# Cobb-Douglas Function: Inverse function and marginal cost

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• Mar 15th 2013, 02:36 AM
EconNor
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})\$
• Mar 15th 2013, 06:55 PM
chiro
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?
• Mar 16th 2013, 12:38 AM
EconNor
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.