(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.

5. Explain the main reasons why a competitive equilibrium may fail to exist. Illustrate using an Edgeworth box.

6."In a one-consumer exchange economy a competitive equilibrium, if it exists, is unique." True or False.

7. 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.

8.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?

9. Now consider the case where the agent's effort is observable only to the agent (the second-best).

10.What incentive schemes get the agent to work hard?

11. 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?

12. Does the agent sometimes receive more than his outside option in the second-best? Discuss why this might be.

13.Give an example of a two-person game in normal form with multiple Nash equilibria that are Pareto ranked.

14. 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?

15.Define the concept of a correlated equilibrium of a normal form game.

16. Give an example of a game in which there exists a correlated equilibrium that is dierent from any of the Nash equilibria.

17.Define the concept of rationalizability for two person games in normal form.

18. 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

19.Show that any Nash equilibrium survives iterated removal of strictly dominated strategies, and is therefore rationalizable.

20.ain the main reasons why a competitive equilibrium may fail to exist. Illustrate using an Edgeworth box.

21."In a one-consumer exchange economy a competitive equilibrium, if it exists, is unique." True or False.

2) State the assumption of independence of irrelevant alternatives as used in Social Choice Theory, and give a defense of it.

23.3) Explain the concept of ordinal comparability across people in the context of Social Choice Theory.

24. A consumer with Utility function over certain prospect U(x) = x 1 2 , where x is the income. Suppose he is offered a lottery ticket with a prize of 100. There will be exactly 10 tickets sold on this lottery. How much would he be willing to pay for one ticket?

25. Lucy has preferences over lotteries that satisfy the standard axioms of consumer choice under uncertainty. Her utility function over certain prospects is given by U(x) = x 1 2 , and her total wealth is $4. Brian offers her a lottery ticket which gives a half chance of winning $15 and a half chance of winning nothing. The price of the ticket is $3. Obviously, Lucy will not buy the ticket since she is risk averse.

26. Supposes a Consumer has a wealth of $10,000. There is a probability of 0.1 that all but $1 000 will be lost. Her utility function over wealth is given by U(w) = w 1 2 . Suppose an insurance company offers insurance against loss of wealth. What is the maximum expected revenue the insurance company can earn from this customer?

27. Suppose a consumer whose preferences over lotteries satisfy the standard axioms of consumer choice under uncertainty has an Utility function over certain prospect U(x) = x 2 . Suppose now that he is offered the following lottery: