ECON Sample Questions Chapter 9 Spring 2024 PDF

ECON 351x — Microeconomics for Business Sample Questions Chapter 9 Spring 2024 [This version: Feb/29/2024] The answer key is at the end of this file. The objective of this workbook is to complement...

ECON 351x — Microeconomics for Business Sample Questions Chapter 9 Spring 2024 [This version: Feb/29/2024] The answer key is at the end of this file. The objective of this workbook is to complement the material from class. This is not an exhaustive list of the topics or types of questions that will appear on the exams. Students should carefully read and study the book, all the material on Blackboard (including the homework), and their own class notes, including all the examples, exercises, and graphs from class. Copyright © 2024. This content is protected and may not be shared, uploaded, or distributed. Chapter 9 Question 2: Jim has $100 to invest today, and he will Question 1: need the money back in 30 days. He is con- sidering 3 investment options. Consumers Niobe and Dione have prefer- First, he could buy shares of ACME Inc. ences that can be represented by the following The current market price of this stock is $10 utility functions: per share. After some research, Jim believes √ Niobe: uNiobe (x) = 2 x that in 30 days there is a 50% chance that Dione: uDione (x) = 3x2. this price will go up to $14 per share; a 25% Each consumer has to choose how to in- chance that the price will stay at $10 per vest her own money. There are two invest- share; and a 25% chance that the price will ment options, A and B. go down to $5 per share. Investment option A pays x=9 with prob- The second investment option is to buy ability 75%, and it pays x=4 with probability shares of Tabajara Inc. This stock currently 25%. trades at $5 per share. Jim believes that in Investment option B pays x=16 with 30 days there is a 50% chance that this price probability 60%, and it pays x=1 with prob- will go up to $9 per share; a 25% chance that ability 40%. the price will stay at $5 per share; and a 25% chance that the price will go down to $0.50 a) Compute the expected value of investment per share. A and the expected value of investment B. Finally, Jim can choose a safe investment and open a USC Credit Union savings ac- b) For Niobe, compute the expected utility count. This savings account pays 1% per from investment A and B. Does Niobe pre- month interest on deposits.√Jim’s utility from fer investment A or B? income is given by u(I) = I, and he has to c) For Dione, compute the expected utility invest all his money in only one of the three from investment A and B. Does Dione pre- investment options. fer investment A or B? 1 a) Compute the expected value of each in- the Panthers win, then he ends up with an vestment option. income IW in. If he bets all his money and the Panthers lose, then he ends up with nothing, b) Compute the expected utility of each in- I Lose = $0. If he does not bet, he keeps his vestment option. Find the best (and $36 for sure. worse) investment option for Jim. The Los Angeles Panthers will win the championship game with probability 75%, Question 3: and will lose the game with probability 25%. Compute the minimum winning prize Pedro Carroll has√the following utility IW in such that Cam would be willing to bet from income: u(I) = I. Currently, he has all his money on the Panthers. an income I=$169 in cash (no risk). He is de- ciding whether or not to bet all his money on (A) IW in = $36 the Los Angeles Seahawks to win the cham- pionship game. If he bets all his money and (B) IW in = $48 the Seahawks win, then he ends up with an income I=$400. If he bets all his money and (C) IW in = $56 the Seahawks lose, then he ends up with noth- (D) IW in = $64 ing, I=$0. If he does not bet, he keeps his $169 for sure. (E) IW in = $72 The Los Angeles Seahawks will win the championship game with probability P, and will lose the game with probability 1-P. Question 5: Compute the minimum winning probabil- ity P such that Pedro would be willing to bet A firm wants to maximize its expected all his money on the Seahawks. profit. The firm must choose the production quantity q. The production cost is C = 2q 2. However, the firm is uncertain about the (A) P = 50% price consumers will be willing to pay for the product. With probability 75% the con- (B) P = 65% sumers will be willing to pay $24 per unit of (C) P = 70% output q. With probability 25% consumers will only be willing to pay $8. (D) P = 55% The firm must choose q before it learns if the price will be high or low. (E) P = 60% Compute the quantity q that maximizes the firm’s expected profit. Find the expected Question 4: profit given this optimal quantity. Cam has the√ following utility from in- Question 6: come: u(I) = I. Currently he has an in- come I=$36 in cash (no risk). He is decid- Anna’s utility of income function is ing whether or not to bet all his money on U (I) = I, while Elsa’s utility of income func- the Los Angeles Panthers to win the cham- tion is U (I) = I 1.5. Therefore, Anna is pionship game. If he bets all his money and and Elsa is. 2 (A) risk neutral, risk loving. Utility u( I ) (B) risk averse, risk averse. Marta (C) risk loving, risk loving. (D) risk averse, risk loving. Income Utility u( I ) (E) risk loving, risk averse. Hannah Question 7: Anna’s marginal utility of income func- tion is M U (I) = I, while Elsa’s marginal Income utility of income function is M U (I) = I 1.5. Therefore, Anna is and Elsa is Using the information from the graphs,. (A) risk neutral, risk loving. (A) Both Marta and Hannah are risk loving (B) Both Marta and Hannah are risk averse (B) risk averse, risk averse. (C) Marta is risk averse; Hannah is risk lov- (C) risk loving, risk loving. ing (D) Marta is risk loving; Hannah is risk (D) risk averse, risk loving. averse (E) risk loving, risk averse. (E) Not enough information from the graphs Question 9: Question 8: The following graphs represent the The following graphs represent the utility marginal utility of income functions of con- function of consumers Marta and Hannah: sumers Henrique and Juliano. 3  (B) reject the bet, independently of I Marginal Utility MU( I ) (C) accept the bet only if I is high enough (D) accept the bet, independently of I Henrique (E) always be indifferent between accepting and rejecting the bet Income Question 11: Marginal Utility MU( I ) The following graph represents Mike’s marginal utility of income: Marginal Utility Juliano Income MU( I ) Using the information from the graphs, Hen- rique is and Juliano is. Income (A) risk neutral, risk loving. We know that Mike’s income I is greater than (B) risk averse, risk averse. $10, but we do not know his exact income. (C) risk loving, risk loving. Adam offers Mike the following bet: Adam will throw a fair coin (that is, heads and tails (D) risk averse, risk loving. have equal probability). If it is heads, Mike pays Adam $10. If it is tails, Adam pays (E) risk loving, risk averse. Mike $10. Mike will Question 10: (A) reject the bet only if I is high enough Mike has a constant marginal utility of (B) reject the bet, independently of I income. We know that Mike’s income I is greater than $10, but we do not know his ex- (C) accept the bet only if I is high enough act income. Adam offers Mike the following bet: Adam will throw a fair coin (that is, (D) accept the bet, independently of I heads and tails have equal probability). If it is heads, Mike pays Adam $10. If it is tails, (E) always be indifferent between accepting Adam pays Mike $10. Mike will and rejecting the bet (A) reject the bet only if I is high enough 4 Question 12: Question 14: A consumer owns shares of a company. Using the information from the previous With probability 80% the value of the invest- question, compute the associated risk pre- ment will go up to x = $144, and with prob- mium. ability 20% the value of the investment will go down to x = $81. The consumer’s utility (A) 3 from income is given by the function (B) 1 √ u(x) = x. (C) 7 a) Compute the consumer’s expected utility from holding this investment. (D) 10 b) Compute the expected value of this invest- (E) 4 ment. c) Compute the certainty equivalent of this Question 15: investment. Tom has the following utility from income: d) Compute the risk premium associated u(I) = √I. He bought a house in Brent- with this investment. wood for $846,400. With probability 25%, high winds will cause $174,000 in damages to Question 13: the house (in this case, the house value be- comes $672,400). AAA offers to fully insure Cuban is an investor who used all his in- the house for a price P. What is the maximum come to buy a startup company called Shark insurance premium P that Tom is willing to Aquarium. With probability 50% the value pay? of the company will increase to $144. With (A) 23,010 probability 50% the value of the company will go down to $64. Cuban’s utility from income (B) 45,375 is √ u(I) = I. (C) 74,000 He has no other source of income. Compute (D) 32,375 the minimum price P such that Cuban would be willing to sell all his shares of Shark Aquar- (E) 15,600 ium. (A) P = 94 (B) P = 130 (C) P = 121 (D) P = 104 (E) P = 100 5 Question 16: Question 19: Using the information from the previous Joe’s utility from income is given by question, how much is the fair insurance pre- √ mium? u(I) = 8 I. (A) 174,000 He has a house in LA County. With prob- (B) 72,000 ability 1/4, the rain will cause a mudslide (C) 130,500 and damage his house, decreasing its value to $250,000. With probability 3/4, no mud- (D) 43,500 slide occurs and the house retains its origi- nal $1,000,000 value. AAA offers to fully in- (E) 168,100 sure the house for a price P. What is the maximum insurance premium P that Joe is Question 17: willing to pay? What is the associated risk premium? Using the information from Question 15 , find Tom’s associated risk premium: Question 20: (A) 1,200 Marcelo has √the following utility from in- (B) 2,100 come: u(I) = I. He has a $10,000 car. (C) 1,550 With probability 25% he will crash his car, and will have to pay $6,348 to the repair (D) 2,000 shop to fix it. With probability 75% no crash (E) 1,875 occurs, and he does not have to repair the car. Progressive Insurance offers to partially insure the car for a price P=124. That is, Question 18: Marcelo pays insurance premium P=124, and in case of a crash, Progressive will only pay Sherman has the following utility from for half of the repair cost (in case of a crash, wealth: √ Progressive will only pay $3,174 to Marcelo). u(x) = x. If Marcelo buys this partial insurance plan, Wealth is the sum of cash and the value of his expected utility will be: his car. He currently has a Ferrari worth $230,900 and $50,000 in cash (risk free). (A) 94 With probability 25%, a paparazzi will cause $158,400 in damages to the car (in this case, (B) 98 the car value becomes $72,500). With proba- bility 75% nothing happens, and the car keeps (C) 96 its original value. AAA offers to fully insure the car for a price P. What is the maximum (D) 95 insurance premium P that Sherman is will- ing to pay? What is the associated risk pre- (E) 97 mium? 6 Question 21: will rain a lot, in which case the value of her investment in Company A will go up to $100 Arthur has the following utility from in- while her investment in Company B will go up come: √ to $90. There is a 50% chance of a drought, u(I) = I. in which case the value of her investment in Company A will go down to $80, while her He has a collection of rare comic books that investment in Company B will go down to is worth $10,000. With probability 50% a $70. silverfish invasion will destroy all his books, Nan has her wealth invested in Company bringing the value of his collection to zero. C and in Company D. Company C is an air- With probability 50% nothing occurs, and his line company, while Company D is an oil com- collection keeps its full value. A company pany. Their value can go up or down depend- called Insect Invasion Insurance (III) offers ing on the international price of oil. There is to partially insure the collection for a price a 50% chance that the price of oil will go up, P=$1,900. That is, Arthur first pays the in- in which case her investment in Company D surance premium P=$1,900; and then, in case will go up to $100 while her investment in of a silverfish invasion, III will send to Arthur Company C will go down to $70. There is a a $2,044 check. If Arthur buys this partial in- 50% chance that the price of oil will go down, surance plan, then his expected utility will be in which case the value of her investment in. Therefore, Arthur. Company D will go down to $80, while her [Be careful with your computation. Keep investment in Company C will go up to $90. as many decimal places as possible along all the steps of the problem. Once you are done a) Are the values of Company A and Com- will all computations, round your final an- pany B positively or negatively corre- swer to the nearest integer. It should be clear lated? which one.] b) Are the values of Company C and Com- (A) 53 ; should buy the insurance pany D positively or negatively corre- lated? (B) 51 ; should buy the insurance c) Who has the riskier investment portfolio, (C) 51 ; should not buy the insurance Feng or Nan? (D) 53 ; should not buy the insurance (E) None of the above, because Arthur is in- difference between buying the insurance or not. Question 22: Feng has all her wealth invested in Com- pany A and in Company B. These are agri- cultural companies, and their value can go up or down depending on the amount of rainfall this season. There is a 50% chance that it 7 Solutions to the Questions 1: a) Compute the expected value: Investment A: E[x] = 0.75 × 9 + 0.25 × 4 = 7.75 Investment B: E[x] = 0.6 × 16 + 0.4 × 1 = 10. b) Compute the expected utility √ √ Investment A for Niobe: E[uNiobe (x)] = 0.75 × 2 9 + 0.25 × 2 4 = 5.5 √ √ Investment B for Niobe: E[uNiobe (x)] = 0.6 × 2 16 + 0.4 × 2 1 = 5.6. Since option B yields a higher expected utility than option A, Niobe prefers option B. c) Compute the expected utility Investment A for Dione: E[uDione (x)] = 0.75 × (3 × 92 ) + 0.25 × (3 × 42 ) = 194.25 Investment B for Dione: E[uDione (x)] = 0.6 × (3 × 162 ) + 0.4 × (3 × 12 ) = 462. Since option B yields a higher expected utility than option A, Dione prefers option B. 2: a) If Jim uses all his money to buy shares of ACME, he buys 10 shares (= 100/10). If the price goes up to $14, he can sell his shares and receive $140 (= 10 ∗ 14); if the price stays at $10, he sells his shares and receives $100 (= 10 ∗ 10); if the price goes down to $5 he receives $50 (= 10 ∗ 5). Given the probabilities, investment option 1 has an expected value E[x1 ] = 0.50 ∗ 140 + 0.25 ∗ 100 + 0.25 ∗ 50 = $107.5. That is, an expected 7.5% return on the initial $100 investment. If Jim uses all his money to buy shares of Tabajara, he buys 20 shares (= 100/5). If the price goes up to $9 he receives $180 (= 20 ∗ 9); if the price stays at $5 he receives $100 (= 20 ∗ 5); if the price goes down to $0.50 he receives $10 (= 20∗0.50). Given the probabilities, investment option 2 has an expected value E[x2 ] = 0.50 ∗ 180 + 0.25 ∗ 100 + 0.25 ∗ 10 = $117.5. That is, an expected 17.5% return on the initial $100 investment. If Jim invests all his money ($100) on the savings account that has a 1% return (there is no risk), the expected value of this investment is E[x3 ] = $101. In summary, investment option 2 has the highest expected value, while investment option 3 has the lowest. b) Use the information above to compute the expected utility from each investment option: √ √ √ E[u1 ] = 0.50 ∗ 140 + 0.25 ∗ 100 + 0.25 ∗ 50 ≈ 10.18 √ √ √ E[u2 ] = 0.50 ∗ 180 + 0.25 ∗ 100 + 0.25 ∗ 10 ≈ 10.00 √ E[u3 ] = 101 ≈ 10.05 8 Note that, although Jim is risk averse, the best option is not the safe investment (option 3). Investment option 1 (ACME) is the best option for Jim, since it yields the highest expected utility (in this case, the higher expected value more than compensates for the risk). Although investment option 2 (Tabajara) has the highest expected value, it is the worse investment option because it yields the lowest expected utility (this option is too risky for Jim). 3: (B) P = 65%. The expected utility from the risk free option (holding cash) is √ E[uRF ] = 169 = 13. The expected utility from the risky option (bet) is √ √ E[uR ] = P × 400 + (1 − P ) × 0 = 20P. We are looking for the probability P such that both options yield the same expected utility, E[uR ] = E[uRF ] 20P = 13 P = 0.65 4: (D) IW in = $64. The expected utility from the risk free option (holding cash) is √ E[uRF ] = 36 = 6. The expected utility from the risky option (bet) is p √ p E[uR ] = 0.75 × Iwin + 0.25 × 0 = 0.75 × Iwin. We are looking for prize Iwin such that both options yield the same expected utility, E[uR ] = E[uRF ] p 0.75 × Iwin = 6 p Iwin = 8 Iwin = 64. 5: q ∗ = 5 and expected profit is $50. Given price p and quantity q, the firm’s profit is pq − 2q 2. The firm chooses quantity q but it is uncertain about the price p. Therefore, expected profit is E(Profit(q)) = (Prob. p=24) × (Profit if p=24) + (Prob. p=8) × (Profit if p=8) = 0.75(24q − 2q 2 ) + 0.25(8q − 2q 2 ) = 20q − 2q 2 (1) ∂E(Profit(q)) = 20 − 4q. ∂q 9 The first-order condition implies ∂E(Profit(q)) = 0 ∂q 20 − 4q = 0 q ∗ = 5. Using equation (1), when the firm optimally produces q ∗ = 5, expected profit is E(Profit(q)) = 20q − 2q 2 = 20 × 5 − 2 × 52 = 50. 6: (A) risk neutral, risk loving. We know that if a consumer has a utility of income function of the form U (I) = I a , then he is risk loving if a > 1, risk neutral if a = 1, and risk averse if 0 < a < 1. 7: (C) risk loving, risk loving. We know that if a consumer has a utility of income function of the form U (I) = I a , then he is risk loving if a > 1, risk neutral if a = 1, and risk averse if 0 < a < 1. However, the question does not give us the utility, the question gives us the marginal utility of income function. We know that if M U (I) is an increasing function of I, then the consumer is risk loving; if M U (I) is constant, then the consumer is risk neutral; if M U (I) is a decreasing function of I, then the consumer is risk averse. In this question, both M U (I) = I and M U (I) = I 1.5 are increasing functions of I. In contrast, if instead we had for example M U (I) = I3 , then this marginal utility is a decreasing function of I. Finally, if instead we had something like M U (I) = 5, then this marginal utility would be constant (not a function of I). 8: (C) Marta is risk averse; Hannah is risk loving. 9: (C) risk loving, risk loving. This question does not give us the graph of the utility of income function, as in Question 8. This question gives us the graph of the marginal utility of income. Since both consumers have marginal utilities that are increasing functions of income, they are risk loving. In summary, if the graph of the marginal utility is an increasing function of I (has a positive slope), then the consumer is risk loving. If the graph of the marginal utility is a constant function of I (horizontal line), then the consumer is risk neutral. If the graph of the marginal utility is a decreasing function of I (negative slope), then the consumer is risk averse. In this particular question, both consumers have a marginal utility that is an increasing function of I (positive slope), therefore they are risk loving. 10: (E) always be indifferent between accepting and rejecting the bet. The consumer’s marginal utility of income is constant, therefore he is risk neutral. A risk neutral consumer 10 is indifferent between accepting and rejecting a fair bet. Note that this is a fair bet, that is, it has an expected value of zero (with probability 0.5 he wins $10, with probability 0.5 he loses $10, hence E[X] = 0.5 × (−10) + 0.5 × (+10) = 0). A fair bet does not change the expected income, but it creates risk. Therefore, a risk averse consumer never takes a fair bet, while a risk loving consumer aways takes a fair bet. A risk neutral consumer is indifferent. 11: (B) reject the bet, independently of I. The consumer has a decreasing marginal utility of income, hence he is risk averse. A risk averse consumer always prefers to reject a fair bet. 12: a) Expected utility is √ √ E(u(x)) = 0.8 144 + 0.2 81 = 11.4. b) The expected value is E(x) = 0.8 × 144 + 0.2 × 81 = 131.4. c) The certainty equivalent is the amount xC such that the consumer is indifferent between holding the risky investment and having amount xC for sure: E(u(x)) = u(xC ) √ 11.4 = xC 129.96 = xC. d) The risk premium is the difference between the expected value of the investment and the certainty equivalent, RP = x̄ − xC = 131.4 − 129.96 = 1.44. 13: (E) P = 100. The minimum price is simply the price such that the consumer is indifferent between holding the risky investment and selling the investment for the price P. The expected utility from holding the investment is √ √ E(u(x)) = 0.5 144 + 0.5 64 = 10. √ If the consumer sells his investment for a price p, his utility becomes u(P ) = P. Therefore, we are looking for the price P such that the consumer is indifferent between the two options, E(u(x)) = u(P ) √ 10 = P 100 = P. Note that this price is simply the certainty equivalent, xC = 100 (the investor is exchanging the risky investment for a risk-free cash). 11 14: (E) 4. The risk premium is the difference between the expected value of the investment E(x) and the certainty equivalent xC. The expected value is E(x) = 0.5 × 144 + 0.5 × 64 = 104. We computed the certainty equivalent in the previous question, xC = 100. Therefore RP = E(x) − xC = 104 − 100 = 4. 15: (B) 45,375. Expected utility without insurance: p p E[uN I ] = 0.75 846, 400 + 0.25 672, 400 = 895. Expected utility with insurance: p E[uI ] = 846, 400 − P. Find P such that the consumer is indifferent between purchasing the insurance or not, E[uN I ] = E[uI ] p 895 = 846, 400 − P 2 895 = 846, 400 − P P = 846, 400 − 8952 P = 45, 375. 16: (D) 43,500. The fair insurance premium is the expected loss, 0.25 ∗ 174, 000 = 43, 500. 17: (E) 1, 875. The risk premium is the difference between the premium the consumer is willing to pay and the fair insurance premium, RP = 45, 375 − 43, 500 = 1, 875. 18: First, find the expected utility without insurance E[uN I ] and the expected utility with insurance E[uI ] at price P. p p E[uN I ] = 0.25 50, 000 + 72, 500 + 0.75 50, 000 + 230, 900 = 485 p E[uI ] = 50, 000 + 230, 900 − P Then find P such that the consumer is just indifferent between purchasing the insurance or not, E[uN I ] = E[uI ] p 485 = 50, 000 + 230, 900 − P P = 45, 675 12 The consumer is willing to pay at most $45,675 for insurance. The fair insurance premium is the expected loss: with probability 0.25 the consumer losses $158,400, therefore the expected loss is 0.25 × 158, 400 = 39, 600. The risk premium is the difference RP = 45, 675 − 39, 600 = 6, 075. 19: First, find the expected utility without insurance E[uN I ] and the expected utility with insurance E[uI ] at price P. 3 p 1 p E[uN I ] = 8 1, 000, 000 + 8 250, 000 = 7, 000 4p 4 E[uI ] = 8 1, 000, 000 − P Then find P such that the consumer is just indifferent between purchasing the insurance or not, E[uN I ] = E[uI ] p 7, 000 = 8 1, 000, 000 − P P = 234, 375 The consumer is willing to pay at most $234,375 for insurance. The fair insurance premium is the expected loss. With probability 1/4 the consumer losses $750,000 (the value of the house goes from 1,000,000 to 250,000). Therefore, the expected loss is 41 (750, 000) = 187, 500. The risk premium is the difference RP = 234, 375 − 187, 500 = 46, 875. 20: (D) 95. In this question, if the consumer buys insurance, then the consumer has to pay P = 124 no matter what. If a crash occurs, the insurance company only pays for half of the damage. That is, in the event of a crash, out of the $6,348 repair cost, the consumer will have to pay $3,174, while the insurance company pays the rest. p p E[u(Partial Insurance)] = 0.25 ∗ 10, 000 − 3, 174 − 124 + 0.75 ∗ 10, 000 − 124 = 95 21: (B) 51; should buy the insurance. Under the partial insurance, the consumer always needs to pay $1,900. If he looses his money, he only gets back $2,044. Expected utility of this partial insurance is p p E[u(Partial Insurance)] = 0.50 ∗ 10, 000 − 1, 900 + 0.50 ∗ 0 − 1, 900 + 2, 044 = 51 To know if the consumer is willing to buy this insurance, we need to know the expected utility of not buying insurance: p √ E[u(No Insurance)] = 0.50 ∗ 10, 000 + 0.50 ∗ 0 = 50. 13 The consumer should buy insurance because it delivers a higher expected utility. 22: a) Companies A and B are positively correlated. b) Companies C and D are negatively correlated. c) Feng has a riskier investment portfolio. If it rains a lot, then the value of her investment will go up to $190, but if we have a drought the value will go down to $150. The expected value is $170, but there is a lot of risk because the companies have positively correlated values. In contrast, Nan diversified her portfolio by investing in negatively correlated companies. The value of her investment will always be $170, independently of oil prices. 14