Part 2 / Section 6: Examples and Applications of the Task-Based Model Section 6.2

A special case: Cobb--Douglas from capital--labor assignment

We can also get a Cobb-Douglas out of an environment where tasks are hard to substitute (\(\eta<1\)). This is another case in which the task assignment makes inputs more substitutable. For this we follow Hubmer and Restrepo (2026) who are studying automation and changes in the labor share. Therefore, we go back to our notation of capital and labor with task production \(\,y(i) \,=\, a_K(i)k(i) \,+\, a_L(i)\ell(i)\).

The key to get the substitution right is to choose task productivity appropriately. Hubmer and Restrepo show this is done by setting

\[\begin{align} a_K(i)&\;=\;i^{\frac{\gamma_K-1}{\gamma_K(1-\eta)}} (1-i)^{\frac{\gamma_K+1}{\gamma_K(1-\eta)}}, & a_L(i)&\;=\;i^{\frac{\gamma_L+1}{\gamma_L(1-\eta)}} (1-i)^{\frac{\gamma_L-1}{\gamma_L(1-\eta)}}, \qquad \gamma_K,\gamma_L>0. \tag{6.12} \end{align}\]

The expressions look complicated, but they are chosen so that the task-cost integrals have simple solutions. Dividing the productivity schedules and collecting exponents gives

\[\begin{equation} \frac{a_L(i)}{a_K(i)} \;=\;\left(\frac{i}{1-i}\right)^{\frac{\gamma_K+\gamma_L}{(1-\eta)\gamma_K\gamma_L}}. \tag{6.13} \end{equation}\]

So, capital has a comparative advantage in low-index tasks and we are once more looking for a threshold index \(I\), below which capital performs tasks and above which labor performs tasks. This threshold satisfies

\[\begin{equation} \frac{w}{r}\;=\;\frac{a_L(I)}{a_K(I)} \;=\;\left(\frac{I}{1-I}\right)^{\frac{\gamma_K+\gamma_L}{(1-\eta)\gamma_K\gamma_L}}. \tag{6.14} \end{equation}\]

Cheaper capital raises, or more expensive labor, raises \(I\), substituting labor for capital through the reassignment of tasks.

The solution for \(I\) is more easily express as the ratio \(t\equiv I/(1-I)\), so that

\[\begin{equation} t\;=\;\left(\frac{w}{r}\right)^ {\frac{(1-\eta)\gamma_K\gamma_L}{\gamma_K+\gamma_L}}. \tag{6.15} \end{equation}\]

Integrating expenditure over tasks

We know from our previous results that the total demand for capital and labor satisfy (5.15). Multiplying by the respective price of each input gives total input expenditure:

\[\begin{align} rK&\;=\;YP^\eta r^{1-\eta} \int_0^I a_K(i)^{\eta-1}\dd i, & wL&\;=\;YP^\eta w^{1-\eta} \int_I^1 a_L(i)^{\eta-1}\dd i. \tag{6.16} \end{align}\]

Here, is where the seemingly complicated functional form of the task productivities starts to pay off. Raising the schedules to \(\eta-1=-(1-\eta)\) cancels their denominators:

\[\begin{align} a_K(i)^{\eta-1}&\;=\;i^{\frac{1}{\gamma_K}-1} (1-i)^{-1-\frac{1}{\gamma_K}}, & a_L(i)^{\eta-1}&\;=\;i^{-1-\frac{1}{\gamma_L}} (1-i)^{\frac{1}{\gamma_L}-1}. \tag{6.17} \end{align}\]

We can now solve the integrals directly using the change of variable \(u=i/(1-i)\). Then \(i=u/(1+u)\), \(1-i=1/(1+u)\), and \(\dd i=(1+u)^{-2}\dd u\). The factors in \(1+u\) cancel in both integrands, and the boundary becomes \(u=t\):

\[\begin{align} \int_0^I a_K(i)^{\eta-1}\dd i &\;=\;\int_0^t u^{\frac{1}{\gamma_K}-1}\dd u \;=\;\gamma_Kt^{\frac{1}{\gamma_K}}, \tag{6.18}\\ \int_I^1 a_L(i)^{\eta-1}\dd i &\;=\;\int_t^\infty u^{-1-\frac{1}{\gamma_L}}\dd u \;=\;\gamma_Lt^{-\frac{1}{\gamma_L}}. \tag{6.19} \end{align}\]

Constant shares and unit elasticity

We now show that the results above imply the characteristic properties of a Cobb-Douglas. Substitute these integrals into factor payments and take their ratio:

\[\begin{equation} \frac{rK}{wL} \;=\;\frac{\gamma_K}{\gamma_L} \left(\frac{r}{w}\right)^{1-\eta} t^{\frac{1}{\gamma_K}+\frac{1}{\gamma_L}}. \tag{6.20} \end{equation}\]

The cutoff condition (6.15) implies \(t^{\frac{1}{\gamma_K}+\frac{1}{\gamma_L}}=(w/r)^{1-\eta}\). The price terms cancel, so

\[\begin{equation} \frac{rK}{wL}\;=\;\frac{\gamma_K}{\gamma_L}, \qquad \alpha_K\;=\;\frac{\gamma_K}{\gamma_K+\gamma_L}, \qquad \alpha_L\;=\;\frac{\gamma_L}{\gamma_K+\gamma_L}. \tag{6.21} \end{equation}\]

These are constant factor shares, exactly as in our earlier Cobb--Douglas model. Rearranging the payment ratio and taking logs gives

\[\begin{equation} \log\frac{K}{L}\;=\;\log\frac{\gamma_K}{\gamma_L} +\log\frac{w}{r} \quad\longrightarrow\quad \sigma\;=\;\frac{\dd\log(K/L)}{\dd\log(w/r)}\;=\;1. \tag{6.22} \end{equation}\]

Taking a step back helps us see what is going on. There are actually two substitution responses taking place when we change relative prices. We can write the capital to labor ratio as

\[\begin{equation} \frac{K}{L}\;=\;\frac{\gamma_K}{\gamma_L} \left(\frac{w}{r}\right)^\eta t^{\frac{1}{\gamma_K}+\frac{1}{\gamma_L}}. \tag{6.23} \end{equation}\]

If \(I\) (and hence \(t\)) were fixed, the elasticity would be \(\eta\). Allowing assignment to adjust gives

\[\begin{equation} \sigma\;=\;\underbrace{\eta}_{\text{across tasks}} +\underbrace{\left(\frac{1}{\gamma_K}+\frac{1}{\gamma_L}\right) \frac{\dd\log t}{\dd\log(w/r)}}_{\text{task reassignment}} \;=\;\eta+(1-\eta)\;=\;1. \tag{6.24} \end{equation}\]

Tasks are hard to substitute for each other, but the firm can replace the producer of a task. Here that additional response is exactly large enough to give unit aggregate elasticity.