Part 2 / Section 5: Task-Based Model of Production Section 5.2

Canonical task-based model: capital and labor

To set up this model we need to describe two levels of production: how tasks combine into final output and how inputs (capital and labor) produce each task. There are many tasks involved in production, which we capture by having tasks be indexed by \(i\in[0,1]\). The quality or level of a task depends on who or what performs it.

Final output is

\[\begin{equation} Y \;=\; \left(\int_0^1 y(i)^{\frac{\eta-1}{\eta}}\dd i\right)^{\frac{\eta}{\eta-1}}, \qquad \eta>0,\quad\eta\neq1, \tag{5.1} \end{equation}\]

where \(y(i)\) denote the quality or quantity of task service \(i\) and \(\eta\) is the elasticity of substitution across tasks. We use \(\eta\) rather than \(\sigma\) because the elasticity between aggregate inputs and tasks need not be the same. When \(\eta<1\), tasks are hard to substitute for each other: producing much more of some activities does not easily compensate for producing little of others. When \(\eta>1\), this substitution is easier. When \(\eta\rightarrow1\) we obtain the continuous Cobb--Douglas:

\[\begin{equation} \log Y\;=\;\int_0^1\log y(i) \, \dd i. \tag{5.2} \end{equation}\]

Tasks are performed using capita \(\,K\,\) or labor \(\,L\,\), which cost \(\,r\,\) and \(\,w\,\) per unit, respectively. The capital used for task \(\,i\,\) is \(\,k(i)\,\) and the labor used in that task is \(\,\ell(i)\,\). The task quality depends on how it is performed and is

\[\begin{equation} y(i) \;=\; a_K(i)k(i) \;+\;a_L(i)\ell(i) \,. \tag{5.3} \end{equation}\]

Here \(a_K(i)>0\) is the productivity of capital in task \(i\) and \(a_L(i)>0\) the prodcutivity of labor. These functions encapsulate the mismatch between the skills of the input factor and the demands of the task when being performed.

Crucially, the two inputs are perfect substitutes within each task. The substitutability of inputs in production depends on how tasks are combined and assigned, not on their substitutability in a given task.