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.
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?
