Part 2 / Section 5: Task-Based Model of Production Section 5.6

The link to aggregate production functions

For a given assignment, holding fixed \(I\), 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. For each task assigned to capital, \(i\in[0,I]\), we can write the amount of capital assigned to that task as

\[\begin{align} k(i)\;=\; a_K(i)^{\eta-1} \frac{K}{H_K(I)}. \tag{5.29} \end{align}\]

We can then use this to write capital's total contribution to the task aggregator as

\[\begin{equation} \int_0^I[a_K(i)k(i)]^{(\eta-1)/\eta}\,\dd i \;=\;K^{(\eta-1)/\eta}H_K(I)^{1/\eta} \;=\;[A_K(I)K]^{(\eta-1)/\eta}. \tag{5.30} \end{equation}\]

where we define

\[\begin{equation} A_K(I)\;=\;H_K(I)^{\frac{1}{\eta-1}},\qquad A_L(I)\;=\;H_L(I)^{\frac{1}{\eta-1}}. \tag{5.31} \end{equation}\]

We use \(A_L\) to follow the same steps and obtain

\[\begin{equation} Y\;=\;\left([A_K(I)K]^{\frac{\eta-1}{\eta}} +[A_L(I)L]^{\frac{\eta-1}{\eta}}\right)^{\frac{\eta}{\eta-1}}, \qquad \alpha_K\;=\;\left(\frac{A_K(I)K}{Y}\right)^{\frac{\eta-1}{\eta}}. \tag{5.32} \end{equation}\]

This looks like the CES function from Section 3, but \(A_K(I)\) and \(A_L(I)\) depend on the task assignment. Holding \(I\) fixed gives elasticity \(\eta\); allowing the cutoff to move adds substitution through reassignment. Thus the aggregate elasticity \(\sigma\) need not equal \(\eta\). The special cases below show this explicitly. Standard factor-share formulas remain valid at the optimal assignment: the marginal task has equal unit costs, so a small boundary reallocation has no first-order effect on optimized output by itself.