Cost minimisation using Lagrangian (substitution): Min: wL + rK s.t: x=AL^bK^b

[JohnB]

Hi all,

This is the first time i have posted on the forum so my hope is i'm not posting this is the incorrect place.

The question comes from the field of advanced microeconomics but the area im struggling with is mathematical.

I am attempting to solve a Lagrangian cost minimization problem given the following:
Min: wL + rK
s.t: x=AL^bK^b

where x is the output, L is the labour input, K is the capital input, and A, b are positive constants.

This is where i have gotten thus far:


The Lagrangian based on the above seems to be as follows:

I solve for the 1st order derivatives:

1.

2.

3.

A bit of re-arranging using equations 1 and 2 seems to give the following (please correct if wrong):

=K = w/r*L


This then needs to get substituted into equation 3 above which is:

3.

Creating:

x-AL^b(w/r*L)^b=0
​

This is where i seem to keep tripping up. I worked through the problem previously but after expanding the brackets and then trying to isolate L i think i made several algebraic mistakes. Could anyone provide some step by steps on where to go from here?


The things i need to do from this point are the following:
a). Solve for L
b) Repeat to solve for K
c) Use L and K to complete cost minimization problem:

Again apologies if this isnt the correct catagory, im having some issues with the algebra but i imagine that may not make it an algebra question.

Thanks in advance for any help,
John

 

[HallsofIvy]

I had first interpreted your last equation as \(\displaystyle x- AL^b\left(\frac{w}{rL}\right)^b= 0\) in which the L cancels! But, based on your previous work, you mean \(\displaystyle x- AL^b\left[\left(\frac{w}{r}L\right)^b\right]\)\(\displaystyle = x- A\left(\frac{w}{r}\right)^bL^{2b}= 0\).

Then \(\displaystyle L^{2b}= \frac{x}{A}\left(\frac{r}{w}\right)^b\) and \(\displaystyle L= \left(\frac{x}{A}\right)^{1/2b}\left(\frac{r}{w}\right)^{1/2}\).

 

[JeffM]

Hi all,

This is the first time i have posted on the forum so my hope is i'm not posting this is the incorrect place. [

Probably better in calculus or business math.

You have a Cobb-Douglas production function with two inputs priced in fully competitive markets. With you so far?

So cost function is indeed

\(\displaystyle c = w L + rK.\)

The production function apparently is

\(\displaystyle q = aL^bK^b.\)

Is that correct? In a typical Cobb-Douglas function the exponents would add to 1 and be b and (1 - b) such that
0 < b < 1.

And you want to minimize cost if q = x, a constant? Is that correct?