Economics 713: Assignment 2
Q1. Thirty-six people live and work on the island of Beesare. There are 17 fishermen, 6 dairy farmers and thirteen bakery workers. The bakery is a monopoly which is owned by the fishermen: there are 170 shares of stock outstanding, and each fisherman owns 10 shares. The only economic activities on the island are the production and consumption of fish, bread and milk.
The Beesare unit of currency is the clam, and 520 clams are in circulation.
The bakery workers will supply a full day's work for 20 clams. If the wage is less than 20 clams they will not work at all.
When the bakery employs 13 workers it produces 70 pounds of bread per day, and uses 6 gallons of milk in the production process.
The bakery is regulated by the government. It is required to produce and sell 70 pounds of bread every day. Otherwise it is free to maximize profit.
Bakery workers spend 50% of their income on bread, 25% on fish and 25% on milk. Fishermen spend half of their income on bread and the other half on milk. Dairy farmers spend half on bread and half on fish.
Each fisherman catches and sells 5 pounds of fish per day. Each farmer produces and sells 7 gallons of milk per day.
Daily transactions occur as follows. Every morning the milk market opens at 8 a.m. and closes at 9 a.m. Each farmer sells 7 gallons of milk at the market price to the various buyers, and receives clams in exchange.
The bread market is open from 10 a.m. to 11 a.m. The bakery sells 70 pounds of bread at the market price, and receives clams in exchange.
The fish market is open from 4 p.m. to 5 p.m. Each fisherman sells 5 pounds of fish at the market price, and receives clams in exchange.
At 6 p.m. each bakery worker is paid 20 clams, and the bakery's profits for the day are distributed evenly to the shareholders.
Next day, everything is repeated. Each day is an exact copy of the day before.
Clams are held solely for transactions purposes. There is no saving, and no speculation.
Questions-
(a) Analyze the demand for money. How many clams does each fisherman hold overnight? Give a complete account of money holdings, showing where all 520 clams are held overnight, and how they change hands during the day.
(b) Find the equilibrium prices of fish, bread and milk.
Q2. Consider a two-person pure exchange economy in which the initial endowments are (1,0) and (0,1), and each person has Cobb-Douglas preferences.
(a) Find the Walrasian equilibrium for this economy.
(b) Now suppose the first person's endowment increases to (2,0). One feasible allocation would be to start from the initial equilibrium and increase the first person's consumption of the first good by one unit. Find the first person's utility at this allocation, and compare this with the utility level at the new Walrasian equilibrium allocation. Explain why one of these utility levels is higher than the other.
Q3. 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.
Q4. 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?