101A_midterm2fa - Econ 101A Midterm 2 Th 2 November 2006....

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Econ 101A — Midterm 2 Th 2 November 2006. You have approximately 1 hour and 20 minutes to answer the questions in the midterm. I will collect the exams at 11.00 sharp. Show your work, and good luck! Problem 1. Pro f t Maximization with Uncertainty (65points)AllisonownsacompanyupinNapa that produces wine. Her income is given by the pro f ts of the company that she runs. Allison is risk-averse, with utility function u ( c ) satisfying u 0 ( c ) > 0 for all c, and u 00 ( c ) < 0 for all c. Allison maximizes her utility from the pro f ts of the company. Wine y is a labor-intensive operation. It is produced using labor with the production function y = l α , with 0 <α< 1 . Workers in the vine are paid the wage w. Allison maximizes utility from her only income source, the pro f ts from wine sales: max l u [ pl α wl ] . (1) 1. De f ne the concept of returns to scale, that is, what we mean by constant/increasing/decreasing returns to scale. Under what conditions for α the production of wine has decreasing returns to scale? Show the steps of your reasoning. (Here assume only α> 0 , from the next point on assume 0 <α< 1 )(4 points) 2. Consider the maximization of utility of Allison, and obtain f rst-order conditions.(3 points) 3. Are the second order conditions satis f ed? (3 points) 4. Solve for the utility-maximizing l . How does the optimal employment 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 envelope theorem, compute the e f ect on the utility of Allison at the optimum u [ p ( l ) α wl ] of an increase in price p. Does the f nding make sense? Discuss. (4 points) 6. In a nearby terrain, Wei runs his operation with a standard pro f t-maximization. He has the same production function and costs as Allison, and hence maximizes max l pl α wl. (2) So lveforthepro f t-maximizing l for Wei. (3 points) 7. Allison and Wei are neighbors to a chatty economist trained in Stanford. (a concession to Prasad) The economist, after drinking a little too much of their wine, is on-the-record on the local TV as saying “it is a pity that Allison is so risk-averse. Unlike Allison, Wei just maximizes pro f ts, which is much better for the company”. Discuss this assertion in light of what you found so far. Provide economic and mathematical intuition. (6 points) 8. The world of wine-growing is a treacherous one, with much uncertainty. In particular, the price of wine F uctuates from year-to-year, and the decisions on how much to produce and how many workers to hire have to be made before the level of price is known. The price p of a bottle is 10 with probability q and 5 with probability 1 q. Let’s consider f rst the problem of Wei, (2). Wei now maximizes expected pro f ts. Set up the maximization of expected pro f ts by Wei, explaining why you are setting up the problem the way you are. (6 points) 9. Derive Wei’s f
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101A_midterm2fa - Econ 101A Midterm 2 Th 2 November 2006....

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