Lecture 2 in ContractsEmployment 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, and service contracts for consumers. This leads to a discussion of the optimal size of the firm.

Read Chapters 17 and 18 of Strategic Play.

Employment contracts

An implication of the examples in the last lecture is the manifest value from setting the rules and conventions that determine how contracts should be written.

Almost by definition, managers have a hand in writing employment contracts for their workers, and often determine upstream supply and downstream service contracts as well.

This lecture analyzes 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.

An example

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 + 9log(16 - h)

where 16 is the maximum number of hours she would consider working, and 9/(16 - h) is the marginal rate of substitution between income and leisure time.

We also assume that if she is not employed with the firm her income equivalent is v.

Firm value

Suppose firm profits are:

3h - y

where 3 is the output price (or value of the worker’s product in terms of hourly units).

The firm chooses h and y to maximize profits subject to the participation constraint that the worker chooses to be employed:

y + 9log(16 - h) v

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 – 9log(16 – h)

The firm maximizes:

3h + 9log(16 – h) – v

Dividing through by 3, and differentiating we obtain the first order condition:

1 = 3/(16 – h) or h = 13

Solution

Substituting this equation for hours into the profit function we obtain:

39 + 9log(3) – v

This is the interior optimum, which only applies if and only if profits are strictly positive.

Therefore the firm sets

h = 13

if profits are positive, meaning

39 + 9log(3) > 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 + 9log(16 - h)

subject to the constraint that the firm accepts her proposal (does not make losses):

y6 3h

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 = 3h.

Hours are now chosen by the outsourcer to maximize

3h + 9log(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 sales commission, denoted by s, on her output.

In this case the worker chooses h to maximize

sh + 9log(16 – h).

Analogous to the previous problem, the solution to this maximization problem is the labor supply function h(s):

h = 16 – 9/s

if 16s – 9 + 9log(9) – 9log(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:

(3 – s)h(s) = (3 – s)(16 – 9/s)

= 57 – 16s – 27/s

This solution to this maximization problem is found by solving the first order condition to the firm’s optimization problem: 16 = 27/s2

Solving for s gives: s = 3(3)1/2/4.

Comparing the schemesTotal profit under the sales commission is:

57 – 12*31/2 – 36*3-1/2

Total profit under the optimal wage contract is:

39 + 9log(3) – v

Since the share is less than the output price, the participation constraints imply there are values of v 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.

Complications and complexity

Complicated or complex contracts are likely to bring the bargaining parties under the umbrella of the firm.

As the complexity of the optimal contract increases, so does the propensity for error, leading to responders:

1. rejecting the contract

2. accepting the contract but taking an action the manager did not intend

3. accepting the contract, taking the intended action, but leaving the manager without any rent.

Size and scope of firms

Simpler contracts (such as uniform prices or letting another player make an offer) yield less rent than optimal contracts, but are easier to implement without unintended consequences.

This led us to consider how strategic interactions help shape the firm’s boundaries.

High value activities within the value chain demand the strategist’s attention as the firm seeks to capture the rents. Here the firm is proactive in approaching its bargaining partners and making contract offers.

Conversely opportunities that offer little rent are best dealt with at arm’s length, for example within the market place.

Lecture SummaryOptimal contracting provides an opportunity for the contractor to extract rents from his business partners, employees, customers and clients.

Private information and outside options available to each party are explicitly modeled through the incentive compatibility and participation constraints.

Extracting maximal rent may require relatively complicated contracts, which if written incorrectly, carry the prospect of loss.

If the rent opportunities are meager, surrendering the rent, and using the market, might be preferable.

These factors form the basis for defining where firm boundaries should be relative to the market.