Firms use more than one input. Choosing the mix is the same problem as the shopper's in Marginal Utility per Dollar: get the most for each dollar. Here the "utility" is output, and the inputs are labor and capital, valued by the MRP idea of Marginal Revenue Product and Hiring.
Key idea
The least-cost rule: a firm produces any given output at the lowest cost when the last dollar spent on each input adds the same output, $$\frac{MP_L}{w} = \frac{MP_K}{r},$$ where $w$ is the wage and $r$ is the rental price of capital. The profit-maximizing rule adds that each input's MRP equals its price: $MRP_L = w$ and $MRP_K = r$.
Why it works
If the last dollar on labor adds 3 units and the last dollar on machines adds 2, move a dollar from machines to labor: output rises by about 1 unit at no extra cost. Keep moving dollars until the two ratios are equal. Least cost doesn't guarantee the best scale, though: a firm could make the wrong amount cheaply. Profit maximization also needs each input hired until its MRP equals its price, which makes both ratios $MRP/\text{price}$ equal 1.
Worked example: Which input to add?
A factory's last worker adds 20 units a day and costs \$100; its last machine adds 30 units and costs \$200 a day to rent. Is it using the least-cost mix?
- Labor: $20 \div 100 = 0.2$ units per dollar. Capital: $30 \div 200 = 0.15$ units per dollar.
- A dollar does more on labor, so shift spending toward labor and away from machines.
- As it hires more workers, $MP_L$ falls; with fewer machines, $MP_K$ rises, until the ratios meet.
Answer: No: use more labor and less capital.
Worked example: Profit maximizing as well
Product price \$2. At the current mix, $MP_L = 15$ with $w = \$30$, and $MP_K = 40$ with $r = \$80$. Is the firm at the least-cost mix? Is it maximizing profit?
- $MP_L/w = 15/30 = 0.5$ and $MP_K/r = 40/80 = 0.5$: equal, so it's the least-cost mix.
- $MRP_L = 15 \times 2 = \$30 = w$ and $MRP_K = 40 \times 2 = \$80 = r$: both equal their prices.
Answer: Yes to both.
Common mistake
Hiring more of the input with the higher marginal product without dividing by its price. A machine that adds 30 units beats a worker who adds 20, but not if the machine costs twice as much. Check: always compare $MP \div \text{price}$.