# Finding the production function in the given case

The Production Function for Wilson Company

Economists at the Wilson Company are interested in developing a production function for fertilizer plants. They collected data on 15 different plants that produce fertilizer (see the following page).

Questions

1. Estimate the Cobb-Douglas production function Q= αL^β 1K_2^β, where Q = output; L=labor input; K =capital input; and α,β_1,and β_2 are the parameters to be estimated.

2. Test whether the coefficients of capital and labor are statistically significant.

3. Determine the percentage of the variation of output that is "explained" by the regression equation.

4. Determine the labor and capital estimated parameters and give an economic interpretation of each value.

5. Determine whether this production function exhibits increasing, decreasing or constant returns to scale. (Ignore the issue of statistical significance).

Plant Output (000 Tons) Capital ($000) Labor (000 Worker Hours)

1 605.3 18,891 700.2

2 566.1 19,201 651.8

3 647.1 20,655 822.9

4 523.7 15,082 650.3

5 712.3 20,300 859.0

6 487.5 16,079 613.0

7 761.6 24,194 851.3

8 442.5 11,504 655.4

9 821.1 25,970 900.6

10 397.8 10,127 550.4

11 896.7 25,622 842.2

12 359.3 12,477 540.5

13 979.1 24,002 949.4

14 331.7 8,042 575.7

15 1064.9 23,972 925.8

https://brainmass.com/economics/cost-benefit-analysis/finding-the-production-function-in-the-given-case-566042

#### Solution Summary

Solution depicts the steps to estimate the production function in the given case. It also provides the economic interpretation of the parameters so obtained.

Definitional formula given to find the derivative functions

Problem 1

Given y = f(x) = x2 + 2x +3

a) Use the definitional formula given below to find the derivative of the function.

b) Find the value of the derivative at x = 3.

Problem 2

Given, y = f(x) = 2 x3 - 3x2 + 4x +5

a) Use the Power function to find derivative of the function.

b) Find the value of the derivative at x = 4.

Problem 3

The revenue and cost functions for producing and selling quantity x for a certain production facility are given below.

R(x) = 16x - x2

C(x) = 20 + 4x

a) Determine the profit function P(x).

b) Use Excel to graph the functions R(x), C(x) and P(x) for the interval 0? x ? 12. Copy and paste the graph below. Note: Use Scatter plot with smooth lines and markers.

c) Compute the break-even quantities.

d) Determine the average cost at the break-even quantities.

e) Determine the marginal revenue R'(x).

f) Determine the marginal cost C'(x)

g) At what quantity is the profit maximized?