Cobb-Douglas Function: Inverse function and marginal cost

Ok, so I wonder wether or not I have done this correctly:

It would be of great help to me.

I have the Cobb-Douglas function $\displaystyle x=An^e$

Were x is produced quantum and n is the input labour.

First question: Is the inverse function $\displaystyle n=(x/A)^{1/e}$ ? and does that equal: $\displaystyle n=(1/A^{1/e})(x^{1/e})$ ?

So the cost function then, were w is the price of labour: wn = $\displaystyle (w/A^{1/e})(x^{1/e})$

Second Question: Is the derivative of the inverse function (if Ive done correctly on the first question):

$\displaystyle (w/eA^{1/e})(x^{(1/e)-1})$

Re: Cobb-Douglas Function: Inverse function and marginal cost

Hey EconNor.

As a general rule, we find the inverse function (if it exists) by swapping the independent and dependent variable and re-solving for an expression of the new independent variable in terms of the dependent variable. In other words,

y = f(x) [Original function] Swapping gives us

x' = f(y').

Now re-arrange to get

y' = g(x') and g is our inverse function.

In your example y = x and x = n. Doing the above gives us:

n = Ax^e

(n/A)^(1/e) = x so our inverse function is

x^(-1)(n) = (n/A)^(1/e) just as you predicted.

For the second we use the normal power rule which gives:

d/dn[(n/A)^(1/e)]

= (1/A)^(1/e) * (1/e) * n^(1/e - 1)

Where did you get your w from?

Re: Cobb-Douglas Function: Inverse function and marginal cost

Ok, thank you.

Oh, the w is just for the cost function.... w = wage n=Hours of labour ... so the cost function is wn.