uu assignment

Within a small nation, there is a fixed population choosing between two locationsGoodburg and Plainville. The population consumes a numeraire good and housing. Within each area, there is free entry of firms producing the numeraire good using both land and labor. The productivity level in Goodburg is a function of Goodburg's population. The productivity level in Plainville is fixed. There is a fixed amount of land in each area which can be used to produce either housing or the numeraire good. There is also free entry of firms producing housing, also using land and labor, and the productivity of firms in the housing sector is the same in each of the two areas. The numeraire good is traded freely on international markets and thus has a normalized market price of one. Housing, however, can only be purchased locally and its price is determined in equilibrium. Individual welfare is a function of consumption of the numeraire good, housing, and a public amenity. The amenity is produced by the public sector, where the level of the amenity is a function only of labor devoted to the public amenity divided by total population. This spending is financed by a lump sum tax on residents and potentially also a housing tax. (1) Assume that government spending is exogenous and funded with area-specific lump sum taxes. Characterize the competitive equilibrium. Note that the above setup provides many, but not all, of the assumptions necessary to solve the model. Be sure to specify any other assumptions you need. (2) Derive the conditions for a socially optimal spatial distribution of population. Describe a minimal set of policy interventions that will produce such an optimum. (3) Assume that localities can set tax levels using lump sum taxes. Derive the tax rates if they are trying to maximize area population and compare with the setting where they are maximizing average welfare of their citizens. (4) Assume localities are providing the local public good using local lump sum taxes to maximize population as in (3). Is it ever Pareto improving to tax housing? Explain. Questions for Part B 1) (i) Give an example of a two-person game in normal form with multiple Nash equilibria that are Pareto ranked. (5) (ii) Take any extensive form game of perfect infomation which is represented by the normal form game you have chosen. Is the Pareto dominating equilibrium in your answer to (i) a subgame perfect equilibrium of this extensive form game? (10) 2) (i) Dene the concept of a correlated equilibrium of a normal form game. (10) (ii) Give an example of a game in which there exists a correlated equilibrium that is dierent from any of the Nash equilibria. (5) 3) Dene the concept of rationalizability for two person games in normal form. (10) 4) Show that for all two person games in normal form the set of rationalizable equilibria is identical to the set of strategies that remain after iterated removal of strictly dominated strategies (10). 5) Show that any Nash equilibrium survives iterated removal of strictly dominated strategies, and is therefore rationalizable (by question 4). (10)

1. (i) Explain the main reasons why a competitive equilibrium may fail to exist. Illustrate using an Edgeworth box. (ii) "In a one-consumer exchange economy a competitive equilibrium, if it exists, is unique." True or False. (iii) Consider the following one consumer-one producer (private ownership) economy. The consumer has utility function 12 12 1 2 x x , where 1 x represents leisure and 2 x consumption. The firm's production function is 1 2 q z , where z is labor input and q is output. The consumer's endowment of labor (i.e., leisure) = L . Compute competitive equilibrium prices, profit, and consumption. 2. A principal hires an agent to work for her. The agent can work hard or be lazy. Working hard imposes a disutility of c on the agent; being lazy imposes no disutility. There are two possible outcomes of the agent's effort: success or failure. The principal's revenue is R if success occurs and r if failure occurs. If the agent works hard, success occurs with probability 3 4 and failure with probability1 4 . If the agent is lazy, success occurs with probability 1 4 and failure with probability3 4 . The principal and agent are both risk neutral, but the agent has limited wealth; that is, the agent cannot be paid less than 0 in any state. The principal has all the bargaining power ex ante, and the agent's ex ante outside option is 0. (a) Solve for the optimal contract when the agent's effort is observable and can be contracted on (the first-best). Under what conditions does the principal want the agent to work hard? (b) Now consider the case where the agent's effort is observable only to the agent (the second-best). What incentive schemes get the agent to work hard? (c) What is the lowest-cost incentive scheme for the principal that gets the agent to work hard? Under what conditions will the principal want the agent to work hard? (d) Does the agent sometimes receive more than his outside option in the second-best? Discuss why this might be.

There are two people i = 1; 2 and two alternatives k = a; b. One alternative must be chosen and it will aect both people. Each individuals preference is characterized by a parmeter i that is uniformly distributed on the interval [1; 1]. The i are statistically independent of each other. Thus the density of (1; 2) is 1 4 everywhere on [1; +1] [1; +1]. In addition to selecting the alternative the two individuals may receive (or pay, if negative) a monetary transfer ti . An outcome is a triple (k; t1; t2). The Bernoulli utilties for the outcomes are given by ui(k; ti ; i) = i + ti if x = a = i + ti if x = b In addition to these consumers there is a government that can serve as a sink for any net negative transfers that might be collected. Each i is is private information. a) What is the Pivotal (or Clarke) mechanism in this example? What outcome does it select as a function of (1; 2)? Does it implement this outcome in dominant strategies, or in Bayesian optimal strategies? Why? (5) b) Comment on the e ciency of Pivotal mechanism. (3) c) Conditional on observing i , what is the expected utility that i would have from playing this mechanism? (5) d) Suppose that after learning i an individual had the option to cancel the mechanism unilaterally, in which case the government would have no information, would select the alternative at random (50-50 probability), and would set both ti = 0. Show that there is no value of i at which the people would prefer to cancel the mechanism in this way instead of going through with it? (3) Now consider a simple Voting Mechanism as follows. After observing i each person announces one of the two alternatives, a or b. If they announce the same alternative, it is implemented. If their announcements disagree, the government chooses an alternative at random (50-50 probability). No monetary transfers are ever made. e) What outcome (or randomized outcome) does it select as a function of (1; 2)? (2) f) Compute the interim expected utilities of the two players in this mechanism. (5) g) Which mechanism is more e cient from an ex ante viewpoint? (5) h) Is there a better mechanism than either of these? What considerations other than the expected utilities produced are relevant when you decide whether or not to use a particular alternative mechanism? (Discuss...no computations required.) (10) Short questions for part D(10 points each) 2) State the assumption of independence of irrelevant alternatives as used in Social Choice Theory, and give a defense of it. 3) Explain the concept of ordinal comparability across people in the context of Social Choice The