Many assignments will be drawn from a collection that is available in either pdf (Acrobat) or postscript form. Many of the questions are at an intermediate level (but you won't be told which ones until it is too late). In most cases, it is a good idea to draw diagrams (if you find yourself differentiating a Lagrangian you are probably lost).
Assignment 1: Problems 2, 12, 16, 21
Due: Tuesday, April 1 (no fooling)
Assignment 2:
Due: Tuesday, April 8. Odd-Numbered problems to Adams, even-numbered
to Arcidiacono.
Problems 10, 11, 57, and
72. Design a contract to maximize the expected
profits received by a risk-neutral principal who will hire a risk-averse
agent. The agent's utility function is u = log(w) - e, where e is effort
(high or low), and w is the wage payment. The agent has an outside option
that is a sure thing worth -½. The low effort level is zero, and
the high effort level is ½. Gross revenue depends on the agent's
effort level. If effort is high, revenue R is uniformly distributed on
the interval [0,1]. If effort is low, R is also distributed on [0,1], with
density f(R) = 2(1-R). The principal cannot observe the agent's effort.
Analyze how the optimal contract changes
as the cost of effort decreases.
Assignment 3:
Due: Tuesday, April 15. Odd-Numbered problems to Adams, even-numbered
to Arcidiacono.
Problems 6, 55, and
71. Consider a simple economy in which there are just two occupations, coal mining and auto repair. The mining and auto repair industries are perfectly competitive, and they happen to have identical labor demand curves, given by
w = 400 - L
where w is the daily wage (net of any training costs borne by workers), and L is the number of workers employed in the industry.
There are 420 workers in the economy, 294 men and 126 women, all equally productive in both jobs. All workers prefer auto repair work to coal mining, but the extent of this preference varies from one worker to another. The distribution of equalizing differences over workers is uniform between 0 and $42. Sex and occupational preferences are independently distributed.
a. Find the equilibrium wage differential and occupational distribution for this economy.
b. Suppose Fred is an "average" worker, who considers the equalizing differential to be $21 a day. Does Fred gain or lose from the diversity of preferences in the economy? That is, would Fred be better or worse off if everyone else in the economy had the same preferences as he does?
c. Suppose that women are excluded from coal mining jobs. How will this affect the equilibrium? What will happen to the average wages of men and women? Who will gain under this restriction, and who will lose?
d. Suppose that employers who have excluded women are found liable for damages. How would you compute the damages?
e. Does it make sense to interpret the exclusion of women from mining jobs as rational exploitation of the minority by the majority?
74. Let F and G be distribution functions defined on the real interval [a,b], with density functions f and g.
a. Can you find an example in which F and G are ordered in the sense of first-order stochastic dominance, but the likelihood ratio is not monotonic? If you can't find such an example, can you prove that no one else can find one either?
b. Can you find an example in which the likelihood ratio is monotonic, but F and G are not ordered in the sense of first-order stochastic dominance? If you can't find such an example, can you prove that no one else can find one either?
Assignment 4:
Due: Tuesday, April 22. Problems 23 and 53 to Adams, 59 and 73 to
Arcidiacono.
Problems 23, 53, 59 and
73. 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. Some workers are qualified for the good job, and some are not. Employers believe that the proportion of a-workers who are qualified is 2/3 and the proportion of b-workers who are qualified is 1/3. If a qualified worker is assigned to the good job the employer gains $1000, and if an unqualified worker is assigned to the good job the employer loses $1000. 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 100. The probability that a qualified worker will have a test score less than t is t. The probability that an unqualified worker will have a test score less than t is t(2-t). Employers are subject to a rule that requires the proportion of a-workers assigned to the good job to be the same as the proportion of b-workers. Otherwise employers maximize expected profits.
Find the profit-maximizing policy for an employer.
Test your policy as follows. If you are told that a worker has just barely passed the test (and you are not told whether the worker is an a-type or a b-type), what is the probability that the worker is qualified? Is it the case that such a worker is a fair bet from the employer's point of view? If not, should the policy be changed?
Assignment 5:
Due: Tuesday, April 29. Problems 24 and 27 to Adams, 21D6 and 75
to Arcidiacono.
Problems 24, 27, 21D6 (Mas-Colell, page 813) and
75. A firm has a large accumulated inventory of a storable good. There are no competing sellers of this good, and there is a linear relationship between the quantity sold in each period and the price that the firm sets. Inventory holding costs are negligible, but the cost of production is higher than any buyer would ever pay. The firm can borrow and lend freely at a fixed discount rate, and acts to maximize the present discounted value of profits. How will the firm set prices?
Assignment 6:
Due: Tuesday, May 6. Problems 36 and 23B2 to Adams, 58 and 76 to
Arcidiacono.
Problems 36 (k),(o),(r),(ee), 58, 23B2 (Mas-Colell, page 918) and
76. Suppose a union and an employer start to negotiate
on January 1, 1998 over wages to be paid for the year 1998. There will
be 50 paid weeks in the year (the rest being unpaid vacation time). No
work will be done until they reach an agreement.
The employer's net revenue, after paying all costs other than wages (including
a normal return on capital), is $500 per worker per week. The workers can
earn $240 per week if they leave this employer and go to work elsewhere.
While negotiations continue workers can collect $130 per week in unemployment
benefits (this is a straight subsidy that does not have to be repaid).
The employer has retained an expert negotiator who charges $200 per week
per worker.
The negotiations proceed as follows. At the beginning of each week the
employer's and the union's negotiators meet, and one side proposes a wage
for the rest of the year. If this proposal is accepted work begins immediately.
If not, the workers collect unemployment benefits for the week, the employer's
negotiator is paid for the week, and nothing happens for the rest of the
week; next week there is a new meeting and a new proposal. At the first
meeting a coin is tossed to determine which side makes the proposal for
that week. In subsequent meetings they take turns: first one side makes
a proposal, then the following week the other side makes a proposal, and
so on.
(a) How long will these negotiations take? What wage agreement will be reached?
(b) Find the Nash bargaining solution for this situation.
The End