# Factor-augmenting technologies (Acemoglu and Restrepo 2018c)

In that paper, they consider the following production function:

$$Y = F(A_KK,A_LL)$$.

Where:

K denotes capital, L is labor, and $$A_K$$ and $$A_L$$ denote capital-augmenting and labor-augmenting technology, respectively. We assume throughout that F is continuously differentiable, concave, and exhibits constant returns to scale. Let $$F_K$$ and $$F_L$$ denote the derivatives of F with respect to capital and labor. We focus on competitive labor markets, which implies that the equilibrium wages is equal to the marginal product of labor:

$$W = A_L F_L(A_KK,A_LL)$$.

Why? Shouldn't simply be:

$$W = F_L(A_KK,A_LL)$$.

Without the $$A_L$$ multiplying the derivative?

Chain rule! To make it simple call $$\tilde{L} = A_L L$$ and $$\tilde{K} = A_K K$$, so that
$$Y = F(\tilde{K}, \tilde{L})$$
$$\frac{\partial Y}{\partial L} = \frac{\partial \tilde{L}}{\partial L}\frac{\partial F}{\partial \tilde{L}} = A_L F_L(A_K K, A_L L)$$
the trick here is that you should read $$F_L$$ as the derivative of $$F$$ with respect to its second argument.
• Thanks! I understand. But don't you think it's confusing to call $F_L$ what is actually $F_{(A_LL)}$? Is there a reason for doing so? – Tecon Jan 22 '19 at 15:34
• @Tecon Yes it is :) But labor is the result of applying $A_L$ to $L$, so in that sense the marginal product of labor is $F_{(A_L L)}$ – caverac Jan 22 '19 at 15:37