As the firm expands its output, the cost minimization points trace out the expansion path. The expansion path connects optimal input combinations as the scale of production expands. It is a line that connects the combination of inputs that minimize the cost of producing each amount of output at the given relative prices.
An expansion path is a curve in a graph with quantities of two inputs, typically physical capital and labor, plotted on the axes. The path connects optimal input combinations as the scale of production expands. A producer seeking to produce a given number of units of a product in the cheapest possible way chooses the point on the expansion path that is also on the isoquant associated with that output level.
The points on an expansion path occur where the firm's iso-cost curves, each showing fixed total input cost, and its isoquants, each showing a particular level of output, are tangent; each tangency point determines the firm's conditional factor demands. As a producer's level of output increases, the firm moves from one of these tangency points to the next; the curve joining the tangency points is called the expansion path.
If the production function is homothetic, the Expansion Path will be linear. The shape of the isoquants determines the shape of the Expansion Path. An Expansion Path that slopes toward an axis indicates an inferior input on the other axis; i.e., use of the input actually declines as output increases. A Cobb–Douglas production function is an example of a production function that has an expansion path which is a straight line through the origin.
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