University of Toronto Department of Economics ECO 204 Summer 2013 Ajaz Hussain TEST 2 SOLUTIONS GOOD LUCK!

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1 University of Toronto Department of Economics ECO 204 Summer 2013 Ajaz Hussain TEST 2 SOLUTIONS TIME: 1 HOUR AND 50 MINUTES DO NOT HAVE A CELL PHONE ON YOUR DESK OR ON YOUR PERSON. ONLY AID ALLOWED: A CALCULATOR YOU CANNOT LEAVE THE EXAM ROOM DURING THE LAST 10 MINUTES OF THE TEST STAY SEATED UNTIL ALL TESTS HAVE BEEN COLLECTED AND THE PROCTORS ANNOUNCE THAT YOU CAN LEAVE THE ROOM IF YOU DETACH ANY PAGES FROM THE TEST, THEN YOU MUST RE-STAPLE THESE LOOSE PAGES TO THE TEST GOOD LUCK! LAST NAME (AS IT APPEARS IN ROSI): FIRST NAME (AS IT APPEARS IN ROSI): MIDDLE NAME (AS IT APPEARS IN ROSI) 9-DIGIT STUDENT ID # (AS IT APPEARS IN ROSI) SIGNATURE: DO NOT WRITE BELOW. FOR GRADER S USE ONLY Question Maximum Possible Points Score Total Points = 100 Page 1 of 16

2 Question 1 [45 POINTS] Ajax Airlines (AA) operates contractual and commercial cargo flights between Toronto and Montreal. Exhibit 1 contains the number of hours AA operated contractual and commercial cargo flights each month in 2013Q1. AA charges commercial customers $1,000/hour and under the terms of the contracted flights, the average monthly revenues are capped at $50,000. AA Contractual and Commercial Hours in 2013Q1 January 2013 February 2013 March 2013 April 2013 Contractual Hours/month Commercial Hours/month (a) [5 POINTS] Given that AA s average monthly contractual revenues are capped at $50,000 calculate the uniform price ($/hour) AA charges its contractual customers. Show all calculations. Average monthly revenues in 2013Q1 are: { } Since the contractual price is always a uniform $ per hour we have: { } { } per hour Page 2 of 16

3 (b) [5 POINTS] In AA s management opinion, raising (lowering) the commercial price by 5% in April 2013 would ve decreased (increased) commercial monthly demand by 2.5%. Use these opinions to compute the price elasticity, commercial demand function, and the commercial demand curve. Justify the functional form (i.e. linear vs. non-linear) of the demand function/curve. Show all calculations. Given the information we see that: { { Using either of these scenarios the price elasticity is: It is OK if students compute by arc elasticity method Notice that the new price and quantities are symmetrical around the April price and output which means that the demand function is linear: We can solve for and by solving these two equations obtained by subbing in the higher and lower new price and output combination (any two price-output pairs will do): This yields: Thus the demand function is: From this we have the demand curve: Page 3 of 16

4 (c) [10 POINTS] Assume that in May 2013, AA s commercial demand function/curve is what you calculated in part (b) and that AA s commercial capacity is of the unconstrained commercial revenue maximizing output. Calculate the optimal price and output in May 2013 for the case where optimal output is less than or equal to capacity. Use the appropriate constrained optimization method (there s no need to have the constraint that ) Show all calculations. Let commercial hours in a month and commercial price per hour. In May 2013, the commercial demand curve is: The commercial revenue function is: Noting that the revenue function is strictly concave: We see that the if we find a stationary point then it must be the solution to the unconstrained revenue maximizing problem: Thus, in May 2013, commercial capacity is: In May 2013, AA s real commercial revenue maximization problem is: The Langrangian equation is [ ] The FOC and KT conditions are: Start by checking the animal which is true whenever: Case A Case B Page 4 of 16

5 The FOC becomes: Case A This violates the condition that and therefore Case A cannot be the solution. Case B The FOC becomes: This satisfies the condition that and therefore Case B is the solution. In this case, the price will be: The complete solution is: Page 5 of 16

6 (d) [10 POINTS] What is the impact of a 1% excise tax on AA s commercial price, output, AA s revenues, tax revenues, and commercial customers welfare? State all assumptions and show all calculations. Following the 1% excise tax, the new commercial price will be: The new commercial output will be: The impact on commercial revenues is: Excise tax revenue will be: As for impact on consumer welfare: suppose we model all commercial consumers behaving as a single representative consumer with the following quasi-linear utility function over commercial hours and everything else: Where is some equation in terms of with and. If we treat all other goods as the base good (i.e. ) and assume an interior solution to the UMP, then the from the demand curve of is equal to. As such, the change in is equal to the change in optimal utility due to the excise tax: Page 6 of 16

7 (e) [10 POINTS] Should AA try to persuade the government not to tax AA s commercial price and to instead tax AA s commercial customers income? If so, explain your answer using a mathematical model and calculate the amount of the income tax; if not, explain your answer using a mathematical model and show why AA might be better off with an excise tax on its commercial price. State all assumptions and show all calculations. We have seen that a 1% excise tax on AA s commercial customers reduces consumer welfare by $1,980. The total tax revenue raised from this excise tax is: There are two reasons AA should try to convince the government to impose an income tax of $1,960 on AA s commercial customers: one, the loss in consumer welfare from the income tax is less than from the excise tax, and second, the income tax has no impact on AA s commercial demand. Here s the proof of the loss in consumer welfare from the income tax is less than from the excise tax: the representative consumer s UMP is: The Lagrangian is: [ ] By the envelope theorem, the change in utility due to an income tax is: By the envelope theorem, the change in utility due to an excise tax is: Since we assumed that the consumer has monotone preferences so that Then regardless of the value of we see that: Hence, a revenue equivalent income tax inflicts less pain on consumers than an excise tax. Here s the proof of that the income tax has no impact on AA s commercial demand: we had assumed an interior solution to the UMP which means that at the optimal bundle: Therefore regardless of the actual equation for demand for commercial flights is independent of the income level. Thus, an income tax of $1,960 would not only raise the same tax revenue as the excise tax, it would inflict less pain on consumers and not impact commercial demand. Page 7 of 16

8 (f) [5 POINTS] Assuming that all commercial customers can be modeled as a single representative consumer, derive the this representative consumer s utility function. State all assumptions and show all calculations. Suppose the representative consumer has the following quasi-linear utility function over commercial hours and everything else: Here is some equation in terms of with and. If we treat all other goods as the base good and assume an interior solution we d have: Now sub so that: Thus, the representative consumer has the following quasi-linear utility function over commercial hours and everything else: Page 8 of 16

9 Question 2 [35 POINTS] Consider an investor with a mean-variance utility function. (a) [5 POINTS] A risk free asset has no risk. Put another way, it offers a guaranteed return. That said, government bonds such as US 3 month T-Bills are often labeled risk free. However, as the following graph shows, the interest rate on US 3 month T-Bills (the return on the bonds ) have been anything but certain -- how do you explain this apparent contradiction? While there is variance in historical interest rates, there is no variance in future interest rate provided the investor does not sell the bond before it matures and she cashes it in the same currency as it was issued. Page 9 of 16

10 (b) [5 POINTS] Why does finance model investors preferences over returns and risk by the mean-variance utility function? Investors have preferences over the entire distribution of returns. It has been argued that most the returns of most risky assets have a normal distribution (see example below) which is completely characterized by two parameters: mean and variance. Thus the mean-variance utility model is capturing preferences over the entire distribution of returns. Page 10 of 16

11 (c) [5 POINTS] Suppose the investor has invested in a risky asset with expected return [ ] and risk. True or false: as the risky asset s risk rises, to remain indifferent to the initial return-risk bundle, the investor must be compensated with higher and higher compensating returns? Show all calculations. The mean-variance utility function is: Note that: [ ] [ ] This implies that: [ ] This expression is proportional to risk and and implies that as risk increases, to leave the investor indifferent to the initial return-risk bundle, we would have to compensate her with higher and higher expected returns in proportion to the risk and the parameter ( risk aversion parameter ). Page 11 of 16

12 (d) [10 POINTS] An investor with a mean-variance utility function allocates a fraction of her portfolio funds to a risky asset with expected return [ ] and variance and the remaining fraction to a risk free asset with return. All else equal, what happens to as the investor becomes less risk averse, the risk premium and the price of risk? Show all calculations. You must derive the expression for. The investor chooses by solving this UMP: [ ] Now: [ ] [ ] [ ] [ ] { [ ] } [ ] [ ] [ ] Sub into: [ ] { [ ] } [ ] ( [ ] ) We saw that the parameter reflects the investor s attitude towards risk; thus, as we have that. As risk premium we see that. Finally, notice that: ( [ ] ) Notice that as price of risk falls (without risk rising) then Page 12 of 16

13 (e) [10 POINTS] In project 2 you were asked to synthesize three risky assets called let s say A, B and C. Write down the expression that was used to choose the weights of these three assets in the synthetic asset. Show all calculations. Let the expected returns of the three risky assets be [ ] [ ] [ ] and the risks of the three risky assets be. Do NOT solve for the weights and show all calculations. The synthetic asset was blended by choosing to maximize the price of risk: Now: [ ] In turn: [ ] [ ] [ ] [ ] In turn: ( ) You saw on project 2 that: Thus: { } Thus: In sum, the synthetic asset was blended by choosing to maximize the price of risk: [ ] [ ] [ ] You used Solver to solve for the weights. Page 13 of 16

14 Question 3 [20 POINTS] (a) [5 POINTS] Consider a production function with as inputs. True or false: if the production function exhibits increasing returns to scale then it must exhibit increasing returns to labor? Prove your answer by an example and show all calculations. False. Consider the Cobb-Douglas production function: This production function has increasing returns to scale whenever and increasing returns to labor whenever However, just because does not mean that as in the following example: Page 14 of 16

15 (b) [5 POINTS] Keystoned Corporation operates a pipeline carrying oil from Canada to the US. Keystoned wants to expand pipeline capacity and can do so by either doubling the size (radius) of the existing pipeline or build another identical pipeline. Assuming both options have the same cost should Keystoned double the current pipeline or build a new identical pipeline? What would happen to Keystoned s under your proposed option? Give a brief explanation. Building a new identical pipeline would double capacity while doubling the radius of the existing pipeline would quadruple capacity since: Doubling the radius means that: Thus, doubling the radius of the existing pipeline results in 4 times the original output or increasing returns to scale (so that as ). Page 15 of 16

16 (c) [10 POINTS] In June 2013, Jet-Airways purchased an aircraft for $200m with a useful life of 5 years. The opportunity cost rate of return for the first 3 years is 10% and the last two years 5%. Calculate the value of the aircraft and the price of using the aircraft in Jet-Airways operations in years and. Show all calculations and use the declining balance depreciation method. The price of using capital is: Now, let s derive an expression for in terms of purchase price and useful life : { } From this we have: Thus, using: We have at And at Page 16 of 16