TACKLE ALL PARTSP5

Problem 1 The Airfare Problem1. You are trying to get the cheapest airfare that you can. You just called up and found that the ticket home will cost $400, and it cannot be refunded or exchanged. You can also buy a ticket for $450, which can be refunded for $430 (and thus costs you $20). The price of tickets will change in one week, and you will have one more chance to buy a ticket. There is a 50% chance that the ticket would cost $300, and a 50% chance that it would cost $600. What should you do to minimize the expected expense? Write the Airfare problem as a decision tree, and solve it. Problem 2 Value of Information Suppose Tom is at a carnival and while there he sees a green monkey stuffed animal that he would like to have. In order to win the stuffed monkey Tom needs to play a game that costs $1. In the game there are 5 shells and a ball that is hidden behind one of the shells. In order to win the monkey, Tom needs to pick the shell that the ball is under. The monkey overall is worth $3. Assume in all parts of this problem the probability the ball is behind a give shell is .2 (i.e. 1/# of shells) Part 2.A Determine using a decision tree if it is in Toms best interest to play the game. Part 2.B After deciding whether to play the game, Tom asks the carnie if the third shell has the ball under it. Since we all know carnies never try to deceive anyone, you know that what he tells you will be correct. What is the greatest amount that Tom should be willing to pay for this perfect information? Part 2.C Suppose instead that Tom asks the carnie before deciding whether to play the game. How does Toms answer change? 1This problem is also discussed in the class. Picture of infant holding green stuffed monkey removed due to copyright restrictions. Gerber logo removed due to copyright restrictions. Problem 3 Gerber Product, Inc. (True story from 1998-99. ) Gerber Products, Inc. needs help in deciding whether to continue using the plastic known as PVC in pacifiers and feeding products. PVC, a composite plastic, is made pliable with a softening chemical called phthalates. Although phthalates have been used in plastic for over 30 years without any cases of health problems, Greenpeace announced that their scientific testing on the chemical has been found it to be carcinogenic in lab rats. Due to this announcement the Consumer Product Safety Commission has decided to study the matter further and issue a press release with their results. This is the point at which Gerber decides to implement a decision tree. Gerber faces two choices: be reactive, wait for the announcement, and gauge consumer response before deciding on a course of action; or be proactive and pursue resolution regardless of the publics response to the report. The CSPC will either issue an unfavorable report recalling of all products with phthalates, or a more favorable one expressing minimal concern over the issue. The two reports have equal probability. If Gerber chooses to be proactive, they can choose to discontinue all products with PVC. In this case if the report is favorable, there is an 80% chance that the public would react favorably causing sales to increase by $1 million, but also there is a 20% chance that sales would decline by $1 million. If the report is negative there is a 25% likelihood that Gerber could preserve current sales, but also a 75 % probability that a recall would hurt sales by $1.25 million. In the event that Gerber waits for the CSPC report before taking action. With a favorable report and a delayed response, there is a 25% chance that sales would remain flat, along with a 75% chance that sales would decline by $2 million. The worst-case scenario is if Gerber remains passive and there is a recall. In that case, there is still a 20 % probability that Gerber could increase sales by $.5 million. However, it was considered an 80% probability that significant volume would be lost ($10 million). Part 3.A Set up the decision tree and find the optimal decision for Gerber. Part 3.B In the worst case scenario, Gerber would be compensated for some of their losses. How much would Gerber need to be compensated so that Gerber would act passively?

Problem 3 Each year, Data Corporal produces as many as 400 computers in Boston and 300 computers in Raleigh. Los Angeles customers must receive 400 computers, and 300 computers must be supplied to Austin customers. Producing a computer costs $800 in Boston and $900 in Raleigh. Computers are transported by plane and may be sent through Chicago. The costs of sending a computer between pairs of cities are shown in the following table. To From Chicago Austin Los Angeles Boston 80 220 280 Raleigh 100 140 170 Chicago - 40 50 (a) Formulate a Minimum Cost Network Flow problem that can be used to minimize the total (production + distribution) cost of meeting Data Corporal's annual demand. (b) How would you modify the Part A formulation if at most 200 units could be shipped through Chicago? (c) How would you modify the network if 500 computers could be produced in Raleigh instead of 300? This illustrates a general strategy for dealing with excess supply. Problem 3 Short answer (True or False). 1. For a Minimum Cost Network Flow Problem, a supply of -1 for a node is equivalent to a demand of 1. 2. For a Minimum Cost Network Flow problem, if the specified supply is not equal to the demand (that is, the sum of the bi's does not equal 0), the corresponding Linear Program with flow balance equality constraints must be infeasible. 3. It is possible for an undirected graph to have an Eulerian Path but not an Eulerian Cycle. 4. Suppose all arcs in a network have different lengths. Then the network has a unique shortest path. 2 Problem 5 Multiple answer 1. Suppose we delete an arc of length k from a graph. Which of the following possibilities can happen to the shortest path from s to t? (a) It could increase by any number between 0 and k. (b) It could increase by a number greater than k. (c) It could possibly become infinite. (d) It could decrease. 2. Let G = (N, A) be a network withn nodes and m arcs. (i) The maximum number of distinct s-t paths can be greater than 2m. (ii) If m < n 1, then G is not connected. (iii) If m is odd, then the graph can not contain an Eulerian Cycle. (iv) If m is even, then every s-t cut has an even number of arcs.

Problem 1 Consider the linear programming feasible region and the integer program feasible region given in the diagram to the right. Figure 1: Sketch of the feasible region of the LP relaxation for Problem 1 Which of the following are valid inequalities for the integer program? (a) x 4 (b) x 2 (c) y 1 (d) y 6 Yes (e) x + y 3. Problem 2 Consider the feasible region of an Integer Program defined by the following constraints: 3x1 + 5x2 12 4x1 + 3x2 20 x1 + x2 11/2 j = 1, 2 xj 0 j = 1, 2 xj Z.

(IP)

(a) Sketch the feasible region of the LP relaxation of the problem. Then, determine the convex hull of the feasible integer points of (IP), first graphically, and then algebraically, using your sketch to determine the inequalities that define the convex hull. (Note: the first constraint passes through the points (4, 0) and (1, 3), and the second constraint passes through the points (5, 0) and (2, 4). This should allow you to draw the feasible region very quickly.) (b) Let F be the feasible region of (IP). (Note: F is a set of integer points, it is not the feasible region of the LP relaxation of (IP).) Consider the set X defined as: X = {x1 + x2 11/2 : x1, x2 Z}. What is the relationship between F and X, in terms of inclusion? Now determine a valid inequality for the set X. Is this inequality also valid for (IP)? Justify your answer based on the relationship between F and X. (Recall the definition of valid inequality for a set: it is an inequality that does not cut off any integer feasible point of the set.) [1:35 PM, 10/25/2021] Flo: You work for a family-owned company that has four Italian restaurants in a mid-sized city. A year ago, they opened a new restaurant in a smaller suburb ten miles away. The restaurant has not prospered, and the owners have asked for your input, furnishing you with information that includes the fact that help is more expensive in this area, there are two other well-established Italian restaurants in the town, the small paper plant that employs many residents has laid off 200 workers, the restaurant has not advertised except in the newspaper, and rent on the building where the restaurant is will go up next month by $300. Prepar an analytical report on this restaurant. In this report, examine what problems have impacted the restaurant and whether these problems ... [1:36 PM, 10/25/2021] Flo: A sheep rancher agreed, in writing, to sell all the wool shorn during the shearing season to a weaver. The contract failed to establish the price and a minimum quantity of wool. After the shearing season, the rancher refused to deliver the wool. The weaver sued the rancher for breach of contract. Under UCC Article 2, will the weaver win?

1 You are the investment manager of a UK pension scheme. Following the results of an asset liability study of the scheme, the trustees have adopted a fixed benchmark. The investment objective is to outperform the benchmark by 0.75% p.a. The fixed benchmark is as follows:

There are tolerances around the benchmark weights which allow the manager to take active positions in order to deliver the required outperformance. As a consequence of a number of asset allocation decisions you have taken (all of which are satisfactorily within acceptable tolerances), a number of stock and asset class positions have arisen. One of the trustees has asked you to provide a written response to the following three questions about these positions. Draft your responses.

(a) BP Amoco is the largest stock in the FTSE All Share Index with a benchmark weight of 7.0% in that index. Your current holding in BP Amoco is 9% of the UK equity weighting which is 4.5% of the total fund. At the same time, you are underweight in US equities relative to the benchmark, so that your total exposure to US equities is only 4% of the total fund. How do you justify holding more in one UK stock than you hold in all of the US which is the world's largest stock market?

b) As part of your Index Linked exposure, you have chosen to hold 6% of the total fund in US Treasury Inflation Protected Securities (TIPS). TIPS are US Treasury Bonds which provide US inflation proofed returns, both at the income and capital level. They currently yield 3.9%, whereas the corresponding UK Index Linked securities are yielding 1.8%. You have hedged the US dollar currency exposure of the TIPS back to sterling. Why have you hedged the currency of the US Index-Linked position, when you have never hedged the currency exposure of any of your overseas equity holdings?

(c) The Japanese economy has been suffering from a series of structural problems such as abandonment of the former economic growth model, major financial restructuring, a continued decline in traditional industries, an ageing population, the strength of the Yen, the problems of dealing with global pressures of competition, and the country's balance of trade. In order to prevent these problems from plunging the economy into deep recession, the government has been obliged to provide large scale public spending plans. This would appear to be a poor environment for the market. Can you explain why you currently hold an overweight position in Japanese equities relative to benchmark?

2 You are an investment consultant recently appointed to advise the trustees of a pension scheme. For its UK equity content, the scheme employs a passive manager and two active managers, with each manager responsible for approximately one third of the UK equity assets. The first active manager uses a sector neutral approach, i.e. it holds index weight in each industry sector and seeks to achieve outperformance by stock selection alone. The second active manager uses the following process for portfolio construction. It restricts equity selection to FTSE 350 stocks. It uses a factor-based screening program based on several accounting ratios to identify 70 stocks for deeper analysis. It then researches these 70 stocks intensively, using investment research techniques, in order to determine the 30 "best" stocks. Its UK equity portfolio holds an equal weight in each of these 30 stocks, irrespective of the index weight of the stock. The research process is continuous and the holdings change to reflect the manager's research opinions. The combined portfolio of the three managers has underperformed the FTSE All Share Index by 1.5% p.a. over the last three years, after allowing for investment management fees. The trustees of the pension scheme have asked you to conduct an evaluation of the managers individually and in combination. Explain in detail how you would conduct your investigation, identifying the additional information you will require. You may assume that you will have the opportunity to interview the managers and will have access to complete performance data on the pension scheme. [1:38 PM, 10/25/2021] Flo: . In the aftermath of the financial crisis of 2007 and 2008, many US policymakers have suggested that banks and other financial intermediaries should be subjected to tighter regulations. a. Suppose that the main effect of these tighter regulations is to restore savers' confidence in the financial system and that, as a result, they work to increase the amount that savers are willing to save at any given interest rate. According to the loanable funds framework, what effect will this change in saving behavior have on the amounts of saving and investment taking place in the economy: will saving and investment rise or fall? b. Still assuming that the main effect of these tighter regulations is to restore savers' confidence in the financial system and that, as a result, they work to increase the amount that savers are willing to save at any given interest rate, what does the loanable funds framework predict about the change in the interest rate: will it rise or fall? c. Suppose instead that the main effect of these tighter regulations is to make it more difficult for firms to borrow and that, as a result, they work to decrease the amount of investment that firms are willing to undertake at any given interest rate. According to the loanable funds framework, what effect will this change in firm behavior have on the amounts of saving and investment taking place in the economy: will saving and investment rise or fall? d. Still assuming that the main effect of these tighter regulations is to make it more difficult for firms to borrow and that, as a result, they work to decrease the amount of investment that firms are willing to undertake at any given interest rate, what does the loanable funds framework predict about the change in the interest rate: will it rise or fall? 7. Ann just graduated from college; she's looking for a job, but hasn't found one yet. Dan had been looking for a job; but after having no luck finding one for more than a year, he gives up and stops looking. a. When Ann graduates and starts looking for a job, does the number of employed people go up, down, or stay the same? b. When Ann graduates and starts looking for a job, does the number of unemployed people go up, down, or stay the same? c. When Ann graduates and starts looking for a job, does the number of people in the labor force go up, down, or stay the same? d. When Dan stops looking for a job, does the number of unemployed people go up, down, or stay the same? e. When Dan stops looking for a job, does the number of people in the labor force go up, down, or stay the same? 8. According to a microeconomic model of labor supply and demand, when the government increases the minimum wage: a. Does the number of employed people go up, down, or stay the same? b. Does the number of unemployed people go up, down, or stay the same? c. Does the number of people in the labor force go up, down, or stay the same? d. Does the natural rate of unemployment go up, down, or stay the sa

1. Suppose that the market for ice cream cones starts out in an initial equilibrium in which the quantities of ice cream cones demanded and supplied both equal 6 and the price of an ice cream cone is $1.50. a. Suppose next that a hotter than normal summer causes more people to want to eat ice cream cones. In a standard microeconomic supply and demand diagram, will this event work to shift the demand curve or the supply curve? b. Will the curve you mentioned above - demand or supply - shift to the left or to the right? c. Suppose for a moment that the price of an ice cream cone remains unchanged at $1.50 after the curve you mentioned above shifts. At this old price, will there be a shortage or a surplus of ice cream cones? d. Now suppose instead that the price of an ice cream cone changes to bring the demand for and supply of ice cream cones back into balance after the curve shifts. Will this new equilibrium price be higher or lower than $1.50? e. In the new equilibrium with the new price, will the quantities of ice cream cones demanded and supplied be larger or smaller than 6? 2. Please indicate whether each statement is true or false (you don't need to explain why). a. If firms in an economy produce luxury automobiles that sell for $50,000 each and apples that sell for $1 each, then each automobile contributes the same amount as 50,000 apples to nominal GDP. b. It is possible for real GDP to rise more rapidly than nominal GDP; this happens if an economy is experiencing deflation. c. It is possible for the CPI to fall over time; this happens if an economy is experiencing deflation. d. US GDP includes the value of goods purchased by the federal government, but not by state and local governments. e. If a US citizen works temporarily in Canada, the market value of the goods he or she produces while in Canada still count as part of US GDP. 2 3. Please indicate by how much, in dollar terms, each of the follow transactions or set of transactions contributes to US nominal GDP. If GDP does not change, just write down $0. For simplicity, assume that all goods are sold during the same period in which they are produced. a. A farmer in the US sells a bag of oranges to a juice company for $5; the juice company uses the oranges to make bottles of juice in the US that then get purchased by an individual consumer in the US for $10. b. The same farmer in the US sells a bag of oranges to an individual consumer in the US for $5. c. A retired person in the US cashes his or her social security check and spends $50 on groceries, all of which were produced in the US. d. A US consumer takes $100 that he or she has saved and deposits it in the bank. e. A small business in the US manufactures and sells $1,000 worth of goods to a foreign customer; the business owner uses $500 to pay his or her rent, $250 to pay his or her workers, and keeps the ... [1:39 PM, 10/25/2021] Flo: Suppose that after observing the short-run effects of the increase in the money supply, the Federal Reserve decides not to reverse that policy action and, instead, leaves the money supply at its new, higher level permanently. a. Given that the Federal Reserve does not reverse its initial policy action, how will the economy move from the short-run equilibrium you described in question 7, above, to a new long-run equilibrium: through a shift in the aggregate demand curve, through a shift in the short-run aggregate supply curve, or through a shift in the long-run aggregate supply curve? (Note: focusing now just on the transition from the short run to the long run, only one of these curves will shift.) b. In which direction will the curve you mentioned above shift: to the left or to the right? c. Compared to its level in the initial long-run equilibrium, will the economywide level of prices in the new long-run equilibrium be higher, lower, or the same? d. Compared to its level in the initial long-run equilibrium, will real GDP in the new long-run equilibrium be higher, lower, or the same?

Suppose that the Federal Reserve acts to increase the money supply. a. In the aggregate demand/aggregate supply diagram, will this monetary policy action work initially to shift the aggregate demand curve, the short-run aggregate supply curve, or the long-run aggregate supply curve? (Note: focusing for now on just the short-run effects of the change in policy, only one of these curves will shift.) b. In which direction will the curve you mentioned above shift: to the left or to the right? c. When the curve you mentioned above shifts, what will the short-run effect on the economywide level of prices be: with it rise, fall, or stay the same? d. When the curve you mentioned above shifts, what will the short-run effect on real GDP be: will it rise, fall, or stay the same? e. When the curve you mentioned above shifts, what will the short-run effect on unemployment be: will it rise, fall, or stay the same?

3. Please answer the following questions, regarding the Federal Reserve's federal funds rate targeting strategy. a. Suppose that the Federal Reserve wants to hold its federal funds rate target constant but banks' demand for reserves decreases at any given interest rate. When faced with this shift in demand, what does the Fed have to do to keep the federal funds rate near its target: does it have to conduct an open market operation in which it buys US Treasury bonds or an open market operation in which it sells US Treasury bonds?

Problem 3 We want to solve an integer program (which is a maximization problem) using the Gomory cutting plane technique. We first consider the linear programming relaxation with the following initial LP tableau. Basic x1 x2 x3 s1 s2 s3 Rhs (z) -2 4 1 0 s1 2 1 1 1 8 s2 -4 4 2 1 8 s3 1 2 3 1 9 After three pivots, we obtain the following optimal tableau for the LP relaxation:

Basic x1 x2 x3 s1 s2 s3 Rhs (z) 0 0 -2 1 3 - 2 3 - 2 3 - 11 1 3 s1 x2 x1 1 1 -1 1 2 1 1 6 1 2 3 1 1 4 1 12 1 6 -1 1 3 1 3 1 3 2 3 1 2 3 (a) Derive two Gomory cuts based on the final two constraints of the second tableau. (b) Does the basic feasible solution from final tableau satisfy either of the two constraints from Part (a)?

Consider the knapsack set: K = {x {0, 1}7 : 11x1 + 6x2 + 6x3 + 5x4 + 5x5 + 4x6 + x7 19}. We want to derive valid inequalities for this set. (a) For each of the following inequalities, identify whether or not they are valid knapsack covers, and explain why. (a) x4 + x5 + x6 2

(b) x1 + x2 + x6 2

(c) x2 + x3 + x6 + x7 3

(d) x2 + x4 + x5 + x6 3

(e) x1 + x3 + x4 + x5 3

(f) x2 + x3 + x4 + x5 + x6 4

(b) For all valid knapsack covers in Part (a), identify whether or not they are minimal. (c) (Advanced material: understanding this question is not required for the Midterm.) Consider the knapsack cover C = {2, 3, 4, 5} that yields the inequality x2 +x3 +x4 +x5 3. Do you see a way of enlarging the cover while keeping the same right-hand side of the inequality? That is: can you find a valid inequality of the form x2 +x3 +x4 +x5 +[. . . ] 3, where the [. . . ] stands for one or more variables? If you can find such an inequality (called an extended cover), do you think that it is stronger or weaker than the initial cover? Justify your answer.

Question #1. PlannerCo's software incorporates assumptions about tax treatment in a variety of countries. In the first version of the software, these assumptions were much simpler than in the current version. The first version was used by ConsultingCo, a consulting firm who advised DevourCo, a publicly-held U.S. company, on an aggressive series of transnational acquisitions. DevourCo has since gone into bankruptcy. Those acquisitions are the focus of some messy lawsuits brought by different shareholders and creditors alleging that the acquisitions were mishandled, including allegations of misrepresentations by DevourCo under the securities laws. There is a lot of money involved and, with DevourCo in bankruptcy, there is a zealous hunt for other possible defendants. ConsultingCo was named as a defendant in several of the suits. ConsultingCo takes the position that it acted properly, but that if there were any errors in its advice or estimates these were due to defects in PlannerCo's software. ConsultingCo is now suing PlannerCo for breach of contract. The contract between ConsultingCo and PlannerCo provided that PlannerCo was not responsible for any inaccuracies in any estimates generated by the software, and that all users should consult a tax attorney. However, PlannerCo's sales force often emphasizes the sophistication and skill in tax matters of the people who designed the software. A. How would you assess the risk to PlannerCo under the contract claim, and what other facts would you want to know to better assess that risk? B. In addition, several of the shareholder groups have added PlannerCo as a defendant in their lawsuits, claiming PlannerCo's negligent design of the software contributed to their loss. How would you assess the risk to PlannerCo under these negligence claims, and what other facts would you want to know to better assess that risk? Question 2: Two years later (after all the lawsuits relating to Question #1 have been settled), BigCo has approached you to express an interest in buying PlannerCo. You want to explore this, but you want these discussions to go forward in complete secrecy until a deal is reached. Your main concern is that PlannerCo has a group of key employees who are hard at work on another product which would be dropped if BigCo acquires PlannerCo, and these employees will be hard to keep if there is any public announcement about the negotiations. A. Under what circumstances might BigCo be required to disclose the discussions with PlannerCo? B. When you worked at BigCo you acquired a substantial block of BigCo stock, and you have since occasionally bought and sold BigCo shares. Can you trade your BigCo stock now, and on what other facts might this depend?