Prelim%201%20Solutions - Economics 3010 Fall 2008 Professor...

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Unformatted text preview: Economics 3010 Fall 2008 Professor Daniel Benjamin Cornell University EXAM #1 SOLUTIONS I. True / False / Uncertain Your grade on these questions will depend on the generality, completeness, and persuasiveness of your explanation, not simply on whether the “true” or “false” is correct. The objective here is to provide an answer that convinces; not merely an answer that is “not wrong.” At the very least, make sure that you give clear definitions for the relevant economic terms used in the question. Please use a separate blue book for this section. (5 points) 1) The opportunity cost of going to college includes not only the foregone income from working full‐time, but also all the time and effort spent going to high school. False. 2 points: The opportunity cost is the foregone benefit of the next‐best alternative. 2 points: So the foregone income from working full‐time is part of the opportunity cost of going to college. 1 point: But the time and effort spent going to high school is a cost incurred whether or not you go to college. (It is a “sunk cost.”) (5 points) 2) The demand for narrow categories of goods (like milk) tends to be more elastic than the demand for broader categories (like food). True. 2 points: The elasticity of demand is the percent change in quantity demanded per one‐percent change in price. 3 points: Narrow categories of goods (like milk) have more substitutes (e.g., other beverages such as OJ) than broader categories (like food). When the price increases, quantity demanded of the good in question will fall by more because consumers will substitute consumption of other goods. (5 points) 3) The fact that the marginal rate of substitution between two goods equals the price ratio at a consumer’s optimum means that every consumer gets exactly the same level of utility from those two goods. False. 2 points: The formula is MRS = MU(x1) / MU(x2) = p1 / p2. 2 points: The correct statement is that every consumer has the same MRS (or ratio of MUs) from the two goods. 1 1 point: This means that on the margin (i.e., for the last unit purchased), every consumer has the same relative increase in utility from the two goods. (5 points) 4) If Madeleine’s demand function for croissants is x*(p,m) = (m / p)2 , then croissants are a luxury good for Madeleine. True. 2 points: A luxury good is a good whose income elasticity of demand exceeds 1 in magnitude (or, a good whose budget share is increasing in income). 2 points: Madeleine’s income elasticity of demand for croissants is (∂x*/∂m) * (m/x*) = (2m / p2) * (m / ((m / p)2 ) = 2. (An alternative calculation is: ∂ln(x*)/∂ln(m) = 2.) 1 points: Since (∂x*/∂m) * (m/x*) = 2 > 1, croissants are indeed a luxury good for Madeleine. (5 points) 5) If the price of bread falls, then the quantity of bread demanded will increase. Uncertain. 2 points: The Law of Demand states that if a good is normal (if quantity demanded is increasing in income), then quantity demanded is negatively related to price. (Alternatively, the Slutsky equation states that a change in price has an income effect and a substitution effect. If the good is normal, then the two effects reinforce each other, and quantity demanded is negatively related to price.) 2 points: Even if a good is inferior, if expenditure on the good is a small share of the budget, then the Slutsky equation implies that the substitution effect will dominate the income effect, so quantity demanded will be negatively related to price. 1 point: But if a good is inferior and the income effect dominates the substitution effect, then quantity demanded can be positively related to price. (Such a good is called a Giffen good.) II. Brief Problem (5 points) Bob, Ted, and Alice are trying to decide where to go for dinner. The three possibilities are McDonald’s (MD), Kentucky Fried Chicken (KFC), and Pizza Hut (PH). The preferences of each person are given below: Ted Alice Bob st MD PH KFC 1 Choice KFC MD PH 2nd Choice rd 3 Choice PH KFC MD They have decided to use majority voting to select a restaurant. For each pair of restaurants, they vote and say that the group prefers the restaurant that gets the most votes. Explain why it is impossible to write down a utility function that describes the group’s preferences. 2 1 point: If the choice set is {MD, KFC}, then MD will get the most votes. If the choice set is {MD, PH}, then PH will get the most votes. If the choice set is {KFC, PH}, then KFC will get the most votes. 2 points: Since the group prefers MD over KFC, KFC over PH, and PH over MD, the group’s preferences are intransitive. 1 point: If preferences are complete, transitive, and continuous, then they can be represented by a utility function. 1 point: Since the group’s preferences are intransitive, it is impossible to write down a utility function that represents those preferences. III. Multi‐Part Problem Note: This question is designed so that if you skip part of the question, you still have enough information (stated explicitly earlier in the question) to answer later parts of the question. Please answer as many parts as you can. Suppose Mom’s utility function for gasoline (x) and money (y) is U(x, y) = x(‐1/σ)+1 / ((‐1/σ)+1) + y, where σ > 0 is a constant, and her budget constraint is Px + y ≤ m. (4 points) (a) Are her preferences well‐behaved? Explain. Draw a diagram of Mom’s indifference curves. 1 point: Preferences are well‐behaved if they are monotonic and convex. 2 points: These preferences are monotonic and convex. (Either because we know they are quasi‐linear, or because we know from a diagram.) Hence they are well‐behaved. 1 point: [Accurate diagram of indifference curves.] (4 points) (b) Show that Mom’s demand function for gasoline is x*(P, m) = P‐σ if m ≥ P1‐σ, and x*(P, m) = (m / P) otherwise. (From now on, we will assume that m ≥ P1‐σ.) 1 point: If the solution is interior, it satisfies MU(x)/MU(y) = P/1. 1 point: This simplifies to x* = P‐σ. 1 point: This amount of gasoline is affordable as long as P(P‐σ) ≤ m, which simplifies to m ≥ P1‐σ. 1 point: If P(P‐σ) > m, then Mom cannot buy as much gasoline as she would like, so we are at a corner solution where Mom spends all of her money on gasoline: x*(P, m) = m / P. (5 points) (c) Suppose the demand side of the market for gasoline is composed of 100 Moms, each of whom has the preferences given above. Let the (base 10) log of price be denoted by p = 3 log10(P), and let the (base 10) log of market demand for gasoline be denoted by xd = log10(Xd). Explain why the (log of) market demand for gasoline is xd(p) = 2 – σ p. Show that the price elasticity of demand for gasoline in this market equals ‐σ. 2 points: The market demand function is Xd(P) = 100P‐σ. 2 points: The (base 10) log of market demand is xd(p) = log10(100) – σ log10(P) = 2 – σ p. 1 point: The price elasticity of demand is (∂Xd /∂P) * (P/Xd) = (‐σ) * 100P‐σ‐1 * (P/(100P‐σ)) = ‐σ. (An alternative calculation is: ∂xd(p) /∂p = ‐σ.) (5 points) (d) Suppose the (log of) market supply for gasoline is xs(p) = 1 + p. Show that the equilibrium (log) price, p, is equal to 1/(1+σ), and the equilibrium (log) quantity, x, is equal to (2+σ)/(1+σ). 2 points: The equilibrium (log) price solves xs(p) = 1 + p = 2 – σ p = xd(p). 1 point: This simplifies to p = 1/(1+σ). 2 points: Plugging in to find the equilibrium (log) quantity, x = 1 + 1/(1+σ) = (2+σ)/(1+σ). (5 points) (e) In 1973, the Organization of Petroleum Exporting Countries (OPEC), a major international cartel, successfully colluded in restricting the supply of gasoline. As a result, suppose the (log of) market supply for gasoline is now xs(p) = p. Show that the new equilibrium (log) price, p, is equal to 2/(1+σ), and the equilibrium (log) quantity, x, is equal to 2/(1+σ). How do the new equilibrium price and quantity compare to the old equilibrium? 2 points: The equilibrium (log) price solves xs(p) = p = 2 – σ p = xd(p). 1 point: This simplifies to p = 2/(1+σ). 1 point: Plugging in to find the equilibrium (log) quantity, x = 2/(1+σ). 1 point: The new equilibrium price is higher, and the new equilibrium quantity is lower. This is as expected when supply falls. (5 points) (f) Show that revenues of gasoline exporters increase as a result of the supply restriction if and only if σ < 1. Explain intuitively. 2 points: Revenues of gasoline exporters are PX. 1 point: Log‐revenues equal p + x = (3+σ)/(1+σ) at the old equilibrium and p + x = 4/(1+σ) at the new equilibrium. Hence revenues increase if and only if 4 > 3+σ¸which is to say σ < 1. 2 points: If demand is inelastic, then a 1 percent increase in price reduces quantity supplied by less than 1 percent. Therefore, an increase in price will increase PX. 4 (7 points) (g) The U.S. government responded to the upward pressure on prices by imposing “price controls,” making it illegal to charge more than a certain price for gasoline (in an effort to prevent inflation). Suppose that σ = 2, and suppose the price ceiling mandated that p ≤ ½. Draw a supply/demand diagram illustrating the price ceiling. Calculate the new equilibrium (log) quantity of gasoline supplied and demanded. Show that the quantity traded is x = ½. Explain why there were long lines at gas stations in 1973. 2 points: [Accurate diagram of a binding price ceiling] 2 point: At the price ceiling of ½, the (log) quantity supplied is xs(½) = ½, and the (log) quantity demanded is xd(½) = 2 – 2(½) = 1. 1 point: The new equilibrium (log) quantity traded will be min{xs(½), xd(½)} = min{½, 1} = ½. 2 point: There were long lines because there was excess demand. The limited supply was rationed according to who got in line earliest. 5 ...
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This note was uploaded on 12/28/2011 for the course ECON 3010 at Cornell University (Engineering School).

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