Economics 711: Final Exam 2000-
Q1. Consider an economy with n agents and two goods: a private good, x and a public good, g. Consumer i has an endowment of ωi units of the private good, and there is a technology that transforms the private good into the public good.
a. Suppose there are m firms that have access to the public good technology, and each consumer owns equal shares of each firm. How would you define a "Walrasian" (competitive) equilibrium for this two-good economy?
b. Now suppose the public good technology has constant returns to scale, at a rate of two units of the private good per unit of the public good.
i. What is the Walrasian equilibrium price ratio?
ii. Are the Walrasian allocations Pareto efficient? Explain.
iii. Relate your answer to the First Welfare Theorem.
Q2. An economy contains many identical consumers, with utility functions u(x) = log(x0) + log(i=1∑N√xi). Each consumer is endowed with some quantity of good 0, and the other goods are produced using identical technologies which require units of x0 to get started, and c units of x0 for each unit of xi produced. Good i is produced by a single firm that maximizes profits. The number of possible goods, N, is big relative to the number of consumers. There is free entry in the production of all goods.
a. How many goods will be produced in equilibrium?
b. Is the equilibrium Pareto optimal?
3. a. State and prove the First Welfare Theorem.
b. State the Second Welfare Theorem. Give an example in which one of the assumptions of the theorem does not hold, and the conclusion of the theorem is false.
Q4. Suppose there are two consumption goods with production functions √Q1 = √K1 + √L1, 1Q2 = 1/K2 + 1/L2, where K1 is the amount of capital used in the production of good 1, etc. There are two small countries, A and B. A is endowed with 4 units of K and 50 units of L, and B is endowed with 20 units of K and 3 units of L. Both economies are competitive, and there is free trade in the consumer goods, but the factors of production cannot move between countries. The prices of the consumer goods, established in the world market, are p1 = 0.8 and p2 = 9. Find the competitive equilibrium in each country.
In your equilibrium, are factor prices equal across these two countries? If so, explain why; if not, explain why not.
Q5. Modify the Bertrand duopoly model to allow different marginal costs for the two firms. Find an equilibrium, and determine whether it is unique.
Q6. Consider an economy in which there are equal numbers of two kinds of workers, a and b, and two kinds of jobs, good and bad. Each employer has an unlimited number of vacancies in both kinds of jobs. Some workers are qualified for the good job, and some are not. If a qualified worker is assigned to the good job the employer gains $2,000, and if an unqualified worker is assigned to the good job the employer loses $14,000. When any worker is assigned to the bad job, the employer breaks even.
Workers who apply for jobs are tested and assigned to the good job if they do well on the test. Test scores range from 0 to 1. The probability that a qualified worker will have a test score less than t is t². The probability that an unqualified worker will have a test score less than t is 1-(1-t)².
There is a fixed wage premium of $42,000 attached to the good job. Workers can become qualified by paying an investment cost, and this cost is higher for some workers than for others: the distribution of costs is uniform between 0 and $20,000. This distribution is the same for a-workers and b-workers. Workers make investment decisions so as to maximize earnings, net of the investment cost (all of these amounts are expressed as present values).
a. Can you find an equilibrium in which there are more a-workers than b-workers in the good jobs?
b. Now suppose that employers are required to assign the same proportion or a-workers and b-workers to the good jobs. If there is no change in the workers' investment behavior, what standards will the employers use?