# Optimization - Lagrange multipliers : minimum cost/maximum production

• MHB
• mathmari
In summary: For (b) it is the rate of change of the maximum possible level of production as the production quantity constraint changes.
mathmari
Gold Member
MHB
Hey! :giggle:

Business operates on the basis of the production function $Q=25\cdot K^{1/3}\cdot L^{2/3}$ (where $L$ = units of work and $K$ = units of capital).

If the prices of inputs $K$ and $L$ are respectively $3$ euros and $6$ euros per unit, then find :

a) the optimal combination of inputs that the company must occupy to minimize its costs, producing $Q = 600$ production units. What is the minimum production cost?

b) the optimal combination of inputs that the company must occupy to maximize production, if the amount of money available for the purchase of inputs is $450$ euros. What is the maximum possible level of production?
I have done the following:

a) We consider the function $f(K,L)=600\cdot 25\cdot K^{1/3}\cdot L^{2/3}$. Do we apply now Lagrange multipliers?

b) We consider the cost function $C(K,L)=K+L$. Do we apply now Lagrange multipliers? But for which function?

:unsure:

mathmari said:
a) We consider the function $f(K,L)=600\cdot 25\cdot K^{1/3}\cdot L^{2/3}$. Do we apply now Lagrange multipliers?

Hey mathmari!

How did you get $f(K,L)$? (Wondering)

It seems to me that we need to minimize the total price $3K+6L$ under the constraint $Q=25\cdot K^{1/3}\cdot L^{2/3}=600$.
We can do that with Lagrange multipliers, but we can also simply solve the constraint for $K$ and substitute that in the price function.

mathmari said:
b) We consider the cost function $C(K,L)=K+L$. Do we apply now Lagrange multipliers? But for which function?
The units don't match. $K$ is in units of capital, and $L$ is in units of work. They are not compatible for summation, and we won't get a cost. (Worried)
Instead I believe we should have $C(K,L)=3K+6L$.
We want to maximize production, which is $Q(K,L)=25\cdot K^{1/3}\cdot L^{2/3}$, under the constraint $C(K,L)=3K+6L=450$.
Again we can do it either with Lagrange multipliers or we can solve the constraint for 1 variable and substitute it in the function we want to maximize.

Ok! I will try that using Lagrange multipliers!
What is the financial interpretation of the Lagrange multiplier in this case? :unsure:

mathmari said:
Ok! I will try that using Lagrange multipliers!
What is the financial interpretation of the Lagrange multiplier in this case?
For (a) it is the rate of change of the minimum cost as the production quantity constraint changes.

Last edited:

## 1. What is the purpose of using Lagrange multipliers in optimization?

Lagrange multipliers are used in optimization to find the minimum or maximum value of a function subject to a set of constraints. This method allows us to take into account both the objective function and the constraints simultaneously, leading to more accurate and efficient solutions.

## 2. How do Lagrange multipliers work in finding the minimum cost?

In order to find the minimum cost, we set up a Lagrangian function that combines the cost function and the constraints using Lagrange multipliers. We then take the partial derivatives of the Lagrangian with respect to each variable and set them equal to zero. This will give us a system of equations that can be solved to find the optimal values for the variables.

## 3. Can Lagrange multipliers be used to find the maximum production?

Yes, Lagrange multipliers can be used to find both the minimum and maximum values of a function. The same process described above can be applied to find the optimal values for the variables that will result in maximum production.

## 4. What are the limitations of using Lagrange multipliers in optimization?

One limitation of using Lagrange multipliers is that it may not always give the global minimum or maximum. It is possible for the method to converge to a local minimum or maximum, which may not be the most optimal solution. Additionally, the method can become computationally expensive for problems with a large number of variables and constraints.

## 5. How can Lagrange multipliers be applied in real-life scenarios?

Lagrange multipliers can be applied in various real-life scenarios such as minimizing production costs for a company, maximizing profits for a business, or optimizing resource allocation in a project. It can also be used in fields such as engineering, economics, and physics to find optimal solutions for complex problems.

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