Optimization - Lagrange multipliers : minimum cost/maximum production

[mathmari]

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:

 

[I like Serena]

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.

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.

 

[mathmari]

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

 

[I like Serena]

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.