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### Production Function

Unit – 3: Production Function
Law of Variable Proportion
Meaning: The law of variable proportion is one of the fundamental laws of economics. It is also known as the 'Law of Diminishing Marginal Returns' or the 'Law of Diminishing Marginal Productivity.' This Law of variable proportion shows the input-output relationship or production function with one variable factor, i.e., a factor, which can be changed, while other factors of production are kept constant.
In short-period when the output of a good is sought to be increased by way of additional application of the variable factor, law of variable proportions comes into operation. When the number of one factor is increased while all other factors remain constant, then the proportion between the factors is altered. On account of change in the proportion of factors there will also be a change in total output at different rates. In economics, this tendency is called Law of Variable Proportions. The law stats that as the proportion of factors is changed, the total production at first increases more than proportionately, then equi-proportionately and finally less than proportionately.
According to Samuelson, “The law states than an increase in some inputs relative to other fixed input will, in a given state of technology, cause total output to increase, but after a point the extra output resulting from the same addition of extra inputs is likely to become less and less.”
Assumptions: The law of variable proportions functions is based on following assumptions:
a)      Constant technology: The technology is assumed to be constant because technological changes will result into rise of marginal and average product.
b)      Snort-run: The law operates in the short-run because it is here that some factors are fixed and others are variable. In the long-run, all factors are variable.

c)       Homogeneous input: The variable input employed is homogeneous or identical in amount and quality.
d)      Use of varying amount of variable factor: It is possible to use various amounts of a variable factor on the fixed factors of production.
Explanation of the Law: Law of variable proportion can be explained with the help of following table and diagram:
 Units of Land Units of Labour Total Product Marginal Product Average Product 1 1 2 2 2 1 2 5 3 2.5 1 3 9 4 3 1 4 12 3 3 End of the first State Beginning of the Second Stage 1 5 14 2 2.8 1 6 15 1 2.5 1 7 15 0 2.1 End of the Second Stage Beginning of the Third Stage 1 8 14 -1 1.7

Explanation: From the above Table and Diagrams drawn on the assumption that production obeys the law of variable proportions, one can easily discern three stages of production. These are elucidated in the following table:
Three States of Production
 States Total Product Marginal Product Average Product 1. Stage Initially it increases at an increasing rate. Later at diminishing rate. Initially increases and reaches the maximum point. The starts decreasing. Increases and reaches its maximum point 2. Stage Increases at diminishing rate and reaches its maximum point. Decreases and becomes zero. After reaching its maximum begins to decrease. 3. Stage Begins to fall Becomes Negative. Continues to diminish.

Causes of Applicability: Main causes accounting for the application of the law of variable proportions are as follows:
1)      Under utilization of Fixed Factor: In the initial stage of production, fixed factor of production like land or machine is under-utilized. More units of variable factor, like labour are needed for its proper utilization. Thus, as a result of employment of additional units of variable factor there is proper utilization of fixe factor. Consequently, total production begins to increase.
2)      Fixed Factors of production: The principal cause of the operation of this law is that some of the factors of production are fixed during the short period. When the fixed factor is used with variable factor, then its ratio if compared to variable factor falls. Production is the result of the cooperation of all factors. Consequently, marginal return of the variable factor begins to diminish.
3)      Optimum Production: After making the optimum use of a fixed factor if it is combined with increasing units of variable factor, then the marginal return of such variable factor begins to diminish.
4)      Imperfect Substitute: It is the imperfect substitution of factors that is mainly responsible for the operation of the law of diminishing returns. One factor cannot be used in place of the other factor. Consequently, when fixed and variable factors are not combined in an appropriate ratio, the marginal return of the variable factors begins to diminish.
Postponement of the Law: Postponement of the law of variable proportions is possible under the following conditions:
1)      Improvement in Technique of Production: Operation of the law can be postponed if along with the increase in variable factors technique of production is improved.
2)      Perfect Substitute: The law can also be postponed if factors of production are made perfect substitutes, i.e. when one factor can be substituted for the other.
Isoquants and its Properties
The word an isoquant is a locus of points, representing different combinations labour and capital .An isoquant Curve. ‘ISO’ is of Greek origin and means equal or same and ‘quant’ means quantity. An isoquant may be defined as a curve showing all the various combinations of two factors that can produce a given level of output. The isoquant shows- the whole range of alternative ways of producing- the same level of output. The modern economists are using isoquant, or ‘ISO’ product curves for determining the optimum factor combination to produce certain units of a commodity at the least cost.
Properties or Features of Isoquant
The following are the important properties of isoquants:
1. Isoquant is downward sloping to the right. This means that if more of one factor is used less of the other is needed for producing the same output.
2. A higher isoquant represents larger output.
3. No isoquants intersect or touch each other. If so it will mean that there will be a common point on the two curves. This further means that same amount of labour and capital can produce the two levels of output which is meaningless.
4. Isoquants need not be parallel to each other. It so happens because the rate of substitution in different isoquant schedules need not necessarily be equal. Usually they are found different and therefore, isoquants may not be parallel.
5. Isoquant is convex to the origin. This implies that the slope of the isoquant diminishes from left to right along the curve. This is because of the operation of the principle of diminishing marginal rate of technical substitution.
6. No isoquant can touch either axis. If an isoquant touches X axis then it would mean that without using any labour the firm can produce output with the help of capital alone. If an isoquant touches Y axis, it would mean that without using any capital the firm can produce output with the help of labour alone. This is impossible.
7.Isoquants have negative slope. This is so because when the quantity of one factor (labour) is increased the quantity of other factor (capital) must be reduced, so that total output remains the same.
Ridge Lines - The Economic Region of Production:
An isoquant represents combinations of two inputs that yield the same level of output. However, not all points of an isoquant are relevant for production. Such points may be called infeasible points. One should consider only feasible portions of an isoquant. This is because of the fact that no rational producer will produce where marginal product of an input is either zero or negative.
If the isoquant is backward bending and upward sloping, marginal product of any input will be negative, and, hence, this portion of the isoquant may be considered as economically non-sensible region of production. Only the negatively sloped segment of the isoquant is relevant for production or economically feasible.
This is shown in Fig. 3.5 where we have drawn three isoquants showing different levels of output for different labour-capital combinations. This diagram separates economic region of production from uneconomic region of production. Region in which marginal products of all inputs are positive constitutes economic region of production.
Or the region in which input substitution takes place may be called economic region of production. In an uneconomic region, as marginal product of an input becomes either zero or negative, the question of input substitution does not arise. Production in such region is, for obvious reasons, unprofitable or infeasible.
At point A on IQ1, the firm employs certain units of labour and capital. Since the tangent to IQ1 at point A is parallel to the vertical axis, marginal product of capital (MPK) is zero. If more capital is used, marginal product of capital should be negative. In other words, beyond point A, MPK is negative. At point B on IQ1, MPL is zero and beyond point B on IQ1, MPL is negative.
Thus, points between A and B represent positive marginal productivities of both labour and capital. Here substitution between two inputs takes place. Similarly, points A1and A2 on IQ2 and IQ3describe zero MPL while points beyond A1 and A2 describe negative MPK. Points B1and B2 on IQ2and IQ3 represent zero MPK and beyond B1and B2 describe negative MPL.
A rational producer will produce in that region where marginal productivities of inputs are positive. By joining points A, A1 and A2 (i.e., points of zero marginal products) we get OR line and by joining points B, B1 and B2 (points of zero marginal products) we get OL line. These lines are called ridge lines. They give the boundaries of the economic region of production where input substitution takes place.
Any point on the Isoquants outside the upper ridge line OR and the lower ridge line OL constitute uneconomic region of production. Production must take place inside the ridge lines. Note that the ridge lines separate the relevant (i.e., negatively sloped) from the irrelevant portions (i.e., positively or zero sloped) of the Isoquants.
Producer’s Equilibrium or Optimum Combi­nation of Factors or Least Cost Combination
The producer’s equilibrium refers to the situation in which a producer maximizes his profits. In other words the producer is producing given amount of output with least cost combination of factors. The least cost combination of factor’s also called optimum combination of the factor or input. Optimum combination is that combination at which either:
a) The output derived from a given level of inputs is maximum OR
b) The cost of producing a given output is minimum.
For producer’s equilibrium or optimum combination, it must fulfill following two conditions as:
(i) At the point of equilibrium the iso-cost line must be tangent to isoquant curve.
(ii) At point of tangency i.e., iso-quant curve must be convex to the origin or MRTSLk must be falling.
The iso-cost line gives information regarding factor prices and financial resources of the firm.
With a given outlay and prices of two factors, the firm obtains least cost combination of factors, when the iso-cost line becomes tangent to an iso-product curve. Let us explain it with the following Fig. 15.
In Figure 15, P1L1 iso-cost line has become tangent to iso-product curve (representing 500 units of output) at point E. At this point, the slope of the iso-cost line is equal to the iso-product curve. The slope of the iso- product curve represents MRTS of labour for capital. The slope of the iso-cost line represents the price ratio of the two factors.
Slope of Iso-quant curve = Slope of Iso-cost curve
MRTSLk = – ∆L/∆L = MPL/MPK = PL/PK
[where ∆K → change in capital, ∆L → change in labour, MPL → Marginal Physical Product of Labour, MPk – Marginal Physical Product of capital, PL Price of Labour, and PK → Price of capital, MRTSLK =Marginal Rate of Technical Substitution of labour and capital.]
The firm employs OM units of labour and ON units of capital. The producing firm is in equilibrium. It obtains least cost combination of the two factors to produce 5 00 units of the commodity.The points such as H, K, R and S lie on higher iso-cost lines. They require a larger outlay, which is beyond the financial resources of the firm.
Expansion Path
As financial resources of a firm increase, it would like to increase its output. The output can only be increased if there is no increase in the cost of the factors. In other words, the level of total output of a firm increases with increase in its financial resources.
By using different combinations of factors a firm can produce different levels of output. Which of the optimum combinations of factors will be used by the firm is known as Expansion Path. It is also called Scale-line.
In the words of Stonier and Hague, “Expansion path is that line which reflects least cost method of producing different levels of output.”
Expansion path can be explained with the help of Fig. 16. On OX-axis units of labour and on OY-axis units of capital are given.
The initial iso-cost line of the firm is AB. It is tangent to IQ at point E which is the initial equilibrium of the firm. Supposing the cost per unit of labour and capital remains unchanged and the financial resources of the firm increase.
As a result, firm’s new iso-cost-line shifts to the right as CD. New iso-cost line CD will be parallel to the initial iso-cost line. CD touches IQ1 at point E1 which will constitute the new equilibrium point. If the financial resources of the firm further increase, but cost of factors remaining the same, the new iso-cost line will be GH.
It will be tangent to Isoquants curve IQ2 at point E2 which will be the new equilibrium point of the firm. By joining together equilibrium points E, E1 and E2, one gets a line called scale-line or Expansion Path. It is because a firm expands its output or scale of production in conformity with this line
It is a Long run concept. All factors of production are variable in the long period. No factor is a fixed factor. Accordingly, scale of production can be changed by changing the quantity of all factors.
According to Koutsoyiannis “The term returns to scale refers to the changes in output as all factors change by the same proportion.”
b)      Constant Returns to Scale.
c)       Diminishing Returns to Scale.
Increasing Returns to Scale: When inputs are increased in a given proportion and output increases in a greater proportion, the returns to scale are said to be increasing. In other words, proportionate increase in all factors of production results in a more than proportionate increase in output It is a case of increasing returns to scale. Thus, if by 100 percent increase in factors of production, output increases by 120 percent or more, it will be an instance of increasing returns to scale.
If the industry is enjoying increasing returns, then its marginal product increases. As the output expands, marginal costs come down. The price of the product also comes down.
Constant Return to Scale: When inputs are increased in a given proportion and output increases in the same proportion, constant return to scale is said to prevail. For example, if inputs are increased by 25% and output also increases by 25%, the return to scale are said to be constant ( = 1). This may be called homogeneous production function of the first degree. In case of constant returns to scale the average output remains constant. Constant returns to scale operate when the economies of the large scale production balance with the diseconomies.
Decreasing Returns to Sale: Decreasing returns to scale is otherwise known as the law of diminishing returns. This is an important law of production. If the firm continues to expand beyond the stage of constant returns, the stage of diminishing returns to scale will start operate. A proportionate increase in all inputs results in less than proportionate increase in output, the returns to scale is said to be decreasing. For example, if inputs are increased by 20%, but output increases by only 10%, ( = < 1), it is a case of decreasing return to scale. Decreasing return to scale implies increasing costs to scale.
Internal and External Economies
Now-a-days, goods are produced on a very large scale in modern factories. When the production is carried on a large scale the producer derives a number of advantages or economies. These advantages of large scale production are called economies of scale. This is the reason why entrepreneurs try to expand the size of their factories. Marshall divides the economies of scale into groups – (i) internal economies and (ii) external economies.
Economies of Scale

Internal Economies                                                         External Economies

Real Economies                                                Pecuniary Economies
1. Economies of Concentration
2. Economies of Information
3. Economies of Disintegration
1. Labour Economies
2. Technical Economies
3. Inventory Economies
4. Selling or Marketing Economies
5. Managerial Economies
6. Transport and Storage Economies
A.      Internal Economies: Internal Economies: A producer drives a number of advantages when he expands the size of his factory. These advantages are called internal economies. They arise because of increase in the scale of production (i.e. output that can be produced). These are secured only by the firm expanding its size. They are dependent on the efficiency of the organizer and his resources. So internal economies are those advantages which are obtained by a producer when he increases or expands the size of his firm. Internal economies are divided into various classes as follows. When a firm increases its scale of production it enjoys several economies. These economies are called internal economies.
Types of Internal Economies: There are two types of internal economies:
a)      Real Economies: Real economies are those which are associated with a reduction in the physical quantity of inputs, raw materials, various types of labour and various types of capital. Real economies can be of six types –
1)      Labour Economies or Specialization: Specialization means to perform just one task repeatedly which makes the labour highly efficient in its performance. This adds to the productivity and efficiency of the labour.
2)      Technical Economies: Technical economies are those economies which are related with the fixed capital that includes all types of machines & plants. Technical economies are of three types:
a)      Economies of Increased Dimension.
c)       Economies of the use of By-Product.
3)      Inventory Economies: A large size firm can enjoy several types of inventory economies; a big firm possesses large stocks of raw material.
4)      Selling or Marketing Economies: A firm producing a large scale also enjoys several marketing economies in respect of scale of this large output.
5)      Managerial Economies: A firm producing on large scale can engage efficient & talented managers.
6)      Transport and Storage Economies: A firm producing on large scale enjoys economies of transport & storage.

b)      Pecuniary Economies: Pecuniary economies are economies realized from playing lower prices for the factors used in the production and distribution of the product due to bulk-buying by the firm as its size increases.
B.      External Economies: When the number of factories producing the same commodity like sugar increases, we say that the particular industry (sugar industry) has developed. When the industry as a whole develops, every firm in the industry derives man advantages. These advantages are called external economies. They are enjoyed by all the firms in the industry. They are not the property or monopoly of any firm. The following are the main types of external economies:
1)      Economies of Concentration: When several firms of an industry establish themselves at one place, then they enjoy many benefits together, e.g. availability of developed means of communications and transport, trained labour, by products, development of new inventions pertaining to that industry etc.
2)      Economies of Information: When the number of firms in an industry increase, then it becomes possible for them to have concerted efforts and collective activities.
3)      Economies of Disintegration: when an industry develops, the firms engaged in its mutually agree to divide the production process among themselves.
Internal and External Diseconomies of Scale
The term diseconomies of scale refer to a situation where an increase in the size of the firm leads to a rising average cost. Diseconomies of scale may be classified into internal diseconomies and external diseconomies of scale.
The major internal diseconomies of scale arise from its size of the firm, technical causes and managerial problems. When a firm achieves a size where it is producing at the lowest possible average cost it is said to be at its optimum size. The optimum size will very over time as technological progress change the technique of production. In addition to this, more loaded men and machinery leads to machine fault and human failure cause breakdown of production.  When the size increases management becomes more complex and difficult. Managerial function of co-ordination, consultation and interdepartmental decision making will get delayed due to the size.  There will be possibility of delay in implementation of decision within the organization. Delay in communication will reduce the involvement of the employees.
There are some external diseconomies of scale in the form of disadvantages:
Ã˜  There is shortage of labour which causes a wage rise.
Ã˜  Increase in the demand for raw materials will also bid up prices.
Ã˜  When there is heavy localization of industries, the land for expansion will become increasingly scarce.  Scarcity will cause an increase in the price to purchase land or to rent.
Ã˜  Transport costs may also rise because of increased congestion.

The change in output will cause a movement along the long run average cost curve. One of the most significant influences is external economies of scale.  If external economies are experienced, the long run average cost will shift down (output will be now be cheaper to produce).  Whereas external diseconomies of scale are encountered the long run average cost curve will move up (output will now be costlier to produce).  Improved technology would lower the long run average cost curve.