Unformatted text preview: University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain ECO 204 2008‐2009 Ajaz Hussain HW 20 Solutions From Lecture 22 recall that a square node is a decision node, while a circular node is a chance node. In the former, the decision maker has complete control on the path, while in the latter, chance determines the path. Question 1 In Lecture 22 we discussed the oil driller’s problem in which the choices were to {Drill, Not Drill}. By definition, these are mutually exclusive choices: that is, you can do one or the other, but not both. In this problem, you will make a decision under uncertainty where the choices are not mutually exclusive. I hope it makes you think harder about how to setup a problem. You are the CEO of a pharmaceuticals company “Economics‐Man Genetics”. It has replicated Economics‐Man’s genes 1 . The company must decide whether to commercialize Economics‐Man genes ‐‐ aimed at students studying for Economics tests and the final exam (just like you!) ‐‐ using a Biochemical or Biogenetic R&D approach. Suppose the profits and probabilities of the competing approaches are: 1 No relation to Eco‐man. 1 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain R&D Approach Biochemical Biogenetic Investment $10m $20m Outcome Large Success Small Success Success Failure Gross Profit $90m $50m $200m $0m Probabilities 0.7 0.3 0.2 0.8 Observe how the Biogenetic approach is more “risky”: it has a high upside but also a low downside. In contrast, the Biochemical approach is less risky but also less lucrative. Find the best approach to commercialize the Economics‐Man’s genes. Assume you can commercialize only one approach. Hint: the choices {Biochemical, Biogenetics} are not mutually exclusive. Answer: Because the approaches are not mutually exclusive, the options available are: • • • • • • Do not invest at all Only Biochemical Only Biogenetic Biochemical and Biogenetic simultaneously Biochemical first followed by Biogenetic Biogenetic first followed by Biochemical. The decision tree is: 2 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain We need to give a value for each of these branches and choose the highest value. This entails drawing trees for each of these branches. “No R&D” Branch: The value is $0. “Biochemical Only” Branch: I will skip the decision tree for the biochemical branch: EV(Biochemical only) = 0.7($90m) + 0.3($50m) – $10m = $68m “Biogenetic Only” Branch: I will skip the decision tree for the biogenetic branch: EV(Biogenetic only) = 0.2($200m) – $20 = $20m Biogenetic & Biochemical Simultaneously Branch: There are four uncertain possibilities with simultaneous R&D programs, each of which cost a total of $30m: Biochemical Large Success Small Success Large Success Small Success Biogenetic Success Success Failure Failure Probabilities 0.7*0.2 = 0.14 0.3*0.2 = 0.06 0.7*0.8 = 0.56 0.3*0.8 = 0.24 Here we have used the fact that for two independent events A and B: P(Event A & Event B) = P(A) P(B). These possibilities are drawn in order below: 3 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain The numbers are net profits (gross profits ‐ total cost). Here is what’s happening branch by branch: Biochemical Large Success & Biogenetic Success: If both programs succeed we have to decide which one to commercialize (recall you can only commercialize one). If we commercialize Biochemical we get $60m and if we commercialize Biogenetic we get $170m. Thus, we choose to commercialize Biogenetic‐‐ notice how we fill the value of the decision in the decision node. Biochemical Small Success & Biogenetic Success: With two successes, we again face a decision. We choose to commercialize Biogenetics because it has the greater value ‐‐ notice how we fill the value of the decision in the decision node. Biochemical Large Success & Biogenetic Failure: There is no decision here‐‐ with a failure in Biogenetic R&D (with gross profits $0) and a large success in Biochemical, we commercialize biochemical. The value of this $90m ‐ $30m = $60m. Biochemical Small Success & Biogenetic Failure: There is no decision here‐‐ with a failure in Biogenetic R&D (with gross profits $0) and a small success in Biochemical, we commercialize biochemical. The value of this $50m ‐ $30m = $20m. Since these four possibilities are uncertain, the EV of simultaneous development is: EV(Biochemical and Biogenetic R&D) = 0.14($170m) + 0.06($170m) + 0.56*($60m) + 4 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain 0.24*($20m) = $72.4m. “Sequential R&D: Biochemical First” Branch Start with biochemical. If it is a large success, you can go to market and make a net profit of $80m. Or, you can pursue biogenetic (because if it succeeds, you could make $170m!) and see what happens. If so, biogenetic could succeed and you make net profit of $170m ($200 ‐ $10 ‐ $20), or, biogenetic could fail, in which case you can “go back” to the biochemical large success and net $80 ‐ $20 = $60m. Note that even though the outcome of a biogenetic failure is $0m, because you have the option of commercializing biochemical the value of biogenetic failure is not 0. The remainder of the tree follows the same logic. Thus: EV(Biochemical first) = $72.4m Indeed, there is no added value by doing biochemical first versus doing biochemical and biogenetic simultaneously. (Maybe you want to think why this is). 5 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain “Sequential R&D: Biogenetic First” Branch The tree follows the same logic as biochemical first. The big difference is that if you have a biogenetic success, you don’t need to explore the option of biochemical R&D because the value of a large success of a biochemical product is always less than value of biogenetic success. Thus: EV(Biogenetic first) = $74.4m In sum: 6 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain Of all the options, Sequential R&D with biogenetics is the best R&D decision. If it succeeds, take product to market. If it fails, pursue biochemical R&D. Question 2 Consider this variation of the oil driller problem we saw in Lecture 22. A company must decide whether to drill for oil. The outcome of not drilling is $0. The outcomes of drilling ‐‐ with a cost of $120m ‐‐ are many fold: net profits from oil can be: $600m, $200m, $0m, or ‐$120m (think of these as different amounts of oil; in Lecture 22 there was either oil or no oil) with probabilities 0.2, 0.18, 0.32 and 0.30 respectively. (a) Draw the decision tree for this problem. Answer: You should first review the oil drilling example in Lecture 22. In that example, the outcomes from drilling were $600 and ‐$200 with probabilities 0.6 and 0.4 respectively. In this problem we have: (b) If the company is risk neutral, what is the decision? Answer: A risk neutral decision maker uses the EV rule for decision making. You should understand why the risk neutral driller will use EV 2 . The driller chooses the decision with the higher EV. If she 2 Recall from Lecture 22 that a risk neutral can either use the EV or EU rule. If you understand risk neutrality you should see that a risk neutral person will make the same decision using EV or 7 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain decides not to drill, the EV will be: EV(no drill) = 1($0m) = $0m On the other hand, if she drills: EV(drill) = 0.2($600m) + 0.18($200m) + 0.32($0m) + 0.30(‐$120m) = $120m Since EV(drill) > EV(no drill), the risk neutral driller chooses to drill. (c) Now suppose the company is risk averse. Denote a gamble with outcomes X1 and X2 with probabilities p1 and p2 by: (X1, X2; p1, p2). Suppose the board of directors decides that: U($600, ‐$200; 1, 0) = U($600) U($600, ‐$200; 0.7, 0.3) = U($200) U($600, ‐$200; 0.5, 0.5) = U($0) U($600, ‐$200; 0.25, 0.75) = U(‐$120) Suppose you’re now risk averse (maybe it’s the economy!). Should the company drill for oil? Answer: The risk averse driller should use the EU rule for decision making. Her decision tree is below. EU. In practice, we use EV because it’s easier than EU, since the latter requires knowing the utility function. 8 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain We need to know her utilities. In class, we saw that any individual can compute the utility of any amount by using the principle of certainty equivalence. In this case, the question gives us the information‐ note how the “gamble” outcomes are from Lecture 22’s oil drilling problem: U($600, ‐$200; 1, 0) = U($600) U($600, ‐$200; 0.7, 0.3) = U($200) U($600, ‐$200; 0.5, 0.5) = U($0) U($600, ‐$200; 0.25, 0.75) = U(‐$120) Let’s take each item and see what it says: U($600, ‐$200; 1, 0) = U($600) says that the utility of $600m is equal to the utility of getting $600m with probability 1 and ‐$200m with probability 0. Thus: U($600) = U($600, ‐$200; 1, 0) = EU(($600, ‐$200; 1, 0) = 1U($600) + 0U(‐$200) But since $600 is the maximum amount, and, ‐$200 is the minimum amount (even though it’s not on the tree, the decision maker is deriving utilities by considering gambles between $600 and ‐$200), we can scale utility so that: U($600m) = 100, U(‐$200) = 0. So: U($600) = U($600, ‐$200; 1, 0) = EU(($600, ‐$200; 1, 0) = 1U($600) + 0U(‐$200) = 100 Similarly: U($600, ‐$200; 0.7, 0.3) = U($200) = 70 9 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain U($600, ‐$200; 0.5, 0.5) = U($0) = 50 U($600, ‐$200; 0.25, 0.75) = U(‐$120) = 25 Hence: Now: EU not drill = 50 EU drill = 0.2 U($600) + 0.18 U($200) + 0.32 U($0) + 0.30 U(‐$120) → EU drill = 0.2(100) + 0.18(70) + 0.32(50) + 0.30(25) = 56.1 The risk averse driller will drill. Needless to say, a more risk averse driller, with different utilities for the same outcome may decide not to drill. (d) (Difficult) Can you reduce the uncertainty for drilling for oil ($600, $200, $0, ‐$120; 0.2, 0.18, 0.32, 0.30) to a gamble between $600 and ‐$200? By doing this question, you’re reducing the drilling option ($600, $200, $0, ‐$120; 0.2, 0.18, 0.32, 0.30) to an “equivalent risk”. This is a technique used by risk managers to transform and understand risk in terms of some benchmark uncertainty. Answer: This question is asking you to equate this oil drilling problem in terms of the lectures oil drilling problem. Intuitively, you are expressing an uncertainty in terms of another uncertainty. The EU 10 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain from above is 56.1. If the oil drilling problem above was equivalent to Lecture 22 oil drilling problem then what we want is: 56.1 = U($600, ‐$200; p, 1‐p) = p U($600) + (1‐p) U(‐$200) = p100 → 56.1 = 100 p → p = 0.561 That is, this oil drilling problem is equivalent to the lectures oil drilling problem for a probability of striking oil equal to 0.561. In general, you can take any uncertain event and express it in terms of another using this “equivalent” risk technique. This is useful for gauging various uncertainties in terms of a “benchmark” uncertainty. Nice. 11 ...
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 Fall '08
 HUSSEIN
 Economics, Microeconomics, Department of Economics, S. Ajaz Hussain

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