Firms often use multiple inputs (like labor and capital) to produce output. Cost minimization means choosing the input mix that produces a given output at the lowest total cost by reallocating spending toward more productive uses.

Core idea: equalize marginal product per dollar

When a firm can adjust two variable inputs, it should compare what each additional dollar spent contributes to production. If one input yields more extra output per dollar, the firm can lower cost by shifting spending toward that input (and away from the other) until the advantage disappears.

This rule is the logic behind the syllabus statement: the “last dollar spent” on each input should yield the same marginal product (per dollar) at the cost-minimizing allocation.

The cost-minimizing condition (two-input case)

What the firm equalizes

For inputs such as labor (L) with wage $w$ and capital (K) with rental rate $r$, the firm compares $MP_L/w$ and $MP_K/r$. Cost minimization requires these to be equal at the chosen input combination.

In words: if the last dollar spent on labor adds more output than the last dollar spent on capital, the firm should spend relatively more on labor and relatively less on capital.

Why equality is stable

• As the firm increases use of one input (holding technology constant), diminishing marginal returns typically reduce that input’s marginal product.

• Shifting spending toward the input with higher marginal product per dollar tends to push its marginal product per dollar down (and the other input’s up), moving the firm toward equality.

How firms adjust when the condition is not met

Reallocation logic (no calculations required)

• If $\frac{MP_L}{w} > \frac{MP_K}{r}$:

  • Labor is producing more output per dollar than capital.

  • The firm can reduce total cost for the same output by using more labor and less capital.

• If $\frac{MP_L}{w} < \frac{MP_K}{r}$:

  • Capital is producing more output per dollar than labor.

  • The firm can lower cost by using more capital and less labor.

The key is that the firm is not trying to maximize output for a fixed cost; it is trying to minimize cost for a required output level (or, equivalently, reach a target production level as cheaply as possible).

Connection to profit maximization (why AP Micro cares)

Even though this rule is stated as cost minimization, it supports profit maximization:

• Lower cost at any given output increases profit for the firm.

• When output choice and input choice are both flexible, firms typically:

  • pick an output level consistent with profit-maximizing production decisions, and

  • choose the least-cost input mix for producing that output.

Practical interpretation for AP graphs (isoquant/isocost intuition)

You do not need heavy math to use the rule, but it helps to know what “best mix” means visually:

• An isoquant represents combinations of inputs that produce the same output.

• An isocost line represents combinations of inputs that have the same total cost.

• The least-cost way to produce a given output occurs where the isoquant just touches the lowest possible isocost line, which corresponds to the “equal marginal product per dollar” condition.

An isoquant ($Q=Q^</em>$) shows all input combinations that produce the same output, while parallel isocost lines represent different total-cost levels. The cost-minimizing choice is the tangency point, where the firm reaches $Q^<em>$ on the lowest attainable isocost line. At that tangency, the slopes match, which corresponds to the equal-per-dollar condition $\frac{MP_L}{w}=\frac{MP_K}{r}$. Source

Common pitfalls to avoid

• Equalizing marginal products ($MP_L = MP_K$) is not the rule; input prices matter, so you equalize marginal product per dollar.

• The rule applies to variable inputs the firm can adjust; if an input is fixed in the short run, the firm cannot fully rebalance.

• “Same marginal product per dollar” is about the last dollar spent at the optimum, not average productivity.