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RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2011 – 2012 / IA 350, Intermediate Microeconomics / Problem Set 1 1. [March 6] You have an income of $40 to spend on two commodities. Commodity 1 costs $10 per unit and commodity 2 costs $5 per unit. a. Write down your budget equation: ______________________________________________. b. If you spend all of your income on commodity 1, how much of it could you buy? _________. c. If you spend all of your income on commodity 2, how much of it could you buy? _________. d. Use blue ink to draw your budget line in the graph below: x2 8 6 4 2 x1 0 e. 2 4 6 8 Suppose that the price of commodity 1 falls to $5 while everything else stays the same. Write down your new budget equation: _______________________________________________. On the graph above, use red ink to draw your new budget line. f. Suppose that the amount you are allowed to spend falls to $30, while the prices of both commodities remain at $5. Write down your budget equation: ________________________. Use black ink to draw this budget line. 2. [March 8] Louis Foulmont consumes two commodities, garbage and old computer games. He doesn't actually consume the former, but dumps it in his back yard where it is eaten by mountain goats while Louis sits in his hovel and plays games on his computer. People pay him $4 per sack for taking the garbage off their hands. Louis can accept as much garbage as he wishes at that price, and has no other source of income. Old computer games cost him $12 each. a. If Louis accepts zero sacks of garbage, how many computer games can he buy? __________. b. If Louis accepts 15 sacks of garbage, how many computer games can he buy? ___________. c. Write down an equation for his budget constraint: __________________________________. RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2011 – 2012 / IA 350, Intermediate Microeconomics / Problem Set 1 d. Draw his budget line and shade in his feasible budget set. Garbage 20 15 10 5 0 3. 5 10 15 20 Computer Games [March 8] Marcel consumes only nuts and berries. Both commodities are economic goods for Marcel. The set of consumption bundles (x1, x2) such that Marcel is indifferent between (x1, x2) and (1, 16) is the set of bundles such that x1, x2 > 0 and x2 20 4 x1 . The set of consumption bundles (x1, x2) such that Marcel is indifferent between (x1, x2) and (36, 0) is the set of bundles such that x1, x2 > 0 and x2 24 4 x1 . a. On the diagram below, plot several points that lie on the indifference curve that passes through the point (1, 16) and sketch this curve, using blue ink. Do the same, using red ink, for the indifference curve passing through the point (36, 0). b. Is the set of bundles that Marcel prefers to (1, 16) a convex set? c. What is the slope of Marcel's indifference curve at the point (9, 8)? d. What is the slope of Marcel's indifference curve at the point (4, 12)? _________. e. What is the slope of Marcel's indifference curve at the point (9, 12)? _________. What is the slope of Marcel's indifference curve at the point (4, 16)? __________. f. Do the indifference curves you have drawn exhibit diminishing marginal rate of substitution? _________. g. Does Marcel have convex preferences? __________. RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2011 – 2012 / IA 350, Intermediate Microeconomics / Problem Set 1 x2 40 , Berries 30 20 10 0 4. 10 20 30 40 x1 , Nuts [March 9] Suppose that Phil likes to consume only apples and oranges. He has $3.00 per week to spend on these goods, the price of apples is $0.20 each, and the most oranges he could consume is 10. a. Write out Phil’s budget constraint and graph it on the diagram below. _____________________. b. Assume that Phil is indifferent between the following bundles, where the first number in parenthesis is a quantity of apples, while the second number is a quantity of oranges: (5,10); (7,6); (9,4); (12,3). Plot these bundles and draw Phil’s indifference curve. Are Phil’s preferences convex? c. What is Phil’s marginal rate of substitution at the bundle (9,4)? _________________. d. If someone offered Phil 2 oranges for 1 apple, would he accept the trade? _________________. Oranges Apples RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2011 – 2012 / IA 350, Intermediate Microeconomics / Problem Set 1 5. [March 9] Adrian's utility function is U(x, y) = (x + 2)(y + 1). a. Write an equation, y = f(x), for Adrian's indifference curve that goes through the point (x, y) = 2, 8). _________________________________. On the diagram below, sketch Adrian's indifference curve for U = 36. b. Suppose that the price of each good is 1 and that Adrian has an income of 11. Write out an expression for the budget constraint: ___________________________. Based on this information, add an appropriate budget line to the diagram. Can Adrian achieve U = 36 with this budget? ___________. c. Write out a mathematical expression for Adrian's marginal rate of substitution in terms of x and y: _____________________. (Remember, d. MRS MU x .) MU y Using this information, what is Adrian's optimal consumption bundle? x = _____; y = ______. y 16 12 8 4 x 0 4 8 12 16 RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2011 – 2012 / IA 350, Intermediate Microeconomics / Problem Set 1 6. [March 12] Consider this following utility function: ( ) a. Derive demand functions for x and y (in terms of prices and income). b. Derive the indirect utility function. c. What are the price elasticities of demand for goods x and y? For each good, is demand elastic, unit elastic, or inelastic? 7. [March 12] Suppose that U(x, y, z) = (x – a)(y – b)(z – c) and that = 1. Solve for the Marshallian demand functions. Remember that you can transform the utility function. 8. [March 12] Suppose that an individual’s utility function is U x12 x28 . a. Write an expression for this individual’s marginal rate of substitution: ___________________. b. Making use of your result in part (a) and of the general formulation of the budget equation ( derive a general statement for this individual’s demand for good p1 x1 p2 x2 m ), x1 as a function of prices and income. ____________________________________________________. c. Now, if p1 2 , p 2 4 , and m 240 , how much of good x1 will the individual demand and what portion of his income will he spend on it? __________________________. d If p1 doubles (to 4), p 2 increases by 50% (to 6), and income remains unchanged, how much of good x1 will the individual demand and what portion of his income will he spend on it? ___________________________. e. Compare the portion of income spent on good x1 in parts (c) and (d). How could you have predicted this result before you even began calculating demand functions? RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2011 – 2012 / IA 350, Intermediate Microeconomics / Problem Set 1 9. [March 13] Modify the two budget and indifference curves below in the following ways: a. Assume that x1 is a normal good and that its price increases. Show the new budget line, a compensated budget line (using the Slutsky approach), and indicate the magnitudes and directions of the substitution and income effects, identifying each, along the horizontal axis. b. Assume that x1 is an inferior good (but not a Giffen good!) and that its price decreases. Show the new budget line, a compensated budget line (using the Slutsky approach), and indicate the magnitudes and directions of the substitution and income effects, identifying each, along the horizontal axis. x2 x2 x2 0 x2 x1 0 x1 0 x1 x1 0 10. [March 15] Suppose a consumer's utility function is given by U(x1, x2) = x1x2, that p1 =1, p 2 =2, and m = 40, and that p 2 suddenly falls to 1. a. Before the price change, this consumer will consume ______ units of x1 and ______ units of x 2 . On the diagram below draw an indifference curve/budget diagram that illustrates this initial choice point, labeling the bundle that the consumer chooses under these conditions with the letter A. b. If, after the price change, the consumer's income had changed so that she could exactly afford her original consumption bundle, her new income would have to be __________. With this new income and at the new prices, however, this consumer would consumer ______ units of x1 and ______ units of x 2 . On the diagram below, use red ink to draw the budget line corresponding to this income and these prices. Label the bundle that the consumer chooses with the letter B. c. Does the substitution effect of the fall in p 2 cause the consumer to buy more or less of much more or less? ______________________________________________. d. After the price change, the consumer actually buys ______ units of x1 and ______ units of x 2 . Use blue ink to draw the actual budget line after the price change. Put the label C on the bundle that she actually chooses after the price change. e. Draw three horizontal lines on the diagram, one from point 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 and substitution effects on the demand for x 2 . x 2 ? _______. How RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2011 – 2012 / IA 350, Intermediate Microeconomics / Problem Set 1 f. Draw three vertical lines on the diagram, one from point A to the horizontal axis, one from B to the horizontal axis, and one from C to the horizontal axis. Along the horizontal axis, label the income and substitution effects on the demand for x1 . x2 40 30 20 10 0 10 20 30 40 x1 11. [March 15] A certain economist spends his free time consuming fine French wine. When the prices of all other goods are fixed at current levels, his demand function for rich, red bordeaux wine is q 0.005m 2 p , where m is his income, p is the price of the wine in dollars, and q is the number of bottles that he demands. His income is $30,000 (obviously he is overpaid), and the dollar price of a bottle of suitable wine is $30. a. At his current level of income and with current prices, how many bottles will he demand? _________. b. If the depreciation of the dollar against the euro causes the dollar price of a bottle of bordeaux to rise to $50, what would his income need to be in order to be exactly able to afford the amount of bordeaux and the amount of other goods that he bought before the price change? ____________. At this income and the price of $50, how many bottles would he demand? ______________. c. At his original income of $30,000 and a price of $50, how many bottles would he demand? _________. d. When the dollar price of bordeaux rises from $30 to $50, the number of bottles that this economist demanded decreased by _________. The substitution effect [increased, decreased] his demand by _________ bottles, and the income effect [increased, decreased] his demand by _________. RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2011 – 2012 / IA 350, Intermediate Microeconomics / Problem Set 1 12. [March 15] Madeline spends all of her income on pansies and petunias, which she regards as perfect substitutes. Pansies cost $4 per unit, and petunias cost $5 per unit. a. If the price of pansies decreases to $3 per unit, will Madeline buy more of them? _________. What part of this change in consumption is due to the income effect and what part is due to the substitution effect? ___________________________________________________________. b. If the price of pansies was $4, the price of petunias was $5, and Madeline's income for flowers is $120, draw her budget line in blue ink on the diagram below. Draw the highest indifference curve that she can attain in red ink, and label the point that she chooses as A. c. Now suppose that the price of petunias falls to $3 per unit while the price of pansies does not change. Draw her new budget line in black ink. Draw the highest indifference curve that she can now reach with red ink, and label her optimum choice point as B. d. How much would Madeline's income have to be after the price of petunias fell, so that she could just exactly afford her old commodity bundle at A? _____________________. e. When the price of petunias fell to $3, what part of the change in Madeline's demand was due to the income effect and what part was due to the substitution effect? ___________________________. RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2011 – 2012 / IA 350, Intermediate Microeconomics / Problem Set 1 13. [March 15] Given U(x1, x2) = x1x22 and p1 = 10, p2 = 10, and m = 3,000 … a. Use this information and your knowledge of demand functions for Cobb-Douglass utility functions to calculate the total demand for x and y and the total expenditures on each good (p1x1 and p2x2). You do not need to show the derivation of the demand functions. b. Suppose the price of x1 increases to 25. Calculate the new demands for x1 and x2. c. Decompose the change in demand for x1 due to this price change into a Slutsky substitution and income effect. In other words, calculate the portion of the total change in demand for x1 that is attributable to a substitution effect and the portion of the total change in demand for x1 that is attributable to an income effect. For full credit, show the steps in your work, including computation of m’, compensated income. d. Carefully and neatly construct a diagram illustrating your solution, indicating all key values along the horizontal and vertical axes. RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2010 – 2011 / IA 350, Intermediate Microeconomics / Problem Set 1 14. [March 16] Suppose that a new political party, The Pigovians, gains control of Congress and their first move is to impose a steep carbon tax on gasoline to encourage conservation and “do something about climate change.” For simplicity, assume that the new tax on gasoline causes the retail price to increase from $3 per gallon to $5 per gallon (yikes!). Now assume that a typical consumer with annual income of $40,000 and convex preferences consumes 1,200 gallons of gas before imposition of the tax 900 gallons of gas after imposition of the tax. a. Carefully and neatly construct a diagram that illustrates this consumer's equilibrium before and after imposition of the tax. Specify equilibrium amounts (before and after the tax) on both axes. Your diagram need not be drawn to scale -- in fact it will be easier if you do not attempt to draw to scale along the vertical axis. b. It took a while for the Pigovians to figure out why everyone was so mad at them after the imposition of the tax. But then you showed them your diagram and explained it to them. What did you say? c. After talking with you, the Pigovians have a new idea: they will send everyone a rebate for the amount of new tax on gasoline they spent. Their argument is very simple: "We didn't want to make people worse off, we just wanted them to use less gas. So if we rebate the tax they pay, they'll be as well off as they were before the tax but will use less gasoline." Critics of this proposal argue that this tax and rebate plan will have no effect on gasoline demand since consumers will just use the rebated money to purchase more gasoline. What does economic analysis say about this? d. The Pigovians claim that their rebate plan will make everyone as well off as they were before the tax. Modify the diagram above to show the flaw in the Pigovian’s rebate logic and explain that flaw here. Specifically, add a new, post-rebate, budget constraint that illustrates the flaw in their rebate plan. (Hint: Think about how the rebate shifts the budget constraint.) RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2010 – 2011 / IA 350, Intermediate Microeconomics / Problem Set 1 15. [March 16] a. In the figure below, the consumer’s gross demands of x1 are _________, while his net demands of x1 are _________. His gross demands of x 2 are _________, while his net demands of x 2 are _________. Modify the diagram to illustrate the effect on the consumer’s budget constraint of an increase in the price of x1 . x2 200 80 A . U0 E 4 x1 200 b. 400 . In the figure above, if the price of x1 rises and his status as a buyer or seller of these goods does not change, will his welfare improve, deteriorate, or is it impossible to tell? 16. [March 19] Herman consumes only two goods, A and B, and his only source of income is gifts of these commodities from neighbors who are concerned about his welfare. He doesn't always get these goods in the proportions in which he wants to consume them, but he can always buy or sell them at prevailing market prices of p A 1 and p B 2 . Herman's utility function for these goods is U A, B AB . a. Suppose that Herman's neighbors give him 100 units of A and 200 units of B. On the diagram below, use red ink to draw his budget line and label his initial endowment E. b. What are Herman's gross demands for A and for B? ____________________________________. c. What are Herman's net demands? ________________________________________. d. Suppose that before Herman has made any trades, the price of good B falls to 1, and the price of good A remains unchanged at 1. Modify the diagram, using blue ink to draw Herman's new budget line at these prices. e. Does Herman's consumption of good B rise or fall? ___________. By how much? __________. What happens to Herman's consumption of good A? ________________________________. RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2010 – 2011 / IA 350, Intermediate Microeconomics / Problem Set 1 f. Suppose that before the price of good B fell, Herman had exchange all of his gifts for money, planning to use the money to buy his consumption bundle later. How much of good B will he choose to consume? _______________. How much of good A? __________________. 17. [March 19] Darrell is a serious college student who has a $40 per week stipend from a trust fund. He has 15 hours of classes per week, religiously devotes 3 hours to study for every hour in class, and spends 68 hours per week sleeping, eating, etc. There are 168 hours in a week, so Darrell’s is left with 40 hours per week of free time to do with as he pleases. He has a job that pays $4 per hour after taxes. This is the result of an $8 per hour wage rate and a 50% average tax rate. Darrell’s boss has told him that he can work as many hours each week as he wants. Finally, assume that Darrell derives utility from consumption and leisure, according to the following utility function: U c, l c1 / 2 l 1 / 2 Recall that this implies Darrell’s demand functions are: l m 2 pl and c m 2 pc where m is the market value of Darrell’s endowment. a. What is the market value of Darrell’s endowment? b. Use the diagram below to Draw Darrell’s budget line. c. Use the given utility function and demand functions to determine Darrell’s optimal consumption bundle and level of utility. Add this point to the diagram and label it A. d. How many hours does Darrell choose to work? What is his gross salary, his net salary, and how much does he pay in income taxes? e. Suppose that the average tax rates are cut to 37.5%, so that Darrell’s after-tax wage rises from $4 to $5. What does this do to the market value of Darrell’s endowment? Show this change on the diagram by adding Darrell’s new budget line. f. Use the given utility function and demand functions to determine Darrell’s optimal consumption bundle and level of utility with the new prices. Add this point to the diagram and label it B. g. How many hours does Darrell choose to work? What is his gross salary, his net salary, and how much does he pay in income taxes? h. What value endowment would Darrel have needed at the new prices to make his original consumption bundle just affordable? i. Use the demand equations and the amount from part (h) to divide Darrell’s movement from A to B into a substitution effect and an income effect. Does the income effect cause him to consume more or less leisure? (Conversely, does the income effect cause him to work more or less?) j. In this problem, the amount of tax revenue generated from Darrell goes down after a tax cut, even though he works more hours. What is it about the problem that contributes to this RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2010 – 2011 / IA 350, Intermediate Microeconomics / Problem Set 1 result? Can you think of conditions that would lead to a situation in which Darrell works more and pays more in taxes? 18. [March 19] Wendell and Troy work in fast food restaurants. Wendell gets $4 an hour for the first 40 hours he works and $6 an hour for every hour beyond 40 hours a week. Troy gets $5 an hour no matter how many hours he works. Each as 80 hours a week to allocate between work and leisure, and neither has any income from sources other than labor. Each has a utility function of the form U c, l c * l , where c is consumption and l is leisure. Each can choose the number of hours to work. a. How many hours will Troy chose to work? b. Wendell’s budget constraint has a kink in it at the point where l = ______ and c = _______. On the diagram below, use blue ink for the part of his budget line where he would be if he does not work overtime. Use red ink for the part where he would be if he worked overtime. c. Regarding Wendell’s budget constraint, write out the equation for the blue line segment: ______________________________________________________________________ Regarding Wendell’s budget constraint, write out the equation for the bred line segment: ______________________________________________________________________ d. If Wendell was paid $4 an hour no matter how many hours he worked, how many hours would he work? Modify the diagram below with a black “well-behaved” indifference curve that is tangent to Wendell’s budget constraint at this point. RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2010 – 2011 / IA 350, Intermediate Microeconomics / Problem Set 1 a. Will Wendell choose to work overtime? _______. What is the optimal choice for Wendell? (c, l) = _________________________. How many hours per week will he work? _______. f. Suppose that the jobs are equally agreeable in all other respects. Since Wendell and Troy have identical preferences, they will be able to agree about who has the better job. Who has the better job? _________________. 19. [March 19] For most people, there are 168 hours in a week. Sarah is one such person, and her wellbeing is based on consumption and leisure according to U(c,l) = c0.25*l0.75. Assume that she has investments that yield $50 per week of unearned income, thus allowing her some level of consumption without working. Finally, assume that she earns a gross wage of $25 per hour as a Fox News commentator, but faces a 25% average tax rate. a. Demonstrate mathematically that Sarah works 40 hours per week. Show your work and explain it. b. Now suppose that politicians cannot resolve a dispute over tax rates, so they eliminate income taxes entirely. Does Sarah work more or fewer hours per week? Be specific and show your work. c. Modify the diagram below to illustrate your solution from part (b). Don’t worry about correct scaling; in fact, it will be helpful if you exaggerate the magnitude of the change associated with the reduced tax rate. Decompose the change in leisure demand (hence, labor supply) into all relevant effects – SE, OIE, and EIE. RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2010 – 2011 / IA 350, Intermediate Microeconomics / Problem Set 1 20. [March 19] In the United States, real wage rates in manufacturing rose steadily from 1890 to the early 1980s. In the period from 1890 to 1930, the length of the workweek was reduced dramatically. But after 1930, despite the continuing growth of real wage rates, the length of the work week has stayed remarkably constant at about 40 hours per week. The table below presents some relevant data related to this topic: Hourly Wages and Length of Work Week In U.S. Manufacturing, 1890 – 1983 Year Wage Hours Worked 1890 1.89 59.0 1909 2.63 51.0 1920 3.11 47.4 1930 3.69 42.1 1940 5.27 38.1 1950 6.86 40.5 1960 8.56 39.7 1970 9.66 39.8 1983 10.74 40.1 Sources: Handbook of Labor Statistics, 1983, and Albert Niemi, U.S. Economic History (p. 274). All wages are expressed in 1983 dollars. a. Use these data to plot a "labor supply” curve on the diagram below. b. At wage rates below $4 per hour, does the workweek get longer or shorter as the wage rate rises? c. The data in this table could be consistent with workers choosing various hours per week to work, given the wage rate. An increase in wages has both an endowment income effect and a substitution effect. Would the substitution effect alone make for a longer or shorter workweek? ____________. If leisure is a normal good, would the endowment income effect tend to make people choose more or less leisure and a longer or shorter workweek? ______________________________________. Which effect appears to dominate at wages rates RHIT / Department of Humanities & Social Sciences / K. Christ Spring 2010 – 2011 / IA 350, Intermediate Microeconomics / Problem Set 1 below $4? ________________________________. How would you explain what happens at wages above $4 per hour? _______________________________________________________.