Below are the attached macroeconomic questions please assist with step by step solutions

Problem 1. (Consumer Choice) Lionel watches movies 21 while drinking beer, 22. His utility function from consuming the two types of goods is given by U(x1, 12) = 2 In(ri) + In(12) a) Plot Lionel's indifference curve map (graph). Find his MRS analytically (give formula). Find the value of MRS at the consumption bundle (2, 4) and depict it in the graph. b) Using "magic formulas," find the optimal level of consumption of 21 and 12 if pi = p2 = 4 and m = 15. Plot carefully the optimal point along with the budget line and the indifference curve passing through the optimal point (a graph + two numbers). c) Argue that the commodities are neither gross complements nor gross substitutes (argue using the magic formula, one sentence). For the rest of the problem suppose that Lionel's preferences change and now they are given by U(x1, 12) = 3x] + 312. d) In a separate graph plot his indifference curves, and find MRS for consumption bundle (2, 1). (a graph tone number). e) Find Lionel's optimal choice given prices pi = p2 = 4 and m = 15 (two numbers). Is the optimal choice unique? (yeso answer) Problem 2. (Equilibrium with Intertemporal Choice) Consider an intertemporal choice problem with Ambrosia and Fergus. Ambrosia is an athlete with income of 4 when young and 1 when old, i.e, w" = (4, 1). Fergus is a manager who earns 1 when young and 4 when old, i.e., his endowment is w = (1,4). Ambrosia and Fergus have the same utility function given by U'(z z) = ; In(z) + ; In(z;) where i = A, F. a) Plot the Edgeworth box and mark the point corresponding to endowments of Ambrosia and Fergus (graph). b) Give a general economic definition of Pareto efficiency (one sentence). Write the equivalent condition for Pareto efficiency in terms of MRS (equation). Use your equation to verify whether the endowment is Pareto efficient. (argue formally using your condition) c) Find all the allocations that are Pareto efficient (derive formula for the contract curve). Plot the contract curve in the Edgeworth box. d) Find the competitive equilibrium. Calculate borrowing and savings for both agents in your equilibrium. (Hint: use that for the intertemporal choice 1 + r = = )- Problem 3. (Technology) Suppose a producer has access to the technology given by the Cobb-Douglass production function y = 2K1 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 = 2 (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 = 9 and hence the total cost is given by TC = 9 + 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) = 24 -p (one number). Problem 4. (Short Questions) a) A Bernoulli utility function is w(r) = r and there are two states of the world which are equally likely. Find the certainty equivalent and the expected value of lottery (1, 7) (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 ten 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 9 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 one year to complete e = 1 a credible signal if the cost of effort for the not talented agent is c (e) = 3e (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) =6 -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 fucntions 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. 2 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, 2 = 205 + 2*. Thus research is a public good. The "profit" of the USA, net scientific expenses is given by "'S = 12 In(1"+x*) _1". The net profit of country R is less sensitive to the scientific advancements, and is given by, R = 3In(x"5 +x") -x* 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 x's, r". (Make sure you show optimal choice " for r > 12). 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 reserch? Is it greater, smaller or equal to the one observed in markets (part c)? Explain intuitively why is it so?Problem 1. (Consumer Choice) a) Plot Lionel's indifference curve map (graph). Find his MRS analytically (give formula). Find the value of MRS at the consumption bundle (1, 1) and depict it in the graph. (4pt) MRS = - MU1 5/x1 12 MU2 10/x2 2r1 MRS(1, 1) =- The MRS at (1, 1) is the slope of the indifference curve at that point: *2 1 b) Using "magic formulas," find the optimal level of consumption of x, and x2 if p1 = p2 = 5 and m = 45. Plot carefully the optimal point along with the budget line and the indifference curve passing through the optimal point (a graph + two numbers). (5pt) xi = = a m _ 5 45 a+b'pi 15 5 = = 3 bm 10 45 X2 =- a+b' p2 15 5 = =6 The graph looks like this: x2 3 9 x1 9 c) Argue that the commodities are neither gross complements nor gross substitutes (argue using the magic formula, one sentence). (2pt) From the magic formula, x1 = #+6-pi .", we can see that the optimal consumption of x1 does not depend on the price of the other good, p2. This implies that if p2 goes up or down, the optimal consumption of xi wouldn't change. So, 1 is neither a gross complement, nor a gross substitute for x2. The reverse argument is identical. You can also show this by taking the derivative of the optimal x, with respect to p2 (and the derivative of the optimal x2 w.It. p,) and show that these derivatives equal 0. d) In a separate graph plot his indifference curves, and find MRS for consumption bundle (2, 1). (a graph + one number). (4pt) MRS = - MD- -MY = - = -1 at any bundle. The graph looks like this: x2 e) Find Lionel's optimal choice given prices p1 = p2 = 5 and m = 45 (two numbers). Is the optimal choice unique? (yeso answer) (5pt) IMRS| = 1 = -. Then, any bundle (x1, x2) that satisfies the budget constraint, i.e., 5x1 + 5x2 = 45 is an optimal bundle and so the optimal bundle is not unique.Problem 2. (Equilibrium with Intertemporal Choice) a) Plot the Edgeworth box and mark the point corresponding to endowments of Ambrosia and Fergus (graph). (2pt) The graph looks like this, where the black dot is the endowment allocation: Income when young F 1000 200 O Income Income when old when old 200 -800 0 800 1000 A Income when young 10 b) Give a general economic definition of Pareto efficiency (one sentence). Write the equivalent condition for Pareto efficiency in terms of MRS (equation). Use your equation to verify whether the endowment is Pareto efficient. (argue formally using your condition) (4pt) If an allocation is Pareto efficient, neither of the two agents can be made better off with a different allocation, without making the other agent worse off. Formally, we need MRSA = MRSF From the utility function, we can find the MRS: MRS = -_ At the endowment, MRS^ = MRS(800,200) = - 2-800 = -3 and MRS = MRS(200, 800) = -2:20 = -2, so the marginal rates of substitution are not equal for the two agents and thus the endowment allocation is not Pareto efficient. c) Find all the allocations that are Pareto efficient (derive formula for the contract curve). Plot the contract curve in the Edgeworth box. (4pt) The contract curve contains all allocations for which the MRS of the two agents are equal and which exactly exhaust the total endowment of the two goods. Formally: 12 (7) x1 + X = 1000 = X7 = 1000 - 14 (8) 12 + 12 = 1000 = 1 = 1000 - 12 (9) Plugging in (2) and (3) into (1), we get: 1000 - 14 = 1000 -14 Cross-multiplying and solving gives: x4 = x4 and x6 = 15. The contract curve is the diagonal of the Edgeworth box: 1000 200 0 Contract curve 200 800 800 1000 d) Find the competitive equilibrium. Calculate borrowing and savings for both agents in your equilibrium. (Hint: use that for the intertemporal choice 1 + r = = ) 5pt First, calculate the "incomes" of the two agents based on their endowments, normalizing p2 = 1: mr = 800p1 + 200p2 = 800p1 + 200 m = 200p1 + 800p2 = 200p1 + 800 Using "magic formulas", express the optimal consumption of x, for both agents: a my 1 800p1 + 200 3 (10) a 1 200p1 + 800 (11) 11