Reservation utility

Question

I am self-studying contract theory using Bolton and Dewatripont (2005). It is meant for grad students, which might be why I am having a difficult time understanding basic terminology. Here is the problem:

Consider two agents, an employee and an employer. The employer's utility function is

$$U(l_1,t_1),$$

where $l_1$ is the amount of 'employee time' consumed and $t_1$ is the amount of output consumed by the employer.

The employee's utility function is

$$u(l_2,t_2),$$

where $l_2$ is the amount of `employee time' spent on leisure and $t_2$ is the amount consumed by the employee.

The initial endowments are

$$(\hat{l_{1}}, \hat{t_{1}}) = (0,1) \\ (\hat{l_{2}}, \hat{t_{2}}) = (1,0).$$

Without trading their respective utility levels are

$$\bar{U} = U(0,1) \\ \bar{u} = u(1,0).$$

Thus far I am able to follow.

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Now if we want to maximize joint surplus, the authors set up the following maximization problem:

$$ \begin{align} \text{max} \quad U(l_1,t_1) &+ \mu u(l_2,t_2) \\ \text{s.t.} \quad l_1 + l_2 &= \hat{l_1} + \hat{l_{2}} = 1 \\ t_1 + t_2 &= \hat{t_1} + \hat{t_{2}} = 1 \end{align} $$

My question is about the parameter $\mu$. The book says $\mu$ represents the individual's respective reservation utility levels, $\bar{U}$ and $\bar{u}$, and their relative bargaining strength.

I don't understand what this means intuitively. Why do we even need to introduce $\mu$ if we simply want to maximise joint surplus? Why not just maximise the sum of respective utility's subject to the endowment constraints?

Answer 1

I suspect the $μ$ is introduced to account for the fact that no party can be forced to enter into a contract. Thus if we want to maximize the joint surplus that only works if both parties are willing to trade, and the parties are only willing to trade if it makes them both better off.

Suppose you had a situation where the maximum joint surplus results in a negative utility of the employee, but that is outweighed by the positive utility for the employer. Clearly, the employee would not want to enter into such a contract, because the employee is better off not working in that case.

Answer 2

It easier to understand this intuitively if one looks at μ as relative bargaining strength. Higher the bargaining strength, higher is the weight attached to their utility in the joint maximization problem. μ can be seen as coefficient of employee's bargaining power/(divided by) coefficient of employer's bargaining power. Whoever enjoys a higher say, will have a higher weight attached to their utility in the joint maximization. The optimal solution, thus, would tend to (relatively) attach a higher "importance" to the one with a higher weight (bargaining power).