Question

Consider the production function f(x,y) = 60x^3/4 y^1/4 ,which gives the number of units of goods produced when utilizing x units of labor and y units of capital.

If the amount of capital is held fixed at y = 16 and the amount of labor increases by 1 unit, estimate the increase in the quantity of goods produced.

Answer 1

I found the partial derivatives from the function in terms of x and y. I'm confused with how I can sub in y and x into the partial derivatives to find the number of quantities produced.

Answer 2

Since capital is fixed, plug y=16 into your equation to simplify it to \[f(x)=120x^{3/4}\] I'm assuming the estimation that you want to use it by linearization. which means that you will be using the estimate \[f'(x) \approx\frac{f(x+1)-f(x)}{(x+1)-x} = f(x+1)-f(x)\] With the left hand side representing the increase in quantity of goods produced. Thus, since \[f'(x)=90x^{-1/4}\] You have your answer.

Partial derivatives are not needed, since capital is fixed and you don't have a value for x anyway, since you are finding a general estimate.

Answer 3

Sorry, "right hand side" instead of "left hand side" above.

Answer 4

Is there a way to solve with partial derivatives? Actually the first part of the question tells you to compute the partial derivatives, which I left out, so I'm guessing I have to solve it with partials...

Answer 5

but then I used partials and got the same answer that you posted.. hahah

Answer 6

Worked out well then :)

The reason it works it just because you are holding capital constant. Thus it isn't changing at all, so there is no rate of change (aka partial derivative) to worry about. Thus, the question reduces to a single variable calculus question, where the partial derivative with respect to x just becomes the normal single variable derivative with respect to x.

Answer 7

Ah, alright. Thank you so very much!

Answer 8

No problem, glad to help :)