12-1. Suppose there are 100 workers in an economy with two firms. All workers are worth $35 per
hour to firm A but differ in their productivity at firm B. Worker 1 has a value of marginal product
of $1 per hour at firm B; worker 2 has a value of marginal product of $2 per hour at firm B, and so
on. Firm A pays its workers a time-rate of $35 per hour, while firm B pays its workers a piece rate.
How will the workers sort themselves across firms? Suppose a decrease in demand for both firms’
output reduces the value of every worker to either firm by half. How will workers now sort
themselves across firms?
Workers 1 to 34 work for firm A as a time rate of $35 is more than their value to firm B, while workers
36 to 100 work for firm B. Worker 35 is indifferent. More productive workers, therefore, flock to the
piece rate firm. After the price of output falls, firm A values all workers at $17.50 per hour, while worker
1’s value at firm B falls to 50 cents, worker 2’s value falls to $1 at firm B, etc. The key question is what
happens to the wage in the time-rate firm. Presumably this wage will also fall by half to $17.50 per hour.
If it falls by half, then the sorting of workers to the two firms remains unchanged.
12-2. Taxicab companies in the United States typically own a large number of cabs and licenses;
taxicab drivers then pay a daily fee to the owner to lease a cab for the day. In return, the drivers
keep their fares (so that, in essence, they receive a 100 percent commission on their sales). Why did
this type of compensation system develop in the taxicab industry?
Imagine what would happen if the cab company paid a 50 percent commission on fares. The cab drivers
would have an incentive to misinform the company about the amount of fares they generated in order to
pocket most of the receipts. Because cab companies find it almost impossible to monitor their workers,
they have developed a compensation scheme that leaves the monitoring to the drivers. By charging
drivers a rental fee and letting the drivers keep all the fares, each driver has an incentive to not shirk on
12-3. A firm hires two workers to assemble bicycles. The firm values each assembly at $12.
Charlie’s marginal cost of allocating effort to the production process is MC = 4N, where N is the
number of bicycles assembled per hour. Donna’s marginal cost is MC = 6N.
(a) If the firm pays piece rates, what will be each worker’s hourly wage?
As the firm values each assembly at $12, it will pay $12 for 1 assembly, $24 for 2 assembly’s, etc. when
offering piece rates. As Charlie’s marginal cost of the first assembly is $4, the second is $8, the third is
$12, and the fourth is $16; Charlie assembles 3 bicycles each hour and is paid an hourly wage of $36.
Likewise, as Donna’s marginal cost of the first assembly is $6, the second is $12, and the third is $18;
Donna assembles 2 bicycles each hour and is paid an hourly wage of $24.
84 (b) Suppose the firm pays a time rate of $15 per hour and fires any worker who does not assemble
at least 1.5 bicycles per hour. How many bicycles will each worker assemble in an 8 hour day?
As working is painful to workers, each will work as hard as necessary to prevent being fired, but that is
all. Thus, each worker assembles 1.5 bicycles each hour, for a total of 12 bicycles in an eight hour day.
12-4. All workers start working for a particular firm when they are 20 years old. The value of each
worker’s marginal product is $18 per hour. In order to prevent shirking on the job, a delayed-
compensation scheme is imposed. In particular, the wage level at every level of seniority is
Wage = $10 + (.4 × Years in the firm).
Suppose also that the discount rate is zero for all workers. What will be the mandatory retirement
age under the compensation scheme? (Hint: Use a spreadsheet.)
To simplify the problem, suppose the workers works 1 hour per year. (The answer would be the same
regardless of how many hours are worked, as long as the number of hours worked does not change over
time). Some of the relevant quantities required to determine the optimal length of the contract are:
on the Accumulated Contract Contract
Age Job VMP VMP Wage Wage
21 1 $18 $18 $10.00$10.00
22 2 $18 $36 $10.40$20.40
23 3 $18 $54 $10.80$31.20
24 4 $18 $72 $11.20$42.40
40 20 $18 $360 $17.60$276.00
41 21 $18 $378 $18.00$294.00
42 22 $18 $396 $18.40$312.40
43 23 $18 $414 $18.80$331.20
60 40 $18 $720 $25.60$712.00
61 41 $18 $738 $26.00$738.00
62 42 $18 $756 $26.40$764.40
The VMP is constant at $18 per year. The accumulated VMP gives the total product the worker has
contributed to the firm up to that point in the contract. The wage in the contract follows from the
equation, and the accumulated wage is the total wage payments received by the worker up to that point.
Until the 20hyear in the firm, the worker receives a wage lower than her VMP; after the 21 year the
worker’s wage exceeds the VMP. The contract will be terminated when the total accumulated VMP
equals the total accumulated wage under the delayed compensation contract, which occurs on the
worker’s 41styear on the job. So the optimal retirement age is age 61.
85 12-5. Suppose a firm’s technology requires it to hire 100 workers regardless of the wage level. The
firm, however, has found that worker productivity is greatly affected by its wage. The historical
relationship between the wage level and the firm’s output is given by: