Week 3 – Theory of Production

Part a

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

(a) Q = 24L 2 3K

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 fixed at K = 27.Write down the functional forms for the total product (TPL), marginal product (MPL) and average product (APL) 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 —increasing, decreasing and negative returns. In which stage does the production take place for profit maximization? Why?

(c) Looking at the plots, what can you say about the relationship between TPL and MPL? What can you say about the relationship between APL and MPL? 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 = AKαLβ , A > 0, α > 0, β > 0.]

(e) For a Cobb-Douglas production function with constant returns to scale, what happens to MPL and MPK 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 file, you can find 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 lnQ, lnK and lnL. Then run a regression analysis with lnQ being the Y or dependent variable and lnK, lnL being the X or independent variable. Once you are done with the regression, interpret the β coeffi cients. 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 inputs