f provide solutions

(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 custom T-shirt printing business has many competitors, so that the perfect competition model may be considered a good approximation. Currently, the market demand curve is given by Q = 120 1.5 p, whereas the market supply is given by

Q = 20 + 2 p.

(a) Determine the market equilibrium Suppose there is a T-shirt craze that increases demand by 10% (that is, for each price, demand is now 10% greater than it was before the craze).

(b) Determine the new demand curve.

(c) Determine the change in equilibrium quantity.

(d) If your answer to the previous question is different from 10%, explain the difference in values. Now go back to the initial demand curve and suppose there is an increase in the cost of blank T-shirts, an essential input into the business of selling custom T-shirts. Specifically, for each unit by each supplier, the production cost goes up by 10%.

(e) Determine the new supply curve.

(f) Determine the change in equilibrium price.

(g) If your answer to the previous question is different from 10%, explain the difference in values.

A country is currently in a recession.

a. Illustrate this economy on a fully-labeled aggregate demand-aggregate supply model. Include aggregate demand, short-run aggregate supply, and long-run aggregate supply. I. Label the short-run equilibrium price level PLE and the short-run equilibrium output YE. II. Label the full-employment level of output YF. b. If the government and central bank do not intervene, how would this economy adjust in the long run? Explain.

c. Illustrate the process of part (b) on your graph from part (a). d. The government decides to use fiscal policy to correct the economic situation in part (a). Assume the difference between the short-run and long-run equilibrium output is worth $36 billion, and the marginal propensity to consume is 0.75. Calculate one specific and effective fiscal policy action the government could take. e. What would be the short-run impact of the government's action on the economy's real output? f. What would be the short-run impact of the government's action on the potential output of the economy? g. Will the long-run equilibrium price level if the government intervenes be less than, equal to, or greater than the long-run equilibrium price level without intervention? h. Show the impact of the government intervention from part (d) on the equilibrium real interest rate on a fully labeled loanable funds market graph. i. Will the long-run aggregate supply curve move as a result of the change from part (h)? Explain.