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Problem 1. (Quasilinaer income effect) Mirabella consumes chocolate candy bars z, and fruits 12. The prices of the two goods are p; = 4,p,= 4. respectively and Mirabella's income is m = 20. Her utility function is a) In the commodity space plot Mirabella's budget set. Find the slope of budget line (one number). Provide the economic interpretation of the slope (one sentence). b) Find analytically formula that gives Mirabella's MRS for any bundle (21, 12) (a function). Give the economic and the geometric interpretation of MRS (two sentences). Find the value of MRS at bundle (F1, 22) = (4,4) (one number). At this bundle, which of the two commodities is (locally) more valuable? (chose one) c) Write down two secrets of happiness that determine Mirabellas's optimal choice (two equation). Provide the geometric interpretation of the conditions in the commodity space. d) Find Mirabella's optimal choice (two numbers). Is solution interior (yes-no answer). e) Suppose the price of a chocolate candy bar goes down to p1= 2, while other price py= 4 and income m = 20 are unchanged. Find the new optimal choice (two numbers). Is a chocolate candy bar an ordinary or Giffen good (pick one)? () Decompose the change in demand for a, in points d) and e) into a substitution and income effect. Problem 2. (Equilibrium) Consider an economy with two consumers, Adalia and Briana and two goods: bicycles a and flowers 12. Adalia initial endowment of the commodities is wr = (40,60) and Briana endowment is w = (60, 40). Adalia and Briana utility functions are given by, i = A, B U'(x],1,) = 4lux, +4luz2 a) Plot an Edgeworth box and mark the point that corresponds to initial endowments. b) Give a definition of a Pareto efficient allocation (one sentence). c) Give a (general) equivalent condition for Pareto efficiency in terms of MRS. Provide geometric arguments that demonstrate the necessity and sufficiency of MRS condition for Pareto efficiency. d) Find competitive equilibrium (six numbers). Depict the obtained equilibrium in the Edgeworth box. Using MRS condition verify that the equilibrium is Pareto efficient. () Using (one of) the secrets of happiness prove that a competitive equilibrium is Pareto efficient in any economy- Problem 3. (Short questions) a) Using A argument prove that Cobb-Douglass production function y = 2AL exhibits increasing returns to scale. Without any calculations, sketch total cost function c(y) corresponding to the production function. b) Now consider a firm (different from point a)) with variable cost c(y) = 2y" and fixed cost F = 2. Find ATCHE's and yous (two numbers). In a long-run equilibrium with free entry how many firms should be expect in the industry if inverse demand is D(p) = 10 - p? c) Suppose a Bernoulli utility function is u(2) = I" and two states are equally likely (probability _). Write down the corresponding von Neuman-Morgenstern utility function. Find the certainty equivalent and the expected value of lottery (0. 2) (two numbers). Which of the two is bigger and why? (two numbers and one sentence.) d) Find Herfindall-Hirschman Index (HHI) for industry with / = 50 identical firms (one number). Is the industry concentrated? e) Derive formula for the present value of perpetuity. Problem 4. (Market Power) Consider an industry with inverse demand p (y) = 8 - y, and a monopoly with cost function TC (y) = 0 who cannot discriminate. a) What are the total gains-to-trade (or potential total surplus) in this industry?' (give one number) b) Write down monopoly's profit function. Derive the condition on MR and MC that gives profit maximizing level of production. Provide economic interpretation of this condition. c) Find the level of production, the price, the deadweight loss and the elasticity of the demand at optimum (four numbers). Illustrate the choice in a graph. d) Assuming the same demand function find the individual and the aggregate level of production and the price in the Cournot-Nash equilibrium with A = 3 identical firms (give three numbers). Show the deadweight loss in the graph. Problem 5.(Externality) Lucy is addicted to nicotine. Her utility from smoking c cigarettes (net of their cost) is given by UL (c)=2mc-c Her sister Taja prefers healthy lifestyle, her favorite commodity is orange juice, j. The two sisters live together and Taja is exposed to second-hand smoke and hence her utility is adversely affected by Lucy consumption of cigarettes c. In particular, her utility function (net of cost of orange juice) is given by UP (j.c) = In (j -c)-j. a) Market outcome: Find consumption of cigarettes c that maximizes the utility of Lucy and the amount of orange juice chosen by Taja (assuming c is optimal for Lucy) (two numbers) b) Find the Pareto efficient level of c and j. Is the value of c higher or smaller than in a)? Why? (two numbers + one sentence) Hint: Derivative of In ( - c) with respect to e is - - Problem 6. (Asymmetric information) In Shorewood Hills area there are two types of homes: lemons (bad quality homes) and plums (good quality ones). The fraction of lemons is equal to . The value of a home for the two parties depends on its type and is given by8. Gordon model: Suppose price of fish P = 10, and total cost of fishing effort is TC(E) = 1000E. a. Find the Total Revenue Product Curve. b. Suppose the fishery is managed by an owner with exclusive property rights. Find the level of effort E that maximizes the rent from the fishery, and calculate the rent. c. Find the level of effort that would occur under open access. d. Under open access, is this fishery suffering from economic overfishing? e. Under open access, is this fishery suffering from biological overfishing? Why or why not? TODProblem 7. Schaefer model: suppose that the yield (harvest h) is given by h = 2Ex , where E is the amount of fishing effort and x is population (here q, the productivity parameter, equals 2). a. Find sustainable population x as a function of effort E. b. Find sustainable yield as a function of E. c. Graph the resulting Yield-Effort curve. Which portions of this curve correspond to biological overfishing?Problem 12-1. Ludwig hires Frederic to work on a project that will yield $700 in revenue if it succeeds and $100 in revenue if it fails. Frederic's opportunity cost of working without substantial effort is $160. His additional cost of working hard is $50. If Frederic works without substantial effort, the probability that the project will succeed is 0.3. If he works hard, the probability that the project will succeed is 0.6. Design a contract whereby Ludwig will pay Frederic so that Frederic has an incentive to work hard