In the article "Multitask Principal-Agent Analyses: Incentive Contracts, Asset Ownership, and Job Design" (Holmström and Milgrom, 1991) it is said that two identical agents ($i=1,2$) devote attention $t_i(k)$ to a task $k$, i.e., they allocate $t_i(k)$ across a continuum of tasks indexed by $k\in[0,1]$. Later it is said that total labor input $\overline t_i$ is equal to $\int t_i(k)dk$.
With this in mind, my questions are the next: what is the meaning of labor input in this context? How can I graph attention $t_i(k)$ and labor input $\overline t_i$?
I have an intuitive idea of the general meaning of labor input, but I can't figure out its meaning in this context. For me, there is no difference between labor input and attention. The problem that I have is that I can't clearly understand the meaning of labor input beyond its mathematical definition as the summation of attention.
In particular, I'm more interested in a verbal and familiar (easy to understand for a freshman) interpretation of "labor input". For example, if attention is the derivative of labor input, does this mean that attention is a measure of the productivity of labor input? (this doesn't make much sense to me) How would you define "labor input" with familiar words?
I will appreciate any help.