Lecture 6
Contracts
This lecture studies how those who create and
administer organizations design the incentives
and institutional rules that best serve their ends.
We focus on schemes that are designed to
maximize the manager’s objectives by creating
the appropriate incentives for the people he
deals with at minimal cost to the organization he
manages. We analyze upstream contracts with
suppliers, employment contracts for workers,
service contracts for consumers, and incentive
schemes within the organization.
Designing the bargaining rules
An implication of our studies on bargaining is
the manifest value from setting the rules and
conventions that determine how bargaining
proceeds.
Almost by definition managers are placed in a
strong position to set the rules of bargaining
games they play.
We look at upstream supply contracts,
downstream consumer agreements, and
employment contracts with labor.
A rent extraction problem
Employers seek to minimize their wage
bill, or in the case of sole proprietors loss
in expected utility, subject to two
constraints:
1. They must attract workers they
wish to hire. This is called the
participation constraint.
2. The workers must perform the
tasks to which they are assigned.
This constraint is called incentive
compatibility.
Full information principal agent problem
A firm wishes to
build a new
factory, and will
hire a builder.
How should it
structure the
contract?
Firm:RL-wL
RH-wH
Builder: wL-uL wH-uH
Constraints facing the firm
We can use backwards induction to solve the
problem:
1. The incentive compatibility constraint is:
wH – uH  wL – uL if H
wL – uL  wH – uH if L
2. The participation constraint is:
wH - uH  0 if H
wL- uL  0 if L
w
The
constraints
illustrated
L
wH-wL=uH-uL
wH =uH
uH -uL
uL-uH
(IC)
uH
wH
Minimum cost of achieving L
The minimum cost of achieving L is found by
minimizing wL such that:
1. wL  uL
2. wL – uL  wH – uH
The first constraint bounds wL from below by uL.
Since uL uH the second constraint is satisfied by not
making the wage depend on effort.
Therefore the minimum cost of achieving L is found
by setting
w* = u*L
Minimum cost of achieving H
The minimum cost of achieving H is found by
minimizing wL such that:
1. wH  uH
2. wH – uH  wL – uL
The first constraint bounds wH from below by uH.
Since uL  uH we must penalize the worker to deter
him from choosing L, by setting:
wL < wH – uH + uL
Therefore the minimum cost of achieving H is:
w*H = u*H
w*L = wH – uH + uL - Penalty
Profit maximization
The net profits from achieving L are
RL – uL*
The net profits from achieving H are
RH – uH*
Therefore the firm hires a worker to achieve H if
RH – uH* > RL – uL*
and hires a worker to achieve only L
otherwise.
Service provider
Multipart pricing schemes are commonly found in the
telecommunications industry, amusement parks. sport
clubs, and time sharing vacation houses and small jets.
In this example a provider incurs a fixed cost of c0 to
connect the consumer to the facility, and a marginal cost
of c1 for every unit provided.
It follows that if the consumer purchases x units the
total cost to the provider is:
c0 + c1x.
We assume the monetary benefit to the consumer from
a service level of x is: x1/2.
How should the provider contract with the consumer?
Optimal contracting
To derive the optimal contract, we proceed in two steps:
1. derive the optimal level of service, by asking how
much the consumer would use if she controlled the
facility herself.
2. calculate the equivalent monetary benefit of
providing the optimal level of service to the
consumer, and sell it to the consumer if this covers
the total cost to the provider.
The equivalent monetary benefit can be extracted two
ways, as membership fee with rights to consume up to a
maximal level, or in a two part pricing scheme, where the
consumer pays for use at marginal cost, plus a joining fee.
A parameterization
In our example we maximize
x1/2 - c0 - c1x
with respect to x to obtain interior solution
x = (2c1)-2
It follows that the costs from an interior
solution are:
c0 + 1/4c1
and the monetary equivalent from consuming
the optimal level of service is 1/2c1.
Therefore the provider extracts 1/2c1 if:
4c0c1 <1
Charging a uniform price
If the service provider charges per unit instead, the
consumer would respond by purchasing a level of
service a a function of price.
Anticipating the consumer’s demand, the provider
constructs the consumer’s demand curve, and sets price
where marginal revenue equals marginal cost.
The provider serves the consumer if and only if the
revenue from providing the service at this price exceeds
the total cost.
Since lower levels of service are provided, and since the
consumer achieves a greater level of utility, than in the
two part contract, the provider charging a unit price
realizes less rent than in the two part contract.
The parameterization revisited
In our example the consumer demands
x = (2p)-2
where p is the uniform unit price of the service.
The service provider maximizes:
x1/2/2 - c0 - c1x
with respect to x to obtain the interior solution
x = (4c1)-2
which is the optimal choice if:
16c0c1 <1
Comparing multipart with
uniform pricing schemes
Since lower levels of service are provided,
and since the consumer achieves a greater
level of utility, than in the two part contract,
the provider charging a unit price realizes less
rent than in the two part contract.
Service provider
Multipart pricing schemes are commonly found in the
telecommunications industry, amusement parks. sport
clubs, and time sharing vacation houses and small jets.
In this example a provider incurs a fixed cost of c to
connect the consumer to the facility, and a marginal cost
of 1 for every unit provided.
It follows that if the consumer purchases x units the
total cost to the provider is:
c + x.
We assume the monetary benefit to the consumer from
a service level of x is: 8x1/2.
How should the provider contract with the consumer?
Optimal contracting
To derive the optimal contract, we proceed in two steps:
1. derive the optimal level of service, by asking how
much the consumer would use if she controlled the
facility herself.
2. calculate the equivalent monetary benefit of
providing the optimal level of service to the
consumer, and sell it to the consumer if this covers
the total cost to the provider.
The equivalent monetary benefit can be extracted two
ways, as membership fee with rights to consume up to a
maximal level, or in a two part pricing scheme, where the
consumer pays for use at marginal cost, plus a joining fee.
A parameterization
In our example we maximize
8x1/2 – c – x
with respect to x to obtain interior solution
4x-1/2 = 1 or x = 16
Hence the costs from an interior solution are c + 16,
and the monetary equivalent from consuming the
optimal level of service is 32.
Therefore the provider can extract 16 – c if c < 16.
A two part pricing scheme that achieves this goal is to
charge a joining fee of 16 and a unit price of 1,
achieving profits of 16 – c.
Charging a uniform price
If the service provider charges per unit instead,
the consumer would respond by purchasing a
level of service a a function of price.
Anticipating the consumer’s demand, the
provider constructs the consumer’s demand
curve, and sets price where marginal revenue
equals marginal cost.
The provider serves the consumer if and only if
the revenue from providing the service at this
price exceeds the total cost.
The parameterization revisited
Supposing p is the price charged for a service unit,
the consumer maximizes:
8x1/2 – px
The first order condition yields the consumer demand
4x-1/2 = p = p(x)
or
x = 16p-2
The service provider maximizes:
p(x)x – c – x = 4x1/2 – c – x
with respect to x to obtain the interior solution
2x-1/2 = 1
or
x = 4 and p = 2
In this case the firm extracts a rent of 4 – c if c < 4.
Comparing multipart with
uniform pricing schemes
In a two part contract rents are 16 – c but with a
uniform price the rent is only 4 – c if c < 4.
Furthermore if 4 < c < 16, a uniform price scheme
cannot yield a profit but a two part price scheme
can.
Since lower levels of service are provided in the
uniform price case, and since the consumer
achieves a greater level of utility than in the two
part contract, the provider charging a unit price
realizes less rent than in the two part contract.
Terms of employment
The same principles apply to hiring a worker. For
example let y denote the income the worker receives for
her labor, in other words her wage earnings.
Let h denote her hours of labor supplies to the firm if
she is employed by the firm.
Assume the worker’s utility function takes the form
y + k log(16 - h)
where k is a positive constant that measures her
willingness to trade off goods for leisure and 16 is the
maximum number of hours she would consider working.
We also assume that if she is not employed with the firm
her income equivalent is v.
Firm value
Suppose firm profits are :
ph - y
where p is the output price (or value of the
worker’s product).
The firm chooses h and y to maximize
profits subject to the participation
constraint that the worker chooses to be
employed.
Optimization
If the firm offered more than v, then it could always
reduce y by so that hours remains unchanged.
Therefore the participation constraint is met with
equality and we set:
y = v – k log(16 - h)
The firm maximizes:
ph + k log(16 - h) – v
The first order condition for this problem can be
written as
h = 16 – k/ p
Solution
Substituting this equation for hours into the profit
function we obtain:
16p – k + k log(k/p) – v
Therefore the firm sets
h = 16 - k/ p
if profits are positive, meaning
16p – k + k log(k/p) > v
and otherwise h = 0.
Outsource
A second type of work contract is for the worker to
approach the firm, and propose an arrangement to
the firm, which the firm can either accept or reject.
This is quite close to outsourcing tasks that might
have been undertaken within the firm.
In this case the worker chooses both the payment
y and hours or output h to maximize her utility
y + k log(16 - h)
subject to the constraint that the firm accepts her
proposal (does not make losses):
y 6 ph
Solution to Outsourcing
The solution is almost identical to the employment
contract problem, except that all the rent accrues
to the worker instead of the firm.
The outsourcer sets a contract so that the firm
only just breaks even, meaning y = ph.
Hours are now chosen by the outsourcer to
maximize
ph + k log(16 - h)
yielding the same choice of hours as in the
original problem.
Sales commission:
the worker chooses her hours
An alternative method of payment is for the firm to
pay its employee a commission, denoted by s, on her
output.
In this case the worker chooses h to maximize
sh + k[log(16 – h)].
Analogous to the previous problem, the solution to this
maximization problem is
h = 16 – k/s
if 16s – k + klog(k/s) > v
and h = 0 otherwise.
Sales commission:
the firm chooses the commission
Upon solving for h(s), the worker’s supply of hours
as function her commission, the firm chooses s to
maximize:
(p – s)h(s) = (p – s)(16 – k/s)
= 16p – 16s – pk/s + k
This solution to this maximization problem is found
by solving the first order condition to the firm’s
optimization problem: 16 = pk/s2
Solving for s gives
s = (pk)1/2/4.
Comparing the schemes
Total profit under the sales commission is:
16p + k – 8 (pk)1/2
Total profit under the optimal wage contract is:
16p – k + klog(k/p) – v
Noting p > s, the participation constraints imply there
are (k,p) parameter combinations where participation
occurs under the wage contract but not the sales
commission.
Give participation in both schemes, the worker is better
off under the commission system than under the
contract. Since the contract extracts all the gains from
trade, it is more profitable than the commission.
Contracting with specialists
Often managers know less than their own workers about
the value employees contribute to and take from the firm.
More generally, medical doctors and specialists diagnose
the illnesses for patients, strategic consultants evaluate
firm performance for shareholders, and building
contractors tell property owners what needs to be done.
This leads us to investigate how principals (like managers)
should design contracts for agents (such as workers)
when the information on their employees is incomplete.
Consider a game between company headquarters and its
research division, which is seeking to increase its budget
so that it can proceed with “product development”.
Research and product development
There are two types of discoveries, minor and
major, denoted by j = 1, 2. The probability it is
minor (j = 1) is p, and the probability it is major
one (j = 2) is 1 - p.
It costs cjx to develop a commercial product with
appeal of x, where c1 > c2, which in turn
produces a present value of log(1+x) to the firm.
A budget of bi is allocated to the research division
to develop the product up to a consumer appeal
level of xi when the research division announces a
discovery of type i = 1,2.
If the research division does not announce any
discovery, it gets its standard budget r.
Research funding policy
Headquarters forms a policy on funding product
development, by announcing (b1,x1) and (b2, x2).
After the policy formulation stage at headquarters,
the division announces whether it has made a
major discovery (i=2), a minor (i=1), or none at
all (i=0).
If i = 0, then shareholders net 0 and the research
division nets r to sustain continued operations.
Otherwise shareholders net:
log(1+xi) – bi
and the research division nets:
bi – cjxi
where cj is the true discovery.
Full information solution
In this case headquarters directly sees the discovery,
and sets the budget just high enough to motivate
optimal development. Thus :
bj = r + cjxj
Substituting for bj into headquarters’ objective
function, it chooses xj to maximize
log(1+xj) – bj = log(1+xj) – r – cjxj
Taking the first order condition and solving we obtain
xj = 1/cj – 1
Funding is undertaken only when cj < 1 and profits, as
defined below, are positive
– log(ci) – r – 1 + cj
Participation and incentive compatibility
when there is incomplete information
Suppose headquarters does not directly observe the
discovery, but relies exclusively on the divisional report .
The division will truthfully report the outcome of its
activities if the following two constraints are met:
1. The participation constraint requires for each j:
bj – cjxj  r
2. The incentive compatibility constraint requires:
b2 – c2x2  b1 – c2x1
and vice versa. Note that both inequalities
cannot be satisfied by strict equality since c1 < c2.
Solving for the budgets
The participation constraint binds for the minor
discovery (j = 1), but not for major ones. That is:
b1 – c1x1 = r
b2 – c2x2  b1 – c2x1 > b1 – c1x1 = r
Substituting for b1 in the incentive compatibility
constraint yields :
b2  b1 + c2x2 – c2x1 = r + c1(x1 – x2) – c2x1
Minimizing b2 we conclude the incentive compatibility
constraint binds with strict equality for major
discoveries (j = 2), but not for minor ones.
Optimal product development
Having derived the optimal budget as a function of product
development, we choose x1 and x2 to maximize:
p[log(1+x1) – b1] + (1 – p) [log(1+x2) – b2]
= p[log(1+x1) – r – c1x1]
+ (1 – p) [log(1+x2) – r – c2x2 + c2x1 – c1x1]
= p[log(1+x1) – cx1] + (1 – p) [log(1+x2) – c2x2] – r
In the third line, c is called the virtual cost of x1 and is
defined by the equation:
c = c1 + (c1 – c2) (1 – p)/p
Solution to the full disclosure policy
Mathematically this is almost the same problem as the
full information case.
Taking the first order condition and solving, we obtain:
x1 = 1/c – 1
x2 = 1/c2 – 1
Substituting for x1 and x2 into the profit equation
derived on the previous slide, we obtain:
p[log(1+x1) – cx1] + (1 – p) [log(1+x2) – c2x2] – r
= p[c – log(c)] + (1 – p)[c2 – log(c2)] – r – 1
Comparing the policy options
on research disclosure
To summarize, the profits from having a full
disclosure policy are:
p[c – log(c)] + (1 – p)[c2 – log(c2)] – r – 1
Alternatively if all discoveries are treated as
minor discoveries, the profits are:
c1 – log(c1) – r – 1
Finally if only major discoveries are reported then
shareholder profits are:
(1 – p)[c2 – 1 – log(c2)] – r
Lecture Summary
Optimal contracting provides an opportunity for
the contractor to extract rents from his business
partners, employees, customers and clients.
Extracting maximal rent may require relatively
complicated contracts, which if written incorrectly,
carry the prospect of loss.
If the rent opportunities are too meager,
surrendering the rent, and using the market, may
provide a better solution. These factors form the
basis for defining where firm boundaries should be
relative to the market.