Part 2 / Section 5: Task-Based Model of Production Section 5.4

The link to aggregate production functions

So far we only know the assignment of tasks, characterized by a cutoff value \(I\) below which tasks go to capital and above which they go to labor. The next step is to see how that assignment shapes aggregate production. It turns out there is a tight link between the assignment of tasks and the CES production function. One way to look at what we gain from explicitly modeling the assignment of tasks is that we are endogenizing the levels of the productivity terms \(A_K\) and \(A_L\) as well as of the elasticity of substitution \(\sigma\). It turns out that all these objects depend on the assignment and move with prices or when the productivity of capital and workers across tasks changes.

For a given assignment, that is, holding \(I\) fixed, we can show that the task-based production function has a CES representation following Jones and Tonetti (2026). The assignment divides tasks between capital and labor. But, how are capital and labor distributed among the tasks they perform? We will provide an explicit answer to this question, but for now it is enough to define the share of capital or labor assigned to a task \(i\). We denote these shares by \(\omega_K(i)\) and \(\omega_L(i)\), respectively:

\[\begin{align} \omega_K(i)\;\equiv\;\frac{k(i)}{K}\,;\qquad\omega_L(i)\;\equiv\;\frac{\ell(i)}{L}\,. \tag{5.14} \end{align}\]

The shares are chosen optimally depending on the productivity of each input at the tasks it is assigned to and on how the output of different tasks is combined in production (where \(\eta\) plays a central role).

For \(\eta\neq1\), we can now express aggregate output \(Y\) in its CES form:

\[\begin{align} Y &= \left(\int_0^1 y(i)^{\frac{\eta-1}{\eta}}\dd i\right)^{\frac{\eta}{\eta-1}} \tag{5.15} \\ Y &= \left(\int_0^I \left(a_K(i)k(i)\right)^{\frac{\eta-1}{\eta}}\dd i \;+\; \int_I^1 \left(a_L(i)\ell(i)\right)^{\frac{\eta-1}{\eta}}\dd i\right)^{\frac{\eta}{\eta-1}}\nonumber \\ Y &= \left[K^{\frac{\eta-1}{\eta}}\int_0^I \left(a_K(i)\omega_K(i)\right)^{\frac{\eta-1}{\eta}}\dd i + L^{\frac{\eta-1}{\eta}}\int_I^1 \left(a_L(i)\omega_L(i)\right)^{\frac{\eta-1}{\eta}}\dd i\right]^{\frac{\eta}{\eta-1}}\nonumber \\ Y &= \left([A_K(I)K]^{\frac{\eta-1}{\eta}} \;+\; [A_L(I)L]^{\frac{\eta-1}{\eta}}\right)^{\frac{\eta}{\eta-1}}. \nonumber \end{align}\]

This looks like the CES function from Section 3, but there are two key differences hiding under the hood. First, the factor-augmenting productivities, \(A_K(I)\) and \(A_L(I)\), depend on the task assignment. They depend on \(I\) and on the distribution of capital and labor across the tasks each performs, captured by the task shares \(\omega_K(i)\) and \(\omega_L(i)\):

\[\begin{align} A_K(I)&\;\equiv\;\left[\int_0^I \left(a_K(i)\omega_K(i)\right)^{\frac{\eta-1}{\eta}}\dd i\right]^{\frac{\eta}{\eta-1}}\;; & A_L(I)&\;\equiv\;\left[\int_I^1 \left(a_L(i)\omega_L(i)\right)^{\frac{\eta-1}{\eta}}\dd i\right]^{\frac{\eta}{\eta-1}}. \tag{5.16} \end{align}\]

Thus \(A_K(I)K\) and \(A_L(I)L\) are effective inputs, as in Section 3. The factor augmenting technology terms, \(A_K\) and \(A_L\), are CES aggregates of task productivity weighted by the distribution of inputs across tasks (summarizing each input's effective productivity over its assigned bundle). They depend on both task productivity and the size of the assigned bundle. Second, the elasticity of substitution does not correspond to \(\eta\). It is true that holding \(I\) fixed gives an elasticity of \(\eta\), but allowing the cutoff to move adds substitution through reassignment. Thus the aggregate elasticity \(\sigma\) is necessarily higher than \(\eta\).

Our next steps are to find out how capital and labor are optimally distributed across tasks and what determines the elasticity of substitution.