I am following and trying to fully understand a famous and interesting work of Bentolila and Saint-paul (2003). They try to explain movements of the factor's share in terms of a relationship between the labor share ($LS$) and the capital-output ratio ($k$). To summarize as much as possible they start from a general production function with labor-augmenting techological progress:

$$Y_{i} = F(K_{i},B_{i}L_{i})=K_{i}f(l_{i}),$$

where $l_{i}=\frac{B_{i}L_{i}}{K_{i}}.$ And show that under the usual microeconomic assumptions of equilibrium, there exists a unique function $g(\cdot)$ such that:

$$LS_{i} = g(k_{i}).$$

Then, at page 6 of the paper, they employ the standard definition of elasticity of substitution to production function above, i.e. $\sigma_{i}=\frac{K_i/L_i}{r/w}\cdot \frac{r/w}{K/L}$ to obtain the following result:

$$\sigma_{i}=\frac{f'(l_{i})}{l_{i}f''(l_{i})}\left [1-\frac{l_{i}f'(l_{i})}{f(l_{i})} \right ]$$

I am trying to do all the derivations, but I can't find a way to derive this last formula with the same result. Is there someone who could kindly help me and show me the steps?.


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