Question

Suppose you are given the following production function

Q= 21x + 9x²-x³

Compose a table calculating the Total Product (Q), Marginal Product (MP) and Average Product (AP), for x ranging from 0 to 9 units.

Using the information from the table, graph these individual curves and identify the rate of which diminishing marginal returns are evident. Delineate the stages of production and explain why Stages I and III are considered as irrational whereas Stage II is termed as rational.   What are the practical problems in estimating a production function? What will be the data requirements and statistical knowledge to estimate real-world production functions? Explain.

2. Old McDonald Had a Farm!

Old McDonald had a farm. Assuming that Old McDonald is buying and selling in both the input and output markets in perfect competition, answer the following questions:

1. Complete Table 1
2. At what point will Freddie will be at the most efficient point of production? Why?
3. What is the optimum point of production? How do you prove it? What are the net revenues at which profits are maximized?
4. If costs of production goes up, what choices does Old McDonald has? At what point will Old McDonald will have to shut down (Poor guy)?   Why (hint: use the shutdown rule for which you will have to derive the average fixed and average variable cost from the information provided).
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Answer 1

Answer: SInce production function is given as

Q= 21X + 9 X2 - X3

On the basis of this production the total product (TP), Average Product(AP), and Marginal Product (MP) obtained by employing each unit of input X is shown in the table below

Table: Showing the Total Product, Marginal Product, and Average Product
Unit of X	Total Product	Average Product	Marginal Product
0	0	0	0
1	29	29	29
2	70	35	41
3	117	39	47
4	164	41	47
5	205	41	41
6	234	39	29
7	245	35	11
8	232	29	-13
9	189	21	-43

On the basis of the above table, following figure can be drawn

Stag Stage-tit e-11 Stage-1 1 2 3 4 5 6 7 8 9 10 Units of X used as input

In the figure, the change in the production of output divided into three stages on the basis of marginal product. The first stage that the marginal product (MP) is increasing, hence this stage can be called as the increasing stage. The second stage has shown that MP is constant when units of factor employed between 3 and 4. This second can be called a constant stage. The third stage is marked after the employment of 4th unit of factor X. After the fourth unit of X, the increase in employment of factor X leads to a decrease in MP. Hence this third stage can be called a decreasing stage. Although the MP decreasing after the fourth unit of X, the production process will continue till the MP become Zero (0).

In the figure the stage-I and stage-III are considered as irrational, because in the first stage the marginal product of the factor is increasing and in the third stage the marginal product is decreasing. But, in the stage-II, marginal becomes constant. Hence this stage is considered as rational.

In the above case if Old McDonald had a farm and Old McDonald is buying and selling in both the input and output markets in perfect competition then the optimum point of production will be at the point where MP is equal to AP. According to the table, when the unit input X = 5, then the optimum will be produced. The optimum or equilibrium output is Q=41. This is called the efficient level of output. At that point the AP becomes maximum.

The shut-down point of production of the Old McDonald farm is the point where marginal product is zero. Becasue if the farm wants to produce by employing more units of input after the MP=0, then the farm will incur loss. Hence the point can be called as shut-down point.