Economics 4311: Labor Economics
Professor Cory Koedel
Fall 2009
Problem Set #6
Incentive Pay
1. Consider a piece-rate pay system where workers are paid $x per unit produced. Using
marginal revenue and marginal cost curves, show that high-ability workers, who have
lower costs of producing any given level of output, produce more output than lowability workers.
2. Give two reasons why piece-rate systems result in higher output than time-rate
systems?
3. Name three problems with piece-rate pay systems.
4. Use a graph to show that tournaments that offer higher relative payouts induce more
effort.
5. A competitive firm hires two workers to produce widgets. The firm values each widget
produced at $20. Ricky’s marginal cost of producing output is equal to 5Q, where Q is
the number of widgets produced per day. Donny has a marginal cost of output equal to
2Q.
a. If the firm pays piece rates, how much will each worker make per day in dollars
and widgets? Graph your solution.
Ricky makes $80 (4 widgets) and Donny makes $200 (10 widgets)
b. If the firm instead offers a time rate of $20 and requires workers to produce at
least 2 widgets per day to not get fired, how many widgets will be produced by
each worker and how much money will each worker make?
Each worker will produce 2 widgets and earn $20.
c. In part (b), the firm can no longer be in a competitive market. Why?
The firm is making positive economic profits from the workers. If this market were
competitive, more firms would enter.
6. Consider three firms who pay time rates and monitor their workers. Each firm monitors
workers and catches workers shirking (goofing off) with probability p. If a worker is
caught shirking, that worker is fired. Firms A, B and C have probabilities, p, of catching
workers equal to 0.2, 0.3 and 0.4 respectively. If the workers across firms are identical
and all three firms pay efficiency wages, which firm would we expect to pay the highest
efficiency wage? Graphically support your answer (hint: What will the output curve
look like for firms A, B and C if shirking reduces output?).
Firm A will offer the highest efficiency wage
7. Principal-Agent Problem. An owner needs a manager to operate his business. He knows
that the manager has the following utility function:
U = W(E) – C(E)
Above, W(E) is the worker’s wage, which is a function of worker effort, C(E) is the
manager’s cost of effort and E is a unit of effort. C’(E) > 0 indicating that the manager
dislikes putting forth effort for the business. The worker produces one unit of output
per unit of effort and each unit of output is worth $2 on the market. Revenue (R) can be
written in terms of effort as follows:
R = 2E
The owner offers the following wage structure to the manager:
W(E) = a + b*(E)
Above, a is an effort-invariant fixed payment and b is the worker’s piece-rate payment on
output produced. The worker gets $b per unit of effort provided.
The worker has the following cost-of-effort function, measured in dollars: C(E) =E3/2
a) How much effort will the worker provide in terms of the wage parameter(s)?
Worker will maximize utility where utility is equal to wages minus cost of effort:
U = a + b*(E) – E3/2
dU/dE: b – (3/2)*E1/2 = 0
E* = (4/9)*b2
b) What is the worker’s participation constraint (Utility = 0 constraint)?
a + b*(E) – E3/2 >= 0 so firm operates where: a + b*(E) – E3/2 = 0 to max profits
c) The owner wants to maximize profit. What is the firm’s profit function, exclusively in
terms of E, if its only costs are the manager’s salary?
PROFIT = 2E – (a + b*(E))
Note that:
a + b*E = E3/2
Subbing into the profit function to get profit entirely in terms of E:
PROFIT = 2E – (E3/2)
(each unit of effort produces a unit of output that can sell for $2)
d) Noting that the owner can effectively “choose” E by choosing the compensation
package, what E will the owner choose?
PROFIT = 2E – (E3/2)
dPROFIT/dE = 2 – (3/2)*E1/2 = 0
E* = 16/9
e) What “b” must the firm choose to elicit the effort level, E*, in part (d)?
b = 2 (from part a). The worker is paid the full value for each unit produced
f) What will be the value of the fixed portion of the wage (hint: it will be negative)? That
is, what compensation package will the firm offer to the worker that will maximize
profits but also satisfy the manager’s participation constraint?
Owner can drive manager utility to zero. Firm sets U = 0 for participation:
= a + b*(E) – E3/2
0 = a + 2*(16/9) – 2.37
= a + 3.555– 2.37
= a + 1.185
a = -1.185 in the limit (or, if you prefer, a = (-1.185+ ε)
U
8. A firm is designing a compensation scheme for a worker. Compensation depends on
worker effort: W = a + b*E. Here, W is the total wage and E is worker effort. Each unit
of effort produces 6 units of output, which can be sold for $2 per unit.
Worker utility is given by: U(w, E) = W – 5E2 = (a + b*E) – 5E2. The worker’s utility must
be greater than or equal to zero for him to accept the job. Calculate the profitmaximizing values of a and b, the optimal level of effort provided, E, and firm profits.
Worker’s problem:
Max U = a + b*E – 5E2
dU/dE: b – 10E = 0
E* = b/10
Firm maximizes profits: π = 12E – a – b*E
Subject to worker’s participation constraint:
Worker works if utility is greater than or equal to zero. Firm operates where utility
equals zero to max profits.
Constraint is: U = 0
a + bE – 5E2 = 0
a + bE = 5E2
Max profit:
12E – (a + b*E)
12E – 5E2
12 – 10E = 0
E* = 6/5.
So, b* = 12:
worker is paid full value for output produced
Get a from worker participation constraint
a + bE – 5E2 = 0
a + 12*6/5 – 5*(36/25) = 0
a=
180/25 – 72/5
a* = 7.2 – 14.4 = -7.2
Firm profit is:
Revenue – Cost
12*E – (a + bE)
= 12E – a – 12E
=-a
= -(-7.2) = 7.2