Lecture 1 in Contracts
Nonlinear Pricing
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 and downstream service
contracts for consumers.
Read Chapters 17 and 18 of Strategic Play.
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.
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.
Characterizing the optimal contract
Intuitively managers seek to maximize the size of
the pie and extract as much as possible. In such
contracts, the private information and outside
options available to each party are explicitly
modeled through the:
1. incentive compatibility constraint
2. participation constraint.
The contract designer extracts the maximal rent
from the relationship subject to these constraints.
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:
32 – (c +16) = 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.
Profits as a function of units provided
Plotting the inverse
demand, profits from
charging a uniform
price, and the surplus
the service provides, for
different values of x, we
see that the at high
values x, the surplus is
not very sensitive to
changes in x, but the
uniform profit function
is negative.
15
10
5
0
0
5
10
15
20
-5
-10
Uniform inverse price demand
Uniform price profit function
Social surplus function
25
30