Least-Cost Rule
What is Least-Cost Rule?
The least-cost rule says a firm minimizes the cost of any output when the marginal product per dollar is equal across all inputs: MPL/PL = MPK/PK.
A firm produces a given output most cheaply when the extra output per dollar spent is the same for every input. If MPL/PL exceeds MPK/PK, labor gives more output per dollar, so the firm shifts spending toward labor; doing so lowers MPL (diminishing returns) and raises MPK until the ratios equalize. This is the cost-minimization condition that underlies the firm's cost curves, and it is distinct from the profit-maximizing hiring rule (MRP = MRC). Graphically it is the tangency of an isoquant and an isocost line.
Least-Cost Rule: a worked example
A workshop hires labor at 10 dollars an hour and rents machine time at 30 dollars an hour. At its current mix the last worker adds 30 units of output and the last machine hour adds 60 units. Output per dollar is 30 over 10, or 3 units per dollar from labor, against 60 over 30, or 2 units per dollar from capital. The ratios differ, so this mix is not the cheapest. Test a swap. Drop one machine hour, lose 60 units, and save 30 dollars of rent. Two extra workers restore those 60 units at a cost of 2 times 10, or 20 dollars. Same output, 10 dollars cheaper. Keep swapping and diminishing returns pull labor's marginal product down. Once the last worker adds only 20 units while the last machine hour still adds 60, both inputs deliver 2 units per dollar and no further swap saves anything.
The mistake students make with least-cost rule
The most common error is comparing marginal products alone and hiring whichever input looks more productive. Capital adding 60 units seems to beat labor adding 30 until you divide by price, and the cheaper input wins on output per dollar. The second error is treating this as the hiring rule for profit. The least cost rule only asks that the two ratios equal each other, which pins down the cheapest way to make an output the firm has already chosen. Profit maximization is stricter, requiring each input's marginal revenue product to equal its own price, and that is what fixes how much output to make.
Least-Cost Rule questions
How do you know if a firm is using the least-cost combination of inputs?
Divide each input's marginal product by its price, then compare the two results. When MPL over the wage equals MPK over the rental rate, a dollar buys the same extra output wherever it is spent, so no reshuffling can produce that output for less. The test extends past two inputs. Bring land or energy into the mix and every input's marginal product per dollar has to land on the same common value, so a firm is at least cost only when all of them line up at once.
What should a firm do if MPL/PL is greater than MPK/PK?
Shift spending toward labor. A dollar moved into labor buys more extra output than that dollar bought sitting in capital, so the firm can hold output constant while cutting total cost, or hold cost constant while raising output. Hiring more workers drives MPL down through diminishing returns, and using less capital drives MPK up, so the gap narrows by itself until the two ratios meet.
How does the least-cost rule look on an isoquant and isocost diagram?
The least cost mix sits where an isocost line just touches the isoquant for the chosen output. Rearranging MPL over PL equals MPK over PK gives MPL over MPK equals PL over PK, so the slope of the isoquant, the rate at which labor can substitute for capital in production, matches the slope of the isocost line, the rate at which the market lets the firm trade one input for the other. Anywhere else on the isoquant those slopes differ and a lower isocost line is reachable.
Formula / Example
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