Lecture 1 in ContractsNonlinear 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 Full information principal agent problemproblem

A firm wishes to build a new factory, and will hire a builder.

How should it structure the contract?

Firm:RL-wL

Builder: wL-uL

RH-wH

wH-uH

Constraints facing the firm

We can use backwards induction to solve the use backwards induction to solve the problem:problem:

1.1. The incentive compatibility constraint is:The incentive compatibility constraint is:

wwHH – u – uHH w wLL – u – uL L if Hif H

wwLL – u – uLL w wHH – u – uH H if Lif L

2.2. The participation constraint is: The participation constraint is:

wwH H - u- uHH 0 if H 0 if H

wwLL- u- uLL 0 if L 0 if L

wH

wL

wH =uH

wH-wL=uH-uL

uH -uL

uL-uH

(IC)

uH

Theconstraintsillustrated

Minimum cost of achieving L

The minimum cost of achieving L is found by The minimum cost of achieving L is found by minimizing wminimizing wLL such that: such that:

1.1. wwLL u uLL

2.2. wwLL – u – uLL w wHH – u – uHH

The first constraint bounds wThe first constraint bounds wLL from below by from below by uuLL. .

Since uSince uLL u uHH the second constraint is satisfied the second constraint is satisfied by not making the wage depend on effort. by not making the wage depend on effort.

Therefore the minimum cost of achieving L is Therefore the minimum cost of achieving L is found by settingfound by setting

w* = u*w* = u*LL

Minimum cost of achieving HThe minimum cost of achieving H is found by The minimum cost of achieving H is found by minimizing wminimizing wLL such that: such that:

1.1. wwH H u uHH

2.2. wwHH – u – uHH w wLL – u – uLL

The first constraint bounds wThe first constraint bounds wHH from below by from below by uuHH. .

Since uSince uL L u uHH we must penalize the worker to we must penalize the worker to deter him from choosing L, by setting: deter him from choosing L, by setting:

wwLL < w < wHH – u – uH H + u+ uLL

Therefore the minimum cost of achieving H is:Therefore the minimum cost of achieving H is:

w*w*H H = u*= u*HH

w*w*L L = w= wHH – u – uH H + u+ uL L - Penalty - Penalty

Profit maximizationProfit maximization

The net profits from achieving L are The net profits from achieving L are

RRLL – u – uLL* *

The net profits from achieving H are The net profits from achieving H are

RRHH – u – uHH**

Therefore the firm hires a worker to Therefore the firm hires a worker to achieve H ifachieve H if

RRHH – u – uHH* > R* > RLL – u – uLL* *

and hires a worker to achieve only L and hires a worker to achieve only L otherwise.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 contractingTo 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 revisitedSupposing 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 withuniform 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.