The document discusses different production functions used in economics including the Cobb-Douglas production function, CES production function, and Spillman production function. The Cobb-Douglas production function represents production processes using capital and labor. It has constant returns to scale and diminishing marginal returns. The CES production function displays constant elasticity of substitution between factors. One of the earliest efforts to estimate an agricultural production function was conducted by Spillman using data on fertilizer, feed, and livestock fattening. He proposed a production function of the form Y = M(1-Rx) to represent diminishing returns.

1. Production
Function Analysis
VAIBHAV

2. Cobb-Douglas Production Function
This Lecture describes in detail the most famous of all production
functions used to represent production processes both in and out
of agriculture. First used in 1928 in an empirical study dealing with
the productivity of capital and labor in the United States, the
function has been widely used in agricultural studies because of its
simplicity. However, the function is not an adequate numerical
representation of the neoclassical three stage production function.
One of the key characteristics of a Cobb Douglas type of
production function is that the specific corresponding dual cost
function can be derived by making use of the first order
optimization conditions along the expansion path.

3. Cobb Douglas Function
Q = Aka Lb
The function had three characteristics viewed at that time as desirable
1. It was homogeneous of degree 1 with respect to the input bundle, which
was consistent with the economics of the day that stressed that production
functions for a society should have constant returns to scale.
2. The function exhibited diminishing marginal returns to either capital or
labor, when the other was treated as the fixed input, so the law of variable
proportions held. The parameter A was thought to represent the
technology of the society that generated the observations upon which the
parameters of the function were to be estimated.
3. The function was easily estimated with the tools of the day

5. CES Production Function
The constant E.S assumption is a restriction on the form of production possibilities, and one
can characterize the class of production functions which have this property. This has been
done by Arrow-Chenery – Minhas -Solow for the two-factor production case. The
CES production function is a neoclassical production function that displays
constant elasticity of substitution. In other words, the production technology has a constant
percentage change in factor (e.g. labour and capital) proportions due to a percentage change
in marginal rate of technical substitution. The two factor (capital, labor) CES production
function introduced by Solow, and later made popular by Arrow, Chenery, Minhas,
and Solow is:
Q = A [aKβ + (1-a) L-β]-1/β

6. Q’ = A [a (mK)-β + (1-a) (mL)-β]-1/β
Q’ = A [m-β {aK-β + (1-a) L-β}]-1/β
Q’ = (m-β)-1/β .A [aK-β + (1-a) L-β)-1/β
Because, Q = A [aK-β + (1-a) L-β]-1/β
Therefore, Q’ = mQ
This implies that CES production function is homogeneous with degree one.

7. Spillman Production Function
Despite the widespread use of the Cobb Douglas production function, it was
not the first or the only production function to be used by agricultural
economists for representing production relationships. Agricultural economics
as a formal discipline is relatively new, having had its start as a separate
discipline in the first decade of the twentieth century. The first work in
agricultural economics was conducted by biological scientists who were
interested in providing farmers with useful information with regard to
designing plans for feeding livestock or fertilizing crops. Even these early
efforts, conducted by biological scientists with little or no training in
economics, had a central focus in obtaining estimates of parameters of
agricultural production functions as a basis for the development of
recommendations to farmers.

8. The Spillman
One of the earliest efforts to estimate a production function in
agriculture was conducted by Spillman, and was published in the
newly created Journal of Farm Economics (later to become the
American Journal of Agricultural Economics) in two articles in
1923 and 1924. The first article was titled "Application of the
Law of Diminishing Returns to Some Fertilizer and Feed Data.'
The second was "Law of the Diminishing Increment in the
Fattening of Steers and Hogs." It is not surprising that Spillman
was interested in determining whether or not the law of
diminishing returns had empirical support within some rather
basic agricultural production processes.

9. Y = M(1-Rx )
Where,
Y = amount of growth produced by a given quantity of growth factor X
M = maximum yield possible when all growth factors are present in optimum amounts
X = quantity of growth factors
R = a constant

10. Factor Relationship