can see that from the demand curve you have derived in part (a), or remember this from the workout. So in this special case, there is no wealth effect, only a substitution effect. So you can say that Lionel will definitely consume more of c 1. 5. Tessa has the following utility function over consumption this period (c 1 ) and consumption next period (c 2 ): u(c 1, c 2 ) = ln(c 1 ) + ln (c 2 ). Tessa has income m 1 this period, and m 2 next period, and c 1 costs p 1 per unit (before any tax) and c 2 costs p 2 per unit (before any tax). The (nominal, that is: not adjusted for inflation) interest rate is r. There is also a sales tax of t 1 in period 1 and of t 2 in period 2. (Explanation (sales tax): For instance, if the sales tax in period 1 is 10%, that is, if t 1 = 0.1, and if the price of a good is usually $50, with the tax that good's price is $55.) (a) (10 points) Write down Tessa's intertemporal budget constraint, making sure that your expression contains the inflation rate instead of the (pre-tax) prices p 1 and p 2. The only thing that is different from how we did this in class is that instead of writing p 1 you now write (1 + t 1 )p 1, and similarly (1 + t 2 )p 2, as the explanation implies. Then you do the same thing we did in class: write down the budget constraint, and replace p 1 /p 2 appropriately by an expression involving inflation. Here's how: (1 + t 2 )p 2 c 2 = (1 + t 2 )p 2 m 2 + (1 + r)((1 + t 1 )p 1 m 1 - (1 + t 1 )p 1 c 1 ), or: c 2 = m 2 + (1 + r) [(1 + t 1 )/(1 + t 2 )] [p 1 /p 2 ] (m 1 - c 1 ). Now replace [p 1 /p 2 ] by 1/(1 + π), as in class to get: c 2 = m 2 + [(1 + r)/(1+ π)] [(1 + t 1 )/(1 + t 2 )] (m 1 - c 1 ). cont'd page 5 of 7
(b) If you were to write down, and solve, Tessa's constrained maximization problem (no need to do this), you would find that her demand function for c 1 is: (1 + π) (1 + t2 ) c 1 = 0.5 m1 + m2 (1 + r) (1 + t1) Suppose that the government announces an increase in the tax rate in period 2 (that is, t 2 increases), while the tax rate in period 1 remains the same. (i) (4 points) What happens to Tessa's consumption in period 1 when t 2 increases? it increases (ii) (10 points) If Tessa was a borrower before that tax increase, could you have predicted (from wealth and substitution effects alone, i.e. without knowledge of Tessa's demand function), how she would alter her consumption of c 1? Explain your intuition (no need to perform a calculation). The intuition is just that if the tax in period 2 increases, period 2 consumption becomes relatively more expensive and therefore, period 1 consumption becomes relatively cheaper. Now the intuition about wealth and substitution effects applies: period 1 consumption is relatively cheaper so the substitution effect tells Tessa to consume more in period 1. The wealth effect tells Tessa (since she is a borrower) to consume more of period 1 consumption also. So Tessa will definitely consume more in period 1. 6. (5 points) Suppose that a perpetuity with a $10 coupon currently trades for $200. (Assume that the first coupon payment you get is next year.) If you are willing to buy or sell this bond (that is, if you are partaking in the market for this bond), you must have a view on what the interest rate is over the lifetime of this bond. What is this interest rate? The present value of a perpetuity is c/r (where c is the coupon and r the interest rate). Here we know that the present value, 200 = 10/r, so you can calculate r to be 0.05, or 5%. This is the interest rate that the market expects, and which is implicit in the market's valuation of this perpetuity. (This will be an idea you encounter again in the 26-36-46 sequence when you discuss the time structure of interest rates, or the "yield".) page 6 of 7
7. (20 points) "Given perfect capital markets, a firm's investment decisions are separate from the consumption preferences of the firm's shareholders." (a) Explain this statement (use diagrams as appropriate) and: (b) discuss the importance of the assumption of perfect capital markets (i.e. equal lending and borrowing rates), again using diagrams as appropriate. answer suggestion: You should explain the Fischer separation theorem, using the diagram. Make sure you explain all key concepts, such as the investment opportunity schedule, the way in which firms can make payouts to investors, how that determines the investors' endowment and how from this endowment investors can save and borrow in a perfect capital market. This implies that investors (whatever their preferences are like) will wish the firm to make the payouts that shift their budget constraint out as far as possible, that is the payouts that maximize the present value of the firm. As I discussed in class (on the board, not on the slides), if lending and borrowing rates differ, this result no longer obtains: differing lending and borrowing rates imply a kinked budget constraint (i.e. kinked about the endowment point). If that is the case, a simple diagram (such as the one I used in class) illustrates that shareholders again disagree on what the firm should do, and the separation of investment decisions from consumption preferences no longer obtains. page 7 of 7