# BCom 1st Year Laws of Production and Isoquants Notes Study Material

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## BCom 1st Year Laws of Production and Isoquants Notes Study Material

**Isoquants**

Isoquants or equal-product curves or iso-product curves are similar to the indifference curves of the theory of consumer behaviour. An isoquant is a curve which shows all those combinations of two inputs required to produce a given quantity of a particular product. This curve is thus a contour line which traces the loci of equal output.

Since an isoquant represents those combinations of inputs which will be capable of producing an equal quantity of output, the producer will be indifferent between them as such. Hence another name given to the isoquant is production indifference curve or producer indifference curve.

But there is one important difference between the two which results from the way the problem is formulated : in the theory of consumer choice we focus on the equilibrium conditions that exist if utility is to be maximized subject to income or budget restrictions; in the case of isoquants the cost is to be minimised subject to the constraint offered by the production function.

In order to explain an isoquant, we take the help of Table. It is presumed here that two factors X and Y are being employed to produce a product. Each of the factor combinations A, B, C, D and E produces the same level of output, say, 20 units. It means that the factor combination A consisting of 1 unit of factor X and 12 units of factor Y produces the given units of the output, i.e., 20 units.

Similar is the case with other combinations. These combinations are plotted in fig, which give us the isoquant.

Isoquant is similar to indifference curves, but with one difference. In the case of indifference curves, no attempt is made to measure utility or satisfaction and so indifference curves are labelled 1, 2, 3, etc. indicating that higher indifference curves represent higher level of satisfaction.

But we can label isoquant in the physical units of output without any difficulty. Production of a good is a physical phenomenon. It lends itself easily to absolute measurement in physical units as shown in fig. In the figure three isoquants have been shown. The lowest one is P_{1} which represents 20 units of the product. Isoquant P_{2} represents 40 units, whereas P_{3} stand for 60 units.

**Marginal Rate of Technical Substitution (MRTS)**

The marginal rate of technical substitution in the theory of production is similar to the concept of marginal rate of substitution in the indifference curve analysis of consumer demand. It indicates the rate at which factors can be substituted at the margin without altering the level of output. More precisely, the marginal rate of technical substitution of factor X for factor Y may be defined as the amount of Y which can be replaced by one unit of factor X, the level of output remaining unchanged.

In the above Table, moving down the table from combination A to combination B, 4 units of Y are replaced by one unit of X in the production process without any change in the level of output. So the MRTS is 4 here. This can be noted in the figure as well. In fig, when we move down the isoquant P_{1} from G to H, a small amount of Y, i.e., ∆Y is replaced by an amount ∆X of factor X, without any loss of output. The slope of the isoquant P_{1} at point G is, therefore, equal to ∆Y/∆X. Thus MRTS = slope of the isoquant = ∆Y/∆X.

MRTS can also be known by the slope of the tangent drawn on the isoquant to that point, as shown in fig. In order to know the MRTS at point K on isoquant P_{1}, we draw a tangent TT’ at that point. Now

MRTS at point K = slope of the tangent TT’ = OT/OT’

The equation of an isoquant can be represented by

∆X (MP_{X}) + ∆Y (MP_{Y}) = 0,

where MP_{X} = marginal product of X and MP_{Y} = marginal product of Y.

This follows from the definition of an isoquant. As we move from one point to another along an isoquant, the loss in output from using less of Y must be just offset by the addition to output from using more of X (no change in output can result from the substitution of X for Y along a given isoquant).

Since ∆X (MP_{X}) + ∆Y (MP_{Y}) = 0

So

– ∆Y (MP_{Y}) = ∆X (MP_{X})

Slope ∆Y/∆X = – MP_{X}/MP_{Y}

The slope of an isoquant is called the marginal rate of technical substitution.

An important characteristic of the marginal rate of technical substitution is that it diminishes as more and more of factor Y is substituted by factor X. In other words, as the quantity of factor X is increased and that of factor Y is reduced, the amount of factor Y that is required to be replaced by an additional unit of the factor X so as to keep the output constant will diminish. This is known as the Principle of Diminishing Rate of Technical Substitution.

This principle is merely an extension of the Law of Diminishing Returns to the relation between the marginal physical productivities of the two factors. Along an isoquant, as the quantity of factor X is increased and the quantity of factor Y is reduced, the marginal physical productivity of X diminishes and the marginal physical productivity of Y increases. Therefore, less and less of factor Y is required to be substituted by an additional unit of X so as to maintain the same level of output.

The rate at which the marginal rate of technical substitution diminishes is a measure of the extent to which the two factors can be substituted for each other. The smaller the rate at which MRTS diminishes, the greater the substitutability between the two factors.

If the MRTS between any two factors does not diminish and remains constant, the two factors are perfect substitutes of each other. In this case a given amount of one input could always be exchanged for some given amount of the other input along a given isoquant. The zero substitutability case is one where a given output requires fixed amounts of each input; no additional amount of X can be substituted for Y or vice versa.

**Properties of Isoquants**

The general characteristics of isoquants are the same as those of indifference curves. The following may be noted:

(i) Isoquants slope downward from left to the right.

(ii) No two isoquants will intersect.

(iii) Isoquants are convex to the origin.

The downward slope of an isoquant from left to right depends upon the technical substitutability of one resource for the other. When resources are technical substitutes, the use of the less of one must be compensated by the use of the more of the other so that total product remains constant. This means that the isoquant will slope downward from left to right.

Non-intersection of two isoquants is quite obvious and hardly needs any discussion. If two isoquants intersect, the intersection point would lie on both the isoquants. It means that two different quantities of the product could be produced with the same resource combination. This is not possible since it is assumed that the most efficient techniques are being used.

Convexity of an isoquant to the origin rests on the following factors:

- different resources are technical substitutes for each other;
- but they are not perfect substitutes;
- the marginal rate of technical substitution declines.

Exceptional isoquants (other than those convex to the origin) are found in the case of the two factors being perfect substitutes as also in the situation of perfect complements.

**Isoquants and Laws of Production **

Isoquants can be used to explain and distinguish between proportion and scale. For this we should look at fig. where both scale and proportion have been shown. In fig, constant returns to scale under returns to scale and the short-period law of variable proportions have been shown.

In order to explain constant returns, we draw a diagonal from the point of origin O. P_{20}, P_{40} and P_{60} are isoquants. The diagonal is divided into equal parts by the isoquants so that OA = AB = BC. It means that equal increase in the combinations of labour and capital causes equal increases in output, from 20 to 40 to 60.

In order to analyse the law of variable proportions, we assume that the quantity of capital is fixed at OK. With this fixed amount of capital, more units of labour are being combined. It can be seen in the figure that in order to raise output in equal quantity (by 20 units), larger quantity of labour is required. So the units of labour increases from KR to RS to ST, i.e., KR < RS < ST. Thus increasing units of labour combined with fixed quantity of capital lead to equal increases in output. This is the law of variable proportions.

Increasing returns to scale is shown in fig. In this case the diagonal OR is so divided by isoquants that OA, AB, BC, become smaller and smaller, i.e., OA > AB > BC CD > DE. It means that in order to increase output by equal amounts, units of the factors are increased at diminishing rate.

Decreasing returns to scale have been shown in fig. OR, the diagonal drawn from the point of origin O, is so divided by isoquants that OA < AB < BC < CD. It means that in order to increase output by equal amount, increasing quantities of labour and capital are needed.

Constant, increasing and decreasing returns to scale have been illustrated in the same diagram. In this figure the straight line from the origin, OR, is first crossed by the isoquants at smaller, then at equal and finally at increasing intervals. So we have first the increasing return, then constant returns and finally decreasing returns to scale.

**Economic Region of Production and the Ridge Lines**

In fig, the production function is presented in the form of a set of isoquants. Higher isoquants mean higher level of production. Isoquants do not interest.

In traditional economic theory the efficient range of output, i.e., the economically efficient range of production is stage II of the operation of the law of variable proportion in which the marginal product of the variable factor diminishes but is positive. The locus of points of isoquants where the marginal products of the factors are zero form the ridge lines.

In fig, two ridge lines OC and OL have been drawn. The upper ridge line OC implies that the marginal product of capital is zero. The lower ridge line OL implies that the marginal product of labour is zero. Production techniques are technically efficient only inside the ridge lines. Marginal products of factors are negative outside the ridge lines and the methods of production are inefficient.

It is so because they require more quantity of both factors for producing a given level of output. “Such inefficient methods are not considered by the theory of production, since they simply irrational behaviour of the firm. The condition of positive but declining marginal products of the factors defines the range of efficient production (the range of isoquants over which they are convex to the origin).”. All rational combinations of labour and capital for the firm lie between the ridge lines OC and OL.

**The Optimum Combination of Inputs**

Any desirable level of output can be attained by different combination of inputs. Now we have to analyse what combination the firm will choose.

In order to acquire a commodity price has to be paid. Similarly, prices have to be paid to acquire the factors of production. So in order to produce at the least cost or to get the maximum production from a given cost, a particular combination of inputs has to be selected and this can be done only when the input prices are known. Then only the firm can maximise profit.

**Changes in Production and the Expansion Path**

Baumol says further that the condition that optimal input combinations occur at points of tangency between an isocost line (the price line or budget line of the indifference curve analysis of consumer demand) and an isoquant is a geometric representation of the following basic optimality rule:

“An optimal combination of any two inputs, I and J, requires that the ratio of their marginal products be equal to the ratio of their prices. Symbolically, we must thus have

MP_{i}/MP_{j} = P_{i}/P_{j}”

The expansion path is also called the isocline line. At all points on this line the MRTS is equal.

In the case of constant returns to scale, the expansion path is a straight line through the origin. In other cases of the returns to scale the expansion path will not be a straight line. The expansion path obtained by the points of tangency of successive isocost lines and successive isoquants is the optimal way of expanding output by the firm. So the expansion path is also called the optimal expansion path.

**Changes in Input Prices **

Let us now assume that the prices of labour, i.e., wages fall, but the price of capital remains unchanged. In this situation the lower part of the isocost line LM will shift to the right to the point M_{1} from M, as shown in figure.

Let the initial equilibrium (before fall in wages) point be S. It indicates that in order to produce P_{1} output firm employs OL_{1} of labour and OK_{2} of capital. When wages fall, the isocost line changes from LM to LM_{1} and equilibrium is established at Q. At this point a combination of OL_{3} of labour and OK_{1} of capital produces P_{2} output. The fall in wages leads to increased employment of labour to the extent of L_{1}L_{3}.

The effect of a fall in wages is felt on employment and more labourers are employed. The increased employment of labour can be divided into substitution effect and output effect.

In order to know the substitution effect, we draw an imaginary isocost line L_{1}M_{2} parallel to LM. Suppose there is no decline in wages, rather they remain at the level given by LM. But expenditure is so increased that the firm is able to produce the output at the level of P_{2}. The imaginary increase in outlay is shown by L’M_{2}. Isoquant P_{2} touches the isocost line L’M_{2} at R, which is a point of equilibrium. The movement from R to Q is the substitution effect. As a result of this additional labour of L_{2}L_{3} is employed.

The output effect is given by the movement from S to R. Increase in employment due to output effect is L_{1}L_{2}. It is the output effect because additional units of factors are required in the event of increased outlay to produce at a higher level at the old input prices.

The influence on the use of labour when wages decline is the sum of two effects, namely, substitution effect and output effect. In terms of fig, they are

Effect on Labour = Substitution Effect + Output Effect

(L_{1}L_{3}) = (L_{2}L_{3}) + (L_{1}L_{2})

**Cobb-Douglas Production Function**

Economists have examined many actual production functions. For this they employed statistical analyses to measure relations between changes in physical inputs and physical outputs. One of the notable attempts to estimate statistically a generalized production for manufacturing industries is the Cobb-Douglas production function, associated with the names of two American economists, C. W. Cobb and P. H. Douglas. They propounded a particular production function for the entire American manufacturing industries.

There is a fair amount of empirical evidence that the share of wages in the national income of the United States has remained relatively constant for a long period. There have been a number of attempts to explain this remarkable fact. The usual explanation of the relatively fixed proportion between wage payments and total national income is based on the hypothesis that the production function is linear and homogeneous. In other words, the production function is of the type of constant returns to scale when there is a proportionate increase in the use of all inputs. This type of production function takes the special form:

Y=kL^{∞} C^{(1–∞)}

where,

Y = national output

k and ∞ = positive fixed quantity and ∞ < 1

L = labour input

C = capital input

This production function is called the Cobb-Douglas production function.

Cobb and Douglas found that the non-linear exponential function fit quite well their sample of inputs and outputs drawn from a variety of industries. This function takes the form of

Q = z L^{a} C^{b} u

where Q = output; L = labour input; C = capital input; u = a disturbance term; and z, a, b = parameters. The production elasticities, a and b, representing the output response with respect to labour and capital respectively are assumed constant in this form of the production function.

If each of the exponents (constant elasticities) is less than one, the function reflects diminishing returns (declining marginal product). Depending on the combined values of a and b the production function may exhibit the following:

(a) if a + b = 1, constant returns to scale;

(b) if a + b > 1, increasing returns to scale; and

(c) if a + b < 1, decreasing returns to scale.

Although Cobb-Douglas production function may predict the shares of labour and capital in national income with reasonable accuracy, it cannot be inferred that all production units have a production function of this kind. This function is widely used in economics, not only in the theory of income distribution, but also in the theory of production and in the theory of economic growth.

The Cobb-Douglas production function cannot be accepted as valid on a priori reasoning. It is an empirical hypothesis and so can explain an empirical observation. “But that is a good scientific procedure which always builds and tentatively accepts theoretical constructs only because their implications accord with (explain) observed phenomena. It is to be added that, at least so far, the statistical evidence does not appear to contradict the Cobb-Douglas hypothesis.”

The Cobb-Douglas production function is subject to the following limitations:

(i) It is based on the assumption of homogeneity and divisibility. The practical use of this assumption is not possible in all sectors of the economy.

(ii) It does not say anything about technical knowledge and non-economic changes.

(iii) It cannot explain a situation where the marginal product is negative.

The Cobb-Douglas production function has been used in Indian agriculture. Other areas where it has been used are rationing and milk production. Desai and Sharma have utilised it to know the relation between production and fertiliser.