Which of the following statements about the expansion path is true?

It shows the optimal input mix for a firm to produce various levels of output while holding technology and input prices constant. Here is a breakdown of why this is the correct answer and why the others are incorrect: 

• Expansion Path Defined: An expansion path is a graph that plots the combinations of inputs (like capital and labor) a firm should use to produce different levels of output at the lowest possible cost. It is derived by finding the points where a series of isoquants (curves showing all combinations of inputs that yield the same output) are tangent to a series of isocost lines (lines showing all combinations of inputs that cost the same amount). The tangency points represent the minimum cost for each level of output.

• Key Assumptions: The concept relies on two crucial assumptions: 

1. Constant Input Prices: The prices of the inputs (e.g., the wage rate for labor and the rental rate for capital) must remain constant. If input prices were to change, the slope of the isocost lines would change, and the entire expansion path would shift. 2. Constant Technology: The production technology, which determines the shape and position of the isoquants, is assumed to be fixed. A change in technology would shift the isoquants and, consequently, the expansion path.

• Why the other options are incorrect:

o 1 is incorrect: The expansion path assumes constant input prices, not changing ones. If input prices changed, the expansion path would shift, not simply show the result of those changes. o 2 is incorrect: This describes a different, but related, concept. The goal of a firm in this case is cost minimization, not output maximization. While the expansion path can be used to analyze output maximization at a fixed cost (by finding the highest isoquant a firm can reach with a given isocost line), its primary function is to show the minimum cost for producing a given output. o 4 is incorrect: The expansion path is a long-run concept. In the short run, at least one input is fixed, so a firm can only expand by increasing its variable inputs. This would be represented by a horizontal line (if capital is fixed) on the isocost/isoquant graph, which is not a true expansion path. The expansion path is a long-run tool because it assumes all inputs are variable and can be adjusted.