Show alll the steps

Problem 3. (Technology) Suppose a producer has access to the technology given by the Cobb-Douglass production function y = 3K1 14. a) What can you say about the returns to scale (chose: IRS, CRS or DRS) and MPK (chose: increasing, constant, or decreasing). b) Find the (variable) cost function c(y) given that the prices of inputs are wx = wy = 4.5 (give a function). For the rest of the problem suppose that in order to have access to the technology, the producer first needs to pay fixed cost F = 16 and hence the total cost is given by TC = 16 + c(y). c) Find the supply function of the individual competitive firm and plot it in the graph (give the formula, in the graph mark the prices for which the market will not open)- d) Assume that the producers are competitive and there is free entry. Determine the number of firms operating in the industry in the long run if the demand is D(p) = 20 - p (one number). Problem 4. (Short Questions) a) A Bernoulli utility function is u(r) = r" and there are two states of the world which are equally likely. Find the certainty equivalent and the expected value of lottery (0, 2) (two numbers). Which of the two is bigger (choose one)? Explain why (one sentence). b) Consider an industry in which the market share of the dominant firm is 30%, while the market shares of the five other firms is 10% each. Find HHI index for this industry. Is the industry competitive, moderately concentrated, or concentrated? c) Suppose there are two types of managers: talented with productivity 5 and not talented with productivity 3. The types are unobservable to employers and the competitive wage is given by the expected productivity of the manager. Is an MBA diploma from a program that takes three years to complete e = 3 a credible signal if the cost of effort for the not talented agent is c(e) = e (yeso + one sentence explaining why). Find minimal e for which the MBA diploma is a credible signal. d) Externality: Give two methods through which a government can achieve market efficiency in the presence of a negative externality. (two sentences for each method). Problem 5. (Market Power) Consider an industry with an inverse demand p(y) = 8 -y and the total cost TO = 0. a) Find the level of production and the price chosen by a monopoly who is not allowed to price discriminate (give two numbers). Illustrate the choice using a graph. Find the consumer surplus (CS), the producer surplus (PS) and the deadweight loss (DWL) (give three numbers and mark them on the graph). b) Which pricing strategy of a monopoly gives rise to the Pareto efficient outcome (one sentence)? Find the consumer surplus (CS), the producer surplus (PS) and the deadweight loss (DWL) under your proposed strategy (give three numbers and mark them on the graph)- c) Find the individual level of production, the price and the profit of each firm in the Cournot-Nash equilibrium if there are two identical firms in the industry with cost functions TO = 0 (give three numbers) d) Find the joint profit of the two firms from part c) if they form a cartel. Explain the mechanism that prevents the formation of a (short-run) cartel in a Cournot-Nash equilibrium. Problem 6. (Provision of a Public Good) There are two countries, the USA and a country that represents "the rest of the world" (denoted by R). The national products of both countries are increasing in the world's spending on research, r = 205 + 2*. Thus research is a public good. The "profit" of the USA, net scientific expenses is given by US = 7In(x"5 + x") _ xUS The net profit of country R is less sensitive to the scientific advancements, and is given by, a) Find analytically the best response of the US to any level of spending r" (derive a function) and plot it in the coordinate system p 5, r". (Make sure you show optimal choice ros for r > 7). b) Find analytically the best response function of country R and add it to the graph in part a). c) Find the Nash equilibrium. What is the world's spending on science, I? Is the predicted outcome associated with free riding? If so by which country? d) Find the Pareto efficient level of spending on research? Is it greater, smaller or equal to the one observed in markets (part c)? Explain intuitively why is it so?Problem 1. (Choice with Cobb-Douglas preferences) Sara spends her income on books I, and food 22. The prices of the two commodities are p, = py = 5 and her income is m = 100. Sara's utility function is given by U (n, 12) = (1) (12). a) Find analytically Sara's MRS as a function of (21, 12) (give a function) and determine its value for consumption bundle (11, 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). 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 (w], (2) = ($30, $60) and the utility function is U (m, m) =n+12 a) Propose some other utility function that gives a higher level of utility for any bundle (21, 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 (71, 12) and show it in the graph (give two numbers). Is your solution interior?" Problem 3. (Equilibrium) Consider an economy with two goods: clothing r, and food 72. Onur's initial endowement is wo = (80, 20) and Janet initially has w = (20,30). Utility functions of Onur and Janet are given by U (1, 12) = = In(21) + = In(22). 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, L) = 40-3103 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 ATOMES and yo s. 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 26 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 1. (Choice with Cobb-Douglas preferences) Sara spends her income on books 21 and food 22. The prices of the two commodities are pi = p? = 10 and her income is m = 200. Sara's utility function is given by U (21, 12) = (11) (12). a) Find analytically Sara's MRS as a function of (21, 22) (give a function) and determine its value for consumption bundle (11, 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 (w1, w2) = ($30, $60) and the utility function is U (m, m) =+ -12. a) Propose some other utility function that gives a higher level of utility for any bundle (21, 12), which represents the same preferences. (utility function) b) Plot intertemporal budget set of Josh for interest rate y = 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, 12)and show it in the graph (give two numbers). Is your solution interior? Problem 3. (Equilibrium) Consider an economy with two goods: clothing r, and food 72. 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(mi) + = 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, L) = KL"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 = 4 and variable cost is c(y) = dy. Find ATOMES and youES. 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 8 firms, assuming demand y = 20 - 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 byProblem 1. (Quasilinaer income effect) Mirabella consumes chocolate candy bars I, and fruits 72. The prices of the two goods are p1 = 4, p,= 4, respectively and Mirabella's income is m = 20. Her utility function is U(I1, 12) = 2In1,+12 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 (x1 , 72) = (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 pi= 2, while other price p2= 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)? f) Decompose the change in demand for T1 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 r, and flowers 12. Adalia initial endowment of the commodities is wh= (40, 60) and Briana endowment is w= (60, 40). Adalia and Briana utility functions are given by, i = A, B U'(I1, X2) = 4 In1 +4 In 12 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. e) 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 = 2K L 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 ATCHES and yoS (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(x) = r" 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 Herfindahl-Hirschman Index (HHI) for industry with N = 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 / = 3 identical firms (give three numbers). Show the dead weight 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) = 2Inc-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 UT (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 (j - c) with respect to c is -je 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 by