Do Not Write Below Question Maximum Possible Points Score Total Points = 100

Transcription

1 University of Toronto Department of Economics ECO 204 Summer 2012 Ajaz Hussain TEST 3 SOLUTIONS TIME: 1 HOUR AND 50 MINUTES YOU CANNOT LEAVE THE EXAM ROOM DURING THE LAST 10 MINUTES OF THE TEST. PLEASE REMAIN SEATED UNTIL THE PROCTOR ANNOUNCES THAT YOU CAN LEAVE THE ROOM IF YOU DETACH ANY PAGES FROM THIS TEST THEN IT S YOUR RESPONSIBILITY TO RE-STAPLE THE PAGES DO NOT HAVE A CELL PHONE ON YOUR DESK OR ON YOUR PERSON. ONLY AID ALLOWED: A CALCULATOR THERE IS A WORKSHEET AT THE END OF THE TEST GOOD LUCK! LAST NAME (AS IT APPEARS ON ROSI) FIRST NAME (AS IT APPEARS ON ROSI): MIDDLE NAME (AS IT APPEARS ON ROSI) STUDENT ID # (AS IT APPEARS ON ROSI) SIGNATURE: Do Not Write Below Question Maximum Possible Points Score Total Points = 100 Page 1 of 21

2 ALL QUESTIONS IN THIS TEST ARE BASED ON THE FOLLOWING MINI-CASE -- PLEASE READ IT BEFORE ATTEMPTING ANY QUESTIONS The Canadian airline Ajax Air (AA) has a very long term contract with the Canadian government to transport prisoners across Canada (and sometimes to other countries such as the US and Cuba) 1. AA s entire fleet of 30 aircrafts was purchased in year for each. Each aircraft has a useful life of years and salvage value of. AA s accountants, Khan, Khan & Kha-na, use the straight line depreciation method to amortize the acquisition cost of aircrafts. Currently, in year AA s annual opportunity cost rate of return of using an aircraft for its operations is of the (annual) value of the aircraft. Currently (in year ), AA operates daily flights 100 days of the year. Each daily flight requires the following complementary inputs : an aircraft, 4 guards, and 2 pilots. The annual salary of a guard is $100,000 and the annual salary of a pilot is $400,000. For the purposes of this case, set the state of management and technology parameter in AA s production function to be. Currently, in year, AA charges the Canadian government a price of for each daily flight and at this price every aircraft in AA s fleet travels once a day, i.e. AA is currently operating flights a day (every day for 100 days in year ). AA s managers estimate that lowering the price of a daily flight by 2% will increase demand for daily flights by 8%. AA s managers believe (and so should you) that AA has a linear demand curve. Due to logistical, personnel, and landing rights constraints, AA can operate at most 50 flights a day (for each of the 100 days in year ); i.e.. ALL QUESTIONS BELOW MUST BE ANSWERED SEQUENTIALLY. THAT IS, QUESTION 1 MUST BE DONE BEFORE QUESTION 2 AND SO ON. 1 After the test check out JPATS (also known as Con Air ) owned and operated by the U.S. government to transport prisoners across the US. See after this test. Page 2 of 21

3 Question 1 [25 Points] Please use the information in the mini case to answer the following parts. (1.1) [5 Points] By how much does an aircraft depreciate each year? Show all calculations and state all assumptions. Since AA s accountants Khan, Khan & Kha-na use the straight line depreciation method to amortize the acquisition cost of aircrafts we have: (1.2) [5 Points] Show that the value of an aircraft in year is. Show all calculations and state all assumptions. In general: Solving this recursively: Thus: Page 3 of 21

4 (1.3) [5 Points] Show that AA s annual opportunity cost of using its owned aircrafts for its operations is $0.6m. Show all calculations and state all assumptions. What is the intuition for this opportunity cost? The annual opportunity cost of rate of return is 5% of the current value of capital. In year cost of using its owned aircrafts for its operations is AA s annual opportunity This opportunity cost arises because when AA uses aircrafts for its operations it gives up the profits it would ve earned from leasing its aircrafts to other airlines. Page 4 of 21

5 (1.4) [5 Points] Show that AA s annual price of using an owned aircraft for its operations in year is = $4.1m. Show all calculations and state all assumptions. In general the price of capital in any period is: Thus: Page 5 of 21

6 (1.5) [5 Points] Show that in the year, AA s daily price of using its owned aircraft for its operations is. Show all calculations and state all assumptions. AA s annual price of using owned aircrafts is: From the mini-case recall that AA operates daily flights for a 100 days in the year owned capital is: and so the daily price of using Page 6 of 21

7 Question 2 [10 Points] Please use the information in the mini case to answer the following parts. Let: # of flights flown per day # of aircrafts used per day (remember an aircraft can fly at most once a day) # of guards hired per day. # of pilots hired per day. Hint: None of the inputs is fixed. (2.1) [5 Points] Write down the production function for the number of daily flights in terms of the inputs:. Show any calculations and state all assumptions. Since and are complementary inputs we know that: From the case we know that. Now, we are told that each aircraft requires 4 guards per aircraft so (i.e. ) and that each flight needs two pilots. Since each aircraft flies once a day this means we need 2 pilots per aircraft (i.e. ). Now: These are represented by the production function: This is because at the optimal solution it must be that: That is: AA needs 1 plane, 4 guards and 2 pilots per daily flight. Page 7 of 21

8 (2.2) [5 Points] What kind of returns to scale does the production function in part (2.1) have? Given your answer, will rise, fall, or remain constant as? Show all calculations and state all assumptions. From lectures we know that the complements production function exhibits constant returns to scale. Let s check this: what happens to output if we double all inputs: if more than doubles we have increasing returns to scale; if doubles we have constant returns to scale; if less than doubles we have decreasing returns to scale. The # of daily flights with some arbitrary level of inputs is: The output with doubled inputs is: Doubling all inputs doubles the output: thus, we have constant returns to scale which means not change as. is constant and does Page 8 of 21

9 Question 3 [30 Points] Please use the information in the mini case and questions 1 and 2 to answer the following parts. (3.1) [5 Points] Write down the daily price of capital that you calculated earlier and below calculate the daily price of hiring guards and pilots. Show all calculations and state all assumptions. From above:. From the mini-case we know that The annual salary of a guard is $100,000 and the annual salary of a pilot is $400,000. Since AA operates 100 days in year this means that daily prices of a guard and a pilot are: Page 9 of 21

10 (3.2) [5 Points] Setup and solve AA s cost minimization problem for the optimal number of aircrafts, guards, and pilots needed to operate a target number of flights a day. Show all calculations and state all assumptions. AA s CMP is: Notice that requires so that AA s CMP is: This CMP cannot be solved by calculus but we can exploit the fact that at optimal solution it must be that: Or: Page 10 of 21

11 (3.3) [5 Points] Derive AA s (long run) cost function of operating flights a day and calculate. Show all calculations and state all assumptions. The cost of operating flights a day (in each of the 100 days in year ) is: From above: Hence: ( ) This implies: Page 11 of 21

12 (3.4) [5 Points] What kind of returns to scale does the cost function in part (3.3) have? Does it match your answer to part (2.2)? Show all calculations and state all assumptions. Since it implies that AA has constant returns to scale which is exactly what we found in part (2.2). I like it. Page 12 of 21

13 (3.5) [5 Points] Does AA s cost function exhibit economies of scope? What is the minimum efficient scale? Show all calculations and state all assumptions. To see if AA has economies of scope we need to check if: Now: and so let s check if: In fact, the left side is exactly equal to the right side. AA does not have economies of scope (i.e. there is no benefit from merging with another identical company. Just like relationships). The is the lowest output at which is minimized. Now since is always constant: The is simply. Page 13 of 21

14 (3.6) [5 Points] Currently, What is AA s contribution margin on a daily flight? To operate, what must be the lowest price of a daily flight? Show all calculations and state all assumptions. The contribution margin of a daily flight is: Currently, AA is charging and since we are in the long run so that: If AA is rational it will cease operations is or if price falls below $53,000 (assuming AA can break the contract with the Canadian government). On the other hand if AA is irrational, it will operate at any price. Page 14 of 21

15 Question 4 [35 Points] Please use the information in the mini case and questions 1, 2 and 3 to answer the following parts. (4.1) [5 Points] Derive the equation of AA s demand curve for daily flights (i.e. the parameters in the equation ) where Price of a daily flight and # of daily flights flown. Show all calculations and state all assumptions. We know that currently and flights a day (on each of the 100 days AA operates in year ). Management feels the demand curve is linear so that: Now, we are told that managers estimate that lowering the price of a daily flight by 2% will increase demand for daily flights by 8% so that: This implies that: We can use this to calculate the parameters and. First, from we see that: Or: In the case of AA: Next, from: Substituting: and we have: Page 15 of 21

16 That is, the Canadian government s max willingness to pay for a daily flight carrying prisoners is $75,000. AA s demand curve is: Page 16 of 21

17 (4.2) [15 Points] Calculate the profit maximizing number of daily flights subject to the constraint that AA can operate at most 50 flights a day and must operate at least zero flights a day. You are expected to use the appropriate constrained optimization method. Show all calculations and state all assumptions. In general, a company with market power s PMP is: Here, we have: [ ] This is an inequality constrained problem and must be solved by the KT method: [ ] [ ] [ ] [ ] [ ] We need to check for 4 different cases: Page 17 of 21

18 When is When is? When is When is When is When is Case A Case B Case C Case D When is? When is? When is? When is When is When is When is When is? Case A: This is impossible because you can t have zero and 50 flights a day. Case B: Since the condition is satisfied. Thus, we need to see if. Since we can see the sign of from the only FOC in this problem: Since we rule out case B (how can AA operate 50 flights a day when it has a fleet of 30 planes with each plane flying at most once a day). Case C: Since the condition is satisfied. Thus, we need to see if. Since we can see the sign of from the only FOC in this problem: [ ] Since we rule out case C: it is not in AA s interest to have zero flights. Case D: Since we have ruled out other cases, case D is most likely the solution. Let s confirm. Set to get the good old ECO 100 condition (don t you miss it?): in the FOC Page 18 of 21

19 That is, even though can operate 30 flights a day (its fleet) it s optimal to operate 22 flights a day. The optimal price of a daily flight will be: Page 19 of 21

20 (4.3) [5 Points] Given your answer to part (4.2), is it optimal for AA to use every aircraft in its fleet for transporting prisoners? Show all calculations and state all assumptions. No. The fleet size is 30 planes and with each plane flying at most once a day, AA has the capacity of operating 30 flights daily but should operate 22 flights daily. What should AA do with the surplus 8 planes or 8 daily flights? Well, lease the planes to other airlines (after all, that s why we had an opportunity cost term in the price of capital). Page 20 of 21

21 (4.4) [5 Points] Calculate the price elasticity at the optimal solution. Show all calculations and state all assumptions. So easy and cute. Now given that and and we have: This is elastic as must be the case with any profit maximizing price and quantity. (4.5) [5 Points] Calculate AA s gross profit (what will AA do with any unused aircrafts?). Show all calculations and state all assumptions. AA s gross profits is: Page 21 of 21