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Problem 4 (23 points) Firm X has the following production function (K, I) = (1 + K2/3). 1. (6 points) Find the marginal product of labor and capital and the marginal rate of technical substitution, and state if the returns to capital and labor are increasing, decreasing or constant. Explain what is " weird" about this MRTS. 2. (8 points) Graph the isoquants of the production function and state whether the production function exhibits constant, increasing or decreasing returns to scale. 3. (9 points) Now suppose that the production function is f( K, L) = exp(2t) (1 + K2/3) where t measures years. How does this affect the marginal products of labor and capital, the returns to scale and the MRTS? What is the rate of productivity increase over time? Graph the isoquants for q = 100 for * = 1, 2. Explain.Problem 2 (35 points) Anne consumes only books (a) and video games (y) . Her preferences can be represented by the following utility function: U = ry. The price of books is p,, the price of video games is py, and Anne has an income of m dollars. 1. (5 points) Write down Anne's budget constraint, calculate the Marginal Rate of Substitution (at an arbitrary bundle (z, y)) and compute her demand for books and video games (as a function of Pr+Py and m). 2. (5 points) Compute the price elasticity of the demand for books. Compute the cross-price elasticity of the demand for books with respect to the price of video games. (The cross-price elasticity measures the change of the quantity demanded for a good in response to a change in the price of another good, holding everything else constant. It is measured as the percentage change in quantity demanded for the first good divided by the percentage change in price of the second good.) 3. (5 points) Draw the Engel curve for video games. Are video games an inferior or a normal good? Explain. 4. (5 points) Suppose that initially the prices are pr = Py = 1 and income is m = 90. How many books does Anne buy? Now suppose that the price of books increases to pr = 2, how many books will she buy now? How much of the drop in demand for books is due to the substitution effect and how much is due to the income effect? Calculate this numerically and show it in a graph. 5. (15 points) Suppose that her utility function were u(r, y) = x1/2 + y instead. Also suppose that the prices are now P, = 1, py = 10 and her income is m = 50. (i) Calculate her demand for books and video games in this case. (ii) Now suppose that the price of video games increases to py = 15 how many books and video games will she buy now? What is different in this case compared to question 2.4. Explain. (ini) How much of the drop in demand for video games is due to the substitution effect and how much is due to the income effect? Calculate this numerically and show it in a graph. Problem 3 (18 points) For each of the following production functions: (a) F(L, K) = D'KI (b) F ( L, K ) = L+DOR! (c) F(L, K) = 2L + K 1. (2 points, per production function) Find the marginal product of labor and capital, and state if the returns to capital and labor are increasing, decreasing or constant. 2. (2 points, per production function) Find the marginal rate of technical substitut tion. 3. (2 points, per production function) State whether the production function exhibits constant, increasing or decreasing returns to scale.5 Utility Maximization (30 points) Chloe consumes only books (r) and video games (y) . Her preferences can be represented by the following utility function: U (z, y) = ry'. The price of books is pr, the price of video games is py, and Chloe has an income of m dollars. 1. (4 points) Write down Chloe's budget constraint. 2. (4 points) Calculate the Marginal Rate of Substitution (at an arbitrary bundle (I. y)). 3. (6 points) Find the equations that describe Chloe's demand for books and her demand for videogames for any possible value of pr, Py and m. 4. (6 points) Repeat part 3 when the utility function is U (r,y) = min {z, y) . 5. (4 points) Let us now go back to the original utility function U (x,y) = zy. Suppose that the government imposes a tax on videogames, such that if the price of a videogame is py, the consumer must pay (1 + 7) py. What is the new demand function for videogames? Plot the demand before the tax and after the tax in the same graph (you don't need to assume any particular values for m, 7, it is enough to provide a qualitative graph). Briefly explain the intuition. 6. (6 points) Let's go back to the case without taxes. Suppose that Pr = 2, Py = 2, m = 30. Suppose that Chloe has one "buy-10-get-10-free" coupon for books (that is, she will get 10 free books if she buys at least 10 books). How many books 3 and how many videogames will Chloe consume? Carefully draw the budget set and the highest attainable indifference curve on the same graph.3 Demand for Video Games (16 points) We have the following weekly demand data for the video game Grand Theft Auto in a US town. We also have the price data for a Playstation at the same time. Price of Playstation (in $) |Price of Grand Theft Auto (in $) | Quantity of games demanded 300 10 100 310 10 95 320 10 90 300 11 99 310 96 1. (6 points) Write down the equation for the demand for Grand Theft Auto in the following form: Q, = a+BP, + BpPp where Q and P, are quantity demanded and price of Grand Theft Auto, Pp is the price of Playstation, and c, By, 8, are constants. 2. (4 points) Does the demand function that you found in part (1) satisfy the law of demand? Explain. 3. (6 points) The supply curve for Grand Theft Auto in this town is Q5 = 2P, Solve for the equilibrium price and quantity, as a function of Pp How does the equilbrium price and quantity depend on Pp? Briefly explain the intuition. 2 4 Indifference Curves (18 points) In each of the following examples, the consumer consumes only two goods, a and y. Based on the information given in each statement, sketch a plausible set of indifference curves (draw at least two curves on a set of labeled axes and indicate the direction of higher utility). Then, write down a possible form of the utility function u(r, y) that is consistent with your graph. 1. (6 points) Alan likes wearing both right shoes (x) and left shoes (y). He always needs to wear them as a pair, having a right shoe is useless without the left one and viceversa. 2. (6 points) Emma likes pizza (x) but hates vegetables (y). She is only willing to eat an extra unit of vegetables if she gets to eat an extra unit of pizza. 3. (6 points) Mary likes Coke (r) and Pepsi (y). She is indifferent between them as she is unable to tell the difference between Coke and Pepsi.1 Positive vs. Normative Statements (16 points) Identify whether each of the following statements is positive or normative. Briefly justify your answer. 1. (4 points) The government has a duty to provide basic healthcare and education to every citizen. 2. (4 points) The cost of health insurance is too high. 3. (4 points) The median earnings of a full-time worker with a college degree are almost twice as high as those of a high-school graduate with no college education. 4. (4 points) The current unemployment rate is 3.9%, the lowest it has been since December 2000. 2 True or False (20 points) For each of the following statements, indicate if they are True or False. Justify your answer. 1. Bill is a football coach. He evaluates his players based on three criteria: height, strength, and speed. Bill prefers one player over another if he is better in at least two of these criteria. Assume that there are no players with the exact same height, nor the exact same strength, nor the exact same speed. (a) (4 points) Bill's preferences are complete. (b) (4 points) Bill's preferences are transitive. 1 2. (4 points) Ann and Bob are utility maximizing consumers. Given their income and market prices, Ann chooses a bundle that gives her a utility UAnn = 100, while Bob chooses a bundle that gives him a utility UBot = 110. Therefore, we know that Bob is happier than Ann. 3. (4 points) John's utility function for food (f) and clothes (c) is given by U (f, c) = (fo + ce)a, with a > 0. John's preferences satisfy the principle of diminishing marginal rate of substitution for any value of a. 4. (4 points) Ava has preferences over two goods that satisfy completeness, transitiv- ity, non-satiation and the indifference curves have a strictly diminishing marginal rate of substitution. Suppose that the price of one of these goods increases (and the price of the other one remains the same). Claim: Ava's utlity must be strictly lower after the price increase. How does your answer depend on the composition of the initial consumption bundle