Marginal Product and Marginal Cost With Changing Input Prices

Why Marginal Product Helps Explain Marginal Cost

A firm’s marginal cost depends not only on what it pays for an additional unit of a variable input, but also on how much extra output that input produces. If one more labor-hour adds many units of output, its cost is spread across those units. If it adds only a few, each extra unit of output is costly. The connection between these two quantities makes it possible to compare cost behavior across production settings with different input prices.

The analysis here is short run: capital and other fixed inputs are held constant, while the firm varies one input, such as labor. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Let output be $q=f(L;\bar K)$, where $L$ is the variable input, such as labor, and $\bar K$ represents fixed inputs. The marginal product of labor is $MP_L=\partial q/\partial L$: the additional output associated with a small increase in labor, holding fixed inputs constant. If each unit of labor costs $w$, the marginal cost is the additional total cost associated with a small increase in output, $MC=dTC/dq$. With one variable input and a constant input price, positive marginal product, and a differentiable production function, $MC=w/MP_L$.

The formula is a short-run result for a particular operating point. It does not say that all firms have the same marginal cost, or that marginal cost is determined by the input price alone. It depends on both the price $w$ and the productivity $MP_L$ at the relevant input level.

Deriving the Relationship

Suppose short-run total cost is $TC=F+wL$, where $F$ is fixed cost and $w$ is a constant price per unit of labor. Assume output is differentiable in $L$ and $MP_L>0$, so a small increase in output can be achieved by increasing labor. The chain rule gives the marginal cost.

\frac{dTC}{dq} = \frac{dTC}{dL}\frac{dL}{dq} = w\frac{1}{dq/dL} = \frac{w}{MP_L}.

The fixed cost disappears because $dF/dL=0$. Also, $dL/dq=1/MP_L$: if one additional labor unit produces $MP_L$ additional output units, then producing one additional output unit requires $1/MP_L$ labor units at the margin. Multiplying those labor units by the price per labor unit gives marginal cost.

Find marginal product.
Differentiate the production function with respect to the variable input, holding fixed inputs constant.

Identify the marginal input price.
Use the cost of one additional unit of the variable input, not a fixed cost or an average cost.

Divide input price by marginal product.
When marginal product is positive and the assumptions hold, $MC=w/MP_L$. State the output unit for the result.

For a discrete production schedule, the corresponding calculation uses changes over an interval rather than derivatives. If labor rises by $\Delta L$, output rises by $\Delta q$, and the input price is constant, then the change in variable cost is $w\Delta L$. Thus the interval’s marginal cost is $w\Delta L/\Delta q$, or $w/(\Delta q/\Delta L)$. This is an average over that output interval, not an exact point marginal cost.

Worked Applications

Worked Example: Diminishing Marginal Product Raises Marginal Cost

A hypothetical workshop has fixed equipment and uses labor-hours $L$ to produce output $q=12L-L^2$, for $0\leq L\leq5$. Labor costs $w=\$18$ per hour, and fixed cost is $\$90$. Find marginal product and marginal cost at $L=2$ and $L=4$. Explain the change.

Method. Differentiate the production function to obtain marginal product, then divide the labor price by marginal product. The fixed cost is not part of the marginal cost calculation because it does not change with labor.

The marginal product is $MP_L=12-2L$. At $L=2$, $q=12(2)-2^2=20$, and $MP_L=12-2(2)=8$ output units per labor-hour. Therefore,

MC=\frac{\$18\text{ per labor-hour}}{8\text{ output units per labor-hour}} =\$2.25\text{ per output unit}.

At $L=4$, $q=12(4)-4^2=32$, and $MP_L=12-2(4)=4$ output units per labor-hour. Therefore,

MC=\frac{\$18\text{ per labor-hour}}{4\text{ output units per labor-hour}} =\$4.50\text{ per output unit}.

The calculation checks: at both labor levels the wage is unchanged, but marginal product falls from 8 to 4, so marginal cost doubles from $\$2.25$ to $\$4.50$. A common mistake is to add the $\$90$ fixed cost to the wage before dividing. That would incorrectly treat a cost unchanged by extra production as marginal; the correction is to use $w=\$18$ alone.

Worked Example: Comparing Two Input Prices at the Same Productivity

A hypothetical packing line produces $q=24\sqrt{L}$ packages per shift, with labor as its only variable input. At $L=9$ labor-hours, compare marginal cost when labor costs $\$20$ per hour with marginal cost when the price is $\$30$ per hour. Assume the production technology and operating labor level are unchanged.

Method. First find the productivity at the stated labor level. Then apply each input price to that same marginal product.

At $L=9$, output is $q=24\sqrt{9}=24(3)=72$ packages per shift. Differentiating the production function gives

MP_L=\frac{12}{\sqrt{L}}, \qquad MP_L\big|_{L=9}=\frac{12}{3}=4 \text{ packages per labor-hour}.

With a labor price of $\$20$ per hour, marginal cost is $\$20/4=\$5.00$ per package. With a labor price of $\$30$ per hour, marginal cost is $\$30/4=\$7.50$ per package. The second marginal cost is 1.5 times the first, matching the 1.5-fold increase in the labor price because marginal product has not changed.

As a check, the extra labor-hour costs $\$20$ or $\$30$ and produces 4 additional packages at the margin, giving the same cost per package. A common mistake is to compare the two prices without holding productivity fixed. Here the comparison is valid specifically because technology and $L$, and therefore $MP_L$, are the same.

Worked Example: Marginal Cost From a Discrete Production Schedule

A hypothetical print shop records the following output as labor-hours increase: 0 hours produce 0 batches, 1 hour produces 10 batches, 2 hours produce 18 batches, 3 hours produce 24 batches, and 4 hours produce 28 batches. The labor price is $\$12$ per hour. Find the interval marginal cost for each additional labor-hour.

Method. For each interval, calculate the additional output $\Delta q$, then divide the added labor cost $w\Delta L=\$12(1)$ by that output change.

Labor interval	Output change $\Delta q$	Labor cost change	Interval marginal cost
0 to 1 hour	10 batches	$12	$12 / 10 = $1.20 per batch
1 to 2 hours	8 batches	$12	$12 / 8 = $1.50 per batch
2 to 3 hours	6 batches	$12	$12 / 6 = $2.00 per batch
3 to 4 hours	4 batches	$12	$12 / 4 = $3.00 per batch

For example, between 2 and 3 hours, output rises from 18 to 24 batches, so $\Delta q=6$. The extra hour costs $\$12$; therefore, interval marginal cost is $\$12/6=\$2.00$ per batch. Each row checks by multiplying its marginal cost by its output change to recover the $\$12$ labor cost. The declining output increments make the interval marginal costs rise.

A common mistake is to divide the total labor cost at 3 hours by the 6 batches added in the 2-to-3-hour interval. That mixes an average cost from all three hours with a marginal output change. The correct numerator is only the cost of the additional hour.

Worked Example: A Higher Wage Can Be Offset by Higher Productivity

Two hypothetical production settings make the same product. In setting A, output is $q=8L$, labor costs $\$24$ per hour, and the target output is 120 units. In setting B, output is $q=12L$, labor costs $\$36$ per hour, and the target is also 120 units. Find the labor required and compare marginal costs at the targets.

Method. Solve each production equation for the labor needed to reach the target, calculate marginal product, and then apply $MC=w/MP_L$.

In setting A, $120=8L$, so $L=120/8=15$ hours. Marginal product is $MP_L=8$ units per hour, so $MC=\$24/8=\$3.00$ per unit. In setting B, $120=12L$, so $L=120/12=10$ hours. Marginal product is $MP_L=12$ units per hour, so $MC=\$36/12=\$3.00$ per unit.

The checks are consistent: setting B uses less labor to reach the target, but its labor price is 1.5 times as high and its marginal product is also 1.5 times as high. Those changes offset in the ratio. A common mistake is to conclude that setting B must have higher marginal cost because its wage is higher; the correction is to account for productivity as well as the input price.

Comparisons, Assumptions, and Limits

At a fixed level of marginal product, a proportional change in the input price produces the same proportional change in marginal cost. At a fixed input price, a higher marginal product lowers marginal cost. If both change, compare the ratios rather than considering either quantity alone.

Change, holding the other factor fixed	Effect on $MC=w/MP_L$
Input price $w$ rises	Marginal cost rises in the same proportion
Marginal product $MP_L$ rises	Marginal cost falls
Input price and marginal product rise by the same proportion	Marginal cost is unchanged

The formula requires care in several cases. Marginal product must be positive for the usual interpretation of producing more output by adding labor. If marginal product is zero, a finite positive output increase cannot be generated by a small labor increase, and $w/MP_L$ is undefined. If marginal product is negative, adding labor reduces output, so the positive-marginal-cost interpretation of the formula does not apply. A production function with a kink may also lack a derivative at that point; a discrete change or one-sided calculation may be more appropriate.

The input price in the formula is the marginal cost of acquiring another unit of the variable input. If the price per labor-hour varies with the number of hours hired, the relevant numerator is marginal input expenditure, not necessarily a quoted average wage. For example, if variable cost is $VC(L)$, then $MC=(dVC/dL)/MP_L$, wherever those derivatives exist and $MP_L>0$. With multiple variable inputs, marginal cost generally cannot be found by dividing one input’s price by its marginal product without additional information about how the inputs are chosen.

This relationship is useful when a firm faces a wage change, compares different productivity levels, or translates a production schedule into short-run costs. It also identifies a common interpretive trap: a wage increase does not by itself establish how much marginal cost changes if productivity changes at the same time. State what is held constant, identify the relevant marginal product, and compare the resulting ratios.

Key takeaway: With one variable input, constant input price, and positive marginal product, short-run marginal cost is the variable-input price divided by marginal product. Higher input prices raise marginal cost; higher productivity lowers it.

Check Your Understanding

For each problem, identify the relevant marginal product or output increment before calculating marginal cost.

A workshop produces $q=20L-L^2$ units per day, with labor priced at $\$24$ per hour. At $L=4$, calculate output, marginal product, and marginal cost. (Answer: $q=64$ units; $MP_L=12$ units per hour; $MC=\$2.00$ per unit.)
A packaging line has $q=16\sqrt{L}$. At $L=16$, compare marginal cost when labor costs $\$18$ per hour and $\$27$ per hour, with technology and labor held fixed. (Answer: $MP_L=2$ packages per hour; marginal costs are $\$9.00$ and $\$13.50$ per package.)
A repair shop’s output is 0 jobs at 0 labor-hours, 9 jobs at 1 hour, 15 jobs at 2 hours, and 19 jobs at 3 hours. If labor costs $\$16$ per hour, calculate interval marginal cost from 1 to 2 hours and from 2 to 3 hours. (Answer: $\$16/6=\$2.67$ per job and $\$16/4=\$4.00$ per job, respectively.)
Setting A produces $q=8L$ and pays $\$24$ per labor-hour. Setting B produces $q=12L$ and pays $\$36$ per labor-hour. For a target of 120 units, find the labor required and marginal cost in each setting. (Answer: A uses 15 hours and has $MC=\$3.00$ per unit; B uses 10 hours and has $MC=\$3.00$ per unit.)
A firm has $q=30\sqrt{L}$, and labor costs $\$27$ per hour. At $L=25$, calculate output and marginal cost. If the labor price then falls to $\$18$ with the same technology and labor level, find the new marginal cost. (Answer: output is 150 units, $MP_L=3$ units per hour, and marginal cost falls from $\$9.00$ to $\$6.00$ per unit.)

Change, holding the other factor fixed	Effect on \(MC=w/MP_L\)
Input price \(w\) rises	Marginal cost rises in the same proportion
Marginal product \(MP_L\) rises	Marginal cost falls
Input price and marginal product rise by the same proportion	Marginal cost is unchanged

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