Economic Problem

CHAPTER 4 INDIVIDUAL AND MARKET DEMAND EXERCISES 1. An individual sets aside a certain amount of his income per month to spend on his two hobbies, collecting wine and collecting books. Given the information below, illustrate both the price-consumption curve associated with changes in the price of wine and the demand curve for wine. Price |Price |Quantity |Quantity |Budget | |Wine |Book |Wine |Book | | |$10 |$10 |7 |8 |$150 | |$12 |$10 |5 |9 |$150 | |$15 |$10 |4 |9 |$150 | |$20 |$10 |2 |11 |$150 |
The price-consumption curve connects each of the four optimal bundles given in the table, while the demand curve plots the optimal quantity of wine against the price of wine in each of the four cases. See the diagrams below. 2. An individual consumes two goods, clothing and food. Given the information below, illustrate both the income-consumption curve and the Engel curve for clothing and food. Price |Price |Quantity |Quantity |Income | |Clothing |Food |Clothing |Food | | |$10 |$2 |6 |20 |$100 | |$10 |$2 |8 |35 |$150 | |$10 |$2 |11 |45 |$200 | |$10 |$2 |15 |50 |$250 | The income-consumption curve (see diagram at right) connects each of the four optimal bundles given in the table above.
As the individual’s income increases, the budget line shifts out and the optimal bundles change. The Engel curve for each good illustrates the relationship between the quantity consumed and income (on the vertical axis). Both Engel curves (see diagrams below) are upward sloping, so both goods are normal. 3. Jane always gets twice as much utility from an extra ballet ticket as she does from an extra basketball ticket, regardless of how many tickets of either type she has. Draw Jane’s income-consumption curve and her Engel curve for ballet tickets. Ballet tickets and basketball tickets are perfect substitutes for Jane.

Therefore, she will consume either all ballet tickets or all basketball tickets, depending on the two prices. As long as ballet tickets are less than twice the price of basketball tickets, she will choose all ballet. If ballet tickets are more than twice the price of basketball tickets, she will choose all basketball. This can be determined by comparing the marginal utility per dollar for each type of ticket, where her marginal utility from another ballet ticket is 2 times her marginal utility from another basketball ticket regardless of the number of tickets she has.
Her income-consumption curve will then lie along the axis of the good that she chooses. As income increases and the budget line shifts out, she will buy more of the chosen good and none of the other good. Her Engel curve for the good chosen is an upward-sloping straight line, with the number of tickets equal to her income divided by the price of the ticket. For the good not chosen, her Engel curve lies on the vertical (income) axis because she will never purchase any of those tickets regardless of how large her income becomes. 4. a. Orange juice and apple juice are known to be perfect substitutes.
Draw the appropriate price-consumption curve (for a variable price of orange juice) and income-consumption curve. We know that indifference curves for perfect substitutes are straight lines like the line EF in the price-consumption curve diagram below. In this case, the consumer always purchases the cheaper of the two goods (assuming a one-for-one tradeoff). If the price of orange juice is less than the price of apple juice, the consumer will purchase only orange juice and the price-consumption curve will lie along the orange juice axis of the graph (from point F to the right). pic] If apple juice is cheaper, the consumer will purchase only apple juice and the price-consumption curve will be on the apple juice axis (above point E). If the two goods have the same price, the consumer will be indifferent between the two; the price-consumption curve will coincide with the indifference curve (between E and F). Assuming that the price of orange juice is less than the price of apple juice, the consumer will maximize her utility by consuming only orange juice. As income varies, only the amount of orange juice varies.
Thus, the income-consumption curve will be the orange juice axis in the figure below. If apple juice were cheaper, the income-consumption curve would lie on the apple juice axis. [pic] 4. b. Left shoes and right shoes are perfect complements. Draw the appropriate price-consumption and income-consumption curves. For perfect complements, such as right shoes and left shoes, the indifference curves are L-shaped. The point of utility maximization occurs when the budget constraints, L1 and L2 touch the kink of U1 and U2.
See the following figure. [pic] In the case of perfect complements, the income consumption curve is also a line through the corners of the L-shaped indifference curves. See the figure below. [pic] 5. Each week, Bill, Mary, and Jane select the quantity of two goods, X1 and X2, that they will consume in order to maximize their respective utilities. They each spend their entire weekly income on these two goods. a. Suppose you are given the following information about the choices that Bill makes over a three-week period: |[pic] |[pic] |[pic] |[pic] |I | |Week 1 |10 |20 |2 |1 |40 | |Week 2 |7 |19 |3 |1 |40 | |Week 3 |8 |31 |3 |1 |55 | Did Bill’s utility increase or decrease between week 1 and week 2? Between week 1 and week 3? Explain using a graph to support your answer. Bill’s utility fell between weeks 1 and 2 because he consumed less of both goods in week 2.
Between weeks 1 and 2 the price of good 1 rose and his income remained constant. The budget line pivoted inward and he moved from U1 to a lower indifference curve, U2, as shown in the diagram. Between week 1 and week 3 his utility rose. The increase in income more than compensated him for the rise in the price of good 1. Since the price of good 1 rose by $1, he would need an extra $10 to afford the same bundle of goods he chose in week 1. This can be found by multiplying week 1 quantities times week 2 prices. However, his income went up by $15, so his budget line shifted out beyond his week 1 bundle.
Therefore, his original bundle lies within his new budget set as shown in the diagram, and his new week 3 bundle is on the higher indifference curve U3. b. Now consider the following information about the choices that Mary makes: | |[pic] |[pic] |[pic] |[pic] |I | |Week 1 |10 |20 |2 |1 |40 | |Week 2 |6 |14 |2 |2 |40 | |Week 3 |20 |10 |2 |2 |60 | Did Mary’s utility increase or decrease between week 1 and week 3?
Does Mary consider both goods to be normal goods? Explain. Mary’s utility went up. To afford the week 1 bundle at the new prices, she would need an extra $20, which is exactly what happened to her income. However, since she could have chosen the original bundle at the new prices and income but did not, she must have found a bundle that left her slightly better off. In the graph to the right, the week 1 bundle is at the point where the week 1 budget line is tangent to indifference curve U1, which is also the intersection of the week 1 and week 3 budget lines.
The week 3 bundle is somewhere on the week 3 budget line that lies above the week 1 indifference curve. This bundle will be on a higher indifference curve, U3 in the graph, and hence Mary’s utility increased. A good is normal if more is chosen when income increases. Good 1 is normal because Mary consumed more of it when her income increased (and prices remained constant) between weeks 2 and 3. Good 2 is not normal, however, because when Mary’s income increased from week 2 to week 3 (holding prices the same), she consumed less of good 2.
Thus good 2 in an inferior good for Mary. c. Finally, examine the following information about Jane’s choices: | |[pic] |[pic] |[pic] |[pic] |I | |Week 1 |12 |24 |2 |1 |48 | |Week 2 |16 |32 |1 |1 |48 | |Week 3 |12 |24 |1 |1 |36 | Draw a budget line-indifference curve graph that illustrates Jane’s three chosen bundles. What can you say about Jane’s preferences in this case?
Identify the income and substitution effects that result from a change in the price of good X1. In week 2, the price of good 1 drops, Jane’s budget line pivots outward and she consumes more of both goods. In week 3 the prices remain at the new levels, but Jane’s income is reduced. This leads to a parallel leftward shift of her budget line and causes Jane to consume less of both goods. Notice that Jane always consumes the two goods in a fixed 1:2 ratio. This means that Jane views the two goods as perfect complements, and her indifference curves are L-shaped.
Intuitively if the two goods are complements, there is no reason to substitute one for the other during a price change, because they have to be consumed in a set ratio. Thus the substitution effect is zero. When the price ratio changes and utility is kept at the same level (as happens between weeks 1 and 3), Jane chooses the same bundle (12, 24), so the substitution effect is zero. The income effect can be deduced from the changes between weeks 1 and 2 and also between weeks 2 and 3. Between weeks 2 and 3 the only change is the $12 drop in income. This causes Jane to buy 4 fewer units of good 1 and 8 less units of good 2.
Because prices did not change, this is purely an income effect. Between weeks 1 and 2, the price of good 1 decreased by $1 and income remained the same. Since Jane bought 12 units of good 1 in week 1, the drop in price increased her purchasing power by ($1)(12) = $12. As a result of this $12 increase in real income, Jane bought 4 more units of good 1 and 8 more of good 2. We know there is no substitution effect, so these changes are due solely to the income effect, which is the same (but in the opposite direction) as we observed between weeks 1 and 2. . Two individuals, Sam and Barb, derive utility from the hours of leisure (L) they consume and from the amount of goods (G) they consume. In order to maximize utility, they need to allocate the 24 hours in the day between leisure hours and work hours. Assume that all hours not spent working are leisure hours. The price of a good is equal to $1 and the price of leisure is equal to the hourly wage. We observe the following information about the choices that the two individuals make: | |Sam |Barb |Sam |Barb | |Price of G |Price of L |L (hours) |L (hours) |G ($) |G ($) | |1 |8 |16 |14 |64 |80 | |1 |9 |15 |14 |81 |90 | |1 |10 |14 |15 |100 |90 | |1 |11 |14 |16 |110 |88 | Graphically illustrate Sam’s leisure demand curve and Barb’s leisure demand curve. Place price on the vertical axis and leisure on the horizontal axis. Given that they both maximize utility, how can you explain the difference in their leisure demand curves? It is important to remember that less leisure implies more hours spent working. Sam’s leisure demand curve is downward sloping.
As the price of leisure (the wage) rises, he chooses to consume less leisure and thus spend more time working at a higher wage to buy more goods. Barb’s leisure demand curve is upward sloping. As the price of leisure rises, she chooses to consume more leisure (and work less) since her working hours are generating more income per hour. See the leisure demand curves below. This difference in demand can be explained by examining the income and substitution effects for the two individuals. The substitution effect measures the effect of a change in the price of leisure, keeping utility constant (the budget line rotates along the current indifference curve). Since the substitution effect is always negative, a rise in the price of leisure will cause both individuals to consume less leisure.
The income effect measures the effect of the change in purchasing power brought about by the change in the price of leisure. Here, when the price of leisure (the wage) rises, there is an increase in purchasing power (the new budget line shifts outward). Assuming both individuals consider leisure to be a normal good, the increase in purchasing power will increase demand for leisure. For Sam, the reduction in leisure demand caused by the substitution effect outweighs the increase in demand for leisure caused by the income effect, so his leisure demand curve slopes downward. For Barb, her income effect is larger than her substitution effect, so her leisure demand curve slopes upwards. 7.
The director of a theater company in a small college town is considering changing the way he prices tickets. He has hired an economic consulting firm to estimate the demand for tickets. The firm has classified people who go the theater into two groups, and has come up with two demand functions. The demand curves for the general public ([pic]) and students ([pic]) are given below: [pic] a. Graph the two demand curves on one graph, with P on the vertical axis and Q on the horizontal axis. If the current price of tickets is $35, identify the quantity demanded by each group. Both demand curves are downward sloping and linear. For the general public, Dgp, the vertical intercept is 100 and the horizontal intercept is 500.
For the students, Ds, the vertical intercept is 50 and the horizontal intercept is 200. When the price is $35, the general public demands [pic] tickets and students demand [pic] tickets. b. Find the price elasticity of demand for each group at the current price and quantity. The elasticity for the general public is [pic] and the elasticity for students is [pic]. If the price of tickets increases by ten percent then the general public will demand 5. 4% fewer tickets and students will demand 23. 3% fewer tickets. c. Is the director maximizing the revenue he collects from ticket sales by charging $35 for each ticket? Explain. No he is not maximizing revenue because neither of the calculated elasticities is equal to –1.
The general public’s demand is inelastic at the current price. Thus the director could increase the price for the general public, and the quantity demanded would fall by a smaller percentage, causing revenue to increase. Since the students’ demand is elastic at the current price, the director could decrease the price students pay, and their quantity demanded would increase by a larger amount in percentage terms, causing revenue to increase. d. What price should he charge each group if he wants to maximize revenue collected from ticket sales? To figure this out, use the formula for elasticity, set it equal to –1, and solve for price and quantity. For the general public: [pic] For the students: [pic]
These prices generate a larger total revenue than the $35 price. When price is $35, revenue is (35)(Qgp + Qs) = (35)(325 + 60) = $13,475. With the separate prices, revenue is PgpQgp + PsQs = (50)(250) + (25)(100) = $15,000, which is an increase of $1525, or 11. 3%. 8. Judy has decided to allocate exactly $500 to college textbooks every year, even though she knows that the prices are likely to increase by 5 to 10 percent per year and that she will be getting a substantial monetary gift from her grandparents next year. What is Judy’s price elasticity of demand for textbooks? Income elasticity? Judy will spend the same amount ($500) on textbooks even when prices increase. We know that total revenue (i. e. total spending on a good) remains constant when price changes only if demand is unit elastic. Therefore Judy’s price elasticity of demand for textbooks is –1. Her income elasticity must be zero because she does not plan to purchase more books even though she expects a large monetary gift (i. e. , an increase in income). 9. The ACME Corporation determines that at current prices the demand for its computer chips has a price elasticity of –2 in the short run, while the price elasticity for its disk drives is –1. a. If the corporation decides to raise the price of both products by 10 percent, what will happen to its sales? To its sales revenue? Note: The answer at the end of the book (first printing) for the percent change in disk drive sales revenue is incorrect. The correct answer is given below. We know the formula for the elasticity of demand is EP = %? Q/%? P. For computer chips, EP = –2, so –2 = %? Q/10, and therefore %? Q = –20. Thus a 10 percent increase in price will reduce the quantity sold by 20 percent. For disk drives, EP = –1, so a 10 percent increase in price will reduce sales by 10 percent. Sales revenue will decrease for computer chips because demand is elastic and price has increased. We can estimate the change in revenue as follows. Revenue is equal to price times quantity sold.
Let TR1 = P1Q1 be revenue before the price change and TR2 = P2Q2 be revenue after the price change. Therefore (TR = P2Q2 – P1Q1, and thus (TR = (1. 1P1 )(0. 8Q1 ) – P1Q1 = –0. 12P1Q1, or a 12 percent decline. Sales revenue for disk drives will remain unchanged because demand elasticity is –1. b. Can you tell from the available information which product will generate the most revenue? If yes, why? If not, what additional information do you need? No. Although we know the elasticities of demand, we do not know the prices or quantities sold, so we cannot calculate the revenue for either product. We need to know the prices of chips and disk drives and how many of each ACME sells. 10.
By observing an individual’s behavior in the situations outlined below, determine the relevant income elasticities of demand for each good (i. e. , whether the good is normal or inferior). If you cannot determine the income elasticity, what additional information do you need? a. Bill spends all his income on books and coffee. He finds $20 while rummaging through a used paperback bin at the bookstore. He immediately buys a new hardcover book of poetry. Books are a normal good since his consumption of books increases with income. Coffee is a neutral good since consumption of coffee stayed the same when income increased. b. Bill loses $10 he was going to use to buy a double espresso. He decides to sell his new book at a discount to a friend and use the money to buy coffee.
When Bill’s income decreased by $10 he decided to own fewer books, so books are a normal good. Coffee appears to be a neutral good because Bill’s purchase of the double espresso did not change as his income changed. c. Being bohemian becomes the latest teen fad. As a result, coffee and book prices rise by 25 percent. Bill lowers his consumption of both goods by the same percentage. Books and coffee are both normal goods because Bill’s response to a decline in real income is to decrease consumption of both goods. In addition, the income elasticities for both goods are the same because Bill reduces consumption of both by the same percentage. d.
Bill drops out of art school and gets an M. B. A. instead. He stops reading books and drinking coffee. Now he reads The Wall Street Journal and drinks bottled mineral water. His tastes have changed completely, and we do not know how he would respond to price and income changes. We need to observe how his consumption of the WSJ and bottled water changes as his income changes. 11. Suppose the income elasticity of demand for food is 0. 5 and the price elasticity of demand is –1. 0. Suppose also that Felicia spends $10,000 a year on food, the price of food is $2, and that her income is $25,000. a. If a sales tax on food caused the price of food to increase to $2. 0, what would happen to her consumption of food? (Hint: Since a large price change is involved, you should assume that the price elasticity measures an arc elasticity, rather than a point elasticity. ) The arc elasticity formula is: [pic]. We know that EP = –1, P1 = 2, P2 = 2. 50 (so (P = 0. 50), and Q1 = 5000 units (because Felicia spends $10,000 and each unit of food costs $2). We also know that Q2, the new quantity, is Q2 = Q1 + ? Q. Thus, if there is no change in income, we may solve for (Q: [pic]. By cross-multiplying and rearranging terms, we find that (Q = –1000. This means that she decreases her consumption of food from 5000 to 4000 units.
As a check, recall that total spending should remain the same because the price elasticity is –1. After the price change, Felicia spends ($2. 50)(4000) = $10,000, which is the same as she spent before the price change. b. Suppose that Felicia gets a tax rebate of $2500 to ease the effect of the sales tax. What would her consumption of food be now? A tax rebate of $2500 is an income increase of $2500. To calculate the response of demand to the tax rebate, use the definition of the arc elasticity of income. [pic]. We know that EI = 0. 5, I1 = 25,000, ? I = 2500 (so I2 = 27,500), and Q1 = 4000 (from the answer to 11a). Assuming no change in price, we solve for (Q. [pic].
By cross-multiplying and rearranging terms, we find that (Q = 195 (approximately). This means that she increases her consumption of food from 4000 to 4195 units. c. Is she better or worse off when given a rebate equal to the sales tax payments? Draw a graph and explain. > Note: The answer at the end of the book (first printing) used incorrect quantities and prices. The correct answer is given below. Felicia is better off after the rebate. The amount of the rebate is enough to allow her to purchase her original bundle of food and other goods. Recall that originally she consumed 5000 units of food. When the price went up by fifty cents per unit, she needed an extra (5000)($0. 0) = $2500 to afford the same quantity of food without reducing the quantity of the other goods consumed. This is the exact amount of the rebate. However, she did not choose to return to her original bundle. We can therefore infer that she found a better bundle that gave her a higher level of utility. In the graph below, when the price of food increases, the budget line pivots inward. When the rebate is given, this new budget line shifts out to the right in a parallel fashion. The bundle after the rebate is on that part of the new budget line that was previously unaffordable, and that lies above the original indifference curve. It is on a higher indifference curve, so Felicia is better off after the rebate. [pic] 12.
You run a small business and would like to predict what will happen to the quantity demanded for your product if you raise your price. While you do not know the exact demand curve for your product, you do know that in the first year you charged $45 and sold 1200 units and that in the second year you charged $30 and sold 1800 units. a. If you plan to raise your price by 10 percent, what would be a reasonable estimate of what will happen to quantity demanded in percentage terms? We must first find the price elasticity of demand. Because the price and quantity changes are large in percentage terms, it is best to use the arc elasticity measure. EP = (? Q/? P)(average P/average Q) = (600/–15)(37. 50/1500) = –1.
With an elasticity of –1, a 10 percent increase in price will lead to a 10 percent decrease in quantity. b. If you raise your price by 10 percent, will revenue increase or decrease? When elasticity is –1, revenue will remain constant if price is increased. 13. Suppose you are in charge of a toll bridge that costs essentially nothing to operate. The demand for bridge crossings Q is given by [pic]. a. Draw the demand curve for bridge crossings. The demand curve is linear and downward sloping. The vertical intercept is 15 and the horizontal intercept is 30. b. How many people would cross the bridge if there were no toll? At a price of zero, 0 = 15 – (1/2)Q, so Q = 30. The quantity demanded would be 30. c.
What is the loss of consumer surplus associated with a bridge toll of $5? If the toll is $5 then the quantity demanded is 20. The lost consumer surplus is the difference between the consumer surplus when price is zero and the consumer surplus when price is $5. When the toll is zero, consumer surplus is the entire area under the demand curve, which is (1/2)(30)(15) = 225. When P = 5, consumer surplus is area A + B + C in the graph above. The base of this triangle is 20 and the height is 10, so consumer surplus = (1/2)(20)(10) = 100. The loss of consumer surplus is therefore 225 – 100 = $125. d. The toll-bridge operator is considering an increase in the toll to $7.
At this higher price, how many people would cross the bridge? Would the toll-bridge revenue increase or decrease? What does your answer tell you about the elasticity of demand? At a toll of $7, the quantity demanded would be 16. The initial toll revenue was $5(20) = $100. The new toll revenue is $7(16) = $112. Since the revenue went up when the toll was increased, demand is inelastic (the 40% increase in price outweighed the 20% decline in quantity demanded). e. Find the lost consumer surplus associated with the increase in the price of the toll from $5 to $7. The lost consumer surplus is area B + C in the graph above. Thus, the loss in consumer surplus is (16)(7 – 5) + (1/2)(20 – 16)(7 – 5) = $36. 14.
Vera has decided to upgrade the operating system on her new PC. She hears that the new Linux operating system is technologically superior to Windows and substantially lower in price. However, when she asks her friends, it turns out they all use PCs with Windows. They agree that Linux is more appealing but add that they see relatively few copies of Linux on sale at local stores. Vera chooses Windows. Can you explain her decision? Vera is influenced by a positive network externality (not a bandwagon effect). When she hears that there are limited software choices that are compatible with Linux and that none of her friends use Linux, she decides to go with Windows.
If she had not been interested in acquiring much software and did not think she would need to get advice from her friends, she might have purchased Linux. 15. Suppose that you are the consultant to an agricultural cooperative that is deciding whether members should cut their production of cotton in half next year. The cooperative wants your advice as to whether this action will increase members’ revenues. Knowing that cotton (C) and watermelons (W) both compete for agricultural land in the South, you estimate the demand for cotton to be C = 3. 5 – 1. 0PC + 0. 25PW + 0. 50I, where PC is the price of cotton, PW the price of watermelon, and I income. Should you support or oppose the plan? Is there any additional information that would help you to provide a definitive answer?
If production of cotton is cut in half, then the price of cotton will increase, given that we see from the equation above that demand is downward sloping. With price increasing and quantity demanded decreasing, revenue could go either way. It depends on whether demand is elastic or inelastic. If demand is elastic, a decrease in production and an increase in price would decrease revenue. If demand is inelastic, a decrease in production and an increase in price would increase revenue. You need a lot of information before you can give a definitive answer. First, you must know the current prices for cotton and watermelon plus the level of income; then you can calculate the quantity of cotton demanded, C.
Next, you have to cut C in half and determine the effect that will have on the price of cotton, assuming that income and the price of watermelons are not affected (which is a big assumption). Then you can calculate the original revenue and the new revenue to see whether this action increases members’ revenues or not. CHAPTER 5 UNCERTAINTY AND CONSUMER BEHAVIOR EXERCISES 1. Consider a lottery with three possible outcomes: • $125 will be received with probability . 2 • $100 will be received with probability . 3 • $50 will be received with probability . 5 a. What is the expected value of the lottery? The expected value, EV, of the lottery is equal to the sum of the returns weighted by their probabilities: EV = (0. 2)($125) + (0. 3)($100) + (0. 5)($50) = $80. b. What is the variance of the outcomes?
The variance, (2, is the sum of the squared deviations from the mean, $80, weighted by their probabilities: (2 = (0. 2)(125 – 80)2 + (0. 3)(100 – 80)2 + (0. 5)(50 – 80)2 = $975. c. What would a risk-neutral person pay to play the lottery? A risk-neutral person would pay the expected value of the lottery: $80. 2. Suppose you have invested in a new computer company whose profitability depends on two factors: (1) whether the U. S. Congress passes a tariff raising the cost of Japanese computers and (2) whether the U. S. economy grows slowly or quickly. What are the four mutually exclusive states of the world that you should be concerned about? The four mutually exclusive states may be represented as: |Congress passes tariff |Congress does not pass tariff | |Slow growth rate |State 1: |State 2: | | |Slow growth with tariff |Slow growth without tariff | |Fast growth rate |State 3: |State 4: | | |Fast growth with tariff |Fast growth without tariff | 3. Richard is deciding whether to buy a state lottery ticket. Each ticket costs $1, and the probability of winning payoffs is given as follows: |Probability |Return | |0. 50 |$0. 00 | |0. 25 |$1. 00 | |0. 20 |$2. 00 | |0. 05 |$7. 50 | a.
What is the expected value of Richard’s payoff if he buys a lottery ticket? What is the variance? The expected value of the lottery is equal to the sum of the returns weighted by their probabilities: EV = (0. 5)($0) + (0. 25)($1. 00) + (0. 2)($2. 00) + (0. 05)($7. 50) = $1. 025 The variance is the sum of the squared deviations from the mean, $1. 025, weighted by their probabilities: (2 = (0. 5)(0 – 1. 025)2 + (0. 25)(1 – 1. 025)2 + (0. 2)(2 – 1. 025)2 + (0. 05)(7. 5 – 1. 025)2, or (2 = 2. 812. b. Richard’s nickname is “No-Risk Rick” because he is an extremely risk-averse individual. Would he buy the ticket? An extremely risk-averse individual would probably not buy the ticket.
Even though the expected value is higher than the price of the ticket, $1. 025 > $1. 00, the difference is not enough to compensate Rick for the risk. For example, if his wealth is $10 and he buys a $1. 00 ticket, he would have $9. 00, $10. 00, $11. 00, and $16. 50, respectively, under the four possible outcomes. If his utility function is U = W0. 5, where W is his wealth, then his expected utility is: [pic] This is less than 3. 162, which is his utility if he does not buy the ticket (U(10) = 100. 5 = 3. 162). Therefore, he would not buy the ticket. c. Richard has been given 1000 lottery tickets. Discuss how you would determine the smallest amount for which he would be willing to sell all 1000 tickets.
With 1000 tickets, Richard’s expected payoff is $1025. He does not pay for the tickets, so he cannot lose money, but there is a wide range of possible payoffs he might receive ranging from $0 (in the extremely unlikely case that all 1000 tickets pay nothing) to $7500 (in the even more unlikely case that all 1000 tickets pay the top prize of $7. 50), and everything in between. Given this variability and Richard’s high degree of risk aversion, we know that Richard would be willing to sell all the tickets for less (and perhaps considerably less) than the expected payoff of $1025. More precisely, he would sell the tickets for $1025 minus his risk premium.
To find his selling price, we would first have to calculate his expected utility for the lottery winnings. This would be like point F in Figure 5. 4, except that in Richard’s case there are thousands of possible payoffs, not just two as in the figure. Using his expected utility value, we then would find the certain amount that gives him the same level of utility. This is like the $16,000 income at point C in Figure 5. 4. That certain amount is the smallest amount for which he would be willing to sell all 1000 lottery tickets. d. In the long run, given the price of the lottery tickets and the probability/return table, what do you think the state would do about the lottery?
Given the price of the tickets, the sizes of the payoffs and the probabilities, the lottery is a money loser. The state loses $1. 025 – 1. 00 = $0. 025 (two and a half cents) on every ticket it sells. The state must raise the price of a ticket, reduce some of the payoffs, raise the probability of winning nothing, lower the probabilities of the positive payoffs, or some combination of the above. 4. Suppose an investor is concerned about a business choice in which there are three prospects – the probability and returns are given below: |Probability |Return | |0. 4 |$100 | |0. 3 | 30 | |0. | –30 | What is the expected value of the uncertain investment? What is the variance? The expected value of the return on this investment is EV = (0. 4)(100) + (0. 3)(30) + (0. 3)(–30) = $40. The variance is (2 = (0. 4)(100 – 40)2 + (0. 3)(30 – 40)2 + (0. 3)(–30 – 40)2 = 2940. 5. You are an insurance agent who must write a policy for a new client named Sam. His company, Society for Creative Alternatives to Mayonnaise (SCAM), is working on a low-fat, low-cholesterol mayonnaise substitute for the sandwich-condiment industry. The sandwich industry will pay top dollar to the first inventor to patent such a mayonnaise substitute.
Sam’s SCAM seems like a very risky proposition to you. You have calculated his possible returns table as follows: |Probability |Return |Outcome | |. 999 | –$1,000,000 |(he fails) | |. 001 |$1,000,000,000 |(he succeeds and sells his formula) | a. What is the expected return of Sam’s project? What is the variance? The expected return, ER, of Sam’s investment is ER = (0. 999)(–1,000,000) + (0. 001)(1,000,000,000) = $1000. The variance is 2 = (0. 999)(–1,000,000 – 1000)2 + (0. 001)(1,000,000,000 – 1000)2 , or (2 = 1,000,998,999,000,000. b. What is the most that Sam is willing to pay for insurance? Assume Sam is risk neutral. Suppose the insurance guarantees that Sam will receive the expected return of $1000 with certainty regardless of the outcome of his SCAM project. Because Sam is risk neutral and because his expected return is the same as the guaranteed return with insurance, the insurance has no value to Sam. He is just as happy with the uncertain SCAM profits as with the certain outcome guaranteed by the insurance policy. So Sam will not pay anything for the insurance. c.
Suppose you found out that the Japanese are on the verge of introducing their own mayonnaise substitute next month. Sam does not know this and has just turned down your final offer of $1000 for the insurance. Assume that Sam tells you SCAM is only six months away from perfecting its mayonnaise substitute and that you know what you know about the Japanese. Would you raise or lower your policy premium on any subsequent proposal to Sam? Based on his information, would Sam accept? The entry of the Japanese lowers Sam’s probability of a high payoff. For example, assume that the probability of the billion-dollar payoff is lowered to zero. Then the expected outcome is: ER = (1. 0)(–$1,000,000) + (0. 0)(($1,000,000,000) = –$1,000,000.
Therefore, you should raise the policy premium substantially. But Sam, not knowing about the Japanese entry, will continue to refuse your offers to insure his losses. 6. Suppose that Natasha’s utility function is given by [pic], where I represents annual income in thousands of dollars. a. Is Natasha risk loving, risk neutral, or risk averse? Explain. Natasha is risk averse. To show this, assume that she has $10,000 and is offered a gamble of a $1000 gain with 50 percent probability and a $1000 loss with 50 percent probability. Her utility of $10,000 is u(10) = [pic] = 10. Her expected utility with the gamble is: EU = (0. 5)[pic] + (0. 5)[pic] = 9. 87 < 10. She would avoid the gamble. If she were risk neutral, she would be indifferent between the $10,000 and the gamble, and if she were risk loving, she would prefer the gamble. You can also see that she is risk averse by noting that the square root function increases at a decreasing rate (the second derivative is negative), implying diminishing marginal utility. b. Suppose that Natasha is currently earning an income of $40,000 (I = 40) and can earn that income next year with certainty. She is offered a chance to take a new job that offers a . 6 probability of earning $44,000 and a . 4 probability of earning $33,000. Should she take the new job?
The utility of her current salary is [pic]= 20. The expected utility of the new job’s salary is EU = (0. 6)[pic] + (0. 4)[pic] = 19. 85, which is less than 20. Therefore, she should not take the job. You can also determine that Natasha should reject the job by noting that the expected value of the new job is only $39,600, which is less than her current salary. Since she is risk averse, she should never accept a risky salary with a lower expected value than her current certain salary. c. In (b), would Natasha be willing to buy insurance to protect against the variable income associated with the new job? If so, how much would she be willing to pay for that insurance? Hint: What is the risk premium? ) This question assumes that Natasha takes the new job (for some unexplained reason). Her expected salary is 0. 6(44,000) + 0. 4(33,000) = $39,600. The risk premium is the amount Natasha would be willing to pay so that she receives the expected salary for certain rather than the risky salary in her new job. In part (b) we determined that her new job has an expected utility of 19. 85. We need to find the certain salary that gives Natasha the same utility of 19. 85, so we want to find I such that u(I) = 19. 85. Using her utility function, we want to solve the following equation: [pic]. Squaring both sides, 10I =394. 02, and I = 39. 402.
So Natasha would be equally happy with a certain salary of $39,402 or the uncertain salary with an expected value of $39,600. Her risk premium is $39,600 – 39,402 = $198. Natasha would be willing to pay $198 to guarantee her income would be $39,600 for certain and eliminate the risk associated with her new job. 7. Suppose that two investments have the same three payoffs, but the probability associated with each payoff differs, as illustrated in the table below: | |Probability |Probability | |Payoff |(Investment A) |(Investment B) | |$300 |0. 10 |0. 0 | |$250 |0. 80 |0. 40 | |$200 |0. 10 |0. 30 | a. Find the expected return and standard deviation of each investment. The expected value of the return on investment A is EV = (0. 1)(300) + (0. 8)(250) + (0. 1)(200) = $250. The variance on investment A is (2 = (0. 1)(300 – 250)2 + (0. 8)(250 – 250)2 + (0. 1)(200 – 250)2 = $500, and the standard deviation on investment A is ? = [pic] = $22. 36. The expected value of the return on investment B is EV = (0. 3)(300) + (0. 4)(250) + (0. 3)(200) = $250.
The variance on investment B is (2 = (0. 3)(300 – 250)2 + (0. 4)(250 – 250)2 + (0. 3)(200 – 250)2 = $1500, and the standard deviation on investment B is ? = [pic] = $38. 73. b. Jill has the utility function [pic], where I denotes the payoff. Which investment will she choose? Jill’s expected utility from investment A is EU= (0. 1)[5(300)] + (0. 8)[5(250)] + (0. 1)[5(200)] = 1250. Jill’s expected utility from investment B is EU=(0. 3)[5(300)] + (0. 4)[5(250)] + (0. 3)[5(200)] = 1250. Since both investments give Jill the same expected utility she will be indifferent between the two. Note that Jill is risk neutral, so she cares only about expected values.
Since investments A and B have the same expected values, she is indifferent between them. c. Ken has the utility function [pic]. Which investment will he choose? Ken’s expected utility from investment A is EU = (0. 1)(5[pic]) + (0. 8)(5[pic]) + (0. 1)(5[pic]) = 78. 98. Ken’s expected utility from investment B is EU=(0. 3)(5[pic]) + (0. 4)(5[pic]) + (0. 3)(5[pic]) = 78. 82. Ken will choose investment A because it has a slightly higher expected utility. Notice that Ken is risk averse, so he prefers the investment with less variability. d. Laura has the utility function [pic]. Which investment will she choose? Laura’s expected utility from investment A is EU=(0. 1)[5(3002)] + (0. 8)[5(2502)] + (0. )[5(2002)] = 315,000. Laura’s expected utility from investment B is EU=(0. 3)[5(3002)] + (0. 4)[5(2502)] + (0. 3)[5(2002)] = 320,000. Laura will choose investment B since it has a higher expected utility. Notice that Laura is a risk lover, so she prefers the investment with greater variability. 8. As the owner of a family farm whose wealth is $250,000, you must choose between sitting this season out and investing last year’s earnings ($200,000) in a safe money market fund paying 5. 0 percent or planting summer corn. Planting costs $200,000, with a six-month time to harvest. If there is rain, planting summer corn will yield $500,000 in revenues at harvest.
If there is a drought, planting will yield $50,000 in revenues. As a third choice, you can purchase AgriCorp drought-resistant summer corn at a cost of $250,000 that will yield $500,000 in revenues at harvest if there is rain, and $350,000 in revenues if there is a drought. You are risk averse, and your preference for family wealth (W) is specified by the relationship [pic]. The probability of a summer drought is 0. 30, while the probability of summer rain is 0. 70. Which of the three options should you choose? Explain. Calculate the expected utility of wealth under the three options. Wealth is equal to the initial $250,000 plus whatever is earned growing corn or investing in the safe financial asset.
Expected utility under the safe option, allowing for the fact that your initial wealth is $250,000, is: E(U) = (250,000 + 200,000(1 + . 05)). 5 = 678. 23. Expected utility with regular corn, again including your initial wealth, is: E(U) = . 7(250,000 + (500,000 – 200,000)). 5 + . 3(250,000 + (50,000 – 200,000)). 5 = 519. 13 + 94. 87 = 614. Expected utility with drought-resistant corn is: E(U) = . 7(250,000 + (500,000 – 250,000)). 5 + . 3(250,000 + (350,000 – 250,000)). 5 = 494. 975 + 177. 482 = 672. 46. You should choose the option with the highest expected utility, which is the safe option of not planting corn. Note: There is a subtle time issue in this problem.
The returns from planting corn occur in 6 months while the money market fund pays 5%, which is presumably a yearly interest rate. To put everything on equal footing, we should compare the returns of all three alternatives over a 6-month period. In this case, the money market fund would earn about 2. 5%, so its expected utility is: E(U) = (250,000 + 200,000(1 + . 025)). 5 = 674. 54. This is still the best of the three options, but by a smaller margin than before. 9. Draw a utility function over income u(I) that describes a man who is a risk lover when his income is low but risk averse when his income is high. Can you explain why such a utility function might reasonably describe a person’s preferences? The utility function will be S-shaped as illustrated below.
Preferences might be like this for an individual who needs a certain level of income, I*, in order to stay alive. An increase in income above I* will have diminishing marginal utility. Below I*, the individual will be a risk lover and will take unfavorable gambles in an effort to make large gains in income. Above I*, the individual will purchase insurance against losses and below I* will gamble. 10. A city is considering how much to spend to hire people to monitor its parking meters. The following information is available to the city manager: • Hiring each meter monitor costs $10,000 per year. • With one monitoring person hired, the probability of a driver getting a ticket each time he or she parks illegally is equal to . 25. With two monitors, the probability of getting a ticket is . 5; with three monitors, the probability is . 75; and with four, it’s equal to 1. • With two monitors hired, the current fine for overtime parking is $20. a. Assume first that all drivers are risk neutral. What parking fine would you levy, and how many meter monitors would you hire (1, 2, 3, or 4) to achieve the current level of deterrence against illegal parking at the minimum cost? If drivers are risk neutral, their behavior is influenced only by their expected fine. With two meter monitors, the probability of detection is 0. 5 and the fine is $20. So, the expected fine is (0. 5)($20) + (0. 5)(0) = $10.
To maintain this expected fine, the city can hire one meter monitor and increase the fine to $40, or hire three meter monitors and decrease the fine to $13. 33, or hire four meter monitors and decrease the fine to $10. If the only cost to be minimized is the cost of hiring meter monitors at $10,000 per year you, as the city manager, should minimize the number of meter monitors. Hire only one monitor and increase the fine to $40 to maintain the current level of deterrence. b. Now assume that drivers are highly risk averse. How would your answer to (a) change? If drivers are risk averse, they would want to avoid the possibility of paying parking fines even more than would risk neutral drivers.
Therefore, a fine of less than $40 with one meter monitor should maintain the current level of deterrence. c. (For discussion) What if drivers could insure themselves against the risk of parking fines? Would it make good public policy to permit such insurance? Drivers engage in many forms of behavior to insure themselves against the risk of parking fines, such as checking the time often to be sure they have not parked overtime, parking blocks away from their destination in non-metered spots or taking public transportation. If a private insurance firm offered insurance that paid the fine when a ticket was received, drivers would not worry about getting tickets.
They would not seek out unmetered spots or take public transportation; they would park in metered spaces for as long as they wanted at zero personal cost. Having the insurance would lead drivers to get many more parking tickets. This is referred to as moral hazard and may cause the insurance market to collapse, but that’s another story (see Section 17. 3 in Chapter 17). It probably would not make good public policy to permit such insurance. Parking is usually metered to encourage efficient use of scarce parking space. People with insurance would have no incentive to use public transportation, seek out-of-the-way parking locations or economize on their use of metered spaces. This imposes a cost on others who are not able to find a place to park.
If the parking fines are set to efficiently allocate the scarce amount of parking space available, then the availability of insurance will lead to an inefficient use of the parking space. In this case, it would not be good public policy to permit the insurance. 11. A moderately risk-averse investor has 50 percent of her portfolio invested in stocks and 50 percent in risk-free Treasury bills. Show how each of the following events will affect the investor’s budget line and the proportion of stocks in her portfolio: a. The standard deviation of the return on the stock market increases, but the expected return on the stock market remains the same. From section 5. 4, the equation for the budget line is [pic], here Rp is the expected return on the portfolio, Rm is the expected return from investing in the stock market, Rf is the risk-free return on Treasury bills, ? m is the standard deviation of the return from investing in the stock market, and ? p is the standard deviation of the return on the portfolio. The budget line is linear and shows the positive relationship between the return on the portfolio, Rp, and the standard deviation of the return on the portfolio, ? p, as shown in Figure 5. 6. In this case ? m, the standard deviation of the return on the stock market, increases. The slope of the budget line therefore decreases, and the budget line becomes flatter.
The budget line’s intercept stays the same because Rf does not change. Thus, at any given level of portfolio return, the portfolio now has a higher standard deviation. Since stocks have become riskier without a compensating increase in expected return, the proportion of stocks in the investor’s portfolio will fall. b. b. The expected return on the stock market increases, but the standard deviation of the stock market remains the same. In this case, Rm, the expected return on the stock market, increases, so the slope of the budget line becomes steeper. At any given level of portfolio standard deviation, ? p, there is now a higher expected return, Rp.
Stocks have become relatively more attractive because investors now get greater expected returns with no increase in risk, and the proportion of stocks in the investor’s portfolio will rise as a consequence. c. The return on risk-free Treasury bills increases. In this case there is an increase in Rf, which affects both the intercept and slope of the budget line. The budget line shifts up and become flatter as a result. The proportion of stocks in the portfolio could go either way. On the one hand, Treasury bills now have a higher return and so are more attractive. On the other hand, the investor can now earn a higher return from each Treasury bill and so could hold fewer Treasury bills and still maintain the same level of risk-free return.
In this second case, the investor may be willing to place more of her money in the stock market. It will depend on the particular preferences of the investor as well as the magnitude of the returns to the two asset classes. An analogy would be to consider what happens to savings when the interest rate increases. On the one hand, savings tend to increase because the return is higher, but on the other hand, spending may increase and savings decrease because a person can save less each period and still wind up with the same accumulation of savings at some future date. CHAPTER 6 PRODUCTION EXERCISES 1. The menu at Joe’s coffee shop consists of a variety of coffee drinks, pastries, and sandwiches.
The marginal product of an additional worker can be defined as the number of customers that can be served by that worker in a given time period. Joe has been employing one worker, but is considering hiring a second and a third. Explain why the marginal product of the second and third workers might be higher than the first. Why might you expect the marginal product of additional workers to diminish eventually? The marginal product could well increase for the second and third workers because each would be able to specialize in a different task. If there is only one worker, that person has to take orders and prepare all the food. With 2 or 3, however, one could take orders and the others could do most of the coffee and food preparation.
Eventually, however, as more workers are employed, the marginal product would diminish because there would be a large number of people behind the counter and in the kitchen trying to serve more and more customers with a limited amount of equipment and a fixed building size. 2. Suppose a chair manufacturer is producing in the short run (with its existing plant and equipment). The manufacturer has observed the following levels of production corresponding to different numbers of workers: Number of chairsNumber of workers 110 218 324 428 530 628 7. 25 a. Calculate the marginal and average product of labor for this production function. The average product of labor, APL, is equal to [pic]. The marginal product of labor, MPL, is equal to [pic], the change in output divided by the change in labor input.
For this production process we have: |L |q |APL |MPL | |0 |0 |__ |__ | |1 |10 |10 |10 | |2 |18 |9 |8 | |3 |24 |8 |6 | |4 |28 |7 |4 | |5 |30 |6 |2 | |6 |28 |4. 7 |–2 | |7 |25 |3. 6 |–3 | b. Does this production function exhibit diminishing returns to labor? Explain. Yes, this production process exhibits diminishing returns to labor.
The marginal product of labor, the extra output produced by each additional worker, diminishes as workers are added, and this starts to occur with the second unit of labor. c. Explain intuitively what might cause the marginal product of labor to become negative. Labor’s negative marginal product for L > 5 may arise from congestion in the chair manufacturer’s factory. Since more laborers are using the same fixed amount of capital, it is possible that they could get in each other’s way, decreasing efficiency and the amount of output. Firms also have to control the quality of their output, and the high congestion of labor may produce products that are not of a high enough quality to be offered for sale, which can contribute to a negative marginal product. 3. Fill in the gaps in the table below. Quantity of |Total |Marginal Product |Average Product | |Variable Input |Output |of Variable Input |of Variable Input | |0 | 0 |– |– | |1 |225 | | | |2 | | |300 | |3 | |300 | | |4 |1140 | | | |5 | |225 | | |6 | | |225 | | | | | | | | | | | Quantity of |Total |Marginal Product |Average Product | |Variable Input |Output |of Variable Input |of Variable Input | |0 | 0 |___ |___ | |1 |225 |225 |225 | |2 |600 |375 |300 | |3 |900 |300 |300 | |4 |1140 |240 |285 | |5 |1365 |225 |273 | |6 |1350 |–15 |225 | ———————– [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic] [pic]