Resource Allocation and Decision Analysis (ECON 8010) Spring 2014 Fundamentals of Managerial and Strategic Decision-Making

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1 Resource Allocation and Decision Analysis ECON 800) Spring 0 Fundamentals of Managerial and Strategic Decision-Making Reading: Relevant Costs and Revenues ECON 800 Coursepak, Page ) Definitions and Concepts: Time Value of Money the recognition that, so long as a positive rate of return can be earned, any particular amount of money is worth more the sooner it is received The notions of Present Value and Future Value follow from this recognition The Future Value t periods from now) of receiving $X today, supposing that we could earn t a return of r)% in each period, is FV X ) X r) The Present Value in the current period) of receiving $Y in t periods, supposing we could Y earn a return of r)% in each period, is PV Y ) t r) Certainty Equivalent the certainty equivalent of a gamble, G) is the unique amount of money for which an individual is indifferent between taking the gamble G) versus receiving the money for certain Expected Value the expected value of a random variable is equal to the weighted average of all possible realized values of the random variable Expected Value Criterion the Expected Value Criterion for decision-making in the face of risk dictates that the decision-maker should choose the available option with the greatest expected value Utility of Wealth: Suppose an individual s preferences for wealth can be summarized by the Bernoulli Utility Function u In general, for a random variable V) that can take on N different possible values, each with probability of 0 p, the value of von Neumann-Morgenstern Expected Utility is: U V ) p u N i i V i i Expected Utility Criterion the Expected Utility Criterion for decision-making in the face of risk dictates that the decision-maker should choose the available option with the greatest value of von Neumann-Morgenstern Expected Utility

2 Common assumptions on Bernoulli Utility Functions. More is better or monotonicity or positive marginal utility ) utility should become larger as the amount of wealth a person has is increased => MU u 0 this is always assumed to be true). Diminishing Marginal Utility of Wealth the marginal utility of wealth while always positive) should likely become smaller as the amount of wealth a person has is increased => M U u 0 this need not always be true, but is usually true for most decision makers in most contexts; as noted, this is equivalent to assuming the decision maker is risk averse ) Risk Aversion an individual is risk averse if for any gamble G), the Certainty Equivalent of G) is less than the expected value of G) Risk Averse M U u 0 diminishing marginal utility of wealth Risk Neutrality an individual is risk neutral if for any gamble G), the Certainty Equivalent of G) is exactly equal to the expected value of G) Risk Neutral M U u 0 u Ax with A 0 constant marginal utility of wealth Risk Seeking an individual is risk seeking if for any gamble G), the Certainty Equivalent of G) is greater than the expected value of G) Risk Seeking M U u 0 increasing marginal utility of wealth Updating Probabilities => Bayes Rule General formula for Bayes Rule: PA B i Ai ) B Ai ) A ) B A ) Notation and Terminology: o The original information is called prior information and it conveys different probabilities of different events occurring each denoted P i) ) o Combining these prior probabilities with new information generates posterior probabilities that are conditional on the information obtained denoted P i " Information") notational convention is to enclose any obtained information in quotation marks, in order to indicate that it may or may not in fact be accurate) Contribution the difference between total revenue and total variable costs for a particular product-line Essentially what we previously defined as Producer s Surplus i.e., Profit plus Fixed Costs of production), but for one particular product of several products that the firm is producing Common decision making errors: ) incorrectly allocating Fixed Costs; ) failure to recognize and account for Substitution or Complementarity effects across different products; ) failure to recognize and account for alternative uses of resources i.e., Opportunity Costs). j j j

3 Illustration of a Bernoulli Utility Function with MU u 0 and M U u 0 : Relatively small increase in value of utility u u Relatively large increase in value of utility 0 x Example: consider u x U u x x MU u 0 so monotonicity is satisfied M 0 so the person has a diminishing marginal utility of wealth 0 Example of computing the value of Certainty Equivalent: Returning to the gamble of p HH., V HH, 000 and p T. 7, V T, 000, what is the value of the certainty equivalent to this gamble for someone with u x? To answer this question we need to find a dollar amount C) for which u C) is exactly equal to U G) i.e., the expected utility of the gamble) U G) is equal to. ) u$,000) ) u$,000) Thus, the certainty equivalent must satisfy: u C).) u$,000) ) u$,000) C.)),000 )), 000 C. ),000 ), 000 C. ),000 ),000 C 8, Increase in Wealth of identical magnitudes

4 Example of updating beliefs via Bayes Rule: Suppose your company is contemplating the development and introduction of a new product Demand can either be high or low: based on past experience your initial estimates of the probabilities of these two possible outcomes are: P and P Suppose you hire a marketing firm to conduct some focus groups and surveys with potential customers in order to gauge future demand => will report either High or Low. Based upon past experiences, suppose that you believe 9 P " H" 0 i.e., if demand is truly high, the marketing firm will tell you High 9 out of 0 times) P " L" 0 i.e., if demand is truly high, the marketing firm will tell you Low out of 0 times) P " L" i.e., if demand is truly low, the marketing firm will tell you Low out of times) P " H" i.e., if demand is truly low, the marketing firm will tell you High out of times) Graphical illustration of given information: Report of High P " H" Actual demand is high P P " L" Report of Low P Actual demand is low Report of Low P " L" By Bayes Rule: P h " H") P P l " H") P l " L") P h " L") Report of High PhP" H" h hp" H" h Pl P" H" l P l P" H" l P l P" H" l PhP" H" h P l P" L" l P l P" L" l PhP" L" h PhP" L" h PhP" L" h Pl P" L" l P " H" ) 0 9 ) 9 ) ) ) ) 0 ) ) ) ) ) 0 ) ) ) 9 ) 0) ) ) ) 0 ) ) 0 ) ) )

5 Example of updating beliefs via Bayes Rule, continued) For further intuition on how to derive the posterior probabilities by applying Bayes Rule, conceptually think about this scenario playing itself out,000 times 67 times) Report of High P " H" 9 0 Report of High when actual demand is high Actual demand is high P 70 times) P " L" Report of Low 0 7 times) Report of Low when actual demand is high 0 times) Actual P l demand is low ) Report of Low P " L" 00 times) Report of Low when actual demand is low P " H" Report of High 0 times) Report of High when actual demand is low So, if you are told High, what should you think? There are a total of 67+0=7 instances in which you would be told High That is, P hp" H" h Pl P" H" l Demand is actually high in 67 of these instances That is, P hp" H" h Thus, after being told High, you reasonably update your belief that demand is actually high to a probability of P h " H") Similarly: 0 0 P l " H"). 0690, P l " L"). 77, and 7 7 P h " L")

6 Example of updating beliefs via Bayes Rule, continued) Finally, recognize that we could illustrate these posterior probabilities as follows: Actual demand is high P h " H").90 Report of High Report of Low P " H") P " L") Actual demand is low Actual demand is low P l " H").0690 P l " L").77 Actual demand is high P h " L").77 What we have done here is referred to as flipping the tree Recognize that the conditionals are flipped We started with probabilities over true demand and beliefs about information conditional on demand We ended up with probabilities over information and beliefs about true demand conditional on information

7 Multiple Choice Questions:. The Certainty Equivalent of a gamble, G) is defined as A. the expected value of the gamble. B. the recognition that so long as a positive rate of return can be earned) any particular amount of money is worth more the sooner it is received. C. the unique amount of money for which an individual is indifferent between taking the gamble G) versus receiving the money for certain. D. None of the above answers are correct.. The criterion for decision-making in the face of risk dictates that a decision-maker should choose the available option with the greatest value of von Neumann-Morgenstern Expected Utility. A. Expected Value B. Expected Utility C. Risk Avoidance D. Bayesian Updating. Allison proposes to roll a fair, six-sided die and pays Zack $00 times the number rolled on the die. The expected value of this random payment is A. $70. B. $,00. C. $,00. D. $,800.. Kyle received a payment of $,000 today. Supposing that an annual return of % could be earned, the future value 0 years from now of this payment is A. $,00. B. $7,0.0. C. approximately $7,0.. D. approximately $, Thomas pays $00 today in order to receive $00 in each of the next three years. Supposing that he could earn a return of 7.% per year, the Net Present Value of this investment is A. approximately $.9. B. approximately $60.0. C. approximately $ D. approximately $ Ed is faced with a random payoff with expected value of $70. His Certainty Equivalent for this random payoff is $9. From these observations, it follows that he is A. risk neutral. B. risk averse. C. risk seeking D. irrational.

8 7. Bob promised to pay Jill $,00 exactly three years from today. Supposing that an annual return of % could be earned in each year between now and this future date, Jill s present value for this payment A. is approximately equal to $,9.9. B. is approximately equal to $,80.9. C. is approximately equal to $, D. cannot be determined from the given information. 8. Jimmy s preferences for wealth can be summarized by the Bernoulli Utility function u 0 x. It follows that A. his marginal utility for wealth is always positive. B. his marginal utility for wealth increases as his level of wealth increases. C. his Certainty Equivalent for any random payoff must be exactly equal to the expected value of the random payoff. D. More than one perhaps al of the above answers is correct. 9. Erin s preferences for wealth can be summarized by the Bernoulli Utility function u ln x ) i.e., the natural log of x+) ). Vitali tells Erin that he is going to toss a fair coin and pay Erin: i) $,000 if the coin comes up heads and ii) $,000 if the coin comes up tails. Erin s expected utility for this random payoff A. is approximately B. is approximately 0.6. C. is exactly equal to $,000. D. cannot be determined from the given information. 0. Brian is faced with the following gamble: he will be paid $0,000 with probability p.), he will be paid $,000 with probability p. ), and he will be paid $0 with probability p.). If Brian is risk averse, then we know that his Certainty Equivalent to this gamble must be A. equal to $0. B. greater than $0 but less than $,000. C. exactly equal to $,000. D. greater than $,000 but less than $,000.. refers to the difference between total revenue and total costs for a particular product-line. A. Future Value B. von Neumann-Morgenstern Expected Utility C. Contribution D. Profit

9 Problem Solving or Short Answer Questions:. Determine the value of the Certainty Equivalent for A. an individual with u x, who will receive: $6 with probability p.), $ with probability p. ), $6 with probability p. ), and $6 with probability p. ). B. an individual with u x, who will receive: $00 with probability p. ), $00 with probability p. ), $00 with probability p. ), and $00 with probability p. ). C. an individual with u 00ln x ), who will receive: $00 with probability p.), $800 with probability p. ), and $,00 with probability p.). D. an individual with u x, who will receive: $0 with probability p. ), $0 with probability p. ), and $00 with probability p. ). u x, who will receive: $,00 with probability E. an individual with p.), $,00 with probability p. ), $,00 with probability p. ), $,000 with probability p. ), $8,000 with probability p. ), and $0,000 with probability p 6.0).. Widgets International is contemplating the development of a new product, which would entail incurring a one-time Fixed Cost of $,00 in the present period. If they develop this new product, then they will face demand given by the inverse demand function P D q) 0.00) q in each of the next five periods after the fifth period consumers will no longer be willing to pay any positive amount for the good). Beyond the initial Fixed Costs, the only costs of the firm would be a constant Marginal Cost of $ per unit. Finally suppose that the relevant rate of return is 6% per period. A. Would you advise Widgets International to develop this new product? Explain. B. If the initial Fixed Costs had been $,000 instead of $,00, would you advise Widgets International to develop this new product? Explain. C. If the relevant rate of return had been 8% per period instead of 6% per period with Fixed Costs of the initially given value of $,00), would you advise Widgets International to develop this new product? Explain.. Consider a firm selling a product for which demand is given by the inverse function P D q) 00.) q. The firm currently has production costs of C q) 0q 0, 000. The firm has the option of attempting to develop a new technology that would lower production costs to C ˆ q) 0q 0, 000. Research and development costs are $,000 and if undertaken) must be incurred regardless of whether or not the new technology is successful or a failure. If the firm attempts to develop the new production technology, their innovation will be a success with probability p 0 and will be a

10 9 failure with probability p ) 0. Throughout your analysis, restrict attention to the profit/loss of the firm in only the current period i.e., assume that the firm will not be operating in any future period). A. If this firm applies the Expected Value Criterion for decision-making, should it attempt to develop the new technology? Explain. 0 u x and applies the B. If this firm has a Bernoulli Utility Function of Expected Utility Criterion for decision-making, should it attempt to develop the new technology? Explain.. Ryan suspects that there is a chance that he may have a rare blood disorder. His prior beliefs are that he is sick i.e., that he has the disorder) with probability P s). 0 and that he is healthy i.e., that he does not have the disorder) with probability P. 9. With his current health insurance there are four different doctors that he could see for a diagnosis: Dr. Coin, Dr. Optimist, Dr. Pessimist, and Dr. Demented. The potential diagnoses of these four doctors of Healthy or Sick ) can be summarized as follows: Dr. Coin Dr. Optimist Dr. Pessimist Dr. Demented P " H". 0 0 P ". 0 P " s). 0 0 P " H" s). 0 A. For each doctor, determine the posterior probability that Ryan should attach to being sick and to being healthy for each diagnosis that the doctor might give to Ryan. B. Based upon your answers to part A), which doctor do you think Ryan should see assuming he wants to accurately know if he is sick or healthy)? Explain.. You are the manager of a firm that is considering bringing a new product to market. You anticipate that demand will either be high with probability P. ), average with probability P. ), or low with probability P. ). You have hired a marketing firm to gauge future demand for this product. The marketing firm will provide a report specifying that they anticipate the product being either a Success or a Flop. Based upon past experience, you believe: P ". 9, P "., P ". 6, P "., P "., and P ". 8. Apply Bayes Rule to determine P h " S"), P h " F" ), P a " S" ), P a " F" ), P l " S" ), and P l " F" ).

11 Answers to Multiple Choice Questions:. C. B. B. C. B 6. B 7. A 8. A 9. A 0. B. C Answers to Problem Solving or Short Answer Questions: A. In this case the Certainty Equivalent, denoted C, must satisfy: u C).) u6).) u).) u6).) u6) C.)) 6.)).)) 6.) 6 C. )).)).)6).)8) C.6).6) C.6). 6 B. In this case the Certainty Equivalent, denoted C, must satisfy: u C).) u00).) u00).) u00).) u00) C. )00).)00).)00).)00) C 0 Additionally, recognize that in this case this individual is risk neutral, so that the Certainty Equivalent is simply equal to the expected value of the random payoff. C. In this case the Certainty Equivalent, denoted C, must satisfy: u C).) u00).) u800).) u,00) 00ln C ).)00) ln0).)00) ln80).)00) ln,0) ln C ).) ln0).) ln80).) ln,0) ln C ) expln C ) exp6.89 e C C D. In this case the Certainty Equivalent, denoted C, must satisfy: u C).) u0).) u0).) u00) C.)0 ).)00 ) 6, 0 C Additionally, recognize that since this decision-maker is risk seeking, the value of the Certainty Equivalent is greater than the expected value of.).

12 E. In this case the Certainty Equivalent, denoted C, must satisfy: u C).) u,00).) u,00).) u,00).) u,000).) u8,000).0) u0,000) C ).),00 ).),00 ).),00 ).),000 ).)8,000 ).0)0,000 C ) C ) C 0.07 C, A. If they choose to develop the product, then in each of the five periods they would face demand given by the inverse function P D q) 0.00) q. From here it follows that Marginal Revenue would be equal to MR q) 0.00) q. We are told that Marginal Costs are MC q). Thus, if the firm were to develop the product, then in each period they would maximize profit by producing the unique quantity of output for which 0.00) q. Solving for q yields q *, 000. It follows that the optimal price of the firm would be p * P q * ) 0.00),000) 0 D 6. From here, the profit * of the firm in each of the five periods would be 6 ),000) 8, 000. With an upfront Fixed Cost of $,00 and a rate of return of 6%, the Net Present Value for the firm from developing the product is: 8,000 8,000 8,000 8,000 8,000,00.06).06).06).06).06),00 7,7.7 7,9.97 6,76.9 6,6,978.07,98.9. Since the Net Present Value is positive, the firm should go ahead with the development of this new product. B. If instead the initial Fixed Costs had been $,000, then Net Present Value would have been: 8,000 8,000 8,000 8,000 8,000,000.06).06).06).06).06),000 7,7.7 7,9.97 6,76.9 6,6,978.07,0.09. Since the Net Present Value is negative, the firm should not go ahead with the development of this new product. C. If the relevant rate of return had been 8% along with a Fixed Cost of $,00), then the Net Present Value would have been: 8,000 8,000 8,000 8,000 8,000,00.08).08).08).08).08),00 7,07. 6,88.7 6,0.66,880., Since the Net Present Value is negative, the firm should not go ahead with the development of this new product. )

13 A. First recognize that for the inverse demand function P D q) 00.) q, Marginal Revenue is MR q) 00.) q. If the firm has costs of C q) 0q 0, 000 from either not pursuing the new technology or by research efforts being a failure ), then Marginal Costs are MC q) 0. In this case, the firm would maximize profit by producing the quantity of output for which 00.) q 0, which is q * 00. The corresponding optimal price * is p P D 00) 00.)00) 60. This would lead to a Producer s Surplus of PS 00)60 0) 00)0) 6,000. If instead the firm has costs of C ˆ q) 0q 0, 000 from research efforts being successful ), then Marginal Costs are M Cˆ q) 0. In this case, the firm would maximize profit by producing the quantity of output for which 00.) q 0, which is q ˆ * 0. The corresponding optimal price is p ˆ * P D 0) 00.)0). This would lead to a Producer s Surplus of P Sˆ 0) 0) 0)) 0, 0. If the firm does not pursue research and development of the new product, then they will realize a certain profit of PS F 6,000 0,000 6, 000. If instead they pursue research and development at a cost of C RD, 000 ), then they will realize a profit of ˆ P S F C RD 0,0 0,000,000 8, 0 with probability p 0 and will realize a profit of PS F C RD 6,000 0,000,000, 000 with probability p ). Thus, the expected value of profit from pursing research and development is 8,0),000) 6, Since this is greater than the certain profit from not pursing research and development, it follows that a firm using the Expected Value Criterion for decision-making should attempt to develop the new technology. B. If instead the firm has a Bernoulli Utility function of Expected Utility Criterion for decision-making, then it needs to compare u 6,000) i.e., the expected utility from not pursing research and development) to 9 8,0) u,000) 0 u. For the given function, 0 0 u 6,000) 6,000) u x and applies the 9 8,0),000) 8,0) u u,000) Since the latter value is greater than the former value, it follows that a firm with 0 u x that applies the Expected Utility Criterion for decision-making, should not attempt to develop the new technology. and

14 A. Dr. Coin can potentially give a diagnosis of either Healthy or Sick. Applying Bayes Rule we have P C h " H". 9, P " ". 0 9 C s H, P C s " S". 0, and h " S" P 9 C Dr. Optimist only ever gives a diagnosis of Healthy. Applying Bayes Rule we P O h " H". and P. 0 9 O s " H" Dr. Pessimist only ever gives a diagnosis of sick. Again, applying Bayes Rule P P s " S". and P. 9 9 P h " S" Dr. Demented can potentially give a diagnosis of either Healthy or Sick. 0 0 P D h " H", P " " D s H, 0 0 see that 9 we have 0 Applying Bayes Rule we have P D s " S" 0, and h " S" 9 0 P D B. From here we can see that the diagnoses of Drs. Coin who essentially just flips a coin to determine what information to give to patients), Optimist who tells all patients that they are healthy), and Pessimist who tells all patients that they are sick) are as should be intuitive clear) useless, in that they do not improve the perceived probabilities of the patient whatsoever. In contrast, the diagnosis of Dr. Demented while always wrong ) actually provides the patient with perfect information in that it resolves all uncertainty). Thus, Ryan should see Dr. Demented for his diagnosis.. Applying Bayes Rule, we have P h " S" P a " S" P l " S" P h " F" P a " F" P l " F" ,,,,, and.