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An expansion path is a fundamental concept in economics that illustrates the optimal combination of inputs a firm would choose as it increases its level of production while minimizing costs. It depicts the variations in input usage and the corresponding levels of output as the firm expands its scale of operations. The expansion path shows how a firm's input mix changes while keeping the output constant or increasing it, reflecting the firm's response to changes in its production scale.

In the context of a linear homogeneous production function, the expansion path takes the form of a straight line. A linear homogeneous production function is characterized by constant returns to scale, where doubling all inputs results in a doubling of output. This production function can be represented mathematically as:

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