The Least-Cost Rule & Choosing Between Inputs

You’ll be able to

Apply the least-cost rule to allocate spending between two inputs
Distinguish the least-cost rule from the profit-maximizing rule
Predict input substitution following a change in an input price

Two different questions

The least-cost rule answers "given that I want this much output, what is the cheapest input mix?" The profit-maximizing rule answers "how much output should I make at all?" A firm can satisfy the first and still be producing the wrong quantity. Both must hold at the optimum, and the exam asks for them separately.

The two rules

least-cost: MP_L / P_L = MP_K / P_K · profit-maximizing: MRP_L / P_L = MRP_K / P_K = 1

Least cost equalizes output per dollar. Profit maximization additionally requires each ratio to equal one — each input paying for itself exactly.

Why per dollar, not per unit

Equalize marginal product per dollar, not marginal product. If labor delivers 20 units at $10 (2 units per dollar) while capital delivers 45 units at $30 (1.5 units per dollar), labor is the better buy despite capital being more productive per unit. Shifting spending toward labor raises output at the same total cost, and it should continue until the two ratios meet.

Input substitution

When an input becomes more expensive, its output-per-dollar falls, so the firm substitutes away from it toward the relatively cheaper input. This is why a rising minimum wage encourages automation and why cheap capital encourages capital-intensive production. The substitution is a movement toward the new least-cost combination for the same output, distinct from any change in how much output the firm chooses to make.

Worked example

Labor costs $20 with a marginal product of 60 units; capital costs $50 with a marginal product of 100 units. Is the firm minimizing cost? If not, what should it do?

1.Labor: MP/P = 60/20 = 3 units per dollar.
2.Capital: MP/P = 100/50 = 2 units per dollar.
3.The ratios are unequal, so the firm is not minimizing cost.
4.Labor delivers more output per dollar, so shift spending toward labor and away from capital.
5.As more labor is used its marginal product falls, and as less capital is used capital's marginal product rises, until the ratios converge.

Answer: Not cost-minimizing. The firm should use more labor and less capital. Diminishing returns on labor and rising marginal product on the reduced capital will bring the two ratios to equality, at which point the same output is being produced at lower cost.

Watch out

Satisfying the least-cost rule does not mean the firm is maximizing profit. It means the current output is being produced as cheaply as possible — which is still the wrong thing to do if the output itself is wrong.

Checkpoint

A firm minimizes cost for a given output when:

Checkpoint

The wage rises while the price of capital is unchanged. A cost-minimizing firm will:

Checkpoint

The profit-maximizing input rule differs from the least-cost rule in requiring that:

On the exam

Write both ratios as decimals side by side before concluding anything. The comparison is the whole answer, and rubrics reward showing the two computed values rather than just naming which input to use more of.

Answer the 3 checkpoints as you read.