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2. Consider an economy with two goods, two consumers, and one firm. The firm produces f(L) = min {5L, 15} units of food when it uses L 2 0 units of labor as input. It also can dispose of either good without cost. Consumer 1 initially owns 2 units of time (for labor or leisure) and gets utility of from consuming c 2 0 units of food and / 2 0 units of leisure. Consumer 2 owns the firm and, initially, no goods. Consumer 2 cares only about food and wants as much of it as possible. a. Graph the production function of the firm. Describe its returns to scale. b. Is there a Pareto efficient (Pareto optimal) allocation in which consumer 1 consumes no food and no leisure? Explain. . Find all Pareto efficient allocations in this economy. How do they differ from one another? d Find a competitive equilibrium for this economy. Is the allocation Pareto efficient? Are there any other competitive equilibrium allocations? Suppose now that there is technical progress. The firm has a new production function g(L) = 8VL for L 2 0. The rest of the economy remains the same as before. e. Find a competitive equilibrium and compare the allocation and the welfare of the con- sumers to what they were in the equilibrium of part d. f Is the allocation in part e Pareto efficient for the economy after there is technical progress? g. Does the technical progress lead to Pareto improvement if the economy is competitive? Explain why or why not. 3. A monopoly firm can hire from a population of qualified workers. Half of the workers are of type 1 and half of type 2. Type 0 workers (0 = 1,2) produce 0(1 + () units of output when they are assigned a job of difficulty type t 2 0. A worker of type 0 who is paid w units of output gets utility w - c(t,#) when working at a job of type t, where c is twice continuously differentiable. Denoting partial derivatives by subscripts, ca (t, 0) > 0, Co(t, 0) 2, for all t > 0 and 0 2 0. The monopoly can offer workers jobs of any type t 2 0 and wages depending on the job type. It seeks to maximize its expected profit, where the profit from any particular worker is the value of the worker's output minus the wage the monopoly pays. Workers are free to accept or reject any contracts offered to them. They maximize their utility, getting reservation utility of 0 if they reject all offers. a. Interpret the assumptions about c and its derivatives and describe how the types of workers differ from each other. b. Suppose the monopoly can recognize every worker's type 0 before it makes any contract offers. Formulate the monopoly's optimization problem. Find an optimal job difficulty t for each type 6 (optimal for the monopoly). Find corresponding wages that are optimal for the monopoly for each of these job types. Show how these optimal contracts can look in a graph. Compare the contracts the two worker types receive, being as specific as possible. c. Suppose now that the monopoly cannot recognize any particular worker's type, but knows all the information given above part a. Formulate the monopoly's optimization problem when it can offer a menu of contracts (job difficulty types and corresponding wages). Characterize the optimal contracts that workers accept and show how they can look in a graph. The graph can be the same one you used in part b. Compare the contracts received by the two types of workers to each other and to those they receive in part b. Be as specific as possible with the given information. Compare the expected profit of the monopoly to what it is in part b. What fraction of the workers in the population does the monopoly want to hire? What can be said about the efficiency of the outcomes in parts b and c? If an outcome is Pareto inefficient, could a government that has the same information as the monopoly obtain a Pareto improvement by restricting the set of contracts the monopoly can offer? Explain.s 4. A monopoly rm I knows that rm E is wondering entering its market and knows that the product design team at rm E is either good [G]! or had {B}. Firm I initially believes that the design team is more likely good than had. Firm E knows the quality of its design team. If E decides to enter the market, it can do so with either a high or low investment. If the investment is high and the design team is good1 then E's product quality is high. if the inymtment is low and the dmigu team is had1 the product quality is low. The product quality is medium ifeither the investment is high and the design team is bad or else if the inmtment is low and the design team is good. Firm 1' can see the quality of E's product (high, medium, or low]. but dorm not directly knew the quality of E's design team before deciding whether to ght or aeeommo-date E in the market. If E stays out of the market1 its payoff is I] and Fe is 5. The two tables below show the payoffs to E and I, depending on their decisions. when E enters the market. The left table shows the payoffs when E's dig'n team is good and right table shows the payoffs if the team is bad. The structure of this interaction is common knowledge to E and I. EU ght aocomrn. Bad Dwisn: hiss low m a. The interaction between E and I can be represented by a game with the tree shove. Writing on this sheet, false] the moves, the players that move them. and the payoffs in the tree above. {Turn in this sheet with your blue-hook.) Explain briey how you can tell which player moves in which information set. b. Explain how you can tell that the decisions represented in the tables above are not the players1 pure strategies. List all of E's pure strategies in the game. c. How many pure strategies done I have in the game? Give examples of two of them. d. Does E have any weakly dominated strategies in the game? If so nd one. a Find every sequential equilibrium [SE] in which I plays a pure strategr. Justify your answer. Explain what the outcomes SE are and discuss whether they are plausible. Is there an SE in which I can detect for sure whether or not E's design team is good? i. Is there a Nash eqmiihrium {NE} in which E chooses not to enter the market no matter what? Ifso, discuss the plausibility of this outcome. [f not, explain whj.r not. 1. Consider an individual with preferences, ~, defined over two commodities, money, m, and TVs, z. While m can be consumed in any quantity, 2 is binary: she can either consume 0 or 1 unit. Her preferences are strictly monotonic in both commodities. Assume that for any bundle (1, m) there is an equivalent amount of money, a(m), such that (1, m) ~ (0, (m)). The consumer is initially endowed with M units of money and 0 units of a. The price of a TV at her present location is p(0) > 0. a. Explain why u(m) > m. b. Under what conditions would she choose to purchase a TV at her present location and when would she choose not to do so? c. Generally, suppose the price of a TV at location d is p(d), where d denotes the distance from 0. In addition, the cost of traveling distance d is cd, where c > 0. Suppose the only options are (i) purchase a TV at 0, (ii) purchase a TV at d, or (iii) do not purchase a TV. Under what conditions would she choose to purchase a TV at d? d. Next, suppose p(0) is known, but in order to determine the price at location d it would be necessary to travel to that location and thus incur the cost ed. Suppose the person's beliefs about p(d) are that it is distributed on [p, p] with density f (p). Also assume she has the von Neumann - Morgenstern utility function u(r, m) = 1000r + m. Under what condition would it be worth traveling to d in search of a lower price rather than purchasing a TV at location (? e. Again consider the case where the agent must travel to d to discover p(d) as in part d. Now supposed is an integer and that stores are located at each integer distance up to d. Assume the consumer is presently at (d - 1) having traveled from 0 and having learned the price at each store along the way including (d - 1). Her options at this point are to purchase a TV at any of the stores she has visited so far (without incurring any additional transportation cost) or to continue searching and go on to d. Determine the criterion for continuing her search. (Hint: would she continue if p(d - 1) = p? If p(d -- 1) = p?) f. How would your answer to part e change if she would have to incur an additional cost of old - d') in order to return to the store at location d'? 2. Consider an economy with / 2 3 consumers, / firms, 2 private goods, and no technology for disposing of goods. Each consumer i (i = 1, ..., /) owns firm i and initially owns e, > 0 units of money. Consumer i gets utility me + id(I,) from consuming mi units of money and z, 2 0 units of food, where #' > 0 and @" 0 units of food using K, + c(q) units of money as input, where c(0) = 0, ( > 0, c' > 0, and 0 0 is constant, to produce one unit of the good of quality z. The monopolist's payoff is its expected profit. The buyers get payoffs equal to the value to them of what they buy minus what they pay. a. Suppose the monopolist can directly observe buyer type and can offer contracts contin- gent on type. Characterize the profit maximizing set of contracts for the monopolist. For the rest of the problem, suppose that types are not observable to the monopolist. The monopolist offers a menu of contracts of the form (po, a;) where a type i contract is meant for type i buyers. b. Formulate the monopolist's pricing problem with incentive and participation constraints, assuming each buyer has a reservation payoff equal to zero. c. Consider the relaxed monopoly pricing problem (RP) in which only the following down- ward adjacent incentive constraints (DAIC) and a participation constraint (PO) for type i = 0 are imposed. even - p2 2 02V/21 - Pi. (DAIC2) 01val - P1 2 01V/20 - po, (DAICI) GovID - Po 2 0 (PO) Show that all these constraints bind in a solution to this relaxed problem. d. Show that if the solution obtained in the relaxed problem (RP) satisfies monotonicity, i.e. r; 2 71 2 26, then all of the incentive constraints and participation constraints in the original problem are satisfied and the solution to the relaxed problem is also a solution to the original problem. e. Solve the relaxed problem (RP). Compare the optimal quality levels (25, 21, a;) to the quality levels the monopolist would choose in part a. Discuss any differences. f. Based on the solution to (RP) in part e, provide a sufficient condition on buyers' preferences such that the solution to (R.P) in part e is indeed a solution to the original problem in part b. Interpret this condition. Is the monopoly better off when this condition holds than when a solution to (RP) is not a solution to the original problem in part b