cobb doublas function

course

Problems 41 and 42 in the 7th edition address the Cobb-Douglas function, and there is an example in the text that illustrates such a function.

In general a Cobb-Douglas function is a function of the form A x^p y^q, where p and q are exponents estimated to fit the particular situation.

I wouldn't expect you to know this and if a problem required that you do, I would omit it. However if the function is given, I would expect you to be able to find its optimal value.

You should know how to analyze a given function of this form.

The function for the situation you describe is f(x,y) = 200 x^.6 y^.4. The relevant derivatives are f_x = 120 x^-.4 y^.4, f_xx = -48 x^-1.4 y^.4, f_y = 80 x^.6 y^-.6, f_yy = -48 x^.6 y^-1.6 and f_xy = f_yx = 48 x^-.4 y^-.6.

Because of the negative exponents in the first-derivative expressions, f_x and f_y cannot both equal zero at any point, and the function therefore has no critical points. So it has no relative max or min.

The level curves are of the form x^.6 y^.4 = c, or y = c x^-(3/2). In the first quadrant, which is the domain of the function, these level curves are asymptotic to the y and x axes, shaped much like hyperbolas, with c values increasing as you move away from the origin.

Dave,

I'm working with the Cobb-Douglas production function. The problems on the test asks you to use this formula to calculate the values of X^.6 (labor units valued at $125 ea) and Y^.4 (capital units valued at $100 ea) times a C

(constant) of $200 that yield the max production level.

I can't find any problems in the book or on the CD's that show how to approach solving optimization problems with this equation. All the examples in the book are much simpler and only ask you to plug in values and work the formula diectly.

Can you help? Thanks.

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