Question

This is the question, there is nothing that part of it, this is the only information given for this problem so please do not answer saying it needs more information because this is all I was given like I said. (Solve A and B)

A. Given a general short run production function y=f(x), where x is the single variable input and y is the output. what properties should we reasonably expect y=f(x) to possess so that it is "WELL BEHAVED".?

B. Now consider a quadratic polynomial production function y=2x^a. Assuming the input, x, is essential for production of y, what must be true of the signs and magnitudes of the parameter, a, so that the production function is "WELL BEHAVED"? Provide mathematical proofs and word explanations in your answer.

Answer 1

Answer

A

Mathematically there is a very simple logic behind the WELL BEHAVE characteristic of a production function .The production function is linear i-e y=f(x) where y is dependent of x .

Also x-input

y= output

As we took a random data to mathematically and graphically check that how it is well behaved So in order below are some value of x at which y is computed .And the graph is drawn which clearly depicts that for a production function of y=f(x) , is always in proportion that is y will increase or decrease will the x depending on the sign and magnitude of x .And the magnitude can be from - infinity to + infinity

x	Linear (y=x+1)	Quadratic(y=2x^2)
1	2	2
2	3	8
3	4	18
4	5	32
5	6	50
6	7	72
7	8	98

(B)

In this case , the production function with input and output as x,y respctively of the form y= 2x^2 is a quadratic fucntion whose curve is parabolic in nature and the degree of polynomial equation is 2.Taking the same value of x as in above we get the values of y   

x	Linear (y=x+1)	Quadratic(y=2x^2)
1	2	2
2	3	8
3	4	18
4	5	32
5	6	50
6	7	72
7	8	98

mathematically y = f(x^2)

,therefore both sign that is positive or negative are same for x , because of square of x which make the value positive in either case .

So, the production function is well behaved for dual sign of x and the a mgnitude can be from - infinity to + infinity

have a good day !