The economists frequently assumes that the problem of optimum input combinations has been solved and conducts his analysis of the firm in terms of its revenues and costs expressed as functions of output. The cost function of the firm gives the functional relationship between total cost and total output. If C represents total cost and Q represents the level of the output, then the cost functions is represented as C=C (Q). The same level of output can be produced with the help of different cost combinations. The cost function gives the least cost combinations for the production of different levels of output.
Cost functions are derived functions. They are derived from the production functions, which describes the available efficient methods of production at any particular point of time. The cost function can be deduced from the inputs combinations of the firm. The input prices of the two inputs of production labor (L) and capital (K) are given to be constant as the wage rate (w) and rent (r), respectively. If L and K are the amounts of the two inputs that are used for the production of the output level Q, the firm will always select those combinations of the two inputs, which lie on the expansion path. Along any expansion path the level of output increases as we gradually depart from the origin. Within the non-inferior zone of the factors of production, their total employment will also increase as we move along the expansion path. Therefore we can say that along any expansion path the demand for any factor of production will depend on the level of output to be produced. So, if L and K are the amounts of the factors of production and Q is the level of output then it can be said that L and K are functions of Q.
L = g1(Q)
And, K = g2(Q)
Now, following the equation of the costs line, the total cost (C) for producing the output level Q is given by
C = L. w + K.r
or C = w. g1(Q) + r. g2(Q)
or C = C(Q).
Since, w and r are constant C is only a function of Q. This function is called the total cost functions of the firm. The function shows that the total cost of the firm depends on the output to be produced. The costs function is deduced from the expansion path of the firm.
The cost function derived from the expansion path of the firm represents the cost function in its long run nature as in this case we have assumed that both the factors of production are variable.