How to Find the Least-Cost Combination of Inputs
A firm reaches its least-cost input mix when marginal product per dollar is equal across inputs: MPL ÷ PL = MPK ÷ PK.
The Least-Cost Rule formula
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Compare marginal product per dollar on labor and capital to see whether the input mix is already least-cost.
Extra output from one more unit of labor.
The wage per unit of labor.
Extra output from one more unit of capital.
The rental rate per unit of capital.
A dollar buys 3 units of output through labor against 2 through capital, so use more labor and less capital until MPL falls and MPK rises enough to close the gap.
- Output per dollar on labor
- 3
- Output per dollar on capital
- 2
- Output per dollar gap
- 1
- Labor price that would equalize the ratios
- $15
A marginal product of 30 divided by a labor price of $10.
A marginal product of 60 divided by a capital price of $30.
Labor's output per dollar minus capital's. Zero is the least-cost condition.
Holding the marginal products fixed, a labor price of $15 would make MPL divided by PL equal MPK divided by PK.
How to calculate Least-Cost Rule, step by step
- 1Find each input's marginal product. MPL is the extra output from one more unit of labor; MPK is the extra output from one more unit of capital.
- 2Divide each by its price. MPL ÷ PL and MPK ÷ PK give the extra output bought by the last dollar spent on each input.
- 3Compare the two ratios. If MPL ÷ PL is larger, labor is the better buy, so use more labor and less capital until the ratios move together.
- 4Stop when the ratios are equal. MPL ÷ PL = MPK ÷ PK means the last dollar spent on each input buys the same extra output, so no reshuffling can produce that output more cheaply.
- 5Recheck after any price change. A wage increase lowers MPL ÷ PL, so the firm substitutes toward capital until the two ratios are equal again.
Worked example: Least-Cost Rule
A firm's last worker adds 30 units and costs $10, so MPL ÷ PL = 30 ÷ 10 = 3 units per dollar. Its last machine adds 60 units and costs $30, so MPK ÷ PK = 60 ÷ 30 = 2 units per dollar. Labor buys more output per dollar, so the firm hires more labor and rents less capital. As it does, MPL falls and MPK rises. Once MPL is 24 and MPK is 72, the ratios are 24 ÷ 10 = 2.4 and 72 ÷ 30 = 2.4, so the input mix is now least-cost.
Least-Cost Rule questions
What happens to the least-cost combination when wages rise?
The firm substitutes toward capital. A higher wage shrinks MPL ÷ PL, so the firm uses less labor, which pushes MPL back up, and more capital until the ratios are equal again.
Is the least-cost rule the same as the profit-maximizing rule?
No. The least-cost rule finds the cheapest way to produce a given output, while profit maximization also picks the output level, requiring marginal revenue product ÷ input price = 1 for every input.
What does marginal product per dollar mean?
It is the extra output a dollar spent on an input buys, found by dividing marginal product by the input's price. Spending shifts toward whichever input delivers more output per dollar.
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