Production Function Analysis with Related Concepts .1 Definition and Classification of Production Function

In microeconomic theory, the production function explains then technical or physical relationship between output and inputs. Specifically, it shows the maximum output obtainable from a given set of inputs. Production is the process of combining and coordinating materials and forces (input, factors, resources, or productive service) in the creation of some good and services. The term input and output only have meaning in connection with a particular production process or it can be a final consumer good. The output of a farm depends upon the quantities of input used in production.

Generally, a function showing the maximum output possible with any given set of inputs, assuming these are used efficiently are called the production function. Let a production function is as follow:

Y=f (X, Z)

Here, Y is dependent variable and X & Z are the explanatory or independent variables.

So, the production function shows the relation between the dependent variable and the respective explanatory variables, more specifically it shows the relation between the production as dependent variable and its necessary inputs as the explanatory variables.

In shorts, the production function is mathematical relationship describing the way in which the quantity of particular product depends upon the quantities of particular input used.

The production function could be expressed in different functional forms such as cobb- Douglas, linear, quadratic, polynomials and square root polynomials, semi log and exponential function. When time perspective is introduced into the production function, we have the short- run and long-run production function. However, only two types of production function namely Cobb-Douglas production function and CES production function will be discussed here.

3.3.2 Cobb-Douglas production function

In order to justify the effectiveness of various production laws, mathematician C.W. Cobb and economist D.H. Douglas investigate production scenario of various industries in U.S.A, Canada and Australia from the period of 1899 to 1922. After the investigation they found that labor and capital was the most important factor process. Then giving priority on labor and capital, the production function what they showed was known as Cobb-Douglas production function. Under the condition at returns to scale. This function can be expressed in general form as.

Q = A𝐿^𝛼𝐾^ᵝ Q = Output L = Labor K = Capital

A = Efficiency parameter and

α, 𝛽 =

The factor shares or factor elasticity.

If L and K are infinity, then Q also will be infinity. If we write it as a logarithmic transformation we can write,

LnQ = InA + αInL + 𝛽InK

However, Cobb-Douglas production function holds several properties. Some of them are:

1. Cobb-Douglas production function is homogenous of degree (α+𝛽);

2. its iso-quants are negatively slope and strictly convex;

3. the exponents of each input variable indicate partial elasticity of output with respect to the input;

4. the marginal productivities of factors of production are positive but declining;

5.the expansion path of Cobb-Douglas production function is straight line;

6.Cobb-Douglas production function satisfies the Euler’s theorem;

7.the elasticity of substitution of Cobb-Douglas production function is one; and

8. the sum of α, 𝛽 indicates the returns to scale in the long run.

If α+𝛽>1, the production function indicates the increasing returns to scale, if α+𝛽 <1, the production function implies the decreasing returns to scale and if α+𝛽=1, the production function depicts the constant to scale.

3.3.3 constant elasticity of substitution (CES) Production Function

The production function characterized by a constant elasticity of substitution is known as CES Production function. Here, elasticity of substitution means to refer to the measurement of the extent of input substitution and that is supposed to be constant. The equation of this function can be expressed as follows:

Q = A [ẟ𝐾^−𝑝+ (1 − ẟ)𝐿^−𝑃]^−1/𝑃

Here Q = output; K=Capital input; L = Labor input; A= Efficiency parameter that indicate state of technology; ẟ = Distribution parameter that deals with the relative factor; p = substitution parameter that determines the value of the constant elasticity of substitution.

In case of CES production function, substitution happens between capital and labor. This production function is the desire output of the ACMS model development be KJ-Arrows, H.B.

Chenery, B.S. Minhas and R.M. Solow. This model was published in the journal “Review of economics and statistics” volume 43 in 1961 August and the associated paper was “capital- labor substitution and economic efficiency”. The CES Production function has got a different and isolated due to its inherent properties.

Some major properties of CES production function are:

1. the CES production function is homogenous of degree one;

2. marginal productivities function of inputs in the CES production function are positive but declining although;

3. the iso-quants generated by the CES production function are negatively sloped and strictly convex to the origin;

4. elasticity of substitution of CES production function is 1/1+p; and

5. Cobb-Douglas production function is a special case of CES production function.

However, the Cobb-Douglas functional form is commonly used for its simplicity and flexibility coupled with the empirical support. For this reason, the researcher confined himself with Cobb- Douglas production function.

Agriculture Productivity

Agriculture productivity is measure as the ratio of agriculture output to agriculture inputs., Conventionally, agricultural productivity is measured by an index of output divided by inputs.

Two measure of productivity are frequently used: The partial factor productivity (PFP) and Total factor productivity (TFP).

Partial Factor Productivity (PFP)

PFP is simply the ratio of output and any one of the inputs, typically labor or land. In notation form this can be expressed as:

PFP = ^𝑌

𝑋𝑖

Where Y is output and X is input i. Although it is commonly used, the partial productivity measure has one important weakness in that does not control for the level of other inputs employed.

Total Factor Productivity (TFP)

Total factor productivity measures the efficiency of all inputs to a productive process. Increase in TFP result usually form technological innovations or improvements. The factors, which fixed the level of production, are known as the determinants of agriculture productivity.

The basic concepts in production measurement are average production (AP), marginal production (MP), Elasticity of production (EP) and returns to scale (RTS). The knowledge of these concepts can be used to study the three stage of the production surface.

Total and Average Production (AP)

Total production (TP) is the amount obtained from various units of production. The average production of any input id defines as the total production divided by number or units od input used. For example, average production of labor,

A𝑃_𝐿= ^𝑇𝑃

𝐿

Where,

TP = Total product and L = Labor input.

The Marginal Production (MP)

The marginal production of any input is defined as the change in total production due to change in one unit of input. For example,

M𝑃_𝐿=^∆𝑇𝑃

∆𝐿

Where,

∆𝑇𝑃 = Change in total product and ∆𝐿 = Change in labor input.

Elasticity of Production (𝑬_𝑷)

The elasticity of production refers to the percentage in output in relation to the percentage change in input. The concept of elasticity can be applied to the production function to determine the stage in which farmers are allocating their resources.