# In our last topic, we discussed the meaning and usefulness of production function. We defined production function as an expression of the technical or physical relationship which connects the number of units of inputs that are fed into a production process and the corresponding units of output that emerge. We also noted that production function is useful in almost all aspects of economy ranging from the balance of trade, national income to maximization of profit. In this topic, we shall look into the various ways of expressing production function.

At the end of this unit, you should be able to:

• list five different ways of expressing production function

• explain three means of expressing production function.

Production function can be represented using different approaches. The most common approaches include: written form, functional notations, tabular expression, graphical expression and mathematical expression.

## Functional Notation

Production function can be expressed using symbols. The most popular symbols commonly used in agricultural production economics is stated below:

Y= ƒ(X1, X2, X3, ------Xn)

Where:

Y = the quantity of output

Xs = the quantities of inputs used in production

ƒ = stands for the forms of the relationship that transforms inputs (Xs) into output (Y).

Functional notation can also provide information on which of the inputs are varied and which are fixed. This can be expressed by using vertical line to separate the varied inputs from the fixed inputs. This can be done as follows:

Y = ƒ (X1,/X2, X3)

The above functional notation implies that the quantity of output (Y) is a function of variable input X1 given the quantity of other inputs X2 and X3. This means that inputs X1 is the variable input while inputs X2 and X3 are the fixed inputs.

The major shortcoming of functional notation means of expressing production function is that it does not provide information on the quantity of output expected when X1, X2 and X3 are combined as inputs. Information on the relationship between output and input as a whole is very important to the farmer and other organs of government.

SELF-ASSESSMENT EXERCISE

Identify the output and input(s) in the following functional notation: Y=ƒ(X1, X2, X3.........Xn)

## Tabular Presentation

Production function can be expressed in form of table: this can be illustrated by showing the various quantity of yam in kg obtained from various quantities of fertilizer application.

Table1: Yam output from varying levels of fertilizer

 Yam output (Kg) Fertilizer Application (Kg) 10 0 20 2 30 4 40 6 50 8

Table 1 showed the relationship between yam output and the various quantities of fertilizer applied when other farm inputs like labour, farm size etc are fixed. For example 2 kg of fertilizer is required to produce 20 kg of yam while 8 kg of fertilizer is also required to produce 50 kg of yam.

Nweze (2002) summarize the usefulness of tabular presentation of production function thus:

i. So much information is contained in a small space

ii. It may be computationally easier to work directly with the tabulated production data especially when it involves one or more variable inputs.

iii. The quantitative relationships are better appreciated at a glance.

Graphical Expression

The relationship between yam output and fertilizer application tabulated above can be expressed in graphical form. This can be done by plotting the values of fertilizer application against the corresponding values of yam output. Each point on the curve will correspond to specific level of input and a corresponding level of output.

Mathematical Presentation

Mathematical expression is more explicit than ordinary functional notation. There are many ways of expressing production function mathematically. For example, production function can be expressed explicitly in linear form as presented below:

Y = a + βx

Where

Y = quantity of output

X = quantity of input used

a = a constant

β = coefficient of X

If a and β takes on specific values like 10 and 0.5 respectively, the above can then be expressed as follows:

Y = 10 + 0.5X

Unlike functional notation, this mathematical expression illustrated above has the same content as in the graphical expression. It is however more meaningful than the graphical approach because it allow one to obtain the intermediate values of the variables.

## Written Word

The relationships between dependant and independent variables in production function can be describe or enumerated in words without resulting in to mathematical, graphical or tabular expression: this is a very weak way of showing the relationship between two variables because the magnitude of the relationship cannot be precisely stated by ordinary word. Secondly, it will be difficult to comprehend the statement at a glance. However, expression of production function in word form is necessary to complement other forms of expression.

SELF-ASSESSMENT EXERCISE

Explain five ways of expressing production function

## CONCLUSION

We discussed the various ways by which production function can be expressed. The various ways identified in the unit include; written word, tabular form, graphical form, symbolic form and mathematical form. We can conclude here that combinations of these ways of expressing production function are necessary to adequately specify the relationships between two or more variables.

## SUMMARY

The main points discussed in this unit are as follows:

(i) The following methods can be used to express production function; written word, table, graph, mathematics and symbols.

(ii) Words expression is needed to complement other methods

(iii) In tabular form, we can quickly appreciate the relationship at a glance

(iv) We can mathematically expressed the relationship as follows Y = a + βx

(v) We can also use symbol like Y = ƒ (X1, X2, X3) to represent relationship between Y and X in production function

(vi) We can also plot the graph of Y against X to express this relationship.

(vii) We need combinations of these methods to satisfactorily explain the relationship in production functions.

ASSIGNMENT .

List and explain five ways of expressing production function.

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