<p>I read some models of using Lagrange multiplier to find the optimal decision in my economics class.</p>
<p>Let's assume, a company's objective function is to maximize its profit <span class="math-container">$\Pi(stock\ price, customer\ review...) = revenue(stock\ price, customer\ review...) - cost(stock\ price, customer\ review...) - W(.)$</span> ,where W(.) is the compensation to CEO. The constraint is that CEO needs minimum wage, s.t. <span class="math-container">$W(.) &gt;= W_{min}$</span>.</p>
<p>I assume optimal contract for CEO is solved by Lagrange multiplier as
<span class="math-container">$$W = \lambda \times (stock\ price + customer\ reviews).$$</span>
This is a result I assume, it is not solved by F.O.C.</p>
<p>However, I see people stop at this step, and start to describe that high stock price and good customer feedbacks will lead better compensation.</p>
<p>But my concern is that when the customer feedbacks changing, the <span class="math-container">$\lambda$</span> may also be affected. Why do we ignore this point and regard <span class="math-container">$\lambda$</span> as a constant in this analysis.</p>
<p>Do we need to write down the explicit formula of <span class="math-container">$\lambda(stock\ price, customer\ reviews)$</span> before analyzing?</p>

CC BY-SA 4.0
2024-12-10T09:32:37.923Z
I can't follow the question. Can you write down the actual optimization problem in your example?
Michael Greinecker
CC BY-SA 4.0
2024-12-10T13:55:18.447Z
Yea it's hard to see how to interpret $\lambda$ intuitively without seeing your derivation.
Kitsune Cavalry
CC BY-SA 4.0
2024-12-10T14:33:12.570Z
@KitsuneCavalry, I edited it and hope this clarification may help. Thanks in advance.
Ruming Liu
CC BY-SA 4.0
2024-12-10T14:37:07.867Z
@MichaelGreinecker, thanks for your reply. I edited the question and hope it is more clear.
Ruming Liu
CC BY-SA 4.0
2024-12-10T16:53:28.780Z
Your formula contains terms like stock price and customer reviews that are not in your profit function. This makes no sense.
Michael Greinecker
<microeconomics><mathematical-economics><labor-economics>
Do we need to write down the explicit solution of Lagrange multiplier $\lambda$ in economics model?
Ruming Liu
# Do we need to write down the explicit solution of Lagrange multiplier $\lambda$ in economics model?

(Asked by: username_0 on 2024-12-10)

I read some models of using Lagrange multiplier to find the optimal decision in my economics class.

Let's assume, a company's objective function is to maximize its profit $\Pi(stock\ price, customer\ review...) = revenue(stock\ price, customer\ review...) - cost(stock\ price, customer\ review...) - W(.)$ ,where W(.) is the compensation to CEO. The constraint is that CEO needs minimum wage, s.t. $W(.) >= W\_{min}$.

I assume optimal contract for CEO is solved by Lagrange multiplier as
$$W = \lambda \times (stock\ price + customer\ reviews).$$
This is a result I assume, it is not solved by F.O.C.

However, I see people stop at this step, and start to describe that high stock price and good customer feedbacks will lead better compensation.

But my concern is that when the customer feedbacks changing, the $\lambda$ may also be affected. Why do we ignore this point and regard $\lambda$ as a constant in this analysis.

Do we need to write down the explicit formula of $\lambda(stock\ price, customer\ reviews)$ before analyzing?

**Tags:** microeconomics, mathematical-economics, labor-economics
> **Comments:**
> **username_1**: I can't follow the question. Can you write down the actual optimization problem in your example?  
> **username_2**: Yea it's hard to see how to interpret $\lambda$ intuitively without seeing your derivation.  
> **username_0**: @KitsuneCavalry, I edited it and hope this clarification may help. Thanks in advance.  
> **username_0**: @MichaelGreinecker, thanks for your reply. I edited the question and hope it is more clear.  
> **username_1**: Your formula contains terms like stock price and customer reviews that are not in your profit function. This makes no sense.  

---

## 0 answers

