I'm reading lecture notes by Acemoglu regarding institutions and I'm trying to study his famous model of non-democratic institutions, deeply described in his article "Modeling inefficient institutions" Modeling inefficient institutions but I'm struggling with the "net marginal product of a worker employed by a producer of group j. Maximization problem seem easy and I obtain $k_t^j$ but when I try to compute $l_t^j$ I don't get how to obtain $\displaystyle \frac{\alpha(1-\tau_t^j)^\frac{1}{\alpha}}{(1-\alpha)}A^j$. I tried to derive respect o labor the production function (equation (2) in the article) but I don't obtain the same result, even after a multiplication with taxes). Anyone can explain how to obtain this expression? I'm thinking that is very simple but I really don't get what I'm not taking account of.


1 Answer 1


Calculate the first-order-condition (FOC) wrt labour: $$\begin{align}\frac{\partial}{\partial l_t^j} \left( \frac{1 - \tau_t^j}{1 - \alpha} (A^j)^\alpha (k_t^j)^{1-\alpha} (l_t^j)^\alpha - w_t l_t^j - k_t^j \right) &= 0 \\ \implies \frac{1 - \tau_t^j}{1-\alpha} (A^j)^\alpha \color{red}{(k_t^j)}^{1-\alpha} (\alpha) (l_t^j)^{\alpha-1} &= w_t \\ \implies \frac{1 - \tau_t^j}{1-\alpha} (A^j)^\alpha \color{red}{\left((1-\tau_t^j)^{\frac{1}{\alpha}} A^j l_t^j\right)}^{1-\alpha} (\alpha) (l_t^j)^{\alpha-1} &= w_t \\ \implies \frac{\alpha}{1-\alpha} (1-\tau_t^j)^{1/\alpha} A^j &= w_t \end{align}$$

While calculating the FOC, substitute $k_t^j$ with the optimal value (already) obtained from the other FOC. This is marked in red above.

net marginal product of a worker employed by a producer of group $j$

In case you don't know what that means, it is the partial derivative of the output $(y_t^j)$ wrt labour $(l_t^j)$. In other words, it is $\frac{\partial y_t^j (l_t^j, k_t^j)}{\partial l_t^j} = \frac{\partial}{\partial l_t^j} \left( \frac{1 - \tau_t^j}{1 - \alpha} (A^j)^\alpha (k_t^j)^{1-\alpha} (l_t^j)^\alpha \right)$.

  • $\begingroup$ Thank you very much. I was expecting to find an expression like $l_t^j=...$ and the real answer was given by the equality between wage and the FOC respect to labor given the solution of the problem for $k_t^j$. Yes I know the meaning of "net..." but computing the derivative I was expecting the same expression of the article without substitution of the optimal value for $k_t^j$. Your answer was very clarifying $\endgroup$
    – Pepus
    Mar 31 at 8:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.