Optimal Factor Allocation
Optimal factor allocation refers to the combination of production inputs that minimizes costs while maintaining the same level of output. Every business aims to operate efficiently by keeping production costs as low as possible relative to revenue while maximizing output based on the available production technology. The optimal allocation of inputs can be identified graphically using isocost lines and isoquant curves. An isoquant curve represents different input combinations that yield the same level of output, while an isocost line shows combinations that entail the same total cost. On a Cartesian plane, we can plot the quantities of the two inputs, x1 and x2, along the axes.
The optimal allocation point is found where the isoquant curve and the isocost line are tangent. Point A represents the most economically efficient choice (producer equilibrium) because it minimizes costs while maintaining output at Y. Any other point along the isoquant curve (such as B or C) results in a higher production cost since it lies on a more expensive isocost line. At the optimal point A, the isoquant and isocost lines share the same slope, meaning they are tangent. This tangency has a clear economic interpretation: at this point, the ratio of input prices (- w1 / w2) matches the marginal rate of technical substitution (MRTS) between the two inputs, x1 and x2.
- w1 / w2 = - MRTS
The marginal rate of technical substitution (MRTS) is, in turn, equal to the ratio of the marginal productivities of the two inputs:
MRTS = - dx2 / dx1 = MP1 / MP2
The relative prices of production inputs (- w1 / w2) are determined by the market and are beyond the control of individual firms. Consequently, the slope of the isocost line is dictated by market conditions and cannot be altered by a firm's decisions. Given a specific production technology, firms can only optimize output by reaching the highest feasible isoquant (full capacity utilization) and minimizing costs. Since input prices determine the slope of the isocost line, at any point on the isoquant other than A, a firm has an incentive to adjust its input mix—using more of the input with higher marginal productivity and less of the one with lower marginal productivity. This reallocation further reduces total production costs while maintaining the same output level Y, improving efficiency. The adjustment process continues until the firm reaches the tangency point, where the ratio of input prices matches the ratio of marginal productivities. At this optimal allocation point, there is no further incentive to alter the input combination.
