Least-cost combination of Production

 

Least-Cost Combination

The problem of least-cost combination of factors refers to a firm getting the largest volume of output from a given cost outlay on factors when they are combined in an optimum manner.

In the theory of production, a producer will be in equilibrium when, given the cost-price function, he maximizes his profits on the basis of the least-cost combination of factor. For this he will choose that combination of factors which maximizes his cost of production. This will be the optimum combination for him.

 

Assumptions

The assumptions on which this analysis is based are:

1. There are two factors. Capital and labor.
2. All units of capital and labor are homogeneous.
3. The prices of factors of production are given and constant.
4. Money outlay at any time is also given.
5. Perfect competition is prevailing in the factor market.

On the basis of given prices of factors of production and given money outlay we draw a line A, B.

The firm cannot choose and neither combination beyond line AB nor will it chooses any combination below this line. AB is known as the factor price line or cost outlay line or iso-cost line. It is an iso-cost line because it represents various combinations of inputs that may be purchased for the given amount of money allotted. The slope of AB shows the price ratio of capital and labour, i.e., By combining the isoquants and the factor-price line, we can find out the optimum combination of factors. Fig. illustrates this point.

 

In the Fig. equal product curves IQ₁, IQ₂ and IQ₃ represent outputs of 1,000 units, 2,000 units and 3,000 units respectively. AB is the factor-price line. At point E the factor-price line is tangent to iso-quant IQ_2­ representing 2,000 units of output. Iso-qunat IQ₃ falls outside the factor-price line AB and, therefore, cannot be chosen by the firm. On the other hand, iso-quant IQ, will not be preferred by the firm even though between R and S it falls with in the factor-price line. Points R and S are not suitable because output can be increased without increasing additional cost by the selection of a more appropriate input combination. Point E, therefore, is the ideal combination which maximizes output or minimizes cost per units: it is the point at which the firm is in equilibrium.

What does the point of tangency tell us? At that point the slope of the factor-price line AB and the slope of the iso-quant IQ₂ are equal. The slope of the factor-price line reflects the ratio of prices of the two factors. Viz, capital and labour. The slope of the iso-quant reflects the marginal rate of technical substitution. At point E the ratio of prices of capital and labour is equal to the marginal rate of technical substitution. The condition of optimal combination is, therefore, given by the equality of the ratio of prices between any two factors and the rate of technical substitution between them. This is the point at which and firm is able to produce maximum quantity and at minimum cost.

Every firm, interested in maximising output or minimising cost, must therefore, consider (a) factor-price ratio which tells the firm the rate at which it can substitute one factor for another in purchasing, and (d) the marginal rate of technical substitution which tells the firm the rate at which it can substitute one factor for another in production. So long as the two are not equal, a firm can achieve a greater output or a lower cost by moving in the direction of equality.

 

Possibility Of operation – Production

 

Possibility of operation

The law of variable proportions guides us about the possibility of operation. A rational producer will never choose first and third stage for its production he will always operate in the second stage, i.e. the stage of diminishing returns. The producer will not produce in the first and the third stages because in the first stage the fixed factor of production i.e. capital is underutilized and its marginal return is negative and in third stage the variable factor of production i.e. labor is over utilized and thus its marginal return becomes negative. In other words, the marginal return of fixed factor and variable factor is negative in first stage and third stage respective. It is the second stage where the return on both the fixed factors and variables though diminishing is positive. The producer will always produce in second stage.

Returns to Scale Meaning

In the run all factors are variable, hence the expansion of output may be achieved by varying all factor-inputs. When we change all factor-inputs in the same proportion, the scale of production is also changed. The study of the effect of change in the scale of production on the amount of output comes under the head of returns to scale.

Thus, the term returns to scale refers to the changes in output as all factor-inputs change by the same proportion in the long run.

Or, in other words, the law expressing the relations between varying scales of production and quantities of output is called returns to scale. In short, returns to scale refer to the effects of scale relationship.

Three Types

Now the question is at what rate the output will increase when all factor- inputs are varied in the same proportion. There can be three possibilities in this regard. The increase in output may be more than, equal to, or less than proportional to the increase in factor-inputs. Accordingly, returns to scale are also of three types-increasing returns to scale, constant returns to scale and diminishing returns to scale.

S.N.	Returns to a Variable Factor	Returns to Scale
1	Operates in the short run	Operates in the long run.
2	Only the Quantities of factor are varied	All factor-inputs are varied in the same proportion
3	Changes in the factor-ratio.	No change in the factor-ratio
4	No change in the scale of production	Changes in the scale of production.

Three stages of Productions

 

THE THREE STAGES OF PRODUCTION

According to Cassels, there are three stages in the production process, when we vary one factor of production, the other factor remaining the same. In stage I, there is increasing average returns to the factor of production, i.e. > 0

i.e. MP_L > AP_L. In stage I, the average product is increasing and the marginal product is greater than the average product. If we refer to figure 4.3 we see that up to the point B on the TP curve, stage I exists. In stage I AP is increasing but MP is first increasing up to A and then decreasing. In stage II, the average product is decreasing and the marginal product is also decreasing, but marginal product is positive. This stage may be called the stage of decreasing returns. The portion of the total product curve between B and C represents this stage. In stage III, total product is diminishing and the marginal product is negative. This stage is called the stage of negative returns. The portion of the total product curve, which lies to the right of the point C, represents this stage.

Let us now discuss the rationale behind the operation of the three stages of production. In the beginning the quantity of the fixed factor of production (which is capital in our case) is abundant relative to the variable factor of production, i.e. labor. Therefore, when more and more units of the variable factor is used, the fixed factor is used more intensively and efficiently. This causes the production to increase at a rapid rate implying increasing AP and MP. But once the point A is reached where the variable factor is used at such a rate that ensures the efficient utilization of the fixed factor, any further increase in the variable factor will cause MP and AP to fall because the quantity of the fixed factor has now become limiting compared to the amount of the variable factor. Again in the stage III the quantity of the variable factor is so large compared to the fixed factor that the formed comes in each other’s way, thereby reducing the efficiency of the fixed factor, which results in a fall in the total product instead of rising. This is the reason behind the negativity of the marginal productivity in this stage. Comparing stage I and stage III, it can be said that, stage III is the mirror image of stage I.

Now the question, which immediately comes in our mind, is that, in which stage would the rational entrepreneur like to be? The answer is the rational entrepreneur will always like to operate in stage II of the production function. Let us analyze the reason behind this.

In stage I, MP and AP both are rising, and MP is more than AP. This has two implications:

1. A given increase in the variable factor leads to a more than proportionate increase in the output.
2. The entrepreneur is not making the best possible use of the fixed factor.

In this case the entrepreneur will employ more of the variable factor keeping the fixed factor constant, i.e. a particular portion of the fixed factor remains unutilized.

Considering the stage III we will see that the MP of the variable factor is negative and the TP is also decreasing. Hence the national entrepreneur will not operate in this stage.

However, if we consider stage II, we find that MP and AP are both falling and MP, though positive, is less than AP. Moreover, at this stage, there is less than proportionate change in output due to change in labor. Hence, at this stage the entrepreneur will employ the variable factor in such a manner that the utilization of the fixed factor is most efficient. So this is the stage in which the entrepreneur can use both of the available resources in an optional manner.

Distinction Between returns to a Variable Factor And Returns to Scale

 

Distinction Between returns to a Variable Factor And Returns to Scale

1. Returns to a Variable Factor. The law of returns to a variable factor states that with the increase in the units of a variable factor, keeping other factors constant, the increase in total production becomes, after some point, smaller an smaller

In this case marginal product first increase then becomes constant and finally it declines and becomes negative. Thus, the law has three stages: increasing marginal returns; diminishing marginal returns and negative returns.

1. Returns to Scale. The term returns to scale refers to the change in output as all factor-inputs change by the same proportion in the long run.

The increase in output may be more than, equal to, or less that proportional go the given increase in factor-inputs, hence returns to scale out of three forms-increasing, constant and diminishing.

S.N.	Returns to a Variable Factor	Returns to Scale
1	Operates in the short run	Operates in the long run.
2	Only the Quantities of factor are varied	All factor-inputs are varied in the same proportion
3	Changes in the factor-ratio.	No change in the factor-ratio
4	No change in the scale of production	Changes in the scale of production.

Cause for the operation of diminishing returns to scale

 

Cause for the operation of diminishing returns to scale

As a firm expands its output, after a certain point, it encounters growing diseconomies. These diseconomies, ultimately, more than cancel out the economies of large scale production and lower down the long run average production. The economies of production are swamped by diseconomies of production.

The main diseconomies are

1. Managerial Diseconomies. These diseconomies occur primarily because of increasing managerial difficulties. As the output grows, top management becomes eventually overburdened and hence less efficient in its role as co-coordinator and ultimate decision-maker.
2. Diseconomies due to exhaustible natural resources Another cause for diminishing returns to scale may be found in the exhaustible natural resources: doubling the fishing fleet not lead to a doubling of the catch of fish; or doubling the plant in meaning or on an oil-extraction field may not lead to a doubling of output.

As a result of these diseconomies of firm, long-run average and marginal cost rise with the increase in output and scale of production.

Thus, it is clear from our analysis that returns to scale have three forms increasing, constant and diminishing. The law of returns to scale with its all the three forms can be shown in one single example and diagram.

 

Example

Units scale of Production	Total Production	Marginal Production	Return to Production   scale
1 Labor + 2 Ropani land 2 Labor + 4 Ropani land 3 Labor + 6 Ropani land 4 Labor + 8 Ropani land 5 Labor + 10 Ropani land 6 Labor + 12 Ropani land 7 Labor + 14 Ropani land 8 Labor + 16 Ropani land	8 17 27 38 49 59 68 76	8 9 10 11 11 10 9 8	Increasing Returns   Constant Returns   Diminishing Returns

From A to B in the diagram is the stage of increasing returns; from B to C constant returns, and from C to D is the diminishing returns to scale.

 

The main reason for the operation of the different forms of returns to scale is found in economics and diseconomies. When economies exceed the diseconomies, the stage of increasing returns operates; when economies and diseconomies equals each other, it becomes the stage of constant returns to scale; and when diseconomies exceeds the economies, then comes the stage of diminishing returns to scale.

Returns to Scale

 

Returns to Scale

Meaning

In the run all factors are variable; hence the expansion of output may be achieved by varying all factor-inputs. When we change all factor-inputs in the same proportion, the scale of production is also changed. The study of the effect of change in the scale of production on the amount of output comes under the head of returns to scale.

Thus, the term returns to scale refers to the changes in output as all factor-inputs change by the same proportion in the long run.

Or, in other words, the law expressing the relations between varying scales of production and quantities of output is called returns to scale. In short, returns to scale refer to the effects of scale relationship.

Three Types

Now the question is at what rate the output will increase when all factor- inputs are varied in the same proportion. There can be three possibilities in this regard. The increase in output may be more than, equal to, or less than proportional to the increase in factor-inputs. Accordingly, returns to scale are also of three types-increasing returns to scale, constant returns to scale and diminishing returns to scale.

 

1. Increasing Return to Scale

Increasing Returns to scale refers to a situation where the total output increases in a greater proportion than the increase in units of factor inputs.

When the increase in output is more than proportionate to the given increase in the quantities of all factor-inputs, it is termed as increasing returns to scale.

For instance, if the increase in factor-inputs is 100 percent and the resultant increase in output is 150 percent, it is increasing returns to scale. We give an Illustration of increasing returns of increasing return to scale by a diagram. Form O three lines OS, OQ and OR are drawn cutting is the product curve 2, curve 3 at various points. Increasing returns to scale is shown as:

OR  > RP > PG

 

Or    OR₁  > R₁P₁>  P₁G₁

 

Or    OR₂ > R₂P₂ > P₂G₂

It means in this case, a doubling of inputs results in more than doubling of output. It is explained in the following example–

Scale of Production          Total Output (Machine + Labor)                 (Units)

1Machine + 2 Labor                     100 2Machine + 4 Labor                     250

 

Causes for the operation of increasing returns to scale

Why does increasing returns to scale operate? The reasons for the operation of increasing returns to scale are found in the form of economics of large-scale production. They are:

(i)      Labor Economies. They are also known as the economies the economies of specialization and division of labor. Division of labour and specialization are possible more in large-scale operation. Different types of works can specialize and do the job for which they are more suited. A worker acquires greater skill by devoting his attention to a particular job. Quality and speed of work both improve. This results in a sharp increase in output per man. Thus in short, with growing scale come, increasing specialization and increasing returns to scale.

(ii)     Technical economies. The main technical economies result from the indivisibilities that are characteristic of the modern industrial techniques of production. Several capital goods, because of the strength and weight required, will work only if they are of a certain minimum size. It may be technically possible to build smaller models of them; but it will not always be possible to use such models. Besides this, there is a general principle that as the size of a capital good is increased, its total output capacity increases far more rapidly than the cost of making it. To double the size and output capacity of a blast furnace for instance, we do not have to double the materials required. This is known as the principle of indivisibility

(iii)    Marketing Economies. Advertising space (in newspapers and magazines) and time (on television radio) and the number of salesmen do not have to rise proportionately with the sales. Thus the selling cost per unit of output falls with scale.

(iv)    Managerial Economies. Managerial economies arise from specialization of management and mechanization of managerial function. Large firms make possible the division of managerial tasks. This division of decision-making in large firms has been found very effective in the increase of the efficiency of management. Besides, large firms apply techniques of management involving a high degree of mechanization, such as telephones, telex machines, television screens and computers. These techniques save time and speed up the processing of information’s.

As the business firms continues to expand it gradually exhausts the economies, which cause the operation of, increasing returns to scale. Beyond this point, further increases in the scale of operation are accompanied by constant returns to scale.

(v)      Economies Related to Transport and Storage Costs. Because a large firm uses it’s own transport means and larger vehicles, per units transport costs would fall. Similarly, storage cost will also fall with the size.

As a result of all these economies firm’s long run average and marginal cost decline with the increase in output and scale of production.

2-Constant returns to Scale

Thus the constant returns to scale means that if all factor-inputs are varied at a certain percentage rate, output will change by the same rate.

Or, when the increase in output is proportional to the increase in the quantities of all factor-input; it is termed as constant returns to scale.

The constant returns to scale sometimes referred to by economists in managerial language, a production curve showing constant returns to scale is often called” Linear and homogeneous”. The Cobb-Douglas production function evolved by American economists Paul Douglas and C. W. Cobb is a linear and homogeneous function. Following Figure illustrates content returns to scale.

 

OR = RP = PG

Or        OR₁ = R₁P₁ = P₁G₁

OR₂ = R₂P₂ = P₂G₂

Scale of Production       Total Output      (Machine + Labor)                (Units)

1Machine + 2 Labor                     100 2Machine + 4 Labor                     200

3 Diminishing Returns to Scale

When the increase in output is less than proportionate to the given increase in the quantities of all factor-inputs, it is termed diminishing returns to scale. For instance, if the increase in factor-inputs is 20 percent and the resultant increase in output is less than 20 percent (say 15 percent) or a doubling of inputs causes a less than a doubling of output, it is diminishing returns to scale. This is explained in the following example-

Scale of Production          Total Output (Machine + Labor)                 (Units)

1Machine + 2 Labor                    100 2Machine + 4 Labor                    150

The result is diminishing return to scale. Diminishing returns to scale implies that for a given increase in output factor is required. In other words, proportionate increase in input factors will be more than proportionate increase in output. This Fig illustrates the application of diminishing returns to scale.

 

Here: –

OR < RP < PG

Or        OR₁ < R₁P₁ < P₁G₁

OR₂ < R₂P₂< P₂G₂

 

Isoquants

 

Isoquants

An isoquants is a curve on which the various combinations of labor and capital show the same output. “An is product curve is a curve along which the maximum achievable rate of production is constant.” It is also known as a production indifference curve or a constant production curve Just as an indifference curve shows the various combinations of any two commodities that give the consumer the same amount of satisfaction (iso-utility), similarly an isoquant indicates the carious combinations of two factors of production which give the producer the same level of output per unit of time. The Table below shows a hypothetical isquant schedule of a firm producing 100 units of a good.

Isoquant Schedule

Combination	Capital Input	Labor Input	Total Output (In units)
First	9	5	100
Second	6	10	100
Third	6	15	100
Fourth	3	20	100

 

This table is illustrated on the figure where labor units are measured along the X-axis and capital units on the Y-axis. The first, second, third and the fourth combinations are shown as A., B., C and D respectively. Connect all these points and we have a curve IQ. This is an isoquant. The firms can produces 100 units of output of point A on this curve by having a combination of 9 units of capital and 5 units of labor. Similarly point B shows a combination of 6 units of capital and 10 units of labor; point C, 4 units of capital + 15 units of labor and point D, a combination of 3 units of capital+ 20 units of labor to yield the same output of 100 units. A number of isoquants representing different amounts of output are known as an isoquant map. In the figure curves IQ, IQ1′ IQ2 show an isoquant map. Starting from curve IQ which yields 100 units of product, the curve IQ1 shows 200 units and the Iq2 curve 300 units of the product which can be produced with altogether different combinations of the two factors.

Law of Variable Proportions

 

Law of variable Proportions

The law of variable proportions is also named as the laws of returns or the laws of returns to a variable factor.

The law states that as the quantity of a variable impute is increased by equal doses, keeping the quantities of other inputs constant, total product will increase, but after a point, at diminishing rate. This principle can also be defined thus, when more units of variable factors are used having the quantities of fixed factors constant, a point is reached beyond which the marginal product, then the average and finally the total product will diminish.’

According to Prof. Watson, “The law of variable proportions, also known as the law of diminishing returns, can be stated as follows: when total output or production of a commodity is increased by adding units of a variable input while the quantities of other inputs are held constant then increase in total production becomes, after some point, smaller.”

In simple words, the law of variable proportions (or returns to a variable factor) states that with the increase in variable factor, keeping other factors constant, the marginal product after rising to some extent becomes smaller and smaller.

Why is it called the Law of variable proportions?

It is because of two reasons:

(i)   The factor: proportion (or factor-ratio) varies as one input varies and others are held constant. This can be understood with the help of an example. Suppose in the beginning 10 Ropani of land and 1 unit of labor are taken for production, hence the land- labor ratio was: 10 1. Now if the land remains the same but the unit of labor increases to 2, now the land- labor ratio would become 5: 1. Thus law analyses the effects of change in factor-proportions on the amount of output and therefore called the law of variable proportions:

(ii)  The return also varies non-proportionally with the change in factor-ratio: It means when one factor is varied keeping other factors constant, the input-output ratio also undergoes a change. For example, if 10 Ropani of land and 1 unit of labor give the output of 20 quintals of wheat then the ratio between labor (variable factor) and output of what is 1:20? Now if with 2 units of labor, the output increases to 30 quintals then the labor-output ratio would become 1: 15. Thus, the ratio between the variable factor and the output change and that is why it is termed as the law of variable proportions.

Assumptions

(i)   One is variable factor and others are the fixed factors.

(ii)  It is possible to make chances in the factor-proportions.

(iii) No change in technique of production and organization.

(iv) All units of the variable factors are homogeneous

Causes of the Operation of the Law:

(i)   In the short period all factors of production cannot be varied i.e. some are fixed factors in the short-run.

(ii)  Factor of productions is not perfect substitutes.

(iii) Factors of production are scarce in relation to their demand.

Explanation of Law

This law of variable proportion can be illustrated with the help of the following example and diagram. In this example, we have presumed that land is a fixed factor and labor is a variable factor.

Example

Fixed factor land (Ropani)	Variable Factor Labor (VF) (Units)	TP (Qnt.)	AP=TP/VF (Qnt.)	MP (Qnt.)
1	0	0	0	–
1	1	4	4	4
1	2	12	6	8
1	3	24	8	12
1	4	32	8	8
1	5	36	7.2	4
1	6	36	6	0
1	7	28	4	-8

In this example we have assumed that sand is the fixed factor and labor is a variable factor. The table shows the different amounts of output obtained by applying different units of labor to one acre of labor to one Ropani of land, which is fixed.

Three stages of the Law

The relation between variable factor and physical output has three stages that are shown in the example as well as in the diagram. These are known as the three stages of the law

Stage I.

In this stage total product increases at an increasing rate and the average product of labor (AP) also increases. In the beginning of this stage marginal product (MP) also increased but increases but after a point it starts to decline. Average product continues to increase till marginal product is greater than the average product. But when marginal product becomes equal to average product, the increase in average product is withheld and this is the outer limit of the first stage. In our example, the first stage, of the law runs up to four units are labor and this limit is shown by point A in the diagram. Since in this stage average product increases with the increases in the units the variable factor, it is called the stage of increasing returns.

Why does the law of increasing return operate? Following are given its main reasons:

(i)   Indivisibility of Factors. There are some factors that cannot be purchased in parts. For example, if no generator is available in the market less than the size of 15-horse power then we have to purchase and use this generator. If at present we are not using this generator to its full capacity then with the increase in production marginal cost will decline.

(ii)  Increase in Efficiency. If the efficiency of fixed factor increases with the increase in the quantities of variable factor, the law of increasing returns will apply.

(iii)            Fixed Factors. The cost of fixed factors is also fixed. Hence, with the increase in output per unit fixed cost will decline which in turn will lower the over all average cost also up to a certain limit.

(iv) Division of Labor. If the increase in the units of labor brings greater division of labor and specialization in the production, it will also create the conditions of increasing returns.

(v)  Optimum combination. If to achieve optimum combination of various factors of production, the increase in the quantities of a particular factor is required it will also bring increasing returns.

Stage II.

In this stage also the total product continues to increase but at a diminishing rate. This stage goes to the point when total product reaches the maximum and marginal product becomes zero. In this stage average product goes on diminishing. It means there is decline in the efficiency of labor. In our example second stages runs between 5 units and 6 units of labor and in the diagram it is between point A and point M. In this stage both average product and marginal product decline but remains positive. This stage is known as the stage of diminishing returns also. Why does the law of diminishing returns operate? Following are given its main reasons:

(i)   Fixed Factor. The quantity of the fixed factor-input per unit of the variable input flls as more and more of the later is put to use successive units of the variable input, therefore, must add decreasing amounts to the total output as they have less of the fixed input to work e.g. land is a fixed factor.

(ii)  Scarcity of Factors. Factors of production are scarce and limited in supply. If factors had not been limited, this law would also not have come into existence.

(iii)            Imperfect Substitutes. Factors of production cannot be substituted fully. For example, labor or capital cannot be substituted in place of land.

(iv) Optimum Combination. These are an optimum combination pf different factors that gives the maximum output. When we increase a particular factor of production beyond this optimum combination, marginal product of that variable factor declines naturally.

Importance

Samuelson has regarded the law of diminishing returns as a natural Law. Since this law is applicable in all the fields of production, it is called as the universal law of production. Many principles of economics such as the Malthusian theory of population, Ricardian theory of rent, marginal productivity theory of distribution are based on the assumption of the law of diminishing returns. Though this law is applicable in all fields yet it is more applicable in the fields of agriculture. The main reason behind this is that in agriculture nature plays a greater role than the man. In the words of Wick steed the law of diminishing returns ” is as universal as the law of life itself.” The universal applicability of this law has transcended economics to the realm of science. Above all, it is of fundamental importance for understanding the problems of under developed countries.

Stage III.

In the third stage total product starts to decline and marginal product becomes negative. That is why total product curve starts to decline and marginal product curve goes below the X-axis. In our example this stage comes when 7^Th and 8^th units of labor are employed and according to diagram this stage starts when more than OM units of labor are employed. This stage is called the stage of negative returns.

The law of Diminishing Marginal Product

 

The Law of Diminishing Marginal Product

It has been discussed earlier that in the short run one of the factors of production is variable and the other(s) constant. The marginal product of the variable factor will decrease eventually as more and more of quantities of this factor will be combined with the other constant factor(s). The expansion of output with one factor (at least) constant is described as the Law of Diminishing Marginal Product of the variable factor, which is often referred to as the law of variable proportions.

If the law of diminishing marginal product operates the isoquants will be convex to the origin. A convex isoquant means that the marginal rate of Technical substitution (MRTS) between L and K decreases as L is substituted for K. But the MRTS is equal to the ratio of the marginal productivities of the two factors i.e. MRTS  = f_L/f_K.

 

In figure, as we move from the point P to the point Q along the isoquant Q₀, we employ RQ units more of L and PR units less of K. By moving from P to Q, we increase the labor input by RQ and reduce the capital by PR. That is, we substitute RQ labor for PR. The rate of this substitution is called Marginal Rate of Technical Substitution (MRTS_L,K ). For the submission, MRTS   = = .

What is the MRTS_L,K  for a movement from Q to S on the isoquant? Here, the increase in labor input is TS, which is equal to RQ. The decrease in capital input in QT. Therefore, MRTS_L,K  = . By comparing MRTS_L,K  at these two points, it is obvious that MRTS_L,K  at Q is less than at P. Though RQ = TS, QT < PR. That is, for the same increase in labor input the reduction in capital will be less as we move down the Isoquant. This is because of law of diminishing marginal product. As we shall see later, MRTS_L,K at any point on the isoquant is equal to the slope of a tangent at that point. The slope is nothing but the ratio of MP₁ and MP_k. Therefore, the marginal productivity of L decreases and that of K increases as we move along the isoquant from P to Q., In other words, (f₁/f_k at Q). Therefore, we can say that MRTS_L,K  at P > MRTS_L,K at Q, i.e. MRTS between L and K decreases as we move from P to Q. Thus it can be said that MRTS between factors follows the diminishing marginal productivities of both the factors of production. Hence the convexity of the isoquant follows directly from the law of diminishing marginal productivity.

As we had discussed in the analysis of indifference curves that two indifferences cannot intersect each other, in the same way it can be proved in the case of isoquants that two isoquants cannot intersects each other.

 

A set of curves form a family of isoquants as shown in figure. All the inputs combinations, which lie on an isoquant, will result in an output indicated for that curve. The further an isoquant lies from the origin, the greater the output level which it represents: Therefore, Q₃ > Q₂ >Q₁.

The partial derivative of production function ¶Q / ¶L is the marginal productivity of labor (MP₁). It is the extra amount of output that can be obtained by employing one additional unit of L, K remaining the same. Similarly, ¶Q / ¶K is the marginal productivity of the capital (MP_K). We assume that MP₁ or f₁ > 0 and MP_K or f_K>0.

ISOQUANT

 

ISOQUANT

 

An isoquant is the firm’s matching part of the consumer’s indifference curve. It is the collection of inputs in the form of factors of production labor (L) and capital (K), which yield the same output. For a definite output level the equation of the production function becomes

 

Q⁰   = f (L, K)

Where, Q₀ is a parameter.

The locus of all the combinations of L and K, which satisfy the above equation, forms an isoquant. Since the production function is continuous, an indefinite number of input combinations will lie on each and every isoquant. The two factors of production are substitutable and we can employ more of one factor and less of another factor to get the same level of output. A higher level of output is represented by a higher isoquant. If we assume that the marginal productivities of both the factors of production are positive and decreasing as more of them are used, the isoquant will be downward sloping and convex to the origin.

 

Box Isoquant

A isoquant is a curve showing all combinations of inputs that can be used to produce a given output. The table shows five methods of production with various input combinations. When we plot on a graph, we get an Isoquant. Marginal Rate of Technical Substitution ((MRTSLK) measures the rate at which labor can be substituted for capital, output being constant. This is nothing but the slope of the Isoquant at any point.