TYPES/FORMS OF PRODUCTION FUNCTION

In the last post we looked at the various ways we can express production function. Some of the ways we discussed in that topic include: written word, Table, Graph, Symbol and mathematics. We also discovered that the use of combinations of these methods are necessary to adequately describe the relationship between output and input. In this post, we shall go a step further to assess the various types of production functions used in agricultural production economics. You are strongly advised to carefully study this so that you can follow and understand the subsequent units of this course. 

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

• identify and list the five major types of production function used in agricultural production economics. 

• write the mathematical expression of these production functions 

• explain the application of at least three of these functions 

• state the major difference between these functions in agricultural production economics. 

In the estimation of production function parameters, many equation forms have been fitted. Five of such equations that are commonly used in agricultural production economics will be examine here. Those examined include: linear, quadratic, Cobb-Douglas, square-root and semi-log functions. 

1. Linear Function 

(a) Algebraic Form 

The algebraic form of this equation can be expressed for the single variables case as: Y = a + βX 

Where Y = is the output and X is the variables input 

a = intercept shown n the Y = axis, and 

β = slope or gradient of the curve 

The special features of linear function include 

i. The curve is a straight line 

ii. The slope of the curve must be constant 

iii. The marginal product (MP) is also a straight horizontal line 

(b) Marginal Product of Linear of Function 

Y = a + bX………………….1 

δY = b (MPP)……………….2 

δX 

(c) Elasticity of Production 

If elasticity of production is given by this equation 

ΔY . X …………………..3 

ΔX Y 

Similarly, b which is the slope of the line gives the way in which Y is changing for a unit change in the input X. 

Therefore, if we make b the subject of equation 1, we then have 

B = Y – a --------------4 

X 

Substituting b in equation 3, we have elasticity = (Y – a) × X = (Y – a )……..5 

X Y Y 

From the above equation of elasticity we can deduce that:

1. if a = 0, then elasticity = 1 

2. If a is greater than 0, then elasticity is less than 1 

3. if a is less than 0, then elasticity is greater than I 

SELF-ASSESSMENT EXERCISE 

Try the above equation with two variable inputs: Y = a + bx, + CX2 

2. Quadratic Function 

(a) Algebraic Form 

For Single variables input we have: 

Y = a + bX – Cx2 

Where Y = output and X = variable input 

a = the constant 

b and c = the coefficients 

Quadratic function allows diminishing total product. The coefficients of X2 must have negative signs which implies diminishing marginal returns 

(b) Marginal Product of Quadratic Function 

Y = a + bx – cx2 ………………..6 

δY = b – 2cx …………………..7 

δX 

Note that marginal product of quadratic function declines by a constant absolute amount, secondly, marginal product curve is linear and thirdly, quadratic function allows negative marginal product 

(c) Elasticity in Quadratic Function 

Ep = Δy . X 

Δx Y 

Using equation 6 above we have 

Ep = bx – 2cX2 

a + bx – cX2 

Note here that: 

i. Elasticity in quadratic function declines with input magnitude 

SELF-ASSESSMENT EXERCISE 

Find the marginal products and elasticities of quadratic function involving two variable inputs: y = a + b1X1 + b2X2 – b3X12 – b4 X22 + b5 X1 X2 

3. COBB – Douglas Power Function 

(a) Algebraic Form 

The single variable input of this function is presented as follows 

Y = aXb 

Where Y = output, X = variable input, 

a = constant and b = elasticity of production 

The Cobb – Douglas function is easy to estimate in its logarithmic form. The above general form can be written in log form as follows 

log Y = log a + b log X 

(b) Marginal Product 

Y = aXb ……………….…9 

δY = baXb – 1 (MPP) …….10 

δX 

Note: 

i) Cobb-Douglas power function allows constant, increasing or decreasing marginal productivity 

ii) Cobb – Douglas allows any of the three above but not all the three 

iii) Marginal product declines if all other inputs are held constant. 

(c) Elasticity of Production 

Ep = ΔY X ………………..11 

ΔX Y 

ΔY in equation 10 = baXb- 1 ….12 

ΔX 

and Y = aXb 

Substituting the two equations (12 and 13) in equation 11 

We have 

Ep = baXb . X = b……..………14 

X aXb 

Note the following features 

1. The linearized log function of Cobb-Douglas power function is easier to fit 

2. The coefficients of the function are the direct elasticities, i.e. the partial elasticity are equal to each of the parameters 

3. When coefficient b = 1, we have a case of constant returns to scale. 

4. When coefficient is greater than 1, we have a case of increasing returns to scale 

5. When coefficient is less than 1, we have a case of decreasing returns to scale 

Olukosi and Ogungbile (1989) identified some major shortcomings of using this function as follows: 

1. Cobb-Douglas function assumes a constant elasticity of production over the entire output –input curve and therefore, the function cannot be used for data which indicates both increasing and decreasing marginal productivity. 

2. Similarly, it cannot be used for data with both positive and negative marginal products 

3. It normally exhibits a non-linear relationship and it does not give a defined maximum response at all input levels 

SELF-ASSESSMENT EXERCISE 

Find the marginal products and elasticities of this Cobb-Douglas power function: Y = aX1b1Xb22 

4. Square Root Function 

(a) Algebraic Form 

A single variable algebraic form of square root function is illustrated below: 

Y = a + bX + cX0.5 

Where Y = dependent variable (output), X = independent variable (input), a = constant, b and c are the coefficients. 

Just as in the case of quadratic function, square root function also allows diminishing total product. 

(b) Marginal Product 

Y = a – bX + cX0.5-----------------------------15 

δY = -b + 0.5cX-0.5 (MPP)---------16 

δX 

The marginal product of this function declines at diminishing rate. 

(c) Elasticity of Production 

Y = a + bX + cX0.5 

Ep = ΔY . X 

ΔX Y 

But ΔY in equation 16 = b + 0.5cX-0.5 

ΔX 

Therefore 

Ep = 0.5cX-0.5- b . X 

Y 

= [0.5cX-0.5 – b]X ----------------------17 

Y 

Elasticity in this function declines at high level of input and output. This situation can happen under certain biological conditions. 

SELF-ASSESSMENT EXERCISE 

Determine the marginal products, elasticities and Rate of Products Substitution for this function: 

Y = a – b1X1 – b2X2 + b3X10.5 + b4X20.5 + b5X10.5X20.5 

5. Semi-Log Function 

(a) Algebraic Function 

Y = a + b log X 

This function is very useful in aggregate production function analysis. 

(b) Marginal Product 

Y = a + b log X ………..18 

δY = b (MPP)………..19 

δX X 

The marginal product declines with increase in variable input and vice versa 

(c) Elasticity of Production 

Y = a + b log X 

Ep = ΔY . X 

ΔX Y 

From equation 19, MPP = b 

X 

EP = b . X = b ……………………….20 

X Y Y 

While marginal product varies with input, elasticity of production varies with output. 

SELF-ASSESSMENT EXERCISE 

Determine the MPP, RTS and EP for the Semi-log function of the two variable inputs below: 

Y = a + b1 logX1 + b2 logX2 

CONCLUSION 

In this topic we have examined five types of functional forms commonly used in estimating parameters in agricultural production economics. The functional forms examined include: linear function, quadratic function, Cobb-Douglas power function, square-root function and Semi-log function. We can conclude here that the choice of these functions will depend on situation at hand. 

SUMMARY 

The main points in this unit include the followings: 

• Linear functions is of the form Y = a + bX for single input and Y = a + bX1 + cX2 + dX3 for three variable inputs 

• The algebraic form of quadratic function is Y = a + bX1 – cX12 + dX2 – eX22 + fX3 – gX23 for three variable inputs 

• Cobb – Douglas power function can be presented as Y = aXb1Xc2X3d for three variable inputs 

• Square root function can also be expressed as:  Y = a + bx1 + cX10.5 + dX2+ eX20.5 fX3 + gX30.5 for three variable inputs 

• Semi-log function can be expressed as follows Y = a + log X1+ b2 log X2 + b3 log X3 for three variables inputs 

• Cobb- Douglas function can be linearized into logarithmic form as follows: Y = aX1bXc2Xd3 linearized as Log Y = log a + b logX1 + c logX2 + d logX3 

ASSIGNMENT 

1. List five types of production functions commonly used in agricultural production economics 

2. State their algebraic forms for two variable inputs 

3. With examples, give full description of any three of them. 

REFFERENCES/FURTHER READING 

Abbot, J.C. and J.P. Makeham (1980). Agricultural Economics and Marketing in the Tropics. London, Longman Publishers. 

Adegeye, A.J. and J.S. Dittoh (1985). Essentials of Agricultural Economics. Ibadan, Impact Publishers. 

Nweze, N.J. (2002). Agricultural Production Economics: An Introductory Text. Nsukka. AP Express Publishers. 

Olayide, S.O. and Heady E.O. (1982). Introduction to Agricultural Production Economics. Ibadan. University Press Ltd. 

Olukosi, J.O. and A.O. Ogungbile (1989). Introduction to Agricultural Production Economics: Principles and Applications. Zaria. AGTAB Publishers Ltd. 

Marshall A.C. (1998). Modern Farm Management Techniques. Owerri. Alphabet Nigeria Publishers. 

Reddy, S.S., P.R. Ram, T.V. Sastry and I.B. Devi (2004). Agricultural Economics. New Delhi. Oxford and Ibh Publishers Ltd.