<p>The FOC is an outcome of a profit maximization problem. Profits are</p>
<p><span class="math-container">$$
\Pi = \underbrace{p_iy_i}_{\text{Total Revenue}} - \underbrace{\left(\sum_j p_jx_{ij} + wl_i\right)}_{\text{Total Costs}}
$$</span>
Differentiating the above with respect to <span class="math-container">$x_{ij}$</span> and setting to zero gives you the equation you are looking for. The assumption that the exponents sum to one in your last equation is just assuming that the production function has constant returns to scale, see <a href="https://en.wikipedia.org/wiki/Returns_to_scale" rel="nofollow noreferrer">here</a>.</p>

CC BY-SA 4.0
2023-04-19T13:14:56.830Z
Thank you for your answer, that bring some comprehensible memories back. 
I will try to continue my computation with your input before concluding, but it is already a huge help. Thank you !
PGCD
CC BY-SA 4.0
2023-04-18T17:10:10.897Z
2023-04-18T17:10:10.897Z
Wittgenstein's Poker
<p>I have been reading the paper of Acemoglu et al., 2016 <a href="https://www.journals.uchicago.edu/doi/full/10.1086/685961" rel="nofollow noreferrer">Networks and the macroeconomy : an empirical exploration</a>, and I have been struggling with a maximization programm in the early pages of the article. Even if I get the following parts, I am a bit frustrated I can't prove one of the first results.</p>
<p>The production function is :</p>
<p><a href="https://i.sstatic.net/TC2SM.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/TC2SM.png" alt="enter image description here" /></a></p>
<p>With <span class="math-container">$x_{ij}$</span> the products of the firm <span class="math-container">$j$</span> treated as input for the output of the firm <span class="math-container">$i$</span>.</p>
<p>My question is the following : What should I take as constraint for my Lagrangian function ? I am having difficulties seeing how the authors came to the (4) conclusion :</p>
<p><a href="https://i.sstatic.net/yfdGA.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yfdGA.png" alt="enter image description here" /></a></p>
<p>I think the constraints is given by the relation</p>
<p><a href="https://i.sstatic.net/WY0WR.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/WY0WR.png" alt="enter image description here" /></a></p>
<p>But I must admit I am not sure about my result with these constraints.</p>
<p>Could anyone give me a tip about the constraints and how to proove the <span class="math-container">$(4)$</span> relation ?</p>

<profit-maximization><lagrangian><networks>
Finding the constraints for a profit maximization program in Acemoglu et al, 2016
PGCD
# Finding the constraints for a profit maximization program in Acemoglu et al, 2016

(Asked by: username_0 on 2023-04-18)

I have been reading the paper of Acemoglu et al., 2016 [Networks and the macroeconomy : an empirical exploration](https://www.journals.uchicago.edu/doi/full/10.1086/685961), and I have been struggling with a maximization programm in the early pages of the article. Even if I get the following parts, I am a bit frustrated I can't prove one of the first results.

The production function is :

[![enter image description here](https://i.sstatic.net/TC2SM.png)](https://i.sstatic.net/TC2SM.png)

With $x\_{ij}$ the products of the firm $j$ treated as input for the output of the firm $i$.

My question is the following : What should I take as constraint for my Lagrangian function ? I am having difficulties seeing how the authors came to the (4) conclusion :

[![enter image description here](https://i.sstatic.net/yfdGA.png)](https://i.sstatic.net/yfdGA.png)

I think the constraints is given by the relation

[![enter image description here](https://i.sstatic.net/WY0WR.png)](https://i.sstatic.net/WY0WR.png)

But I must admit I am not sure about my result with these constraints.

Could anyone give me a tip about the constraints and how to proove the $(4)$ relation ?

**Tags:** profit-maximization, lagrangian, networks

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## 1 answers

### Answer by username_1 on 2023-04-18. Score: 3

The FOC is an outcome of a profit maximization problem. Profits are

$$
\Pi = \underbrace{p\_iy\_i}\_{\text{Total Revenue}} - \underbrace{\left(\sum\_j p\_jx\_{ij} + wl\_i\right)}\_{\text{Total Costs}}
$$
Differentiating the above with respect to $x\_{ij}$ and setting to zero gives you the equation you are looking for. The assumption that the exponents sum to one in your last equation is just assuming that the production function has constant returns to scale, see [here](https://en.wikipedia.org/wiki/Returns_to_scale).

> **Comments:**
> **username_0**: Thank you for your answer, that bring some comprehensible memories back.  I will try to continue my computation with your input before concluding, but it is already a huge help. Thank you !  

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