Universidad Carlos III de Madrid June Microeconomics Grade

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1 Universidad Carlos III de Madrid June 206 Microeconomics Name: Group: Grade You have 2 hours and 45 minutes to answer all the questions.. Multiple Choice Questions. (Mark your choice with an x. You get 2 points if your answer is correct, points if it is incorrect, and zero points if you do not answer.).. If a consumer s income increases by 0%, the price of good x increases by 5%, and the price of good y increases by 0%, then his budget line shifts parallel towards the origin shifts parallel away from the origin rotates over its intersection on the y-axis rotates over its intersection on the x-axis..2. If a consumer prefers lexicographically good x over good y, his monetary income is I = 4 euros, and prices are p x = 2; p y = ; then his optimal bundle is (2; 0) his optimal bundle is (; 2) his optimal bundle is (0; 4) there is no optimal bundle... An individual preferences are represented by the utility function u(x; y) = x + 2y; and the prices are p x = p y = 2. A decrease of the price of y to p 0 y = causes an increment of the demand of good x a decrease of the demand of good y a substitution e ect equal to zero an income e ect equal to zero..4.in 204 the prices of goods x and y were (p 204 x ; p 204 y ) = (; 2) and in 205 the prices were (p 205 x ; p 205 y ) = (2; ). Therefore the Laspeyres consumer price index (CPI) for an individual whose consumption in 204 was (x; y) = (; ) is:

2 .5. An individual s risk premium for the lottery l = (x; p); that pays x = (0; 4; 6) with probabilities p = ( 4 ; 2 ; 4 ); is RP (l) = 2. Hence his certainty equivalent is CE(l) = 2 CE(l) = CE(l) = CE(l) = 4:.6. A competitive rm that produces a good with labor (L) and capital (K) according to the production function F (L; K) = p L + 2K has: increasing returns to scale diseconomies of scale constant returns to scale A concave total cost function..7. If a rm has diseconomies of scale (and is competitive in the input markets), then its marginal cost is less than its average cost its average cost is increasing its total cost is concave its marginal cost is increasing..8. In the short run competitive equilibrium of a market each rm s average cost is less than or equal to the price average variable cost is less than or equal to the price marginal cost is less than its average cost pro ts are positive..9. There are two technologies, A and B; that allow to produce a good with total costs C A (q) = q 2 + q + and C B (q) = q 2 +4, respectively. The market demand for the good is D(p) = maxf20 p; 0g. If there is free entry and both technologies can be freely adopted, then in the long run competitive equilibrium the price p and the output q are (p ; q ) = (8; 2) (p ; q ) = (4; 6) (p ; q ) = (5; 5) (p ; q ) = (2; 8):.0. If a rm produces with zero cost a good whose demand is D(p) = maxf0 p; 0g, then in the monopoly equilibrium with rst degree price discrimination the output is q M = 5 the deadweight loss is 25=2 the total surplus is 75=2 the monopoly pro t is 50: 2

3 2. Mia has a daily endowment of 6 hours for leisure and work. Her preferences are represented by the utility function u(h; c) = hc 2 ; where h denotes the number of hours of leisure and c denotes her consumption in euros therefore the price of consumption is p c = : Mia has a daily (non-labor) monetary income of M euros and the salary is w euros/hour. (a) (5 points) Describe Mia s problem, and calculate her demand of consumption c and leisure h, and her labor supply l as a function of M and w. The consumer s problem is max h;c u(h; c) = hc 2 c + wh M + 6w 0 h 6; c 0 Interior solution: in order to nd Mia s demand for consumption and leisure, we calculate the marginal rate of substitution MRS(h; c) = c 2h ; and solve the system, The solution to this system is h = c 2h = w c + wh = M + 6w: M w + 6 ; c = 2 (M + 6w): In order for 0 h 6 to hold, we need w M=2: If w < M=2; then the solution to Mia s problem is h = 6; c = M: Therefore, the demand functions are M h (M; w) = w + 6 if w M=2 6 if w < M=2 ; c (M; w) = 2 (M + 6w) if w M=2 M if w < M=2 : Mia s labor supply function is l(m; w) = 6 h(m; w) = (2 M w ) if w M=2 0 if w < M=2:

4 (b) (0 points) Calculate Mia s optimal consumption-leisure bundle for M = 4 and w = 2: Also, calculate the income and substitution e ects over the demand of leisure of a 50% tax on labor income. Solution: Mia s optimal consumption bundle for (M; w) = (4; 2) is (h ; c ) = (h (4; 2) ; c (4; 2)) = (6; 24) : The tax imposed on labor income is equivalent to a reduction of the wage from w = 2 to w 0 = ( 0:5) 2 = euros per hour. With this wage the optimal bundle is (h ; c ) = (h (4; ) ; c (4; )) = (20=; 80=) : Hence the total e ect is T E = 20 6 = 2 : In order to nd the substitution e ect, we solve the system to get ^h = p 864: Thus, the substitution e ect is hc 2 = 6 (24) 2 c 2h = ; SE = p ' ; 52; Using the equation T E = IF + SE; we calculate the income e ect IE = T E SE = 2 p ' 2:85: 4

5 . (0 points) An individual must decide whether to nance its house with a mortgage at a xed interest rate (FM), or at a variable interest rate (VM). FM involves an annual payment of P thousands of euros, while a VM involves an annual payment of 7 thousand euros with probability 2 ; 20 thousand euros with probability, and 0 thousand euros with probability 6. The individual s annual income is 50 thousand euros, and his welfare depends on his net income x; measured in thousands of euros, which is equal to his income minus his mortgage payment. The individual s preferences over lotteries are represented by the Bernoulli utility function u (x) = 2 p x: For which values of P would he prefer to nance his house with the FM mortgage? Solution: The alternative mortgages may be represented as the individual can choose from. These lotteries are: F M(P ) = (50 P ; ) ; V M = 40; 0; 20; 2 ; ; 6 where the payo s show the individual s possible net income levels. We have Eu(F M(P )) = 2 p 50 P ; and Eu(V M) = 2 2p p p 20: Therefore Solving this equation we get Eu(F M(P )) = Eu(V M) () 2 p 50 P = 2 2p p p 20 P = 50 p p p = 728 euros. For P < P the individual would prefer the mortgage F M(P ); and for P > P he would prefer the mortgage V M. 5

6 4. A rm produces a good with the production function F (L; K) = L (K ). The rm is a price taker in the input markets, in which the prices are w = r = : (a) (5 points) Calculate the total, marginal and average cost function of the rm, as well as its supply. Solution: We have MRT S(L; K) = K L : For q > 0 the condition factor demands solve the system The solution is K L = L (K ) = q: L(q) = q 2 ; K(q) = q 2 + : For q = 0; the solution to the cost minimization problem is K = L = 0: Hence the rm s cost function is ( 2q 2 + if q > 0 C(q) = wl(q) + rk(q) = 0 if q = 0: For q > 0 the average and marginal cost functions are AC(q) = 2q 2 + q MC(q) = q 2 : In order to calculate a rm s supply we solve the equation p = MC(q); that is, p = q 2 ; yields q = p 2 =9: Since the marginal cost is increasing, we need to check the shutting down condition, (q) = pq C(q) = (q 2 )q 2q 2 + = q 2 > (0) = 0; which requires q = p 2 =9 > ; that is p >. Therefore, the supply of the rm is 8 >< 0 if p < s(p) = f0; g if p = >: p 2 9 if p > : 6

7 (b) (0 points) Calculate the long run competitive equilibrium price and number of rms assuming that the demand of the good is D(p) = 240=p: (Assume that this technology is the only one available, and that rms are free to enter the market and adopt the technology.) Solution. In order to calculate the long run competitive equilibrium with free entry and no patents, we know that p = min AC(q): q0 The solution to this problem solves the equation dac(q) dq = 0; whose solution is ql = and AC(q L ) = : Hence the long run competitive price is p L equilibrium quantity is Q L = D( p L) = 240 = 80 and the number of rms is n LqL = Q L, n L = 80: = ; the 7

8 5. (20 points) A rm produces a good at zero cost and monopolizes two markets where the demands are D (p) = maxf0 p; 0g and D 2 (p) = maxf4 p; 0g. Calculate the monopoly equilibrium with and without price discrimination, and determine who are the winners and losers of price discrimination (relative to the equilibrium without price discrimination). where Solution: The monopoly problem with third degree price discrimination is max p (q )q + p 2 (q 2 )q 2 C(q + q 2 ) = p (q )q + p 2 (q 2 )q 2 ; q ;q 2 0 p (q ) = The solution to this problem solves the system 0 q si q si q > 0 ; p q2 si q 2(q 2 ) = si q 2 > 4: 0 2q = 0 4 2q 2 = 0: Solving we obtain q = 5 and q 2 = 2: The equilibrium prices are p = 5 and p 2 = 2, and the consumer surpluses and monopoly pro ts are CS MD = 2:5; CS MD 2 = 2; MD = 29 In order to solve the monopoly problem without price discrimination we need to calculate the total market demand. Observe that for prices greater than 0 euros there is no demand in either market, whereas for prices between 4 and 0 euros there is demand only in market, and for prices below 4 euros there is demand in both markets. Hence the total market demand is 8 < 0 if q > 4 p(q) = 7 : 2q if 6 < q 4 0 q if q 6: The monopoly revenue is R(q) = p(q)q and its marginal revenue is 8 < 0 if q > 4 MR(q) = 7 q if 6 < q 4 : 0 2q if q 6: The monopoly equilibrium output without price discrimination solves the equation MR(q) = MC(q). Assuming that 6 < q 4; this equation is 7 q = 0. Hence q M = 7 2 (6; 4], and the equilibrium price is p M = :5. The consumer surpluses and monopoly pro ts are CS M = 2:25; CS M 2 = 0:25; M = 24:5 We see that the monopoly and the consumers of market 2 are worse o, but he consumers of market are better o without price discrimination than with third degree price discrimination. 8