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:

- There are two factors. Capital and labor.
- All units of capital and labor are homogeneous.
- The prices of factors of production are given and constant.
- Money outlay at any time is also given.
- 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 IQ_{1}, IQ_{2} and IQ_{3} 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 IQ_{3} 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 IQ_{2} 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.

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