The Least-Cost Rule and the Profit-Maximizing Input Rule

A firm buys two kinds of inputs, labor and capital, and each has a price: wage w for labor, rental rate r for capital. The question is never just which input is cheaper. It's which input gives more extra output per dollar. That number is MP_L / w for labor and MP_K / r for capital. If one ratio is bigger, the firm is wasting money — it could get more output by shifting a dollar away from the low-ratio input toward the high-ratio one.

Shifting dollars this way is why the rule is self-correcting. As a firm buys more of the high-ratio input, diminishing returns drag its marginal product down, so its ratio falls. As it buys less of the other input, that input's marginal product rises, so its ratio climbs. The firm keeps shifting until MP_L / w = MP_K / r. At that point no reshuffling of the same budget can raise output any further — this is the least-cost rule. The profit-maximizing input rule adds one more condition on top: the firm keeps hiring an input only while its marginal revenue product exceeds its price, stopping each input where MRP equals price.

This is the producer-side twin of utility maximization in consumer choice, where a shopper equalizes marginal utility per dollar across goods. There the currency is satisfaction; here it's output and revenue. Both rules say the same thing in different clothes: stop treating a budget as fixed slices for each item, and instead chase the best marginal payoff per dollar until every option pays off equally at the margin.