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Final Exam (Group A) You have 2h to complete the exam. The final consists of 6 questions (15+10+15+25+20+15=100). Problem 1. (Choice with Cobb-Douglas preferences) Sara spends her income on books z, and food 22. The prices of the two commodities are pi = pz = 5 and her income is m = 100. Sara's utility function is given by U (n, 1) = (n) (1). a) Find analytically Sara's MRS as a function of (21, 12) (give a function) and determine its value for consumption bundle (F1, 12) = (6,2). Provide economic and geometric interpretation of MRS at this bundle (one sentence + graph). b) Give two secrets of happiness that determine Sara's optimal choice (two equation). Explain why violation of any of them implies that the bundle cannot be optimal (one sentence for each condition). ) Find Sara's optimal choice (two numbers) and mark the optimal bundle in the commodity space. 1) Using magic formulas for Cobb-Douglas preferences argue that both commodities are ordinary com- modities. (formulas and one sentence) Problem 2. (Intertemporal choice with perfect substitutes) Josh chooses a consumption plans for two periods. His income in the two periods is (w], "2) = ($30, $60) and the utility function is U (m. m)=n+ -xz. a) Propose some other utility function that gives a higher level of utility for any bundle (r1, 12), which represents the same preferences. (utility function) b) Plot intertemporal budget set of Josh for interest rate r = 100%. Find PV and FV of the endowment cash flow and depict the two values in the graph. On the budget line mark all consumption plans that involve borrowing. c) Find optimal consumption plan (21, r2)and show it in the graph (give two numbers). Is your solution interior? Problem 3. (Equilibrium) Consider an economy with two goods: clothing I and food 22. Onur's initial endowement is a" = (80. 20) and Janet initially has w= (20,30). Utility functions of Onur and Janet are given by U (1,12) = = In(my) + - In(zz). a) Plot an Edgeworth box and mark the point corresponding to the initial endowments. b) Give the definition of a Pareto efficient allocation (one sentence) and provide its equivalent character- ization in terms of MRS (equation). Verify whether the endowment allocation is Pareto efficient (compare two numbers). c) Find the prices and the allocation in the competitive equilibrium (six numbers) d) Using MRS condition demonstrate that the competitive allocation is Pareto efficient.Problem 4. (Short questions) a) You are renting a home that gives you $1000 each month in form of rent (forever). Find PV of the cashflow if the monthly interest rate is r = 1%. b) Demonstrate that production function f(K, [) = K L" exhibits decreasing returns to scale. (use "lambda" argument). Without any calculations sketch the cost curve associated with this production function. c) Suppose fixed cost is F = 2 and variable cost is c(y) = 2y'. Find ATONES and yos. Give formula for a supply function of individual firm and plot it in a graph. Find equilibrium price and aggregate output in an industry with 4 firms, assuming demand y = 10 - p. d) Give a von Neumann-Morgenstern utility function over lotteries for a Bernoulli utility function is u(c) = Inc and the probability of each state is 0.5 (formula). Is a consumer with this utility function risk loving, risk averse or risk neutral? (choose one) e) In a market for second-hand vehicles there are two types of cars: lemons (bad quality cars) and plums (good quality ones). The value of a car depends on its type and is given by Lemon Plum Seller 0 20 Buyer 10 Are plums going to be traded if the probability of a lemon is =? (compare relevant numbers)- Problem 5.(Market Power) Consider an industry with inverse demand p(y) = 200 - y and total cost TO = 40y. a) What are the total gains to trade in this industry? (number). Find the HHI index of this industry with one firm, a monopoly. (one number) b) Find the optimal level of output and the price of a monopoly assuming uniform pricing (give two numbers). Illustrate its choice in a graph. Mark a DWL. c) Find profit and a DWL if monopoly uses the first-degree price discrimination. d) Find the aggregate output, the price and the markup in a Cournot-Nash equilibrium with N firms (all functions of N ). What is the limit of the markup function as N goes to infinity? Why? Problem 6. (Public good) Alfonsia and Betonia are two countries that are members of the same military alliance. Their security depends positively on joint military spending r" + 2" of the two countries. Thus, Alfonsia's "utility" net of cost of military spending is given by and the analogous function for Bretonia is a) Find the best response functions for Alfonsia and Bretonia (two formulas) and plot them in the coordinate system (1*,1" ) b) Find the Nash Equilibrium (give two numbers). Is one of the two countries free riding? If yes, which one? c) Find the efficient level of military spending of the alliance (one number). Is the efficient spending smaller or bigger than the one observed in the Nash equilibrium? Why? (one sentence)Intermediate Microeconomics Prof. Marek Weretka Final Exam (Group B) You have 2h to complete the exam. The final consists of 6 questions (15+10+15+25+20+15=100). Problem 1. (Choice with Cobb-Douglas preferences) Sara spends her income on books , and food 22. The prices of the two commodities are pi = pz = 10 and her income is m = 200. Sara's utility function is given by U (n, 1) = (1) (12). a) Find analytically Sara's MRS as a function of (21, 12) (give a function) and determine its value for consumption bundle (21, 12) = (2,6). Provide economic and geometric interpretation of MRS at this bundle (one sentence + graph). b) Give two secrets of happiness that determine Sara's optimal choice (two equation). Explain why violation of any of them implies that the bundle cannot be optimal (one sentence for each condition). c) Find Sara's optimal choice (two numbers) and mark the optimal bundle in the commodity space. d) Using magic formulas for Cobb-Douglas preferences argue that both commodities are ordinary com- modities. (formulas and one sentence) Problem 2. (Intertemporal choice with perfect substitutes) Josh chooses a consumption plans for two periods. His income in the two periods is (@1, "2) = ($30, $60) and the utility function is a) Propose some other utility function that gives a higher level of utility for any bundle (r1, r2), which represents the same preferences. (utility function) b) Plot intertemporal budget set of Josh for interest rate r = 100%. Find PV and FV of the endowment cash flow and depict the two values in the graph. On the budget line mark all consumption plans that involve borrowing. c) Find optimal consumption plan (21, r2)and show it in the graph (give two numbers). Is your solution interior? Problem 3. (Equilibrium) Consider an economy with two goods: clothing In and food 22. Onur's initial endowement is a? = (40, 10) and Janet initially has w= (10, 15). Utility functions of Onur and Janet are given by U (1,12) = = In(m) + = In(z2). a) Plot an Edgeworth box and mark the point corresponding to the initial endowments. b) Give the definition of a Pareto efficient allocation (one sentence) and provide its equivalent character- ization in terms of MRS (equation). Verify whether the endowment allocation is Pareto efficient (compare two numbers). ") Find the prices and the allocation in the competitive equilibrium (six numbers) d) Using MRS condition demonstrate that the competitive allocation is Pareto efficient
