What is the law of diminishing returns

Question

Assignment:

Theory of Production

Part A

1. What do we mean by a "production function"? Interpret the following production functions.

(a) Q = 24L 2/3 K 1/3

(b) Q = 810KL² K²L³

2. What is the difference between short-run and long-run production?

3. What is the law of diminishing returns? What is the crucial assumption behind this law? Does the law hold in the long-run? Why or why not?

4. Short-run production:

(a) Suppose capital is Öxed at K = 27: Write down the functional forms for the total product (TP_L), marginal product (MP_L) and average product (AP_L) of labor for the production functions given in Question 1 (a and b).

(b) Plot the functions and identify (the ranges of output) for the three stages of production ñincreasing, decreasing and negative returns. In which stage does the production take place for proÖt maximization? Why?

(c) Looking at the plots, what can you say about the relationship between TP_L and MP_L? What can you say about the relationship between AP_L and MP_L? Prove these relationships mathematically.

5. Long-run production:

(a) What do we mean by an isoquant? What is its slope? Why is it negative? What is the reason behind the convexity of an isoquant?

Part B

(b) Find the equation for an isoquant for the production function given in Question 1(a). Prove that it is downward sloping and convex.

(c) What is marginal rate of technical substitution?

(d) What do we mean by returns to scale? Give examples of Cobb-Douglas production functions exhibiting increasing, decreasing and constant returns to scale. [A Cobb-Douglas production function takes the following form: Q = AK^αL^β ; A > 0,α  > 0,β  > 0.]

(e) For a Cobb-Douglas production function with constant returns to scale, what happens to MP_L and MP_K if we increase L and K by the same proportion? In light of this result, can you say anything about the slope of the isoquants for this type of production functions?

6. Data analysis:

Go to the excel sheet titled 03 - Production Data.xlsx in the modules section. In that Öle, you can Önd 3 columns for Q; K and L: First, use natural log transformation (using the function LN in excel to transform the entire data set to create 3 new columns for ln Q; ln K and ln L:

Then run a regression analysis with ln Q being the Y or dependent variable and ln K; ln L being the X or independent variable. Once you are done with the regression, interpret the β coefficients. What can you say about the returns to scale?

7. Give examples for the following production functions. Interpret them. Draw isoquants. Comment on their returns to scale.

(a) Fixed proportion inputs

(b) Perfectly substitutable input

Attachment:- Production data.zip