# Least Cost Combination of Inputs

17/04/2020The firm may produce a particular quantity of its product at each of the alternative input combinations that lies on the IQ for that quantity. Since the firm’s goal is to maximize profit, the optimum input combination for producing a particular quantity of its product would be one that would produce the output at the minimum possible cost.

The optimum input combination in this case is known as the least cost combination of inputs. In order to explain the firm’s selection of the least cost combination of inputs, let us suppose that some of the firm’s isoquants (IQs) and iso-cost lines (ICLs) are given in Fig. 1.

Let us now suppose that the firm intends to produce a particular quantity q = q_{3} of its product, and the isoquant for this particular quantity is IQ_{3}. In other words, if the firm uses any of the input combinations lying on IQ_{3}, it would be able to produce the output quantity q = q_{3}.

But, since the different points on IQ_{3}, viz., S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, etc. lie on different ICLs, they produce the same output, viz., q = but at different levels of cost, For we know that a higher (or a lower) ICL represents a higher (or a lower) level of cost.

Therefore, in order to produce the output of q_{3} at the least possible cost, the firm would have to select that point on IQ_{3} that would lie on the lowest possible ICL. In Fig. 8.12, we see that the point S_{3} on IQ_{3 }lies on the lowest possible ICL, viz., L_{3}M_{3}. Any other point on IQ_{3} lies on a higher ICL or a higher level of cost than L_{3}M_{3}.

Therefore, at an output of q_{3}, the least cost combination of inputs is S_{3}(x̅, y̅). In other words, if the firm is to produce an output of q_{3}, it would buy and use the quantity x of input X and the quantity y of input Y. Here it is very important for us to observe that the least cost combination of inputs is the point of tangency (here S_{3}) between the particular isoquant (here IQ_{3}) and an iso-cost line (here L_{3}M_{3}).

Similarly, for producing a particular quantity of output, if the firm is to remain on IQ_{2}, then the least cost combination of inputs would be given by the point T_{2}, because this point is the point of tangency between IQ_{2} and an ICL (i.e., L_{2}M_{2}).

**Maximum Output Combination of Inputs**

In iso-cost lines (ICLs), we have seen that if the prices (r_{X} and r_{Y}) of the inputs (X and Y) are given and constant, then by spending a particular amount of money the firm can buy any one of a large number of input combinations that lie on the corresponding ICL.

Since the firm’s goal is to maximize the level of profit, the optimum input combination in this case would be one that would produce the maximum possible output. Therefore, this input combination is called the maximum- output combination of inputs.

We shall now see with the help of Fig. 8.12, how the maximum output-input combination can be obtained by the firm. Let us suppose that the firm has decided to spend a particular amount of money, TVC_{3}, (TVC stands for total variable cost) for the two variable inputs (X and Y), and the ICL for this expenditure is L_{3}M_{3}.

That is, if the firm is to spend the amount of money TVC_{3}, then it would have to buy some combination that lie on the iso-cost line, L_{3}M_{3}.

Now the points like V_{1}, V_{2} S_{3}, V_{4}, V_{5} lying on L_{3}M_{3} are situated on different isoquants. That is, at different points on the line L_{3}M_{3}, the cost of the firm is the same (= TVC_{3}), but the quantities produced are different.

Since a higher IQ represents a higher level of output, of all the points on L_{3}M_{3}, the profit-maximizing firm would select that one as optimum which lies on the highest possible IQ, i.e., which produces the highest possible level of output. This point is S_{3 }(x̅, y̅) on, IQ_{3}—this point is the maximum-output Combination of inputs subject to the cost constraint of TVC = TVC_{3}.

We have to note here that the maximum-output combination of inputs subject to the cost constraint (here S_{3}) is the point of tangency between the ICL corresponding to the given cost level (here TVC_{3}) and an IQ (here IQ_{3}).

Similarly, if the given ICL of the firm is L_{4}M_{4}, then the maximum-output combination of inputs would be the point R_{4}, because this point is the point of tangency between the line L_{4}M_{4} and an IQ which is here IQ_{4}.

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