#### Cost Equations of a Firm:

Cost Equation in the Short Run:

If the firm uses two variable inputs, X and Y, along with one or more fixed inputs, to produce its output, then we may express its total cost of production (TC) as:

TC = rXX + rYY + F                        (8.25)

Here TC = total cost per period

x = quantity used per period of input X

y = quantity used per period of input Y

rX = price of input X (given and constant)

rY = price of input Y (given and constant)

F = total fixed cost (TFC) per period, i.e., cost for the fixed inputs = constant (by definition).

Equation (8.25) is called the cost (TC) equation of the firm.

Here, the firm’s total variable cost (TVC) per period, i.e., the cost for the variable inputs, would be

TVC = TC – F = rXx + rYy                             (8.26)

Eqn. (8.26) may be called the TVC-equation of the firm.

Cost Equation in the Long Run:

The inputs that the firm uses are all variables in the long run. By definition, no inputs are fixed in the long run. Therefore, if the firm uses two inputs, X and Y, then both of them would be variable in the long run, and its (total) cost equation would be

TC = rXx + rYy                                    (8.29)

Also, the firm’s long-run iso-cost equation would be

where TC = total cost = total variable cost and = a fixed amount of TC.

Equations (8.27) and (8.28) are known as the iso-cost equations of the firm. It may be noted that (8.27) and (8.28) are two different forms of the same equation, since F is constant. Therefore, the (input) combinations of x and y for which TC would be constant would be the same as those for which TVC also would be constant (= = – F).