The expansion path in economics refers to the trajectory showing the optimal combination of inputs a firm should use to produce different output levels while minimizing production costs. Recall that isoquants represent various combinations of inputs that yield the same output level, while isocost lines depict the combinations of inputs that can be purchased at a given cost. The expansion path is derived by analyzing the tangency points between isoquant and isocost lines as output expands.

By identifying the points of tangency between isoquants and isocost lines, we can determine the efficient input combinations for different output levels. Connecting these points across all possible output levels forms the expansion path, illustrating how the optimal mix of inputs changes as output levels vary.

The long-run average total cost (LRATC) curve can be derived using cost and output information from the expansion path at different output levels. The LRATC curve represents the lowest possible average cost of production at each output level when all inputs are variable.

Understanding the expansion path and LRATC curve helps firms make strategic decisions about production levels and input usage to minimize production costs and maximize efficiency.