What is the meaning of “labor input” in the context of incentive theories?

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.

Since the authors state that the total labor input is:

$$\int t_i(k)dk$$

the meaning of the total labor input in this case would be that it is the sum of all attention $$t_i$$ allocated over those tasks $$k$$.

For example if we would assume that $$t_i (k) = k$$ then the labor supply across continuum given by $$[0,1]$$ would be equal to $$\frac{1}{2}$$ because $$\int k dk = \frac{1}{2} k^2 + c$$ and when you evaluate it between bounds $$[0,1]$$ the area would sum to $$\frac{1}{2}$$ (note that geometric interpretation of an integral is an area under the curve that you are integrating).

You can graph it by making assumption about the function, for example following my assumption of $$t_i(k)=k$$ the labor supply can be plotted (using tikz in LaTex) as:

Where the total labor supply is the area under the curve bounded by $$[0,1]$$. Of course you might want to impose some less simplistic assumption on the function $$t_i (k)$$.

• Thanks so much. I thought the same, although you went a little bit farther than me. 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? – David Fernando Jiménez Jul 16 at 22:06
• @DavidFernandoJiménez if you care only about purely intuitive explanation then I would say that in this case the labor supply depends on the intensity of effort a person puts into the task. Any student should be familiar with the fact that for example studying is not all or nothing - you can study with more or less concentration put more or less effort. In this case the labor supply is simply the sum of all that effort over tasks – 1muflon1 Jul 16 at 22:09
• I really appreciate your comments. They help me a lot. There are left some gaps, but I think I can fill them up based on your ideas. – David Fernando Jiménez Jul 17 at 21:22