Final Exam: ECON-100A Intermediate Microeconomics Bob Baden Section I: Answer at least two questions (answer all parts of the questions you choose) Question 1: (16 points) Donald Fribble is a stamp collector. The only things other than stamps that Fribble consumes are Hostess Twinkies. It turns out that Fribble s preferences are represented by the utility function u(s,t) = s + ln t, where s is the number of stamps he collects and t is the number of Twinkies he consumes. The price of stamps is p s and the price of Twinkies is p t. Donald s income is m. a. Write an expression that says that the ratio of Fribble s marginal utility for Twinkies to his marginal utility for stamps is equal to the ratio of the price of Twinkies to the price of stamps. b. Use the equation you found in the part (a) to show that if he buys both goods, Donald s demand function for Twinkies depends only on the price ratio and not on his income. c. Notice that for this special utility function, if Fribble buys both goods, then the total amount of money that he spends on Twinkies has the peculiar property that it depends on only one of the three variables m, p s, and p t. Which variable does the total amount of money depend on? d. Since there are only two goods, any money that is not spent on Twinkies must be spent on stamps. Use the budget equation and Donald s demand function for Twinkies to find an expression for the number of stamps, s, he will buy if his income is m, the price of stamps is p s and the price of Twinkies is p t. e. The expression from part (d) is negative if m < p s. Surely, it makes no sense for Donald to be demanding negative amounts of postage stamps. If m < p s, what would Donald s demand for postage stamps be? f. If m < p s, what would Donald s demand for Twinkies be? g. Donald s wife complains that whenever Donald gets an extra dollar, he always spends it on stamps. Is she right (assume that m > p s )? Explain. h. Suppose that the price of Twinkies is $2 and the price of stamps is $1. Using a solid line, draw Fribble s Engel curve for Twinkies. i. With the same assumptions as in part (h), using a dashed line, draw Fribble s Engel curve for stamps. Page 1
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Question 2: (16 points) The demand function for football tickets for a typical game at a large midwestern university is D(p) = 200,000 10,000p. The university has a clever and greedy athletic director who sets his ticket prices so as to maximize revenue. The university's football stadium holds 100,000 spectators. (a) Write down the inverse demand function. (b) Write an expression for total revenue as a function of the number of tickets sold. (c) Write an expression for marginal revenue as a function of the number of tickets sold. (d) Using a solid line, draw the inverse demand function. (e) Using a dashed line, draw the marginal revenue function. (f) Using a solid line, draw a vertical line for stadium capacity and label it. (g) What price will generate the maximum revenue? (h) What quantity will be sold at the price in part (g)? (i) At the quantity in part (h), what is marginal revenue? (j) At the quantity in part (h), what is the price elasticity of demand (ε)? (k) Will the stadium be full at the quantity in part (h)? (l) A series of winning seasons caused the demand curve for football tickets to shift upward. The new demand function is D(p) = 300,000 10, 000p. What is the new inverse demand function? (m) Write an expression for marginal revenue as a function of the number of tickets sold using the new demand function. (n) Using a dotted line, draw the new demand function. (o) Using pluses draw the new marginal revenue function. (p) Ignoring stadium capacity, what price would generate maximum revenue? (q) What quantity would be sold at the price in part (p)? (r) The quantity of seats that would maximize total revenue with the new higher demand curve is greater than the capacity of the stadium. Clever though the athletic director is, he cannot sell seats he doesn t have. He notices that his marginal revenue is positive for any number of seats that he sells up to the capacity of the stadium. Therefore, in order to maximize his revenue, how many tickets should he sell? (s) What price per ticket should the athletic director charge for the quantity sold in part (r)? (t) When he uses the quantity from part (r) and the price from part (s), what is his marginal revenue from selling an extra seat? (u) What is the elasticity of demand for tickets using the quantity from part (r) and the price from part (s)? Page 4
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Question 3: (16 points) 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. Using a dashed line, draw the locus of combinations of first- and second-class tickets that Agatha can just afford to purchase with her $200. b. Using a solid line, draw the locus of combinations of first and second-class tickets that are sufficient to carry her the entire distance from Istanbul to Paris. c. Label the combination of first- and second-class miles Agatha will choose a. d. Let m 1 be the number of miles she travels by first-class coach and m 2 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. e. What is the number of miles that she travels by second-class coach? f. 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. Using pluses draw the combinations of first-class and second-class tickets that she can afford with her $200 at these prices. g. Label the combination of first- and second-class miles from part (f) that Agatha will choose b. h. How many miles does she travel by second class with the changes from part (f)? i. Is second-class travel a normal good for Agatha? Explain. j. Is second-class travel a Giffen good for Agatha? Explain. k. 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 sock, 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? l. Using a dotted line, draw the locus of combinations of first- and second-class tickets that she can just afford after discovering her loss. m. Label the combination of first- and second-class miles from part (l) that Agatha will choose c. n. How many miles will she travel by second-class now? o. 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)? p. 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? q. When the price of second-class tickets fell from $.10 to $.05, how much of a change was due to an income effect? Page 7
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Section II: Answer at least two questions (answer all parts of the questions you choose) Question 1: (16 points) Suppose that the demand function for Japanese cars in the United States is such that annual sales of cars (in thousands) will be 250 2P, where P is the price of Japanese cars in thousands of dollars. a. If the supply schedule is horizontal at a price of $5,000 what will be the equilibrium number of Japanese cars sold in the United States? b. How much money will Americans spend in total on Japanese cars? c. Suppose that in response to pressure from American car manufacturers, the United States imposes an import duty on Japanese cars in such a way that for every car exported to the United States the Japanese manufacturers must pay a tax to the U.S. government of $2,000. At what price will they be sold? d. How many Japanese automobiles will now be sold in the United States? e. How much revenue will the U.S. government collect with this tariff? f. Draw a graph where the price paid by American consumers is measured on the vertical axis. g. Show the demand and supply schedules before the import duty is imposed. h. Draw the new supply schedule. i. Suppose that instead of imposing an import duty, the U.S. government persuades the Japanese government to impose voluntary export restrictions on their exports of cars to the United States. Suppose that the Japanese agree to restrain their exports by requiring that every car exported to the United States must have an export license. Suppose further that the Japanese government agrees to issue only 236,000 export licenses and sells these licenses to the Japanese firms. If the Japanese firms know the American demand curve and if they know that only 236,000 Japanese cars will be sold in America, what price will they be able to charge in America for their cars? j. How much will a Japanese firm be willing to pay the Japanese government for an export license? k. How much is the Japanese government s revenue from the sale of export licenses? l. How much money will Americans spend on Japanese cars? m. Why might the Japanese voluntarily submit to export controls? Page 10
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Question 2: (16 points) In Heifer s Breath, Wisconsin, there are two bakers, Anderson and Carlson. Anderson s bread tastes just like Carlson s nobody can tell the difference. Anderson has constant marginal costs of $1 per loaf of bread. Carlson has constant marginal costs of $2 per loaf. Fixed costs are zero for both of them. The inverse demand function for bread in Heifer s Breath is p(q) = 6.01q, where q is the total number of loaves sold per day. a. What is Anderson s Cournot reaction function? b. What is Carlson s reaction function? c. Find the Cournot equilibrium quantities denoted by q A and q C. d. What is the Cournot equilibrium price? e. What are the profits of each baker? f. Suppose that one of the bakers of Heifer s Breath plays the role of Stackelberg leader. Perhaps this is because Carlson always gets up an hour earlier than Anderson and has his bread in the oven before Anderson gets started. If Anderson always finds out how much bread Carlson has in his oven and if Carlson knows that Anderson knows this, then Carlson can act like a Stackelberg leader. Carlson knows Anderson s reaction function. How many loaves of bread should Carlson bake? g. How many loaves of bread will Anderson bake? h. What will the price of bread be? i. What are the profits of each baker? Page 13
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Question 3: (16 points) Lucy and Melvin share an apartment. They spend some of their income on private goods like food and clothing that they consume separately and some of their income on public goods like the refrigerator, the household heating, and the rent, which they share. Lucy s utility function is 2X L +G and Melvin s utility function is X M G, where X L and X M are the amounts of money spent on private goods for Lucy and for Melvin and where G is the amount of money that they spend on public goods. Lucy and Melvin have a total of $8,000 per year between them to spend on private goods for each of them and on public goods. a. What is the absolute value of Lucy s marginal rate of substitution between public and private goods? b. What is the absolute value of Melvin s marginal rate of substitution between public and private goods? c. Write an equation that expresses the condition for provision of the Pareto efficient quantity of the public good. d. Suppose that Melvin and Lucy each spend $2,000 on private goods for themselves and they spend the remaining $4,000 on public goods. Is this a Pareto efficient outcome? e. Give an example of another Pareto optimal outcome in which Melvin gets more than $2,000 and Lucy gets less than $2,000 worth of private goods. f. Give an example of another Pareto optimum in which Lucy gets more than $2,000. g. Describe the set of Pareto optimal allocations. h. Will the Pareto optima that treat Lucy better and Melvin worse have (more of, less of, the same amount of) public good as the Pareto optimum that treats them equally. Page 16
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Section III: Answer no more than two questions (answer all parts of the questions you choose) Question 1: (8 points) You have an income of $40 to spend on wine and beer. Wine costs $10 per bottle, and beer costs $5 per 6-pack. a. Write down your budget equation. b. If you spent all your income on wine, how much could you buy? c. If you spent all your income on beer, how much could you buy? d. Draw your budget line as a solid line on a graph. e. Suppose that the price of wine falls to $5 while everything else stays the same. Write down your new budget equation. f. Draw your new budget line as a dashed line. g. Suppose that the amount you are allowed to spend falls to $30, while the prices remain at $5. Write down your budget equation. h. Draw your new budget line as a dotted line. i. On your diagram, use horizontal lines to shade in the area representing commodity bundles that you can afford with the budget in part (g) but not in part (a). j. On your diagram, use vertical lines to shade in the area representing commodity bundles that you can afford with the budget in part (a), but not in part (g). Page 19
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Question 2: (8 points) For each of the following demand curves, compute the inverse demand curve: a. D(p) = max{10 2p,0} b. D(p) = 100/p 1/2 c. ln D(p) = 10 4p d. ln D(p) = ln 20 2ln p Page 22
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Question 3: (8 points) In the game of baseball, a pitcher throws a ball towards a batter who tries to hit it. In a simplified version of the game, the pitcher can pitch high or pitch low, and the batter can swing high or swing low. The ball moves so fast that the batter has to commit to swinging high or swinging low before the ball is released. Suppose that if the pitcher throws high and batter swings low, or the pitcher throws low and the batter swings high, the batter misses the ball, so the pitcher wins. If the pitcher throws high and the batter swings high, the batter always connects. If the pitcher throws low and the batter swings low, the batter will connect only half the time. a. What is the payoff matrix, where if the batter hits the ball the batter gets a payoff of 1 and the pitcher gets 0, and if the batter misses, the pitcher gets a payoff of 1 and the batter gets 0. b. What are the Nash equilibria in pure strategies? c. What are the mixed strategy Nash equilibria? d. What is the probability of each payoff? e. What is the pitcher s expect payoff? f. What is the batter s expected payoff? Page 24