Find the long-run cost-minimizing input combination

The last dollar should add the same output across adjustable inputs.

For labor and capital, cost minimization requires MP_L/P_L=MP_K/P_K. If labor yields more additional output per dollar, use relatively more labor and less capital. As input use changes, diminishing marginal productivity adjusts the ratios toward equality.

The condition asks what the last dollar spent on each input contributes. If one dollar on labor produces more than one dollar on capital, the current bundle cannot be least cost: shifting spending toward labor can maintain output at lower cost or produce more with the same budget. The ratios must use input prices, not product price.

Units provide a check. MP_L is output per worker and P_L is dollars per worker, so MP_L/P_L is output per dollar. Capital’s ratio must have the same unit. Comparing a raw MP of 40 with a rental price of $20 would mix output and dollars rather than form a usable efficiency measure.

Reallocate the input budget

Labor’s MP is 24 at a $12 wage, or 2 output units per dollar. Capital’s MP is 30 at a $20 rental rate, or 1.5 per dollar. The current combination is not least cost. Shifting spending toward labor raises output for the same cost until the ratios equalize.

If the firm needs to keep output fixed, it can reduce capital spending and add enough labor to replace the lost capital output. Because labor initially produces more per dollar, the replacement costs less. As labor use rises, its marginal product may fall. As capital use falls, its marginal product may rise. These changes bring the ratios together.

The least-cost condition is not MP_L=MP_K unless input prices are equal. It is also not the profit-maximizing output rule. Cost minimization chooses how to produce a given output. MR=MC chooses how much output to produce.

An equivalent condition is MP_L/MP_K=P_L/P_K, which equates the technical rate at which inputs can substitute with their market price ratio. The per-dollar form is usually easier for tables. Both express the same idea when an interior combination is feasible.

Corner solutions can occur if one input cannot substitute for the other or one ratio remains higher over the feasible range. The textbook equality describes an interior optimum. A factory may still require at least one machine regardless of labor productivity. Use constraints stated in the stem.

After minimizing cost for each possible output, the firm still compares marginal revenue with the resulting marginal cost to choose output. A technically efficient production plan can be unprofitable if buyers’ willingness to pay is too low. Do not let “least cost” become “maximum profit” without the revenue side.

If a question asks for a missing input price, use equality as an equation. If MP_L=18, the wage is $9, and MP_K=30, cost minimization requires 30/P_K=18/9=2, so capital’s rental price must be $15. Substitute back and confirm that both inputs add two output units per dollar.

The condition assumes the firm can substitute at the margin and uses a positive amount of each input. A required machine or other technological minimum can create a corner. Within the basic interior model, however, unequal ratios prove that the same output can be produced more cheaply. Move expenditure away from the lower ratio and toward the higher one.