Expansion path may be defined as the locus of efficient combinations of the factors (the points of tangency between the isoquants and the iso-cost lines). It is the curve along which output or expenditure changes, when factor prices remain constant.
Hence, the optimal proportion of the inputs will remain unchanged. It is also known as scale-line, as it shows how the producer will change the quantities of the two factors, when it raises the scale of production.
The expansion path may have different shapes and slopes depending upon the relative prices of the factors used and shape of the isoquant. In case of constant returns to scale (homogenous production function), the expansion path will be a straight line through the origin, indicating constancy of the optimal proportion of the inputs of the firm, even with changes in the size of the firm’s input budget. (Fig. 7.12 (b)). In short-run, however, the expansion path will be parallel to X-axis (when capital is hold constant at K shown in Fig. 7.12 (b)).
As expansion path depicts least cost combinations for different levels of output, it shows the cheapest way of producing each output, given the relative prices of the factors. It is difficult to tell precisely the particular point of expansion path at which the producer in fact be producing, unless one knows the output which he wants to produce or the size of the cost or outlay it wants to incur.
But, this much is certain that though for a given isoquant map, there can be different expansion paths for different relative prices of the factors. Yet, when prices of the variable factors are given, a rational producer will always try to produce at one or the other point of the expansion path.