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Isoquants are a geometric representation of the production function. Various combinations of factor inputs can produce the same level of output. Assuming continuous variation in the possible combination of labour and capital, we can draw a curve by plotting all these alternative combinations for a given level of output. This curve, which is the locus of all possible combination is called the ‘isoquant’.

Types of Isoquants

The production isoquant may assume various shapes depending on the degree of substitutability of factors.
Linear Isoquant: This type assumes perfect substitutability of factors of production. A given commodity may be produced by using only capital, or only labour, or by an infinite combination of K and L.
Input-output Isoquant: This assumes strict complementarity, that is, zero substitutability of the factors of production. There is only one method of production for any one commodity. The isoquant takes the shape of a right angle. This type of isoquant is called “Leontief isoquant.”
Kinked Isoquant: This assumes limited substitutability of K and L. There are only a few processes for producing any one commodity. Substitutability of factors is possible only at the kinks. It is also called “activity analysis isoquant” or “linear-programming isoquant” because it is basically used in linear programming.
Smooth, Convex Isoquant: This form assumes continuous substitutability of K and L only over a certain range, beyond which factors cannot substitute each other. This isoquant appears as a smooth curve convex to the origin.
Isoquants show the following characteristics:
They slope downward to the right.
It is convex to origin.
It is smooth and continuous.
Two isoquants do not intersect.

Marginal Rate of Technical Substitution

The marginal rate of technical substitution of L for K (denoted by MRTSL, K) is defined as the number of units of input K that a producer is willing to sacrifice for an additional unit for L so as to maintain the same level of output (i.e., remain on the same isoquant). The marginal rate of technical substitution (MRTS) is numerically equal to the negative of the slope of an isoquant at any one point and is geometrically given by the slope of the tangent to the isoquant at that point. The marginal rate of technical substitution of L for K is also equal to the ratio of the marginal product of capital to the marginal product of labour. Thus,
Slope of an Isoquant = –dK/dL = MRTSLK = MPL/MPK
It can be shown that the marginal rate of technical substitution of labour for capital (MRTSLK) is equal to the ratio of marginal productivities of labour and capital. In a change from combination 3 to 4 in the figure above, let DL (=1) be the increase in labour employed and DK (=3) be the decrease in capital employed. Since output remains the same at combinations 3 and 4, the gain in output due to increase in labour employed must neutralise the loss in output due to decrease in capital employed.

Optimal Factor Combinations

The theory of production may be viewed from two angles which are dual to each other. A firm may decide to produce a particular level of output and then attempt to minimise the cost of total inputs or it may attempt to maximise its output subject to a cost constraint.
A firm spends money on two inputs only, X and Y. It decides its budget and knows the price of each of the inputs which remains constant.The slope of the budget line or the isocost line will be – OA , where
OA =Cost/Pr ice of Y and OB =Cost/Pr ice of X
Therefore, slope of AB = – OA/OB
= Price of X/Price of Y =- PX/PY
The negative sign indicates negative slope. In absolute terms, the slope of the budget line is equal to the price ratio of the two inputs.
The budget line of the firm has been superimposed on its isoquant map. The firm would be in equilibrium at a point where an isoquant is tangent to the budget line AB, i.e., point E. Thus in equilibrium, the firm produces on the isoquant Q2 and uses OX1 units of input X and OY1 units of input Y. At point E, the slope of the isoquant Q2 is equal to the slope of the budget line, i.e., the marginal rate of technical substitution of X and Y is equal to the ratio of prices of two inputs.
Thus, to minimise production costs (or to maximise output for a given cost outlay), the extra output or marginal product spent on labour must be equal to the marginal product per unit spent on capital.