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Exercises PT 2 [Please do not distribute] Ch 8: Slutsky Equation • For this chapter, you should be able to do the following: • Find Slutsky income effect and substitution effect of a specific price change if you know the demand function for a good. • Show the Slutsky income and substitution effects of a price change on an indifference curve diagram. • Show the Hicks income and substitution effects of a price change on an indifference curve diagram. • Find the Slutsky income and substitution effects for special utility functions such as perfect substitutes, perfect complements, and Cobb-Douglas. • Use an indifference-curve diagram to show how the case of a Giffen good might arise. • Show that the substitution effect of a price increase unambiguously decreases demand for the good whose price rose. • Apply income and substitution effects to draw some inferences about behavior. 8.1 • As we know, Charlie consumes apples and bananas. Recall that this utility function is U(xA, xB) = xAxB. The price of apples is $1, the price of bananas is $2, and Charlie’s income is $40 a day. Now assume that the price of bananas suddenly falls to $1. • (a) How many apples and bananas did Charlie consume a day before the price change? On a graph with apples on x-axis and bananas on y-axis, use black ink to draw Charlie’s original budget line and put the label A on his chosen consumption bundle. • (b) If, after the price change, Charlie’s income had changed so that he could exactly afford his old consumption bundle, what would his new income be? With this income and the new prices, how many apples and bananas did Charlie consume a day? Use red ink to draw the budget line corresponding to this income and these prices. Label the bundle that Charlie would choose at this income and the new prices with the letter B. • (c) Does the substitution effect of the fall in the price of bananas make him buy more bananas or fewer bananas? How many more or fewer? • (d) How many apples and bananas does Charlie buy actually buy after the price change? Use blue ink to draw Charlie’s actual budget line after the price change. Put the label C on the bundle that he actually chooses after the price change. Draw 3 horizontal lines on your graph, one from A to the vertical axis, one from B to the vertical axis, and one from C to the vertical axis. Along the vertical axis, label the income effect, the substitution effect, and the total effect on the demand for bananas. Is the blue line parallel to the red line or the black line that you drew before? • (e) The income effect of the fall in the price of bananas on Charlie’s demand for bananas is the same as the effect of an (increase, decrease) in his income of $10 per day. Does the income effect make him consume more bananas or fewer? How many more or how many fewer? • (f) Does the substitution effect of the fall in the price of bananas make Charlie consume more apples or fewer? How many more or fewer? Does the income effect of the fall in the price of bananas make Charlie consume more apples or fewer? What is the total effect of the change in the price of bananas on the demand for apples? 8.2 • Neville loves wine. When the prices of all other goods are fixed at current levels, Neville’s demand function for high quality claret is q = .02m − 2p, where m is his income, p is the price of claret (in £), and q is the number of bottles of claret that he demands. Neville’s income is £7,500, and the price of a bottle of suitable claret is £30. • (a) How many bottles of claret will Neville buy? • (b) If the price of claret rose to £40, how much income would Neville have to have in order to be exactly able to afford the amount of claret and the amount of other goods that he bought before the price change? At this income, and a price of £40, how many bottles would Neville buy? • (c) At his original income of 7,500 and a price of 40, how much claret would Neville demand? • (d) When the price of claret rose from 30 to 40, the number of bottles that Neville demanded decreased by 20. The substitution effect (increased, reduced) his demand by s bottles and the income effect (increased, reduced) his demand by i bottles. 8.3 Note: Do this problem only if you have read the section entitled “Another Substitution Effect” that describes the “Hicks substitution effect”. • Consider the figure below, which shows the budget constraint and the indifference curves of Zog. Zog is in equilibrium with an income of $300, facing prices pX = $4 and pY = $10. • (a) How much X does Zog consume? • (b) If the price of X falls to $2.50, while income and the price of Y stay constant, how much X will Zog consume? • (c) How much income must be taken away from Zog to isolate the Hicksian income and substitution effects (i.e., to make him just able to afford to reach his old indifference curve at the new prices)? • (d) From what point to what point does consumption change indicating the total effect of the price change? • (e) From what point to what point corresponds to the income effect? What movement corresponds substitution effect? • (f) Is X a normal good or an inferior good? • (g) On another graph, sketch an Engel curve and a demand curve for Good X that would be reasonable given the information in the graph above. Be sure to label the axes on both your graphs. 8.4 • Maude spends all of her income on delphiniums and hollyhocks. She thinks that delphiniums and hollyhocks are perfect substitutes; one delphinium is just as good as one hollyhock. Delphiniums cost $4 a unit and hollyhocks cost $5 a unit. • (a) If the price of delphiniums decreases to $3 a unit, will Maude buy more of them? What part of the change in consumption is due to the income effect and what part is due to the substitution effect? • (b) If the prices of delphiniums and hollyhocks are respectively pd = $4 and ph = $5 and if Maude has $120 to spend, draw her budget line in blue ink on a graph. Draw also the highest indifference curve that she can attain in red ink, and label the point that she chooses as A. • (c) Now assume the price of hollyhocks fall to $3 a unit, while the price of delphiniums does not change. Draw her new budget line in black ink. Draw the highest indifference curve that she can now reach with red ink. Label the point she chooses now as B. • (d) How much would Maude’s income have to be after the price of hollyhocks fell, so that she could just exactly afford her old commodity bundle A? • (e) When the price of hollyhocks fell to $3, what part of the change in Maude’s demand was due to the income effect and what part was due to the substitution effect? 8.7 • Mr. Consumer spends $100 per month on cigarettes and ice cream. Mr. C’s preferences for cigarettes and ice cream are unaffected by the season of the year. • (a) In January, the price of cigarettes was $1 per pack, while ice cream cost $2 per pint. Faced with these prices, Mr. C bought 30 pints of ice cream and 40 packs of cigarettes. Draw Mr. C’s January budget line with blue ink and label his January consumption bundle with the letter J. • (b) In February, Mr. C again had $100 to spend and ice cream still cost $2 per pint, but the price of cigarettes rose to $1.25 per pack. Mr. C consumed 30 pints of ice cream and 32 packs of cigarettes. Draw Mr. C’s February budget line with red ink and mark his February bundle with the letter F. • The substitution effect of this price change would make him buy (less, more, the same amount of) cigarettes and (less, more, the same amount of) ice cream. Since this is true and the total change in his ice cream consumption was zero, it must be that the income effect of this price change on his consumption of ice cream makes him buy (more, less, the same amount of) ice cream. The income effect of this price change is like the effect of an (increase, decrease) in his income. Therefore the information we have suggests that ice cream is a(n) (normal, inferior, neutral) good. • (c) In March, Mr. C again had $100 to spend. Ice cream was on sale for $1 per pint. Cigarette prices increased to $1.50 per pack. Draw his March budget line with black ink. Is he better off than in January, worse off, or can you not make such a comparison? How does your answer to the last question change if the price of cigarettes had increased to $2 per pack? • (d) In April, cigarette prices rose to $2 per pack and ice cream was still on sale for $1 per pint. Mr. C bought 34 packs of cigarettes and 32 pints of ice cream. Draw his April budget line with pencil and label his April bundle with the letter A. Was he better off or worse off than in January? Was he better off or worse off than in February, or can’t one tell? • (e) In May, cigarettes stayed at $2 per pack and as the sale on ice cream ended, the price returned to $2 per pint. On the way to the store, however, Mr. C found $30 lying in the street. He then had $130 to spend on cigarettes and ice cream. Draw his May budget with a dashed line. Without knowing what he purchased, one can determine whether he is better off than he was in at least one previous month. Which month or months? 8.10 • Agatha must travel on the Orient Express from Istanbul to Paris. The distance is 1,500 miles. A traveler can choose to make any fraction of the journey in a first-class carriage and travel the rest of the way in a second-class carriage. • The price is 10 cents a mile for a second class carriage and 20 cents a mile for a first-class carriage. Agatha much prefers first-class to second-class travel, but because of a misadventure in an Istanbul bazaar, she has only $200 left with which to buy her tickets. • Luckily, she still has her toothbrush and a suitcase full of cucumber sandwiches to eat on the way. Agatha plans to spend her entire $200 on her tickets for her trip. • She will travel first class as much as she can afford to, but she must get all the way to Paris, and $200 is not enough money to get her all the way to Paris in first class. • (a) On a graph, use red ink to show the locus of combinations of first- and second-class tickets that Agatha can just afford to purchase with her $200. Use blue ink to show the locus of combinations of first and second-class tickets that are sufficient to carry her the entire distance from Istanbul to Paris. Locate the combination of first- and second-class miles that Agatha will choose on your graph and label it A. • (b) Let m1 be the number of miles she travels by first -class coach and m2 be the number of miles she travels by second- class coach. Write down two equations that you can solve to find the number of miles she chooses to travel by first -class coach and the number of miles she chooses to travel by second-class coach. • (c) How many miles does she travel by second-class coach? • (d) Just before she was ready to buy her tickets, the price of second-class tickets fell to $.05 while the price of first-class tickets remained at $.20. • On the graph that you drew, use pencil to show the combinations of first-class and second-class tickets that she can afford with her $200 at these prices. Also on the same graph, locate the combination of first-class and second-class tickets that she would now choose. (Remember, she is going to travel as much first-class as she can afford to and still make the 1,500 mile trip on $200.) Label this point B. • How many miles does she travel by second class now? (Hint: For an exact solution you will have to solve two linear equations in two unknowns.) Is second-class travel a normal good for Agatha? If not, what type of good is it? • (e) Just after the price change from $.10 per mile to $.05 per mile for second-class travel, and just before she had bought any tickets, Agatha misplaced her handbag. Although she kept most of her money in her socks, the money she lost was just enough so that at the new prices, she could exactly afford the combination of first- and second-class tickets that she would have purchased at the old prices. How much money did she lose? • On the graph you drew previous, use black ink to draw the locus of combinations of first- and second- class tickets that she can just afford after discovering her loss. Label the point that she chooses with a C. How many miles will she travel by second class now? • (f) Finally, poor Agatha finds her handbag again. How many miles will she travel by second class now (assuming she didn’t buy any tickets before she found her lost handbag)? • When the price of second class tickets fell from $.10 to $.05, how much of a change in Agatha’s demand for second-class tickets was due to a substitution effect? • How much of a change was due to an income effect? Ch 9: Buying and Selling • In previous chapters, you studied the behavior of consumers who start out without owning any goods, but who had some money with which to buy goods. • In this chapter, the consumer has an initial endowment, which is the bundle of goods the consumer owns before any trades are made. • That is, a consumer can trade away from his initial endowment by selling one good and buying the other. 9.1 • Abishag owns 20 quinces and 5 kumquats. She has no income from any other source, but she can buy or sell either quinces or kumquats at their market prices. The price of kumquats is four times the price of quinces. There are no other commodities of interest. • (a) How many quinces could she have if she was willing to do without kumquats? How many kumquats could she have if she was willing to do without quinces? • (b) Draw Abishag’s budget set, using blue ink, and label the endowment bundle with the letter E. If the price of quinces is 1 and the price of kumquats is 4, write Abishag’s budget equation. If the price of quinces is 2 and the price of kumquats is 8, write Abishag’s budget equation. What effect does doubling both prices have on the set of commodity bundles that Abishag can afford? • (c) Suppose that Abishag decides to sell 10 quinces. Label her final consumption bundle in your graph with the letter C. • (d) Now, after she has sold 10 quinces and owns the bundle labeled C, suppose that the price of kumquats falls so that kumquats cost the same as quinces. On the diagram above, draw Abishag’s new budget line, using red ink. • (e) If Abishag obeys the weak axiom of revealed preference, then there are some points on her red budget line that we can be sure Abishag will not choose. On the graph, make a squiggly line over the portion of Abishag’s red budget line that we can be sure she will not choose. 9.2 • Mario has a small garden where he raises eggplant and tomatoes. He consumes some of these vegetables, and he sells some in the market. • Eggplants and tomatoes are perfect complements for Mario, since the only recipes he knows use them together in a 1:1 ratio. One week his garden yielded 30 kg of eggplant and 10 kg of tomatoes. At that time the price of each vegetable was $5/kg. • (a) What is the monetary value of Mario’s endowment of vegetables? • (b) On a graph, use blue ink to draw Mario’s budget line. How much does Mario ends up consuming? Draw the indifference curve through the consumption bundle that Mario chooses and label this bundle A. Also note his endowment point. • (c) Suppose that before Mario makes any trades, the price of tomatoes rises to $15/kg, while the price of eggplant stays at $5/kg. What is the value of Mario’s endowment now? Draw his new budget line, using red ink. What will he choose to consume? • (d) Now suppose that Mario had sold his entire crop at the market for a total of $200, intending to buy back some tomatoes and eggplant for his own consumption. Before he had a chance to buy anything back, the price of tomatoes rose to $15, while the price of eggplant stayed at $5. Draw his budget line, using black ink. What does Mario consume? • (e) Assuming that the price of tomatoes rose to $15 from $5 before Mario made any transactions, what is the change in the demand for tomatoes due to the substitution effect? What is the change in the demand for tomatoes due to the ordinary income effect? What is the change in the demand for tomatoes due to the endowment income effect? And, lastly, what is the total change in the demand for tomatoes? 9.3 • Lucetta consumes only two goods, A and B. Her only source of income is gifts of these commodities from her many admirers. She doesn’t always get these goods in the proportions in which she wants to consume them, but she can always buy or sell A at the price pA = 1 and B at the price pB = 2. Lucetta’s utility function is U(a, b) = ab, where a is the amount of A she consumes and b is the amount of B she consumes. • (a) Suppose that Lucetta’s admirers give her 100 units of A and 200 units of B. In a graph, use red ink to draw her budget line. Label her initial endowment E. • (b) What are Lucetta’s gross demands for A? And for B? • (c) What are Lucetta’s net demands? • (d) Suppose that before Lucetta has made any trades, the price of good B falls to 1, and the price of good A stays at 1. Draw Lucetta’s budget line at these prices on your graph, using blue ink. • (e) Does Lucetta’s consumption of good B rise or fall? By how much? What happens to Lucetta’s consumption of good A? • (f) Suppose that before the price of good B fell, Lucetta had exchanged all of her gifts for money, planning to use the money to buy her consumption bundle later. How much of good B will she choose to consume? How much of good A? • (g) Explain why her consumption is different depending on whether she was holding goods or money at the time of the price change. 9.7 • Mr. Blog works in a machine factory. He can work as many hours per day as he wishes at a wage rate of w. Let C be the number of dollars he spends on consumer goods and let R be the number of hours of leisure that he chooses. • (a) Mr. Blog earns $8 an hour and has 18 hours per day to devote to labor or leisure, and he has $16 of non-labor income per day. Write an equation for his budget between consumption and leisure. • (b) Use blue ink to draw his budget line in a graph. His initial endowment is the point where he does no work and enjoys 18 hours of leisure per day. Mark this point on the graph below with the letter A. • (Remember that although Blog can choose to work and thereby “sell” some of his endowment of leisure, he cannot “buy leisure” by paying somebody else to loaf for him.) • If Mr. Blog has the utility function U(R,C) = CR, how many hours of leisure per day will he choose? How many hours per day will he work? • (c) Suppose that Mr. Blog’s wage rate increases to $12 an hour, draw his new budget line in red ink. If he continued to work exactly as many hours as he did before the wage change, how much more money would he have each day to spend on consumption? With this new budget line, how many hours does he work and how much does his consumption change? • (d) Suppose that Mr. Blog still receives $8 an hour but that his non-labor income rises to $48 per day. Use black ink to draw his budget line. How many hours does he choose to work? • (e) Suppose that Mr. Blog has a wage of $w per hour, a non-labor income of $m, and that he has 18 hours a day to divide between labor and leisure. Blog’s budget line has the equation C + wR = m + 18w. Using the same methods you used in the chapter on demand functions, find the amount of leisure that Blog will demand as a function of wages and of non-labor income. (Hint: Notice that this is the same as finding the demand for R when the price of R is w, the price of C is 1, and income is m + 18w.) • What is Mr. Blog’s demand function for leisure? • And what is Mr. Blog’s supply function for labor? Ch 10. Intertemporal Choice • Peregrine consumes (c1, c2) and earns (m1,m2) in periods 1 and 2 respectively. Suppose the interest rate is r. • (a) Write down Peregrine’s intertemporal budget constraint in present value terms. • (b) If Peregrine does not consume anything in period 1, what is the most he can consume in period 2? • (c) If Peregrine does not consume anything in period 2, what is the most he can consume in period 1? • (d) What is the slope of Peregrine’s budget line? 10.2 • Molly has a Cobb-Douglas utility function U(c1, c2) = c1α c21− α , where 0 < a < 1 and where c1 and c2 are her consumptions in periods 1 and 2 respectively. We saw earlier that if utility has the form u(x1, x2) = x1α x21− α and the budget constraint is of the “standard” form p1x1+p2x2=m, then the demand functions for the goods are x1 = am/p1 and x2 = (1 − a)m/p2. • (a) Suppose that Molly’s income is m1 in period 1 and m2 in period 2. Write down her budget constraint in terms of present values. • (b) We want to compare this budget constraint to one of the standard form. In terms of Molly’s budget constraint, what is p1? What is p2? What is m? • (c) If α = .2, solve for Molly’s demand functions for consumption in each period as a function of m1, m2, and r. That is, what are her demand functions for consumption in period 1 and period 2? 10.3 • Nickleby has an income of $2,000 this year, and he expects an income of $1,100 next year. He can borrow and lend money at an interest rate of 10%. Consumption goods cost $1 per unit this year and there is no inflation. • (a) What is the present value of Nickleby’s endowment? What is the future value of his endowment? With blue ink, show the combinations of consumption this year and consumption next year that he can afford. Label Nickelby’s endowment with the letter E. • (b) Suppose that Nickleby has the utility function U(C1,C2) = C1C2. Write an expression for Nickleby’s marginal rate of substitution between consumption this year and consumption next year. (Your answer will be a function of the variables C1,C2.) • (c) What is the slope of Nickleby’s budget line? Write an equation that states that the slope of Nickleby’s indifference curve is equal to the slope of his budget line when the interest rate is 10%. Also write down Nickleby’s budget equation. • (d) Solve these two equations. What does Nickleby will consume? Label this point A on your diagram. • (e) Will he borrow or save in the first period? How much? • (f) On your graph use red ink to show what Nickleby’s budget line would be if the interest rate rose to 20%. Knowing that Nickleby chose the point A at a 10% interest rate, even without knowing his utility function, you can determine that his new choice cannot be on certain parts of his new budget line. Draw a squiggly mark over the part of his new budget line where that choice can not be. (Hint: Close your eyes and think of WARP.) • (g) Solve for Nickleby’s optimal choice when the interest rate is 20%. What will he consume? • (h) Will he borrow or save in the first period? How much? 10.5 • Laertes has an endowment of $20 each period. He can borrow money at an interest rate of 200%, and he can lend money at a rate of 0%. (Note: If the interest rate is 0%, for every dollar that you save, you get back $1 in the next period. If the interest rate is 200%, then for every dollar you borrow, you have to pay back $3 in the next period.) • (a) Use blue ink to illustrate his budget set in the graph below. (Hint: The boundary of the budget set is not a single straight line.) • (b) Laertes could invest in a project that would leave him with m1 = 30 and m2 = 15. Besides investing in the project, he can still borrow at 200% interest or lend at 0% interest. Use red ink to draw the new budget set in the graph above. Would Laertes be better off or worse off by investing in this project given his possibilities for borrowing or lending? Or can’t one tell without knowing something about his preferences? Explain. • (c) Consider an alternative project that would leave Laertes with the endowment m1 = 15, m2 = 30. Again suppose he can borrow and lend as above. But if he chooses this project, he can’t do the first project. Use pencil or black ink to draw the budget set available to Laertes if he chooses this project. Is Laertes better off or worse off by choosing this project than if he didn’t choose either project? Or can’t one tell without knowing more about his preferences? Explain. 10.8 • Mr. O. B. Kandle will only live for two periods. In the first period he will earn $50,000. In the second period he will retire and live on his savings. His utility function is U(c1, c2) = c1c2, where c1 is consumption in period 1 and c2 is consumption in period 2. He can borrow and lend at the interest rate r = .10. • (a) If the interest rate rises, will his period-1 consumption increase, decrease, or stay the same? • (b) Would an increase in the interest rate make him consume more or less in the second period? • (c) If Mr. Kandle’s income is zero in period 1, and $ 55,000 in period 2, would an increase in the interest rate make him consume more, less, or the same amount in period 1? 10.9 • Harvey’s utility function is U(c1, c2) = min{c1, c2}, where c1 is his consumption of bread in period 1 and c2 is his consumption of bread in period 2. The price of bread is $1 per loaf in period 1. The interest rate is 21%. Harvey earns $2,000 in period 1 and he will earn $1,100 in period 2. • (a) Write Harvey’s budget constraint in terms of future value, assuming no inflation. • (b) How much bread does Harvey consume in the first period and how much money does he save? • (c) Suppose that Harvey’s money income in both periods is the same as before, the interest rate is still 21%, but there is a 10% inflation rate. Then in period 2, a loaf of bread will cost $ 1.10. Write down Harvey’s budget equation for period-1 and period-2 bread, given this new information. 10.10 • In an isolated mountain village, the only crop is corn. Good harvests alternate with bad harvests. This year the harvest will be 1,000 bushels. Next year it will be 150 bushels. There is no trade with the outside world. Corn can be stored from one year to the next, but rats will eat 25% of what is stored in a year. The villagers have Cobb-Douglas utility functions, U(c1, c2) = c1c2 where c1 is consumption this year, and c2 is consumption next year. • (a) Use red ink to draw a “budget line,” showing consumption possibilities for the village, with this year’s consumption on the horizontal axis and next year’s consumption on the vertical axis. Put numbers on your graph to show where the budget line hits the axes. • (b) How much corn will the villagers consume this year? How much will the rats eat? How much corn will the villagers consume next year? • (c) Suppose that a road is built to the village so that now the village is able to trade with the rest of the world. Now the villagers are able to buy and sell corn at the world price, which is $1 per bushel. They are also able to borrow and lend money at an interest rate of 10%. On your graph, use blue ink to draw the new budget line for the villagers. Solve for the amount they would now consume in the first period and the second period. • (d) Suppose that all is as in the last part of the questi on except that there is a transportation cost of $.10 p er bushel for every bushel of grain hauled into or out of the village. On your graph, use black ink or pencil t o draw the budget line for the village under these circ umstances. 10.12 • Marsha doesn’t care whether she consumes in period 1 or in period 2. Her utility function is simply U(c1, c2) = c1 + c2. Her initial endowment is $20 in period 1 and $40 in period 2. In an antique shop, she discovers a cookie jar that is for sale for $12 in period 1 and that she is certain she can sell for $20 in period 2. She derives no consumption benefits from the cookie jar, and it costs her nothing to store it for one period. • (a) On a graph, label her initial endowment, E, and use blue ink to draw the budget line showing combinations of period-1 and period-2 consumption that she can afford if she doesn’t buy the cookie jar. On the same graph, label the consumption bundle, A, that she would have if she did not borrow or lend any money but bought the cookie jar in period 1, sold it in period 2, and used the proceeds to buy period-2 consumption. If she cannot borrow or lend, should Marsha invest in the cookie jar? • (b) Suppose that Marsha can borrow and lend at an interest rate of 50%. On the graph where you labelled her initial endowment, draw the budget line showing all of the bundles she can afford if she invests in the cookie jar and borrows or lends at the interest rate of 50%. On the same graph use red ink to draw one or two of Marsha’s indifference curves. • (c) Suppose that instead of consumption in the two periods being perfect substitutes, they are perfect complements, so that Marsha’s utility function is min{c1, c2}. If she cannot borrow or lend, should she buy the cookie jar? If she can borrow and lend at an interest rate of 50%, should she invest in the cookie jar? If she can borrow or lend as much at an interest rate of 100%, should she invest in the cookie jar?