f (tl) <tf(l) for all L and t>1. + u 0 [p (l ) α wl ] pα (l ) α 1 w =0 l =

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1 Econ 101A Midterm Th November 006. You have approximatey 1 hour and 0 minutes to answer the questions in the midterm. I wi coect the exams at sharp. Show your work, and good uck! Probem 1. Profit Maximization with Uncertainty (65points)AisonownsacompanyupinNapa that produces wine. Her income is given by the profits of the company that she runs. Aison is risk-averse, with utiity function u (c) satisfying u 0 (c) > 0 for a c, and u 00 (c) < 0 for a c. Aison maximizes her utiity from the profits of the company. Wine y is a abor-intensive operation. It is produced using abor with the production function y = α, with 0 <α<1. Workers in the vine are paid the wage w. Aison maximizes utiity from her ony income source, the profits from wine saes: max u [p α w]. (1) 1. Define the concept of returns to scae, that is, what we mean by constant/increasing/decreasing returns to scae. Under what conditions for α the production of wine has decreasing returns to scae? Show the steps of your reasoning. (Here assume ony α>0, from the next point on assume 0 <α<1) (4 points). Consider the maximization of utiity of Aison, and obtain first-order conditions.(3 points) 3. Are the second order conditions satisfied? (3 points) 4. Sove for the utiity-maximizing. How does the optima empoyment of workers in the vineyard vary as the wage w of workers increases? And when the price p increases? Discuss the economic intuition. (3 points) 5. Using the enveope theorem, compute the effect on the utiity of Aison at the optimum u [p ( ) α w ] of an increase in price p. Does the finding make sense? Discuss. (4 points) 6. In a nearby terrain, Wei runs his operation with a standard profit-maximization. He has the same production function and costs as Aison, and hence maximizes max p α w. () Sovefortheprofit-maximizing for Wei. (3 points) 7. Aison and Wei are neighbors to a chatty economist trained in Stanford. (a concession to Prasad) The economist, after drinking a itte too much of their wine, is on-the-record on the oca TV as saying it is a pity that Aison is so risk-averse. Unike Aison, Wei just maximizes profits, which is much better for the company. Discuss this assertion in ight of what you found so far. Provide economic and mathematica intuition. (6 points) 8. The word of wine-growing is a treacherous one, with much uncertainty. In particuar, the price of wine fuctuates from year-to-year, and the decisions on how much to produce and how many workers to hire have to be made before the eve of price is known. The price p of a botte is 10 with probabiity q and 5 with probabiity 1 q. Let s consider first the probem of Wei, (). Wei now maximizes expected profits. Set up the maximization of expected profits by Wei, expaining why you are setting up the probem the way you are. (6 points) 9. Derive Wei s first order conditions and sove for the profit-maximizing eve of abor hired by Wei. (5 points) 10. How does the quantity of abor hired vary as the probabiity of a high price of wine q increases? Provide economic intuition (3 points) 11. Suppose that instead of facing uncertainty over the price of wine, Wei knew that the price of wine was going to be for sure q10 + (1 q)5, which is the expected vaue of the uncertain price. Using the soution you obtained in point 6, compute the quantity of abor hired by Wei in this case, and compare it to the quantity hired when the price is uncertain. How do the two quantities compare? (I mean, the one in point 9 and the one you just derived) Discuss the reason for this resut. (5 points) 1

2 1. Let us now go back to Aison s probem (1). Aison now faces uncertainty over the price of wine as we. As in Wei s case, the price p of a botte is 10 with probabiity q and 5 with probabiity 1 q. Aison now maximizes expected utiity. Set up the maximization of expected utiity by Aison. Derive the first-order conditions. (6 points) 13. Soving this probem formay is tricky. Using your economic intuition, discuss whether you expect that the quantity of abor hired by Aison in this case with uncertainty wi be the same, smaer, or arger than the quantity hired by Wei (that is, what you derived in point 10. Compared to the discussion in point 7, can the utiity function u matter here? (6 points) 14. (Hard) Can you prove formay whether the quantity of abor hired by Aison in this case with uncertainty wi be the same, smaer, or arger than the quantity hired by Wei? (Hint: use the property that u 0 (y) <u 0 (x) for y>xif u is concave) (8 points) Soution to Probem The concepts of returns to scae captures the response of production to an increase of a the inputs (here, just abor). Formay, a production function with ony one input, L, dispays decreasing returns to scae if f (tl) >tf(l) for a L and t>1, constant returns to scae if f (tl) =tf (L) for a L and t>1, increasing returns to scae if f (tl) <tf(l) for a L and t>1. In this case, f (tl) =(tl) α = t α L α = t α f (L). Since t α <tiff α<1, the function dispay decreasing returns to scae iff α<1.. The maximization of profits of Aison eads to the condition ³ u 0 [p ( ) α w ] pα ( ) α 1 w =0. 3. The second order conditions are ³ u 00 [p ( ) α w ] pα ( ) α 1 w + u 0 [p ( ) α w ] ³pα α (α 1) ( ) which are negative since u 00 (x) < 0 for a x and α<1 by assumption. Hence, the s.o.c. is satisfied. 4. Since u 0 (p α w) is aways positive, the f.o.c. above impies or pα ( ) α 1 w =0 = µ 1/(1 α) w αp As wage increases, the empoyment of workers decreases, since abor has become more costy. As the price increases, the empoyment of workers increases, since it has become more profitabe to produce more. 5. Using the enveope theorem, we know that du [p ( ) α w ] dp = u[p ( ) α w ] p = u 0 [p ( ) α w ] ( ) α > 0. An increase in price p increases the profits and hence the utiity of Aison in equiibrium.

3 6. Wei maximizes.thef.o.c.is eading to the same soution as for Aison. max p α w. pα ( ) α 1 w =0, = µ 1/(1 α) w, αp 7. The tipsy economist is not right in this case (though he be coser to right for the case beow). The fact that Aison is risk-averse does not affect in any way the maximization. One way to think of it is that the function u (x) introduces simpy a monotonic transformation of the profit function, and monotonic transformations of a function do not affect the optima soution. (Can you prove this?) This is very simiar to what we saw for the case of utiity maximization: a monotonic transformation of the same utiity function represents the same preferences, and hence wi ead to the same utiity-maximizing soution. This wi no more be true once we introduce uncertainty. 8. Now that there is uncertainty about the price of wine, Wei maximizes max q [10 α w]+(1 q)[5 α w]. Expected utiity tes that that we maximize the sum of the utiities in each state of the word (high price/ow price), weighted by the probabiities of each state. 9. The first-order condition is q 10α α 1 w +(1 q) 5α α 1 w =0 (3) or or [q10 + (1 q)5]α α 1 w =0 µ w = α (q10 + (1 q)5) 1/(1 α) 10. As the probabiity that the price of wine is high increases, the quantity of abor empoyed increases. The higher the expected price at which Wei can se the wine, the more Wei boosts the production. A higher (expected) price justifies hiring more workers and sending them to grow the margina parts (higher margina cost) of the vineyard that Wei woud not use otherwise. 11. Now that there is no more uncertainty about the price of wine, we are back to the case above, and we can use the soution µ 1/(1 α) w = αp to derive µ 1/(1 α) w =, α (q10 + (1 q)5) which is the same soution as in point 9. Wei just maximizes expected profits, he does not care whether the price is certain or uncertain, as ong as the expected vaue of the price is the same. 1. Aison s probem is max qu [10 α w]+(1 q) u [5 α w]. This is because Aison maximizes expected utiity. The first-order condition is qu 0 (10 α w) 10α α 1 w +(1 q) u 0 (5 α w) 5α α 1 w =0. 3

4 13. Aison maximizes the utiity of profits, and the utiity is concave. this means that Aison cares more at the margin about ow profits than she cares about high profits. When setting the optima price, Aison does not know whether prices wi be high or ow. If prices are high, she can make a itte extra profits by increasing the quantity produced, but this wi reduce the profits obtained if the price turns out to be ow. Conversey, is prices are ow, she can make extra profit by reducing the quantity produced, but this wi reduce the profits earned it the price is high. By the concavity of the utiity function, Aison cares more about decisions when profits are ow (and hence utiity is ow) than when profits are high (and utiity is high). Hence, she cares more about the case in which prices are ow, and she reduces the quantity produced, reative to what Wei produces. 14. Compute the derivative of the expected utiity function with respect to the amount invested s: qu 0 (10 α w) 10α α 1 w +(1 q) u 0 (5 α w) 5α α 1 w. (4) This expression, when set equa to zero, generates the first order condition. The trick is to prove that ³ this expression is negative for the vaue of W = w α(q10+(1 q)5) 1/(1 α) that maximizes the expected utiity of Wei. This impies that Aison, unike Wei, can improve utiity by producing ess than W. Together with the concavity of the utiity function of Wei, this impies that the optimum for Aison A is ower than for Wei, that is, A < W. To prove this, note that expression (4) has the same sign as expression (we divided by u 0 (10 α w)+u 0 (5 α w)) Bq 10α α 1 w +(1 B)(1 q) 5α α 1 w, (5) with u 0 (10 α w) B = u 0 (10 α w)+u 0 (5 α w). Expression (5) is the weighted sum of q 10α α 1 w and of (1 q) 5α α 1 w. Evauate now expression (5) at = W which, by definition, satisfies Wei s first order condition (3). If we had B =1/, we woud go back to Wei s first order condition (3), and the expression woud be zero. By the risk-aversion property, however, B<1/, since u 0 (10 α w) <u 0 (5 α w). Since 10α α 1 w> 5α α 1 w, it foows that, for = W, expression (5) is smaer than 0 since it is a convex combination that gives ess weight to the term 10 α w and more weight to 5α α 1 w, compared to expression (3) which is zero. We are done this was a hard proof to do. 4

5 Probem. Investment in Risky Asset (30 points) Consider a standard probem of investment in risky assets, simiar to the one that we covered in cass. The agent can invest in bonds or stocks. Bonds have a return r =0. (that is, they return back your capita at the end of the period) Stocks have a stochastic return,.10 (ten percent) with probabiity 1/, and.05 (minus 5 percent) with probabiity 1/. The agent has income w and utiity function u (x). The agents wants to decide the amount s of weath to invest in stocks; we assume 0 s w. The remaining of his weath, w s, is invested in bonds. (Note: s is not the share of weath, but the amount of weath) The agent s utiity function (defined over weath at the end of the period) is inear, u (x) =a + bx, with b>0. 1. At the beginning, before investing, the agent has weath w. What is the weath at the end, after investing, if the return of the stock is high? What is the weath if the return of the stock is ow?(4 points). Write down the expected utiity Eu of the agent, expaining how the expected utiity is defined. (4 points) 3. Derive the first order condition. (4 points) 4. Derive the second order condition. Is it satisfied? (3 points) 5. Find the soution for s. Argue the steps of the soution as precisey as you can. (Hint: Consider your answer to point 4) (10 points) 6. How does the soution for s change if bonds have a return r =.05 (5 percent)? (5 points) Soution to Probem. 1. The agent invests w s doars into a bond which provides gross return (w s) and s doars into a stock which provides return s (1 +.10) with probabiity 1/ and s (1.05) with probabiity 1 1/. Therefore, if stocks give high returns, the agent s weath at the end is (w s)+s (1.1) = w +.1s; if stocks give ow returns, the agent s weath at the end is (w s)+s (.95) = w.05s.. The expected utiity is the probabiity-weighted utiity under each state. Therefore, the expected utiity U is U = 1 (a + b (w +.1s)) + 1 (a + b (w.05s)). 3. The first order condition with respect to s is 1 0 b 1 40 b = 1 b =0 (6) The second order condition is 0 (not satisfied), which means that we shoud ook for corner soutions. 5. The derivative of the utiity function with respect to the investment s is 1 40b from (6). Since b> 0, 1 40b is aso positive, and hence the utiity is aways increase in the eve of s. The agent wants to increase the s as much as possibe, that is, up to s = w, which is the soution. 6. If r =.05, the utiity function becomes U = 1 (a + b ((w s)(1.05) + 1.1s)) + 1 (a + b ((w s)(1.05) +.95s)) = = 1 (a + b (1.05w +.05s)) + 1 (a + b (1.05w.10s)) = = a + b 1.05w + b.05 s b.10 = a + b 1.05w b 1 40 s. 5

6 Differentiating this with respect to s, we get b/40, which is negative. To the contrary of what we found before, here the agent aways wants to ower s, which wi ead to s =0as the soution. Intuitivey, now that the return to the riskess asset has increased, the expected return on the stock is now ower than the expected return to the bond, and a risk-neutra investor wi aways go for the asset with higher return. 6