Universidad Carlos III de Madrid May 017 Microeconomics Name: Group: 1 3 5 Grade You have hours and 5 minutes to answer all the questions. 1. Multiple Choice Questions. (Mark your choice with an x. You get points if your answer is correct, -0.66 points if it is incorrect, and zero points if you do not answer.) 1.1. According to the preferences % over consumption bundles in R +, (x; y) % (x 0 ; y 0 ) if and only if x x 0 and y y 0. Therefore % do not satisfy axiom A:1 (completeness) do not satisfy axiom A: (transitivity) do not satisfy axiom A:3 (monotonicity) satisfy axioms A:1; A: and A:3. 1.. A consumer s preferences are represented by the utility function u(x; y) = x + y. The prices are (p x ; p y ) = (1; 1): Hence the signs of the substitution e ect (SE) and income e ect (IE) over the demand of good x of an increase in the price of x to p 0 x = 3= are SE < 0; IE = 0 SE = 0; IE > 0 SE = 0; IE < 0 SE = 0; IE = 0: 1.3. If x and y are perfect complements, then a sale tax on x decreases the demand x and increases the demand of y increases the demand x and decreases the demand of y decreases the demand of both, x and y decreases the demand of x and has no e ect on the demand of y. 1.. The prices of goods x; y were (p 015 x ; p 015 y ) = (; 1) in 015, and they are (p 016 x ; p 016 y ) = (1; ) in 016. Hence, Laspeyres consumer price index for an individual whose consumption in 015 was (x; y) = (1; 1) is: 1 1 3. 1
1.5. If the preferences of the consumer of the previous question are represented by the utility function u(x; y) = minfx; yg, then her true consumer price index is: 1 1 3. 1.6. The preferences over lotteries for a consumer are represented by the Bernoulli utility function u(x) = ln x. Identify her expected utility and certainty equivalence for the lottery l = (x; p) that pays x = (1; ; 16) with probabilities p = ( 1 ; 1 3 ; 1 6 ): Eu(l) = ln ; CE(l) = Eu(l) = ln ; CE(l) = Eu(l) = ln ; CE(l) = Eu(l) = ln ; CE(l) = : 1.7. If a rm s output at the short run competitive equilibrium price, p, is positive, then its marginal cost is greater than or equal to its average cost its average cost is less than or equal to p its average variable cost is non decreasing its marginal cost is decreasing. Questions 8, 9 and 10 refer to Lolita, the competitive cow of Holstein that produces milk using oats (O) and hay (H), which she buys at prices p O and p H, respectively, according to the production function Q = minfo; p Hg: 1.8. Lolita s conditional demand of hay, H(p O ; p H ; Q); is Q Q p H Q p O p H Q. 1.9. Lolita produces milk with cost (p O + p H ) Q p O Q + p H Q p H p O Q (p O + p H ) Q, 1.10. and has diseconomies of scale economies of scale constant marginal cost decreasing average cost.
. A consumer s preferences for food (x) and clothing (y) are represented by the utility function u(x; y) = x p y. (a) (15 points) Calculate her demand for food and clothing, x(p x ; p y ; I) and y(p x ; p y ; I). (Verify the possible existence of interior and corner solutions to the consumer s problem.) Graph the consumer s budget set and calculate her optimal bundle and utility at prices and income (p x ; p y ; I) = (; 1; 6). Solution: Since RMS(x; y) = p y x p y = y x ; an interior solution to the consumer s problem solves the system Solving the system we get y = p x x p y p x x + p x y = I: x(p x ; p y ; I) = I 3p x ; y(p x ; p y ; I) = I 3p y : Since u(0; y) = u(x; 0) = 0 and u(x; y) > 0 for (x; y) 0; then for I > 0 there are no corner solutions. For (p x ; p y ; I) = (; 1; 6) the budget constraint is the optimal bundle is and the consumer s utility is x + y = 6; (x ; y ) = ( (6) 3 () ; 6 ) = (; ); 3 (1) u(; ) = p = u The following graph illustrates these calculations. y 6 u(x,y)=u* 0 0 1 3 x 3
(b) (10 points) If at prices and income (p x ; p y ; I) = (; 1; 6); the government introduces a one euro per unit tax on the consumption of clothing, what would be its tax revenue? Calculate the equivalent variation of this tax, and verify that it is greater than the tax revenue. Solution: The tax increase the price of clothing to p 0 y = : At this price, the demands for food and clothing are ( (6) 3 () ; 6 ) = (; 1); 3 () and the government revenue is T = 1(1) = 1 euro. In order to calculate the equivalent variation, we calculate the consumer s utility at the new prices, u(; 1) = p 1 = ; and solve the system x p y = y x = ; which solution is x = y = 3 : Without the tax the cost of the bundle (x; y) is Therefore, the equivalent variation is C = xp x + yp y = 3 ( + 1) : EV = I C = 6 3 ( + 1) ' 1; > 1 = T:
3. (15 points) Elena is a Carlos III student. Her welfare depends of her weekly leisure time (h; measured in days), her weekly consumption (c; measured in euros) and her average grade (y [0; 10]). Her preferences are represented by the utility function u(h; c; y) = 5h + c + y: Her average grade depends on how many days a week she studies of the days a week she has available, x [0; ]; according to the formula y = 5 p x. (Elena attends lectures 3 days of the week.) Obviously, Elena s leisure time is x. Elena has no other income than her parents weekly stipend of w y euros. Use Elena s budget constraints to calculate her utility as a function of the days she studies a week, v(x). Set up and solve Elena s problem to determine the optimal number of days of study as a function of w; x(w): Then use the function x(w) to calculate her weekly leisure and consumption, h(w) and c(w); as well as her average grade, y(w). Solution: Since h + x = ; y = 5 p x and c = wy = 5w p x; we can write Elena s utility as a function x as v(x) = u( x; 5w p x; 5 p x) = 5 ( x) + 5w p x + 5 p x: Elena s problem is max v(x): x[0;] An interior solution to this problem solves the equation v 0 (x) = 5 + 5 p (w + 1) = 0; x whose solution is x (w + 1) = : Since we must have x ; the solution to Elena s problem is ( (w+1) x(w) = if w 3 if w > 3; and her leisure, consumption and average grade are ( (w+1) h(w) = if w 3 0 if w > 3; c(w) = y(w) = 5 w (w + 1) if w 3 10w if w > 3; 5 (w + 1) if w 3 10 if w > 3: 5
. The demand of a good is D(p) = maxf180 5p; 0g: The are two technologies to produce the good, A and B; with cost functions given by C A (q) = 8 + q and C B (q) = 3 + 3q, respectively, for q > 0, and C A (0) = C B (0) = 0: (a) (15 points) Calculate supply functions of competitive rms with technologies A and B, and the competitive equilibrium assuming that there are 0 rms of type A and 30 rms of type B. Solution: Supply of the rms with the technology A. The closing condition is p min AC A (q): The average cost function is AC A (q) = 8 q + q: Since dac A (q) dq = 8 q + = 0, qa = ; then min AC A (q) = 8: MC A (q) = p, q = p: Since for p < 8 we have q = p= < ; which do not satisfy the closing condition, then the rm s supply is ( p s A if p 8 (p) = 0 if p < 8: Supply of the rms with the technology B. The closing condition requires p min AC B (q): The average cost function is AC B (q) = 3 q + 3q: Since dac B (q) dq = 3 3 q = 0, qb = 1: we have min AC B (q) = 6: We get the rm s supply by solving the equation MC B (q) = p, 6q = p: Since for p < 6 we have q = p=6 < 1, which do not satisfy the closing condition, the rm s supply is ( p s B if p 6 (p) = 6 0 if p < 6: The market supply is 8 < 10p if p 8 0s A (p) + 30s B (p) = S(p) = 5p if 6 p < 8 : 0 if p < 6: In the competitive equilibrium supply and demand coincide, that is 180 5p = 10p ) p = 1; q = 10; q A = 3; q A = : 6
(b) (5 points) Determine the price and number of rms with technology A and B in the long run competitive equilibrium. (The functions given provide the long run cost.) Solution. The long run competitive equilibrium only the rms with technology B survive since the rms with technology A are less e cient, min AC A (q) = 8 > 6 = min AC B (q): Moreover, the rms must have zero pro ts, which implies that in equilibrium p = 6; and the demand is D(6) = 16: Since the optimal scale of the rms with technology B is q B = 1; the number of rms of each type is n A = 0 and n B = D(6) q B = 150 1 = 150: 7
5. A pharmaceutical rm must decide whether or not to make an investment to develop a new vaccine for a virus that is common in some developing countries. The cost of developing the vaccine is 700 million euros, and the average variable cost of production and distribution 10 euros. Hence the rm s total cost function is C(q) = 700 + 10q, where q is given in millions of units, if it develops the vaccine, and it is C(0) = 0 otherwise. The demand of the vaccine, in millions of units, is D(p) = maxf60 p; 0g, where p is in euros/unit. (a) (10 points) Determine whether the rm will develop the vaccine. (Note that the rm would have a patent on the vaccine, and therefore would monopolize the market.) Solution: Let us calculate the monopoly equilibrium assuming that the rm develops the vaccine. The rm s output q solves the problem max P (q)q C(q); q>0 where P (q) is the inverse demand, which we can get from D(p) as P (q) = 60 q for q [0; 60] (for q > 60 it is not de ned). The rst order condition for a solution to this problem is that is, Solving this equation we get and q = P 0 (q)q + P (q) = C 0 (q); ( 1)q + (60 q) = 10: 60 10 p = P (q ) = 60 The pro t of the rm is this equilibrium is = 5 million units q = 35 euros/unit. = p q C(q ) = 35 (5) 700 10 (5) = 75 Million de euros: Therefore the rm will not develop the vaccine. 8
(b) (10 points) Determine the e ect over the rm s pro ts and consumer surplus of a subsidy to the rm of 10 euros for each unit of the vaccine it sells. Calculate also the cost of this program. (Maintain the notation p for the price the consumer pays, and keep in mind that the rm receives p + 10 euros for each unit of the vaccine its sells.) Solution: With the subsidy, if the rm develops the vaccine its output would solve the problem max(p (q) + 10)q C(q); q>0 The rst order condition for an interior solution to this problem is that is, Solving this equation we get and P 0 (q)q + (P (q) + 10) = C 0 (q); ( 1)q + (60 q + 10) = 10: q s = 60 p s = P (q s) = 60 = 30 million units, The pro t of the rm in the monopoly equilibrium is now q s = 30 euros/unit. = (p s + 10)q s C(q s) = (30 + 10) (30) 700 10 (30) = 00 million euros: Therefore, with the subsidy the rm will develop the vaccine. The consumer surplus that would generate the subsidy is The cost of the subsidy is EC = 1 (60 p ) q = 50 million euros. 10q = 300 million euros. Therefore the subsidy generates a total surplus, net of the cost of the subsidy, of 00 + 50 300 = 550 million euros. 9