Department of Economics ECO 204 Microeconomic Theory for Commerce Test 2

Transcription

1 Department of Economics ECO 204 Microeconomic Theory for Commerce Test 2 IMPORTANT NOTES: Proceed with this exam only after getting the go-ahead from the Instructor or the proctor Do not leave during the first hour of the exam or the last 15 minutes of the exam You are not allowed to leave the exam room unattended. If you need to go to the washroom, please raise your hand and a proctor will accompany you to the washroom. You are allowed to go to the washroom ONLY. You are required to stop writing and turn your exam face down when asked to stop by the instructor or proctor at the end of the exam Please note that proctors will take down your name for academic offenses, which will be treated in accordance with the policies as published by the Faculty of Arts and Sciences. Exam details: Duration: 2 hours and 20 minutes Total number of questions: 3 (see next page of test for breakdown of points) Total number of pages: 27 (including title page and two worksheets at the back of the test) Total number of points: 100 Please answer all questions. To earn credit you must show all calculations. Exam aids: This is a closed note and closed book exam. You may use a non-programmable calculator. Sharing is not allowed. University of Toronto Academic Code of Conduct: The University s Code of Behavior on Academic Matters ( Code ) applies to all Rotman Commerce students. The Code prohibits all forms of academic dishonesty including, but not limited to, cheating, plagiarism, working on the exam after the proctor has announced the exam has ended and the use of unauthorized aids. Students violating the Code may be subject to penalties up to and including suspension or expulsion from the University. By signing my name and entering my name and student number, I am confirming that I have read and understand the University s Code of Behavior on Academic Matters. I will conduct myself with the utmost integrity and I will neither give nor receive unauthorized aid in tests or examinations. Please provide the following information. Tests without the following information will NOT be graded. Please circle the section in which you are registered (not necessarily the same as the section you attend) Mon 1 3 L0101 Tue 11 1 L0102 Tue 2 4 L0103 Wed 6 8 L5101 Signature PRINT your Last Name PRINT your First Name 9 or 10 Digit Student ID # For use by Proctors only Check here if student does not have U of T STUDENT ID Page 1 of 27

2 THIS TABLE IS FOR GRADERS USE ONLY QUESTION MAXIMUM POINTS SCORE COMMENTS BONUS TOTAL SCORE OUT OF 100 POINTS GOOD LUCK! GIVE BRIEF ANSWERS AND SHOW ALL NECESSARY CALCULATIONS Page 2 of 27

3 Question 1 [25 points] This question consists of two parts A and B Part A (a) [3 points] [This part is independent of all other parts below] Consider production processes which use to produce output. Give a real life example of each of the following production functions: (i) a long run Cobb-Douglas production function (ii) a long run complements production function (iii) a long run linear production function. Provide a brief explanation for your answers. Page 3 of 27

4 (b) [5 points] [This part is independent of all other parts below] A company uses to produce output according to the Cobb-Douglas production function. Let the price of labor be and the price of capital be Set up and solve the Cobb-Douglas long run CMP in parametric form: Assume that the CMP has an interior solution (i.e. Case D ) which means that the non-negativity constraints can be dropped. Show that the optimal labor and capital are: ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ] Do NOT solve for any Lagrange multipliers. Show all necessary calculations. Page 4 of 27

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6 (c) [5 points] The Cobb-Douglas long run cost function can be expressed as: ( ) Use this expression to derive the Lagrange multiplier corresponding to the constraint run CMP: in the following long [ ] Assume that the CMP has an interior solution (i.e. Case D ) which means that the non-negativity constraints can be dropped. : Page 6 of 27

7 Part B In a tutorial, Michelle gave you data on a company s labor ( ), capital ( ), and output ( ) and used regression analysis to estimate the parameters of the company s Cobb-Douglas production function: Here is the regression output (this question does not require you to use statistics such as t-stats etc.): SUMMARY OUTPUT Regression Statistics Multiple R 0.96 R Square 0.92 Adjusted R Square 0.91 Standard Error 0.43 Observations ANOVA df SS MS F Significance F Regression Residual Total Independent Variable Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept (d) [5 points] Use the regression output above to write down the production function form Show all calculations and state all assumptions. in numerical Page 7 of 27

8 (e) [3 points] The Cobb-Douglas long run production function can be expressed as: ( ) Use this expression to prove (not state) whether Michelle s company: Has a strictly concave, linear or strictly convex cost function? and curves rise, fall, or stay constant as?? Make sure to answer this question using parameter values for Michelle s company (you re not being asked to prove general results). Show all necessary calculations and state all assumptions. Page 8 of 27

9 (f) [2 points] Michelle used Excel Solver s to solve the following long run CMP for her company (notice the max below): [ ] Reproduced below is a portion of her Excel sensitivity report : Microsoft Excel 14.0 Sensitivity Report Final Lagrange Cell Name Value Multiplier $C$12 Q Interpret the Lagrange multiplier what does it mean? Give a brief explanation. Page 9 of 27

10 (g) [2 points + 2 bonus points] Suppose that in the short run, Michelle s company has the following production function: True or false: producing more output by always doubling labor results in a U-shaped short run curve? You do not have to prove your answer: rather, state the answer and give a brief explanation. Bonus points awarded for proving your answer? Page 10 of 27

11 Question 2 [20 points] Consider a risk free asset and two risky assets A and B. The following table provides partial information on returns and variances of the three assets: Asset Return Variance A B Risk Free The following table contains partial information on the covariance of asset returns: A B Risk free (a) [3 points] What are Covariance of Returns A B Risk free You do not need to prove your answer. Page 11 of 27

12 (b) [17 points] Portfolio X has been formed by combining the risk free asset and risky assets A and B. Fill in the blanks in the following table. Show all calculations. Portfolio Fraction of portfolio In Risky Asset Fraction of portfolio In Risky Asset Fraction of portfolio in risk-free asset Portfolio Return Portfolio Risk Portfolio Risk Premium Portfolio Price of risk X ???? All assets must be considered in your calculations (i.e. calculations of the portfolio return, risk, risk premium, and price of risk must include all three assets). Enter your calculations in the table below. Page 12 of 27

13 Portfolio Fraction of portfolio In Risky Asset Fraction of portfolio In Risky Asset Fraction of portfolio in risk-free asset X Portfolio Return Portfolio Risk Portfolio Risk Premium Portfolio Price of risk Page 13 of 27

14 Question 3 [55 points] [This question is loosely based on the Ivey Business School case Gold Claims at Sturgeon Lake ] This question consists of three parts A, B, and C. all parts to two decimal places. Andrew McKendry, a geologist, has been hired by a Toronto based mining company to advise them about the following two mutually exclusive decisions regarding gold mining operations at Sturgeon Lake (near Thunder Bay, Ontario): Road Drill : First attempt to build a permanent road to the drill site at Sturgeon Lake and if the road project is successful to then drill for gold. The cost of constructing the road is $33, and there s a 70% chance that the road construction project will be a success. The cost of drilling is $98, and there s a 22% chance of finding gold. If the road construction project and drilling are both successful then the of mining profits (before construction and drilling costs) is $2,194, Drill Road : First build a temporary ice road (with a 100% chance of success) and drill for gold. The ice road costs nothing to build. The cost of drilling is $98, and the probability of drilling for gold is If drilling is successful the mining company will attempt to build a permanent road; the cost of constructing the permanent road is $33, and there s a 70% chance that the road construction project will be a success. If both drilling and road construction are successful, then the of mining profits (before drilling and construction costs) is $2,194, For your convenience, here is a summary of the numbers: ( ) ( ) ( ) Page 14 of 27

15 (a) [5 points] Draw the decision tree for this problem. Do NOT solve the decision tree just yet. HINT: You might want to first sketch the decision tree on a worksheet at the back of this test before drawing the final version below. Page 15 of 27

16 Part A (b) [10 points] Suppose that the Toronto based gold mining company is risk neutral : what are its optimal decision and optimal course of action? Show all calculations and state all assumptions. Page 16 of 27

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18 (c) [5 points] [This part is independent of all other parts below] Let probability that drilling for gold is a success. For what values of will the mining company make the same decision as your answer to part (b)? Show all calculations and state all assumptions. Page 18 of 27

19 Part B [Part B is independent of Part C below] (d) [10 points] Return to the original numbers at the beginning of the question: ( ) ( ) ( ) For this question you should use the decision tree in part (a). Now suppose that the Toronto based gold mining company is risk averse and its board of directors is of the opinion that: ( ) { } { } What are the optimal decision and the optimal course of action? Show all calculations and state all assumptions. Page 19 of 27

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21 (e) [5 points] [This part is required to answer part (f) only] Suppose a risk averse agent has wealth. With probability the agent s wealth will be reduced by. Suppose this agent wants to insure against the risky situation { ( ) }. What is the optimal amount of the insurance policy if the insurance industry is charging actuarially fair prices? Show all calculations and state all assumptions. Page 21 of 27

22 (f) [5 points] Consider the decision you have made in part (d) (not the whole tree; just the decision). Suppose the mining company has purchased actuarially fair insurance against the risk of drilling. Given your answer to part (b), what is the insurance premium? Show all calculations and state all assumptions. Page 22 of 27

23 Part C [To be answered independently of Part B above] (g) [15 points] Return to the original numbers at the beginning of the question: ( ) ( ) ( ) For this question you should use the decision tree in part (a). Assume that the gold mining company is risk neutral and now suppose that the gold mining must choose between the following two mutually exclusive decisions: The optimal decision in part (b) above. The optimal decision in part (b) above but now with the option of testing the site prior to drilling. The test results will come back as either positive or negative. Based on historical data, the following table contains the probabilities of test results and actual outcomes of drilling: Test Result Drilling S F Total Total Draw the decision tree for the optimal decision in part (b) above with the option of testing prior to drilling and recommend whether you will make the decision in part (b) with or without testing. The cost of the test is not known. Page 23 of 27

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26 WORKSHEET [This page will NOT be graded] Page 26 of 27

27 WORKSHEET [This page will NOT be graded] Page 27 of 27