Question

Suppose that a firm has the p production function f(x1; x2) = sqrt(x1) + x2^2.

(a) The marginal product of factor 1 (increases, decreases, stays constant) ------------ as the amount of factor 1 increases. The marginal product of factor 2 (increases, decreases, stays constant) ----------- as the amount of factor 2 increases.

(b) This production function does not satisfy the definition of increasing returns to scale, constant returns to scale, or decreasing returns to scale. How can this be?

(c)Find a combination of inputs such that doubling the amount of both inputs will more than double the amount of output.
___________________________________________________

The demand function for football tickets for a typical game at a large midwestern university is D(p) = 200, 000 -10, 000p. The university's football stadium holds a maximum of 100,000 spectators. On a clearly labelled graph, draw the inverse demand function and the marginal revenue function.

Answer 1

a. The MP of factor 1 is decreasing. The MP of factor 2 is increasing. (Refer to image for the MP of both factors.)

b.The two factors have decreasing and increasing returns. Hence the production function as a whole cannot exhibit IRS,CRS or DRS. Basically the returns from the two factors is moving in the opposite direction. For this let us consider two different input combinations and see what is its effect on the production function. (Refer to Image) In the image, I have taken two input combinations and checked the effect of doubling the inputs on the returns to scale. For the first combination I have taken (x1>x2) and in the second combination (x2>x1). On doubling in both the cases, in the first output less than doubles and in the other output more than doubles.

Hence it can concluded that the production function as a whole cannot follow IRS,CRS or DRS.

c.The question demands that I double the inputs so that the output more than doubles. Refer to the table drawn for ans b. The input combination of (x1,x2)=(0,1) produces output of 1. When i double the input to (0,2), the output more than doubles to 4. Basically the condition requires us to use a input combination where x2>x1 i.e the factor with increasing returns is used in a larger proportion. To check use another input combination of (1,2). Output is 5. On doubling input to (2,4), the output becomes approx 17.4 (i.e more than triples).