# Cost Minimization function with exponent

• Oct 18th 2009, 07:19 AM
ibrox
Cost Minimization function with exponent
OK, so the equation is:

X = K^.25 L^.75, w = 2, r = 4

I know that:

Marginal Product of Labour/Marginal product of Kapital = w/r

So I take the derivative with regards to "l" on top and do the same with respect to "k" on bottom and get...

(3/4K^.25 L^-0.25)/(1/4K^-0.75 L^0.25) = 2/4

The 4's on bottom negate each other...and we know that the negative exponent flips to the other side of the dividing line, which gives us:

3K/L = 1/2

Cross multiply, survey says...

6K = L

Now, I know I have to plug this back into the original equation (ie. for every "L" there is 6"k")...when I do that, I get

X = K^0.25 6K^0.75

My question is where do I go from here, and did I do the math right?

Ibrox
• Oct 18th 2009, 07:30 AM
Prof Paulo
Yes, you did!

X = K^.25 6K^.75 (The same base so you adds the exponents)

X = 6K or X = L.
• Oct 18th 2009, 05:54 PM
ibrox
I thought the 6K^0.75 kicks in first before I add the exponents up
• Oct 19th 2009, 04:46 AM
stapel
Quote:

Originally Posted by ibrox
OK, so the equation is:

X = K^.25 L^.75, w = 2, r = 4

I know that:

Marginal Product of Labour/Marginal product of Kapital = w/r

So I take the derivative with regards to "l" on top and do the same with respect to "k" on bottom...

From what you've posted, K (or k?) is "on top", so I'm not sure what you mean by "to 'k' on bottom"...? Also, you posted this to "pre-algebra", but then refer to taking derivatives (from post-algebra calculus)...?

How do X, K, L, w, r, l, and k relate? With respect to what are you differentiating? What are you differentiating? What are the necessary formulas and relations? What are you trying to accomplish?

Please be complete. Thank you! (Wink)