# Relationship between isoquant and production function

### Isoquant - Wikipedia

The total product function shows the relationship between the quantity of labor (L) and the quantity of .. This is the unit isoquant for this production function. The Production Function measures the maximum possible output that the firm can Note the graphical relationship between average and marginal products in A set of isoquants, or isoquant map, describes the firm's production function. (1) The technical relation between inputs and output (i.e., how outputs vary as The long-run production function of a firm involving the usage of two factors, say, .

As with indifference curves, two isoquants can never cross. Also, every possible combination of inputs is on an isoquant. Finally, any combination of inputs above or to the right of an isoquant results in more output than any point on the isoquant. Although the marginal product of an input decreases as you increase the quantity of the input while holding all other inputs constant, the marginal product is never negative in the empirically observed range since a rational firm would never increase an input to decrease output.

An isoquants shows all those combinations of factors which produce same level of output. An isoquants is also known as equal product curve or iso-product curve. It describes the firm's alternative methods for producing a given level of output.

Shapes of Isoquants[ edit ] If the two inputs are perfect substitutes, the resulting isoquant map generated is represented in fig. A; with a given level of production Q3, input X can be replaced by input Y at an unchanging rate. Moreover, the use of the same optimum factor proportions with constant relative factor prices at different levels of output in homogeneous production function of the first degree is also very useful in input-output analysis.

Homogeneous production function of the first degree, which, as said above, implies constant returns to scale, has been actually found in agriculture as well as in many manufacturing industries.

In India, farm management studies have been made for various states and data have been collected for agricultural inputs and outputs. Analysing the data collected in these farm management studies. Khusro has reached the conclusion that constant 4 returns to scale prevail in Indian agriculture. Likewise, empirical studies conducted in the United States and Britain have found that many manufacturing industries are characterised by a long phase of constant long-run average cost LAC curve which again implies constant returns to scale and homogeneous production function of the first degree.

Many economists have studied actual production functions and have used statistical methods to find out relations between changes in physical inputs and physical outputs. A most familiar empirical production function found out by statistical methods is the Cobb-Douglas production function.

- Production Functions with Two Variable Factors: Isoquants and Isoclines
- Long-Run Production Function (With Diagram)

Originally Cobb-Douglas production function was applied not to the production process of an individual firm but to the whole of the manufacturing industry.

Output in this function was thus manufacturing production. Cobb-Douglas production can be estimated by regression analysis by first converting it into the following log form. Cobb-Douglas production function is used in empirical studies to estimate returns to scale in various industries as to whether they are increasing, constant or decreasing. Further, Cobb-Douglas production function is also frequently used to estimate output elasticities of labour and capital.

Output elasticity of a factor shows the percentage change in output as result of a given percentage change in the quantity of a factor. Cobb-Douglas production has the following useful properties: In the linear homogeneous Cobb- Douglas production function: Note that marginal product of a factor, say labour, is first derivative of the production function with respect to labour.

The marginal product of labour of Cobb-Douglas production can be obtained as under: Linear homogenous Cobb-Douglas Production Function exhibits diminishing returns to variable factor. To prove that the linear cobb-Douglas production function exhibits diminishing marginal product of a variable factor we have to show that the second derivative of the function with respect to a variable factor say labour is negative.

Remember that output elasticity of a factor shows the percentage change in output resulting from a given percentage change in the quantity of a factor, other factors remaining constant. We can easily prove this. Elasticity of Technical Substitution Between Factors: In part 2 concerning the theory of demand, we explained the concept of elasticity of substitution between goo. In the theory of production we are concerned with the elasticity of substitution between factors or inputs in the production of goods.

Thus, in the theory of production we are concerned with what may be called elasticity of technical substitution. As seen above, marginal rate of technical substitution MRTS or factor X for factor Y declines as more of factor X is substituted for factor Y along an isoquant.

In other words, marginal rate of technical substitution is different at different factor-proportions i. This responsiveness of the proportions or ratios in which factors or inputs are used as there is a movement along an isoquant may be compared with the change in substitution possibilities in production as measured by the change in the marginal rate of technical substitution.

If the elasticity of substitution between the two factors is high, one factor can easily be substituted by another. Therefore, we too shall explain the concept of elasticity of substitution with reference to capital and labour as factors of production. In order to understand the concept of elasticity factor of substitution, consider Figure It is, therefore, clear that as we substitute more labour for capital along the isoquant, while labour-capital ratio is rising, capital-labour ratio is falling.

Now, as we have already seen, marginal rate of technical substitution at point A is given by the absolute value of the slope of the isoquant at that point which is equal to the slope of the tangent t1t1 drawn to that point. It will be seen that the slope of tangent t2t2 is less than that of slope of t3t3 is less than that of t2t2, that is, MRTSIK diminishes as more labour is substituted for capital.

The greater the ease with which one factor can be substituted for another to produce a given level of output, the higher will be the elasticity of technical substitution between them. In the extreme case of factors which are perfect substitutes, there are infinite possibilities of substituting one factor for another to produce a given level of output without causing any change in MRTS, the elasticity of factor substitution is equal to infinity and the isoquants between them are straight lines as shown in Fig.

Thus, in case of perfect substitutes: On the other extreme case when the two factors are perfect complements which are used in a fixed ratio in the production of a commodity and hence there is no possibility of substitution between them at all i. Elasticity of Substitution and Factor Price Ratio: Thus, The replacement of factor-price ratio for marginal rate of technical substitution in the formula of elasticity of substitution is greatly helpful in practical applications of the concept of elasticity of substitution.

Further, changes in factor-proportions in which factors are used are generally induced by the relative factor prices. Elasticity of factor substitution occupies an important place in the theory of distribution.

Factor proportions are altered by keeping the quantity of one or some factors fixed and varying the quantity of the other.

An increase in the scale means that all inputs or factors are increased in a given proportion. Increase in the scale thus occurs when all factors or inputs are increased keeping factor proportions unaltered. Changes in Scale and Factor Proportions Distinguished: Before explaining returns to scale it will be instructive to make clear the distinction between changes in the scale and changes in factor proportions. The difference between the changes in scale and changes in factor proportions will become clear from the study of Fig.

We suppose that only labour and capital are required to produce a particular product. An isoquant map has been drawn.

A point s has been taken on the X-axis and the horizontal line ST parallel to X-axis has been drawn. OS represents the amount of capital which remains fixed along the line ST. As we move towards right on the line ST, the amount of labour varies while the amount of capital remains fixed at OS. Thus, the movement along the line ST represents variation in factor proportions. Likewise, a vertical line GH parallel to the Y-axis has been drawn which also indicates changes in factor proportions.

But in this case the quantity of labour remains fixed while the quantity of capital varies. Now, draw a straight line OP passing through the origin. It will be seen that along the line op the inputs of both the factors, labour and capital, vary. Moreover, because the line OP is a straight line through the origin, the ratio between the two factors along OP remains the same throughout.

Thus, the upward movement along the line OP indicates the increase in the absolute amounts of two factors employed with the proportion between two factors remaining unchanged. Assuming that only labour and capital are needed to produce a product, then the increase in the two factors along the line OP represents the increase in the scale since along the line OP both the factors increase in the same proportion and therefore proportion between the two factors remains unaltered.

If any other straight line through the origin such as OQ or OR is drawn, it will show, like the line OP, the changes in the scale but it will represent a different given proportion of factors which remains the same along the line. Validity of the Concept of Returns to Scale: We now proceed to discuss how the returns vary with the changes in scale, that is, when all factors are increased in the same proportion.

But some economists have challenged the concept of returns to scale on the ground that all factors cannot be increased and therefore the proportions between factors cannot be kept constant.

For instance, it has been pointed out that entrepreneurship is a factor of production which cannot be varied in the single firmthough all other factors can be increased. The entrepreneur and his decision-making are indivisible and incapable of being increased.

Thus, the entrepreneur is a fixed factor in all production functions. Thus, the concept of returns to scale involves a puzzle for economists which still remain unresolved.

Thus, when labour and capital are increased in a proportion, the entrepreneurship can be assumed to be increased automatically in the same proportion. In this sense, returns to scale, that is, returns when all factors are varied, can be conceived. But we shall explain below the concept of returns to scale by assuming that only two factors, labour and capital, are needed for production.

This makes our analysis simple and also enables us to proced our analysis in terms of isoquants or equal-product curves. Constant Returns to Scale: Returns to scale may be constant, increasing or decreasing. If we increase all factors i. Thus, if a doubling or trebling of all factors causes a doubling or trebling of outputs, returns to scale are constant.

But, if a given percentage increase in all factors leads to a more than proportionate increase in output, returns to scale are said to be increasing. Thus, if all factors are doubled and output increases by more than a double, then the returns to scale are increasing.

On the other hand, if the increase in all factors leads to a less than proportionate increase in output, returns to scale are decreasing. We shall explain below these various types of returns to scale. As said above, the constant returns to scale means that with a given percentage increase in the scale or the amounts of all factors leads to the same percentage increase in output, that is, doubling of all inputs doubles the output.

In mathematics the case of constant returns to scale is called linear homogeneous production function or homogeneous production function of the first degree. There are a number of special theorems which apply when production function exhibits constant returns to scale. Empirical evidence suggests that production function for the economy as a whole is not too far from being homogeneous of the first degree.

Empirical evidence also suggests that in the production function for an individual firm there is a long phase of constant returns to scale.

### Long-Run Production Function (With Diagram)

Let us illustrate diagrammatically the constant returns to scale with the help of equal product curves i. It is assumed that, in the production of the good, only two factors, labour and capital, are used. In order to judge whether or not returns to scale are constant, we draw some straight lines through the origin.

As shown above, these straight lines passing through the origin indicate the increase in scale as we move upward. It will be seen from the figure that successive isoquants are equidistant from each other along each straight line drawn from the origin.

The distance between the successive equal product curves being the same along any straight line through the origin, means that if both labour and capital are increased in a given proportion, output expands by the same proportion. It is thus argued by them that if, for instance, all factors or inputs are doubled, and then what is there to prevent the output from being doubled.

Suppose we build three exactly same type of factories by using exactly same type of workers, capital equipment and raw materials, will we not produce three times the output of a single factory? They say that if constant returns to scale do not prevail in some industries it is because it is not possible to increase or diminish factors used in them in exactly the same proportion. They advanced two reasons for our inability to vary the factors in the same proportion.

The scarcities of these factors cause diminishing returns to scale. Secondly, it is pointed out that some factors are indivisible and full use of them can be made only when production is done on quite a large scale.

Because of the indivisibility they have to be employed even at a small level of output. Therefore, when output is sought is to be expanded, these indivisible factors will not be increased since they are already not being fully utilised.

## Isoquant and Isocost Lines (With Diagram) | Economics

Thus, with the increase in output, cost per unit will fall because of the better utilisation of indivisible factors. Indivisibilities are a source of a good many economies of large-scale production. According to this view, if the limited supply of some factors and the existence of indivisibilities would not have stood in the way of increasing the amounts of all factors in the same proportion, then there must have been constant returns to scale.

The above explanation of the absence of economies of scale when the factors of production are perfectly divisible, stresses the role of factor proportionality in production. According to this view, for achieving best results in production, there are a certain optimum proportion of factors.

The capacity of the producer is shown by his monetary resources, i. So, like the consumer the producer has also to operate under a budget resource constraint. This is picturised by his budget line called isocost line. To find the least cost combination of inputs to produce a given output, we need to construct such equal cost lines or isocost lines.

An isocost line is a locus of points showing the alternative combinations of factors that can be purchased with a fixed amount of money. In fact, every point on a given isocost line represents the same total cost. To construct isocost lines we need information about the market prices of the two factors. For example, suppose, the price of labour is Re. Then an outlay of Rs.

All these and other various combinations are shown in Fig. These lines are straight lines because factor prices are constant and they have a negative slope equal to the factor-price ratio, i. Here, the firm seeks to minimise its cost of producing a given level of output.

To find the least-cost combination of factors for fixed level of output we combine Fig. Suppose, the producer wants to produce six units of output.

He could do so using the combination represented by points A, B or C in Fig. For example, the cost would be Rs. It looks for that factor combination that is on the lowest of the isocost lines.