Department of Economics ECO 204 Microeconomic Theory for Commerce 2012 2013 Ajaz Hussain Test 2 Solutions IMPORTANT NOTES: Proceed with this exam only after 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, 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: 1 hour and 50 minutes Total number of questions: 4 Total number of pages: 26 Total number of points: 100 Please answer all questions. To earn credit you must show all calculations. Please see last page of this exam for the allocation of points across questions. 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. Signature: Last Name: First Name: 9-Digit Student ID #: Please circle the section in which you are registered (NOT the section you attend): L0101 MON 1 3 L0201 TUE 11 1 L0301 TUE 2 4 L5101 WED 6 8 EXAMS WITHOUT SIGNATURES, ID # AND NAMES WILL NOT BE GRADED. Page 1 of 27
Question 1 [25 POINTS] Consider an agent in an economy with a single good (corn) and where all agents live for exactly two periods At the beginning of each period, the agent is endowed with real incomes and (in units of corn). Let and denote the amounts of corn consumed in and respectively. Assume that the inter-temporal consumption set is { } At agents can save or borrow corn at the real interest rate. Agents do not save to bequeath savings to future generations because in this economy there s no romance and no possibility of amorous encounters between agents as such, agents do not have children. (1.1) [3 POINTS] Suppose an agent tells you that she must consume positive amounts of corn in both periods and that she perceives and as imperfect substitutes with diminishing marginal rate of substitution. Write down the equation of a utility function that represents this agent s preferences and show why this utility function represents the agent s preferences. Show all calculations and state all assumptions. We can represent the agent s preferences over inter-temporal consumption by the Cobb-Douglas utility function: Where are positive parameters. Notice that the for we must have and which means she must consume corn in each period. The marginal rate of substitution (slope of indifference curves) is: As we see that (i.e. diminishing marginal rate of substitution). Page 2 of 27
(1.2) [7 POINTS] State and solve this agent s inter-temporal Utility Maximization Problem (UMP) with a real (Future Value) inter-temporal budget constraint. To earn credit you must use the appropriate constrained optimization method, solve for all variables, and interpret any Lagrange multipliers. Show all calculations. Hint: If applicable, use a convenient positive monotonic transformation of the utility function in part (1.1). We can represent the agent s preferences over inter-temporal consumption by the Cobb-Douglas utility function: Where are positive parameters. For convenience, take the log-linear transformation: The inter-temporal UMP is: Assuming, combined with, implies that at the optimal solution. As such, we can drop the non-negativity constraints: The Lagrangian equation is: { } For convenience, denote so that: { } The FOCs are (there are no KT conditions why?): { } Re-arranging the first two FOCs gives: The 3 rd FOC implies: Page 3 of 27
Solving and simultaneously yields: Substitute these in any of the first two FOCS to get: By the envelope theorem, what is the interpretation of? The change in due to a small change in arising from a change in either or but not is: { } At the optimum, so that: That is, is the marginal utility of future value lifetime income due to a small change in either or but not : Notice that this is strictly positive. Page 4 of 27
(1.3) [5 POINTS] Suppose the agent is given the opportunity of having either one more unit of corn income at or one more unit of corn income at. True or false: the agent will prefer to have one more unit of corn income at? Show all calculations. By the envelope theorem, the change in due to a small change in arising from a change is: { [ ]} At the optimum, so that: This is the marginal utility due to a small change in. Applying the envelope theorem to change in yields: Notice that since : Thus, the agent prefers to have more corn at than a dollar tomorrow.. This is the classic finance principle of a dollar today is worth more Page 5 of 27
ECO 204, 2012-2013, Test 2 (1.4) [3 POINTS] Use your answer to part (1.2) to calculate that value of the real interest at which the agent is indifferent between saving and borrowing at. Show all calculations. Recall that the savings function can be described by the following equation where we substitute for the optimal value of consumption at time, from part (1.2): The agent is indifferent between saving and borrowing at when. Setting in the equation above, solve for the value of the real interest rate Therefore, at the agent is indifferent between saving or borrowing at Page 6 of 27
(1.5) [4 POINTS] Suppose the current real interest rate is the value calculated in part (1.4). If the government levies a small flat income tax (in units of corn) on will the agent become a saver or a borrower at? Show all calculations. : Suppose,, i.e. the agent is indifferent between saving and borrowing at If the government levies a small flat income tax at, then the new future value of income,, is: ( ) And savings at are: ( ) ( ) ( ) Simplifying the expression above yields: since by assumption. Therefore, at the interest rate from part (1.4) such that the agent indifferent between saving and borrowing, the agent becomes a saver if the government levies a small flat income tax at. Page 7 of 27
(1.6) [3 POINTS] Use your answer to part (1.2) to answer this question. True or false: a small flat income tax (in units of corn) on has a larger impact on consumption at than consumption at? Show all calculations. With a small flat income tax, the future value of income is given by: Using the solution from part (1.2) for optimal consumption at and, we can compare the impact of a small income tax on consumption in two periods: Impact of a small flat income tax, on consumption at time is : Impact of a small flat income tax, on consumption at time is : A small flat income tax has a larger impact on consumption at time if Page 8 of 27
Question 2 [30 POINTS] (2.1) [3 POINTS] The following table contains the monthly closing price and monthly dividend yields on Sunoco and JP Morgan stocks for the last 6 months of 2011: Date Sunoco Price Sunoco Dividend Yield JP Morgan Price JP Morgan Dividend Yield 7/29/2011 40.65 0 40.45 0.006107 8/31/2011 38.14 0.00369 37.56 0 9/30/2011 31.01 0 30.12 0 10/31/2011 37.23 0 34.76 0.008301 11/30/2011 38.81 0.004029 30.97 0 12/30/2011 41.02 0 33.25 0 Source: CRSP through U of T s CHASS system If the last digit of your ID # ends in 0, 2, 4, 6, 8 Calculate Sunoco s monthly capital gains and monthly dividend issued and enter these values below. Give a brief explanation on how to calculate capital gains and dividends. If the last digit of your ID # ends in 1, 3, 5, 7, 9 Calculate JP Morgan s monthly capital gains and monthly dividend issued and enter these values below. Give a brief explanation on how to calculate capital gains and dividends. : Numbers in cells correspond to Sunoco/JP Morgan respectively: Circle the stock you are analyzing: Sunoco JP Morgan Date Monthly Capital Gains Dividend Issued July 2011 NA/NA NA/NA August 2011-0.0617/-0.071 0.1499985/0 September 2011-0.1869/-0.1981 0/0 October 2011 0.2006/0.1541 0/0.25003 November 2011 0.0424/-0.1090 0.14999967/0 December 2011 0.0569/0.0736 0/0 Brief explanation on how to calculate capital gains and dividends: To calculate capital gains in period To calculate dividend issued in period Page 9 of 27
(2.2) [3 POINTS] On December 1 st, 2011, the monthly interest rate on 3-month US T-Bills and reported on an annualized basis was 0.01%. Calculate the monthly interest out of 100 on a de-annualized basis. Hint: You were asked to do this in Excel Project #1. Only if you are unable to calculate this number here should you use the value 0.00002 below for the risk free rate (note this is NOT the correct value and you will get a zero in all parts below if you calculate the correct value here but use 0.00002 below instead). Monthly rate calculated on a de-annualized basis is equal to (2.3) [3 POINTS] The following table shows the average monthly returns and risk of Sunoco and JP Morgan stocks as well the covariance of Sunoco and JP Morgan monthly returns over the period May 1969 to December 2011: Sunoco JP Morgan Average Monthly Return 0.0106 0.0114 Risk 0.0863 0.0917 Covariance 0.0016 Briefly explain: how to calculate the risk of a stock why JP Morgan stocks offer higher returns (on average) than Sunoco stocks. : (1) The risk of a stock is the standard deviation of its returns. (2) JP Morgan offers higher returns on average than Sunoco in order to compensate its investors for the higher risk associated with holding JP Morgan stocks. Page 10 of 27
(2.4) [4 POINTS] Write down an investor s mean-variance utility function (be sure to use the correct notation for the utility function parameter(s)). True or false: as the asset becomes riskier, the investor demands higher returns but at a decreasing rate? Show all calculations : An investor s mean-variance utility function over the return and risk is: where: [ ] - is the expected return on portfolio - is the risk-aversion parameter - is the variance/risk of the portfolio An investor is facing a trade-off between the risk and return of the portfolio which can be described by the marginal rate of substitution (MRS) between the risk and return: It is true that the investor demands higher returns for the riskier asset: as seen from the MRS, to keep investor indifferent, if risk increases by one unit, the expected return has to increase by (assume that the investor is risk averse such that the risk-aversion parameter implies that ). To check whether investor requires higher returns at a decreasing rate, take derivative of MRS with respect to risk, and see whether MRS is decreasing in Therefore, investor demands higher returns for riskier asset at an increasing rate (False!). Page 11 of 27
(2.4) [5 POINTS] For the month of January 2012, an investor wishes to construct a portfolio consisting of: If the last digit of your ID # ends in 0, 2, 4, 6, 8 Sunoco stocks and 3-month US T-Bills. If the last digit of your ID # ends in 1, 3, 5, 7, 9 JP Morgan stocks and 3-month US T-Bills. Using the risk free rate calculated in part (2.2), calculate that value of the mean-variance utility function parameter at which the investor will allocate 100% of portfolio funds in the risky asset. You are expected to solve the appropriate UMP. Show all calculations to 4 decimal places. : Let be a fraction of portfolio invested in a risky asset. Notice that the expected return of the investor s portfolio is equal to a weighted average of the returns of the risky and risk free asset: [ ] [ ] (1) The variance of the portfolio is equal to (apply the variance formula from ECO220Y to ): The UMP of the investor is: [ ] (2) Substituting (1) and (2) into UMP, the investor s problem becomes: [ ] Solving for yields: [ ] If the investor allocates 100% of the portfolio funds into the risky assets, then. The value of the mean-variance utility function parameter is: [ ] For Sunoco, For JP Morgan, Page 12 of 27
(2.5) [2 POINTS] For the month of January 2012, construct a portfolio consisting of: If the last digit of your ID # ends in 0, 2, 4, 6, 8 Sunoco stocks and 3-month US T-Bills by using the risk free rate calculated in part (2.2) and 1.05 times the value of the mean-variance utility function parameter calculated in part (2.4). What fraction of the portfolio is in the risky asset, what is expected return and risk of this portfolio? Is the investor leveraging her portfolio and if so, indicate how she does this. If the last digit of your ID # ends in 1, 3, 5, 7, 9 JP Morgan stocks and 3-month US T-Bills by using the risk free rate calculated in part (2.2) and 0.8 times the value of the mean-variance utility function parameter calculated in part (2.4). What fraction of the portfolio is in the risky asset, what is expected return and risk of this portfolio? Is the investor leveraging her portfolio and if so, indicate how she does this. Show all calculations to 4 decimal places. Sunoco JP Morgan [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] Note: In this case, the investor is not leveraging portfolio In this case, the investor is leveraging portfolio by selling 3-month US T-Bills and buying JP Morgan stocks Page 13 of 27
(2.6) [4 POINTS] For the month of January 2012, an investor wants to create a synthetic risky asset consisting of Sunoco stocks and JP Morgan stocks. What fraction of the synthetic risky asset is in each risky asset and what is expected return and risk of this synthetic risky asset? Use the risk free interest rate calculated in part (2.2), and if relevant, feel free to use the utility parameter value from part (2.5) and the formula below. Show all calculations. [ ] Proctors are NOT allowed to tell you anything about this formula. Please indicate which stocks are A and B and show all calculations to 4 decimal places. Let A be Sunoco stocks and let B be JP Morgan stocks. Then stocks. Using the formula above: is a fraction of the synthetic risky asset in Sunoco Therefore, the fraction of the synthetic risky asset in Sunoco stocks is 51% and the fraction in JP Morgan stocks is 49%. The expected return and risk of the synthetic risky asset are: [ ] Page 14 of 27
(2.7) [3 POINTS] For the month of January 2012, construct a portfolio consisting of: If the last digit of your ID # ends in 0, 2, 4, 6, 8 3-month US T-Bills and the synthetic risky asset from part (2.6). Use the risk free rate calculated in part (2.2) and use 0.9 times the value of the mean-variance utility function parameter calculated in part (2.4). What fraction of the portfolio is in the risky asset, what is expected return and risk of this portfolio? Is the investor leveraging her portfolio and if so, indicate how she does this. If the last digit of your ID # ends in 1, 3, 5, 7, 9 3-month US T-Bills and the synthetic risky asset from part (2.6). Use the risk free rate calculated in part (2.2) and use 1.1 times the value of the mean-variance utility function parameter calculated in part (2.4). What fraction of the portfolio is in the risky asset, what is expected return and risk of this portfolio? Is the investor leveraging her portfolio and if so, indicate how she does this. Show all calculations. ID # ends in 0, 2, 4, 6, 8 ID # ends in 1, 3, 5, 7, 9 [ ] [ ] Note: 218% of this portfolio is invested in the synthetic risky asset. The investor is leveraging by shortselling the risk-free asset 3 month US T-Bills [ ] [ ] [ ] 187% of this portfolio is invested in the synthetic risky asset. The investor is leveraging by short-selling the risk-free asset 3 month US T-Bills [ ] [ ] [ ] Page 15 of 27
(2.8) [3 POINTS] Use your answer to part (2.7) to answer this question. An investor has $5,000 to invest. Fill the entries in the following table and show all calculations. If the last digit of your ID # ends in 0, 2, 4, 6, 8 An investor has $5,000 to invest. Fill the entries in the following table and show all calculations below: US 3-Month T-Bills Synthetic Asset Sunoco JP Morgan Fraction of Portfolio in: -1.1754 2.1754 1.10959 1.0658 Dollars in: -5,876.93 10,876.93 5,547.94 5,328.99 Show calculations: If the last digit of your ID # ends in 1, 3, 5, 7, 9 An investor has $8,000 to invest. Fill the entries in the following table and show all calculations below: US 3-Month T-Bills Synthetic Asset Sunoco JP Morgan Fraction of Portfolio in: -0.8687 1.8687 0.9532 0.9156 Dollars in: -6,949.75 14,949.75 7,625.34 7,324.41 Show calculations: Page 16 of 27
Question 3 [30 POINTS] Phew! You passed ECO 204 and cruised through your third and fourth year courses. But you re one course shy of graduation. In desperation, you look for a bird course (like ECO 204) and after consulting your brainy friends you settle on RSM 000 The Business of Nothingness (nope, it s not a marketing course). The trouble is that this course is full (after all, it s a bird course) and you must choose one of the following options: Either offer the RSM 000 professor a $200 bribe to let you in. With a chance the professor accepts the bribe (tsk tsk) and immediately enrolls you in the course, for which you ll have to pay $4,000 in tuition. On the other hand, with a chance the professor is offended that you tried to bribe him and reports you for academic violation (life sucks). The school confiscates your bribe, kicks you out, and fines you $2,000. Or don t bribe the RSM 000 professor and wait to see if you get into the course. With probability you do get into RSM 000 in which case your total cost will be $4,000 in tuition and an opportunity cost of $500. On the other hand, with probability you don t get into RSM 000 in which case you will have to take a course at Ryerson University which costs you $3,000 in tuition and an opportunity cost of $500. (3.1) [7 POINTS] Assuming you are risk neutral, should you bribe the RSM 000 professor to let you into his course? Show all calculations and be sure to interpret your calculations. Hint: It is strongly recommended that you draw a decision tree. Draw decision tree here. Show calculations below. Bribe Bribe accepted with probability (1-p) and you must pay the bribe ($200) + tuition ($4000) = $4,200 Decision You are reported for academic violaitons with probability p and you must pay the bribe ($200) + a fine ($2000 )= $2,200 Do Not Bribe You are enrolled into RSM 000 with probability q = 0.1 and you must pay tuition at ($4000) + an opportunity cost ($500) =$4,500 You taka a course at Ryerson with probability (1-q) = 0.9 and you must pay tuition at Ryerson ($3000) + an opportunity cost ($500) = $3,500 Page 17 of 27
You have to make a choice between Bribe and Do Not Bribe based on the expected values of the corresponding outcomes (see the decision tree). Expected value of the Bribe option is: Expected value of Do Not Bribe option is: The decision rule is to bribe if holds:. For instance, you should bribe if the following inequality This implies that you should bribe if the probability that the professor reports you for academic violations (or simply the probability of being caught) is equal or greater than 0.3 Page 18 of 27
(3.2) [5 POINTS] Assume that the probability the professor will not take the bribe and report you for academic violations is. Will you try to bribe the professor and what is the expected value of your decision? Show all calculations. From part (3.1) we know that it is optimal to bribe if the probability that the professor will not take the bribe is greater or equal to 0.3. Let s show numerically that the expected value of Do Not Bribe when probability of being caught is 0.2 is greater than the expected value of Bribe : Since expected value of Bribe is smaller than expected value of Do Not Bribe, it is optimal not to bribe the professor. Page 19 of 27
(3.3) [10 POINTS] Assume the probability that the professor will not take the bribe and report you for academic violations is Now suppose a friend of yours, as well as the professor s, tells you that he will for a fee test the professor s integrity and honesty. Perhaps your friend will bring up the subject of bribes and get a feel for whether the professor will take the bribe. You need to decide whether to make the decision of whether to bribe or not bribe the professor on the basis of no test information (as you did in part (3.2)) or on the basis of test information. Since your friend has done this sort of stuff in the past you ask him for a breakdown of test results and what actually happened. Here s what your friend gives you: Professor Took Bribe Professor Did not Take Bribe Total Positive (evidence that professor is corrupt) 15 5 20 Negative (evidence that professor is not corrupt) 5 75 80 Total 20 80 100 Based on the information in this table, should you make the decision to bribe versus not bribe the professor by hiring your friend to do tests on the professor? Draw the decision tree below, show all calculations, and indicate what you will do if the test comes back positive or negative. Draw decision tree here. Show calculations below. Bribe Professor took the bribe with probability 0.75 Decision Test result is positive with probability 0.2 Test result is negative with probability 0.8 Do Not Bribe Bribe Do Not Bribe You are reported for academic violations with probability 0.25 RMS 000 with probability q=0.1 Course at Ryerson with probability 0.9 Professor took the bribe with probability 0.0625 You are reported for academic violations with probability 0.9735 RMS 000 with q=0.1 Course at Ryerson with 0.9 Page 20 of 27
First, let s determine the optimal decision (Bribe or Do Not Bribe) conditional on the result of the test. To do that, we need to compute four conditional probabilities from the historical data: ( ) ( ) ( ) ( ) Now, let s determine the optimal decision for each of the positive or negative result of the test. When the test is positive: Since the expected value from the Bribe option when test is smaller is bigger than the expected value from the Do Not Bribe option, you should not bribe the professor. When the test is negative: When the test is negative, you should bribe the professor. Now, the expected value of the decision to bribe or not to bribe the professor by hiring your friend is equal to: The expected value of decision with test (-$2,580) is bigger than the expected value of the decision without the test (- $3,600). So, you should hire your friend to do tests on professor. Page 21 of 27
(3.4) [4 POINTS] [THIS IS INDEPENDENT OF PARTS (3.2) AND (3.3)] Recall that you are risk neutral. From the information above, derive the equation of your utility function and compute the utilities of the four outcomes in your decision problem. Show all calculations and briefly explain the steps used to derive the utility function. Since the investor is risk-neutral, the utility function is linear and in general form can be written as, where is a slope and is a constant. Recall that utility is ordinal, i.e. we can assign arbitrarily numbers to utility levels as long as these numbers satisfy the ranking of preferences. For instance, let s assign value of 100 to the best/highest outcome (- $2,200) and 0 to the worst/lowest outcome (-$4,500). We want to find the parameters of the utility function, corresponds to the lowest and highest outcomes: { and. Consider two equations for two levels of utility which And the utility function is: The levels of utility corresponding to the four outcomes are: -$2,200 100 -$3,500 43.51 -$4,200 13.076 -$4,500 0 Page 22 of 27
(3.5) [4 POINTS] [THIS IS INDEPENDENT OF PARTS (3.2) AND (3.3)] Use your answer to part (3.4) to compute the certainty equivalence to the uncertainty arising from bribing the professor not bribing the professor. The utility function from part (3.4) is Since the investor is risk-neutral, By the definition of the certainty equivalent, : By the same logic, : Page 23 of 27
Question 4 [15 POINTS] [ALL PARTS ARE INDEPENDENT OF EACH OTHER] The original formula and production process for manufacturing Coca-Cola did not permit the use of any other sweetener besides sugar. (4.1) [5 POINTS] Assuming sugar is the only input for producing Coca-Cola, write down the equation of the long run production function, plot an iso-quant with sugar on the x-axis and fructose on the y-axis, and characterize the returns to scale. The production function with only one input,, and productivity/management parameter is: This production function exhibits constant returns to scale. That can be easily checked by doubling all the inputs: Fructose q1 q2 S1 S2 Sugar Page 24 of 27
(4.2) [5 POINTS] Assuming sugar, capital and labor are the only complementary inputs for producing Coca-Cola (where each machine requires 3 units of sugar and 2 workers) write down the equation of the long run production function, plot an iso-quant with sugar on the x-axis and labor on the y-axis, and characterize the returns to scale. Denote machines by, workers by and sugar by. We know that the inputs are combined in the production such that The long run production function with complementary inputs has the functional form: { } This production function exhibits constant returns to scale and we can check it by doubling all the inputs: { } { } Labor Sugar Page 25 of 27
(4.3) [5 POINTS] Eventually Coke was able to reformulate Coca-Cola s formula and manufacturing process to allow the use of sugar and/or fructose as imperfect substitutes (with diminishing technical rate of substation) and permitting the use of no sugar or no fructose. Write down the equation of a long run production function that describes this production process, plot an iso-quant with sugar on the x-axis and fructose on the y-axis, and characterize the returns to scale. There are many examples of production functions that would fit these criteria. One class is quasi-linear production functions. For example, We can characterize the returns to scale by doubling all the inputs: This long run production function exhibits decreasing returns to scale as the scaling parameter at is smaller than 2. In other words, doubling all inputs leads to increase in the output less than by two. The graph of the iso-quant for this production function is described by equation: Fructose Sugar Page 26 of 27
DO NOT WRITE HERE FOR USE BY GRADERS ONLY Question 1 2 3 4 Maximum Possible Points Total Points = 100 Score Page 27 of 27