   Chapter 7.3, Problem 46E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Cobb-Douglas Production Function Show that the Cobb-Douglas production function z = c x a y 1 − a can be rewritten asIn ln z y = ln   C   +   a   ln   x y .

To determine

To prove: Cobb-Douglas production function z=Cxay1a can be written as lnzy=lnC+alnxy

Explanation

Given Information:

A function is given as z=Cxay1a

Formula used:

Basic properties of ln are,

ln(xy)=lnx+lnylnxα=αlnxln(x)ln(y)=ln(xy)

Proof:

The given function is,

z=Cxay1a

Now, taking log on both sides,

logz=log(Cxay1a)

Converting the log at natural base,

lnz=ln(Cxay1a)

As, ln(xy)=lnx+lny

Thus, function becomes:

lnz=lnC+lnxa+lny1a

Now, apply lnxα=αlnx

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