Econ 101A Midterm 2 Th 6 November 2003.

Transcription

1 Econ 101A Midterm 2 Th 6 November You have approximately 1 hour and 20 minutes to anser the questions in the midterm. I ill collect the exams at sharp. Sho your k, and good luck! Problem 1. Production. (43 points) In this exercise, e consider a firm producing a product using only one input, lab L. The production function f is as follos: ½ L L if L L f(l) 0 if 0 L< L, that is, the firm produces a positive quantity of output only if there are at least L kers. Assume that the age of a ker is. Assume L >0 and >0. 1. Dra a picture of the production function assuming.5 and L 1. (1 point) 2. F hich values of (if any) does the function exhibit (eakly) decreasing returns to scale (f (tl) tf (L) f all t>1 and all L 0)? Does the function exhibit (eakly) increasing returns to scale (f (tl) tf (L) f all t>1 and all L 0) f 1? Provide an analytical proof if you can. (7 points) 3. Consider no the first step of the cost minimization problem. The firm solves min L s.t.f (L) y f y>0. What is the solution f L, y L,? (This notation stresses that the solution depends also on the parameters and L. Hint: You are better off not using Lagrangeans...) (5 points) 4. Write don the implied cost function c, y L,.(2points) 5. Derive an expression f the average cost c, y L, /y and the marginal cost c 0 y, y L, f y>0. Graph the average cost and marginal cost f.5, 1and L 1. Graph the supply function f the same values of the parameters. [remember, y is on the hizontal axis]. (7 points) 6. No that e graphically solved f the supply function, e also derive it fmally. Consider the second step of cost minimization max py c, y L,. y Write don the first der condition and the second der conditions. Solve f y, p L,. F hat values of is the second der condition satisfied? (5 points) 7. Assume no <1 and rite the condition under hich firms are making positive profits, that is, under hich py, p L, c, y L, > 0. If you can, solve f the value of price p such that firms produce if the price p is larger than p. (5 points) 8. Taking into account the ansers to points 6 and 7, rite don the supply function, that is, an expression f y, p L,. Keep assuming <1. (3 points) 9. Why do these conditions imply that the supply function is the ption of the marginal cost curve above average cost as long in its upard sloping ption, and zero otherise (4 points)? 10. Keep assuming <1. What happens to the supply function as L increases? What if the age increases? You can respond using the graphs using the solution in point 8. Give an economic interpretation. (4 points) Solution of Problem 1. 1

2 1. See Figure There is no >0 such that the function exhibits decreasing returns to scale. Consider any L 0 such that L 0 < L. If t is large enough (f example, t 2 L/L 0 ), tl 0 ill fall on the upard part of the production function, and therefe f (tl 0 ) > 0tf (L 0 ). We no sho that the function ill exhibit increasing returns to scale as long as 1. First, as e just proved, f (tl) tf (L) f any L L. As f L> L, f (tl) tl L tl t L t f 1 the production function exhibits increasing returns to scale. t L L t >t L L t f (L). Therefe, 3. Given that the quantity to be produced y is positive, the quantity of lab to be employed is at least as large as L. F all such values of L, the production function is increasing in L. Therefe the budget constraint ill be satisfied ith equality. (if it as not, the firm could cut costs by using less lab) Using f (L) L L y, e can invert the function and obtain L L y 1/ L L + y 1/. The solution to the problem therefe is L, y L, L + y 1/. 4. The cost function is c, y L, L, y L, L + y 1/ f y>0. 5. The average cost is c, y L, /y L + y 1/ /y L/y + y ()/. The marginal cost is c, y L, / y L + y 1/ / y 1 y()/. See Figure 2 f the second part of the exercise. 6. The first der condition is This implies p c 0 y, y L, 0 p 1 y()/. y, y L, p The second der condition is This condition is satisfied f <1. c 00 y,y, y L, < y(1 2)/ < The firms make positive profits if py c, y L, p p p 1 p p 1 L p 1 p 1 L 0 L 1 () p. 8. Therefe the supply function is y ½ p if p p 0 if p< p 2

3 9. In point 6, e sa that the first der condition is p c 0 y. Therefe, the optimal quantity y can be found graphically on the marginal cost curve. The second der condition arns us that this is only true if, at the candidate optimum, the second der condition is upard sloping. Finally, point 7 shos that firms ill not find it profitable to produce is they make negative profits, that is, if at the price p is belo average cost. 10. A change in the minimum quantity of lab L does not affect the marginal cost function, but it shifts upard the average cost function. Therefe it reduces the range of prices f hich the firm is illing to produce. This makes sense since an increase in L is equivalent to an increase in fixed costs. An increase in the age shifts up both the marginal cost curve and the average cost curve. It both reduces the range prices f hich the firm produces and the quantity produced f each price. This is because it is an increase in the variable costs incurred by the firm. 1. Problem 2. Uncertainty (17 points). So far, in class e have assumed that the utility function depends only on consumption c. No, e consider the case of Prospectus, hose utility function also on a reference point r (Kahneman and Tversky, 1979). Prospectus has utility function ( U(c r) (c r) 1/2 if c r (r c) 1/2 if c<r You can see this utility function plotted in the attached Figure. Prospectus needs to choose beteen jobs G and I. Job G is in a government agency and guarantees consumption of $40,000 per year. Job I is in an investment bank and is very risky: ith probability.5 the job ill go ell and Prospectus ill consume $70,000 per year, but ith probability.5 Prospects ill be fired and have a yearly consumption of $10,000. Your task is to help Prospectus decide hich job to take. 1. Assume that Prospectus just came out of school and has a lo reference point, that is, r equals $10,000. Write don the expected utility EU (c r) from accepting job G and the expected utility EU (c r) from accepting job I. What ould you recommend that Prospectus should do in der to maximize expected utility? (6 points) 2. Assume that Prospectus just quit a consulting job and has a high reference point, that is, r equals $70,000. Write don the expected utility EU (c r) from accepting job G and the expected utility EU (c r) from accepting job I. In this case, hat ould you recommend that Prospectus should do in der to maximize expected utility? (5 points) 3. Can you give an intuition f hy the best job choice f Prospectus depends on the reference point? (Hint: use the Figure and possibly Jensen s inequality) (6 points) Solution to Problem The expected utility from job G is The expected utility from job I is Eu G (40, 000 r 10, 000) (40, , 000) 1/2 (30, 000) 1/2. Eu I.5u (70, 000 r 10, 000) +.5u (10, 000 r 10, 000).5(70, , 000) 1/2 +.5(10, , 000) 1/2 µ 1/2.5(60, 000) 1/2 1 (60, 000) 1/2 (15, 000) 1/2. 4 Prospects should choose job G in der to maximize expected utility. 3

4 2. The expected utility from job G is Eu G (40, 000 r 70, 000) (70, , 000) 1/2 (30, 000) 1/2. The expected utility from job I is Eu I.5u (70, 000 r 70, 000) +.5u (10, 000 r 70, 000).5 1/2 (70, , 000) +.5 1/2 (70, , 000).5(60, 000) 1/2.5(2 30, 000) 1/2 2 2 (30, 000)1/2. Prospectus should choose job I in der to maximize expected utility. 3. Prospectus is risk averse f gambles involving only outcomes above the reference point, and risk seeking f gambles involving only oucomes belo the reference point. When he just came out of school, therefe, he prefers not to take risk and chooses the government job. When, instead, he has a high reference point, he chooses the investment banking job because he is risk seeking. Fmally, notice that, on outcomes above the reference point, the utility function is concave and therefe Jensen s inequeality implies Eu(X) <u(ex). But job G guarantees exactly the expected value of job I, and therefe Eu G <u I (X). The converse applies f the other case. Problem 3. Altruistic kers. (20 points)consider a ker in a firm that has to decide ho hard to k. The ker s efft e, ith0 e 1, has a disutility f the ker e 2 /2. The ker gets age 0 <<1. The ker s utility is therefe U e 2 /2, that is, the age net of the efft cost. The production of the firm is e and the good is sold at price The ker chooses e to maximize on utility (subject to the constraint 0 e 1). What is the optimal choice of efft e? (3 points) 2. The firm does not like the above outcome and decides to monit the ker ith probability p. With probability p (1 e) the ker is caught shirking and is fired, and therefe gets 0 age. With probability p e the ker is found at k and gets the age. With probability (1 p) there is no moniting and the ker gets age. The ker maximizes the expected age payment minus the efft cost. Write don the expected utility of the ker and solve f the utility maximizing efft e. What is the comparative statics ith respect to p? Provide intuition. (6 points) 3. The firm no attempts to maximize profits given by revenue e minus the expected age payment minus moniting costs cp 2. (Hint: The firm pays the age ith probability 1 p + pe) Solvef the profit-maximizing level of moniting p. (5 points) 4. We no go back to point 1, that is, there is no moniting. Assume no, hoever, that the ker is altruistic toard the firm. That is, the ker maximizes e 2 /2+e, here the last term is the product of the altruism coefficient and the production of the firm. Solve f the utility maximizing efft in this case assuming no moniting. What is the comparative statics ith respect to the altruism coefficient? (3 points) 5. A firm ants to improve productivity e. The firm has the choice beteen increasing the costly moniting selecting altruistic kers. What should the firm do? (3 points) 4

5 Solution to Problem The ker maximizes U e 2 /2 by setting e 0. The ker puts in no efft. 2. The expected utility of the ker is p (1 e) 0+pe +(1 p) e 2 /2pe +(1 p) e 2 /2. The first der condition ith respect to e is p e 0. The ker therefe sets e p. Clearly, e / p >0. The higher is the probbility of moniting, themeefft the ker puts in. 3. The firm maximizes max e (p, ) (1 p) pe (p, ) cp 2 p (1 p) (p) 2 cp 2. p The first der condition is 2 2p 2 2cp 0, hich implies p ( 2 + c). 4. In case of an altruistic ker, the ker maximizes e 2 /2+e, hich yields the first der condition e + 0, e. The efft of the ker is increasign in the altruism. 5. Either an increase in moniting p an increase in altruism lead to increase production. But one is costly, hile the other is free. The firm therefe is better offer attracting altruistic kers. 5