<p>Calculate the first-order-condition (FOC) wrt labour: <span class="math-container">$$\begin{align}\frac{\partial}{\partial l_t^j} \left( \frac{1 - \tau_t^j}{1 - \alpha} (A^j)^\alpha (k_t^j)^{1-\alpha} (l_t^j)^\alpha - w_t l_t^j - k_t^j \right) &amp;= 0 \\ \implies \frac{1 - \tau_t^j}{1-\alpha} (A^j)^\alpha \color{red}{(k_t^j)}^{1-\alpha} (\alpha) (l_t^j)^{\alpha-1} &amp;= w_t \\ \implies \frac{1 - \tau_t^j}{1-\alpha} (A^j)^\alpha \color{red}{\left((1-\tau_t^j)^{\frac{1}{\alpha}} A^j l_t^j\right)}^{1-\alpha} (\alpha) (l_t^j)^{\alpha-1} &amp;= w_t \\ \implies \frac{\alpha}{1-\alpha} (1-\tau_t^j)^{1/\alpha} A^j &amp;= w_t \end{align}$$</span></p>
<p>While calculating the FOC, substitute <span class="math-container">$k_t^j$</span> with the optimal value (already) obtained from the other FOC. This is marked in red above.</p>
<blockquote>
<p>net marginal product of a worker employed by a producer of group <span class="math-container">$j$</span></p>
</blockquote>
<p>In case you don't know what that means, it is the partial derivative of the output <span class="math-container">$(y_t^j)$</span> wrt labour <span class="math-container">$(l_t^j)$</span>. In other words, it is <span class="math-container">$\frac{\partial y_t^j (l_t^j, k_t^j)}{\partial l_t^j} = \frac{\partial}{\partial l_t^j} \left( \frac{1 - \tau_t^j}{1 - \alpha} (A^j)^\alpha (k_t^j)^{1-\alpha} (l_t^j)^\alpha \right)$</span>.</p>

CC BY-SA 4.0
2024-03-31T08:22:15.620Z
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
Pepus
CC BY-SA 4.0
2024-03-30T21:13:11.573Z
2024-03-30T21:21:30.387Z
2024-03-30T21:21:30.387Z
uninterestedacademic
<p>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 &quot;Modeling inefficient institutions&quot; <a href="https://economics.mit.edu/sites/default/files/publications/Modeling%20Inefficient%20Institutions.pdf" rel="nofollow noreferrer">Modeling inefficient institutions</a> but I'm struggling with the &quot;net marginal product of a worker employed by a producer of group j. Maximization problem seem easy and I obtain <span class="math-container">$k_t^j$</span> but when I try to compute <span class="math-container">$l_t^j$</span> I don't get how to obtain <span class="math-container">$\displaystyle \frac{\alpha(1-\tau_t^j)^\frac{1}{\alpha}}{(1-\alpha)}A^j$</span>. 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.</p>

<microeconomics><profit-maximization><institutions>
Net product marginal in Acemoglu's article "Modeling inefficient institutions"
Pepus
# Net product marginal in Acemoglu's article "Modeling inefficient institutions"

(Asked by: username_0 on 2024-03-30)

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](https://economics.mit.edu/sites/default/files/publications/Modeling%20Inefficient%20Institutions.pdf) 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.

**Tags:** microeconomics, profit-maximization, institutions


---

## 1 answers

### Answer by username_1 on 2024-03-30. Score: 4 | Accepted answer

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 $\frac{\partial y\_t^j (l\_t^j, k\_t^j)}{\partial l\_t^j} = \frac{\partial}{\partial l\_t^j} \left( \frac{1 - \tau\_t^j}{1 - \alpha} (A^j)^\alpha (k\_t^j)^{1-\alpha} (l\_t^j)^\alpha \right)$.

> **Comments:**
> **username_0**: 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  

---

