Least-Cost Rule: MPL/PL = MPK/PK With a Worked Table

The least-cost rule says a firm produces any given level of output at the lowest possible cost when the last dollar spent on each input adds the same amount of output: MPL ÷ PL = MPK ÷ PK. If one input gives more output per dollar than the other, the firm can cut cost by shifting spending toward it until the two ratios are equal.

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A competitive labor market, where labor demand (MRP) and labor supply set the wage. Tap it to drag the curves, open the full version, or put it on your own site free, or turn it into a five-minute class activity.

The least-cost rule tells a firm the cheapest mix of labor and capital. It does not tell the firm how much to produce. That is the job of the profit-maximizing rule, which says to hire each input until its marginal revenue product equals its price. This guide works through both rules with one table, shows how they fit together, and covers how AP Microeconomics Topic 5.3 tests them. If marginal revenue product is new to you, read the factor markets and MRP guide first.

The notation

• MPL is the marginal product of labor: the extra output from one more unit of labor.
• PL is the price of labor, usually the wage.
• MPK is the marginal product of capital: the extra output from one more unit of capital, such as one more machine.
• PK is the price of capital, usually the rental price of a machine for the period.

MPL ÷ PL is "output per dollar spent on labor," and MPK ÷ PK is "output per dollar spent on capital." The least-cost rule asks those two numbers to match. The glossary entry on the least-cost rule has the one-line version.

Why the ratios have to be equal

Suppose a firm's last dollar on labor buys 0.4 units of output and its last dollar on capital buys only 0.2 units. Moving one dollar from capital to labor loses 0.2 units and gains 0.4 units, a net gain of 0.2 units with no change in total spending. The firm keeps finding these free gains as long as the ratios differ. Because of diminishing marginal returns, adding labor lowers MPL and cutting capital raises MPK, so the ratios move toward each other. Once they are equal, no reshuffle of spending can raise output, which also means no cheaper way exists to make that output.

You may recognize the logic. It is the same equal-marginal principle as the consumer's utility-maximizing rule, MUx ÷ Px = MUy ÷ Py, from the utility maximization guide, with output in place of utility.

Worked example, part 1: the cheapest mix

A firm hires labor at PL = $20 per worker and rents machines at PK = $40 per machine. The table lists each input's marginal product. As in a typical textbook table, the marginal products are given for each input on its own, and total output is the sum of the marginal products of the units hired.

Question: the firm has $180 to spend. What mix gives the most output?

Spend each block of money where it buys the most output per dollar, in order:

• 1st machine (0.60 per dollar), running total $40
• 1st worker (0.50), running total $60
• 2nd worker (0.40), running total $80
• 2nd machine (0.40), running total $120
• 3rd worker (0.30), running total $140
• 3rd machine (0.30), running total $180

The budget runs out at 3 workers and 3 machines, where MPL ÷ PL = 6 ÷ 20 = 0.30 and MPK ÷ PK = 12 ÷ 40 = 0.30. The ratios are equal, so the least-cost rule holds.

Output = (10 + 8 + 6) + (24 + 16 + 12) = 24 + 52 = 76 units.

Check it against other $180 mixes.

• 5 workers and 2 machines: cost = 5 × $20 + 2 × $40 = $180. Output = (10 + 8 + 6 + 4 + 3) + (24 + 16) = 31 + 40 = 71 units.
• 1 worker and 4 machines: cost = $20 + 4 × $40 = $180. Output = 10 + (24 + 16 + 12 + 8) = 10 + 60 = 70 units.

Both alternatives spend the same $180 and produce less. Look at the 5-worker mix: MPL ÷ PL = 3 ÷ 20 = 0.15 while MPK ÷ PK = 16 ÷ 40 = 0.40. Capital buys more output per dollar, so the rule says shift toward capital. Letting go of the 4th and 5th workers saves $40 and loses 4 + 3 = 7 units; renting the 3rd machine for that $40 adds 12 units. Output rises by 5, from 71 to 76, at the same cost.

The same mix can be read the other way. Since 76 units is the most the firm can make for $180, $180 is the least the firm can spend to make 76 units. "Most output for a budget" and "lowest cost for an output" are two descriptions of the same point. The least-cost input combination calculator runs this check for any numbers.

Worked example, part 2: the profit-maximizing quantities

Now suppose the firm sells its output in a perfectly competitive market at P = $5, so marginal revenue is $5. Each input's marginal revenue product is its marginal product times marginal revenue.

The profit-maximizing rule says to hire each input until MRP = MRC, where MRC is marginal resource cost (some textbooks and calculators call it marginal factor cost, MFC). In competitive input markets, MRC is just the input's price, so the rule becomes MRPL = PL and MRPK = PK.

• Labor: MRPL = $20 = PL at the 4th worker. The 5th worker would add $15 and cost $20, so stop at 4.
• Capital: MRPK = $40 = PK at the 4th machine. The 5th machine would add $20 and cost $40, so stop at 4.

At 4 workers and 4 machines, both versions of the rule hold:

• Profit-maximizing: MRPL ÷ PL = 20 ÷ 20 = 1 and MRPK ÷ PK = 40 ÷ 40 = 1.
• Least-cost: MPL ÷ PL = 4 ÷ 20 = 0.20 and MPK ÷ PK = 8 ÷ 40 = 0.20.

Output = (10 + 8 + 6 + 4) + (24 + 16 + 12 + 8) = 28 + 60 = 88 units. Revenue = 88 × $5 = $440. Spending on the two inputs = 4 × $20 + 4 × $40 = $240. Revenue minus input spending = $200.

The 4th worker adds exactly $20 of revenue and costs exactly $20, and the 4th machine adds exactly $40 and costs exactly $40, so profit would also be $200 at 3 workers and 3 machines (revenue 76 × $5 = $380 minus $180). The usual AP convention is to hire up to and including the unit where MRP equals MRC. The optimal hiring quantity calculator applies that convention to labor.

How the two rules fit together

Compare 2 workers and 2 machines. The ratios are equal (MPL ÷ PL = 8 ÷ 20 = 0.40 and MPK ÷ PK = 16 ÷ 40 = 0.40), so this is the least-cost way to make (10 + 8) + (24 + 16) = 58 units. But revenue is 58 × $5 = $290 and input spending is 2 × $20 + 2 × $40 = $120, leaving $170, which is less than $200. The firm is producing 58 units as cheaply as possible, and it is still producing too little.

Every profit-maximizing combination satisfies the least-cost rule, but most least-cost combinations do not maximize profit. Multiply both sides of MPL ÷ PL = MPK ÷ PK by MR and you get MRPL ÷ PL = MRPK ÷ PK. The profit rule adds one more condition: both ratios must equal 1.

What happens when an input price changes

Suppose the wage rises from $20 to $40 while machines still rent for $40. At 3 workers and 3 machines, MPL ÷ PL falls to 6 ÷ 40 = 0.15 while MPK ÷ PK stays at 0.30. Capital now buys twice as much output per dollar, so the firm substitutes toward machines and away from workers. This substitution toward capital is one reason a higher wage reduces the quantity of labor firms hire, and it is why a rise in the price of one input can raise demand for a substitute input. On a graph, the same result appears as the tangency of an isoquant and an isocost line, covered in the isoquants and isocosts guide.

How it shows up on the AP exam

Questions on this topic usually give a table of marginal products (or total products) for labor and capital along with their prices, then ask you to:

• Compute MP per dollar for each input and say whether the firm is using the least-cost mix.
• State which input to use more of when the ratios differ, and explain why in terms of output per dollar.
• Find the profit-maximizing quantity of each input by comparing MRP with the input price.
• Predict the effect of a change in a wage or rental price on the quantity of each input hired.

Common mistakes:

• Comparing marginal products without dividing by price. A machine that adds 16 units is not better than a worker who adds 8 if the machine costs twice as much. Per dollar, they tie.
• Stopping at the least-cost rule when the question asks for profit-maximizing quantities. Equal ratios are necessary for maximum profit but not enough.
• Using MP × P with a price maker. For a firm that is not a price taker in its output market, MRP = MP × MR, and MR is below price.
• Confusing total and marginal product. If the table gives total product, find each marginal product first by taking differences.

The AP Micro 5.3 topic page lists the related terms, and the factor markets sandbox lets you move the wage and watch the hiring decision change.

Summary

The least-cost rule, MPL ÷ PL = MPK ÷ PK, finds the cheapest way to produce a given output by equalizing output per dollar across inputs. The profit-maximizing rule goes further and sets each input's MRP equal to its price, which also fixes how much the firm produces. A profit-maximizing firm always uses a least-cost mix, but a least-cost mix can still be the wrong size. When an input gets more expensive, its output per dollar falls and the firm shifts toward the other input.