Labor’s Marginal Product With Capital Fixed

Why Hold Capital Fixed?

A firm may add workers without immediately changing its number of machines, workstations, or other productive capacity. To assess what the additional labor contributes, compare output while holding capital fixed. If capital changes at the same time as labor, the output difference cannot be attributed to labor alone.

Labor’s marginal product describes the additional output associated with a change in labor, with capital and other relevant conditions unchanged. Its value can vary as the workforce grows: workers may initially make better use of fixed equipment, but later face congestion or have too little equipment to use. The pattern depends on the production process; it is not automatic that every added worker contributes less than the one before.

Definition: With capital fixed at a particular level \( \bar K \), labor’s marginal product is the change in output attributable to a change in labor, keeping capital and other production conditions fixed. For integer worker changes, it is calculated from the change in output divided by the change in workers. In a differentiable continuous-input model, it is the derivative of output with respect to labor in the production relationship with \( \bar K \) held constant.

Marginal Product From Output Data

When a production schedule reports output for different numbers of workers, the marginal product over an interval is the output difference divided by the labor difference. If labor rises by exactly one worker, this is simply the output produced with the additional worker minus output before that worker was added. Check that the capital level and the period of production are the same for both observations.

\text{Marginal product of labor over an interval} = \frac{\Delta Q}{\Delta L} = \frac{Q_2-Q_1}{L_2-L_1} \qquad \text{with capital fixed}

Here, \(Q\) is output and \(L\) is the number of workers. If the interval is one worker, the result is the output change per additional worker. If the interval covers several workers, it is an average marginal product over that interval; it does not reveal each worker’s individual contribution within the interval.

Worked Example: Finding Each Worker’s Contribution From a Schedule

A small assembly shop has 4 identical workstations and does not change its equipment during a shift. Its output is recorded as follows. Find the marginal product associated with adding each worker, and identify where the contribution begins to decline.

Workers \(L\)	Output \(Q\), units per shift
0	0
1	8
2	19
3	29
4	36

Method. Calculate the difference in output between each pair of consecutive rows. Each labor change is one worker, so each output difference is that worker’s marginal product in this schedule.

From 0 to 1 worker, output changes by \(8-0=8\) units. The first worker’s marginal product is 8 units per shift.

From 1 to 2 workers, output changes by \(19-8=11\) units. The second worker’s marginal product is 11 units per shift.

From 2 to 3 workers, output changes by \(29-19=10\) units. The third worker’s marginal product is 10 units per shift.

From 3 to 4 workers, output changes by \(36-29=7\) units. The fourth worker’s marginal product is 7 units per shift.

The sequence is 8, 11, 10, and 7. The second worker adds more output than the first, but the third and fourth workers add less than the second. Thus, marginal product first rises and then declines in this schedule. Check: the four marginal products sum to \(8+11+10+7=36\), the total output increase from 0 to 4 workers.

Common mistake avoided: Calling the fourth worker’s marginal product \(36/4=9\) confuses marginal product with average product. The fourth worker’s marginal product is \(36-29=7\); \(36/4=9\) is average output per worker with four workers.

Marginal Product From a Production Function

A production function may describe output continuously rather than give a schedule only at whole-worker values. When the function is differentiable in labor, fix capital at \( \bar K \) and treat output as a one-variable function of labor: \(q(L)=f(L,\bar K)\). Labor’s marginal product at a particular labor level is the rate at which this fixed-capital output function changes with labor.

MP_L(L;\bar K)=\frac{dq(L)}{dL}

This derivative is a local rate, measured in units of output per unit of labor. If workers are modeled as a continuous input, it describes the instantaneous change at the stated labor level. If workers must be whole people, the derivative is a model-based local rate; the actual output change from hiring one worker is a finite difference. The derivative formula requires a differentiable production relationship and does not replace the output difference when a question gives only discrete observations.

Why the derivative represents marginal product: At a labor level \(L\), the difference quotient \( \frac{q(L+h)-q(L)}{h} \) measures the average output change per added unit of labor over an increase \(h\), with capital still fixed. As \(h\) approaches zero, this quotient approaches the derivative \( \frac{dq(L)}{dL} \), when that derivative exists. Therefore the derivative is the local rate of output change attributable to labor in the fixed-capital model.

Worked Example: Differentiating a Fixed-Capital Production Function

A workshop’s output per day, with its number of machines fixed, is modeled by \(q(L)=10L+3L^2-0.5L^3\), where \(L\) is labor measured in workers and \(q\) is output units per day. Find labor’s marginal product at \(L=2\) and \(L=4\), and explain the comparison.

Model and method. The machine level and other conditions are held fixed, and the given production function is differentiable. Differentiate the one-variable output function with respect to labor:

MP_L(L)=\frac{dq}{dL} =10+6L-1.5L^2

At \(L=2\), substitute 2 for labor:

$$ MP_L(2)=10+6(2)-1.5(2^2) =10+12-6 =16 $$

At \(L=4\):

$$ MP_L(4)=10+6(4)-1.5(4^2) =10+24-24 =10 $$

The model’s marginal product is 16 output units per additional unit of labor at \(L=2\), and 10 at \(L=4\). It is lower at the higher labor level. For this function, the rate itself changes with labor according to \( \frac{dMP_L}{dL}=6-3L \). This rate is positive below \(L=2\), zero at \(L=2\), and negative above \(L=2\); accordingly, marginal product rises up to \(L=2\) and declines after that point. Check: \(MP_L(2)=16\) and \(MP_L(4)=10\), consistent with a decline between those labor levels.

Common mistake avoided: Substituting labor directly into \(q(L)\) gives total output, not marginal product. The method is to differentiate first, then evaluate the derivative at the requested labor level.

Why Marginal Product Can Rise and Then Fall

With capital fixed, early additions to the workforce can increase output more rapidly if workers can divide tasks, coordinate activities, or keep equipment in use more effectively. Later additions may contribute less if workers must wait for the same machines, share limited space, or interfere with one another’s tasks. These are possible explanations for a pattern in the data, not assumptions that must hold in every firm.

A declining marginal product means that each additional unit of labor adds less output than the preceding unit over the relevant range. It does not, by itself, mean that total output is falling. Total output can still rise while its increments get smaller. Total output falls as labor increases only where the marginal product is negative.

For discrete schedules, compare consecutive output changes to see whether contributions are rising or falling. For a differentiable function, examine how the derivative changes as labor changes. These methods answer questions about labor’s contribution under a fixed-capital condition; they do not establish what would happen if the firm also changed its equipment.

Check what is held fixed.
Confirm that capital, the production period, and other stated conditions are unchanged across the output observations or in the model.

Choose the calculation.
Use output differences divided by labor differences for a schedule. Use the derivative of the fixed-capital output function when a differentiable function is given.

Interpret the result in context.
State the output units per worker or per unit of labor, name the labor level or interval, and distinguish a rising or falling contribution from a fall in total output.

Common Errors and Why the Measure Matters

Marginal product helps describe how output responds to additional labor when existing capacity is held fixed. For example, a hypothetical manager considering an additional shift worker could use a production schedule to estimate the output change under the same equipment conditions. The estimate describes production, not whether hiring is profitable: a revenue and cost comparison would be needed to answer that separate question.

Using average product instead. Total output divided by the number of workers is output per worker on average; it is not the output change associated with the next worker.
Ignoring the fixed-capital condition. If both labor and equipment change, the output difference combines changes in inputs, so it does not isolate labor’s marginal product.
Calling a declining marginal product a decline in total output. A positive but shrinking output increment still raises total output.
Treating a multi-worker interval as one worker’s contribution. If labor increases by several workers, the output difference divided by that increase is an interval average, not necessarily the contribution of each individual worker.
Applying a derivative to a schedule without a function. Discrete output data support finite differences. A derivative requires a differentiable production function or an explicitly justified continuous model.

Key takeaway: Labor’s marginal product measures labor’s contribution to output only when capital and other production conditions are held fixed. Calculate it with an output difference for discrete data or a derivative for a differentiable fixed-capital function, then interpret its units and labor level carefully.

Check Your Understanding

For each problem, keep the stated capital condition in view and distinguish marginal product from average output per worker.

A packing line has fixed equipment and produces 0, 9, 21, 30, and 34 crates per hour with 0, 1, 2, 3, and 4 workers, respectively. Find the marginal product of the third worker and say whether it is lower than the second worker’s marginal product. (Answer: 9 crates per hour; yes, it is lower than the second worker’s 12.)
With capital fixed, a differentiable production function is \(q(L)=4L+2L^2-0.25L^3\). Find labor’s marginal product at \(L=2\). (Answer: 9 output units per unit of labor.)
A fixed-capital production schedule reports 42 units at 3 workers and 58 units at 5 workers. What is the average marginal product over this labor interval, and can it be identified as the fourth worker’s individual marginal product? (Answer: 8 output units per worker over the interval; no, the interval average does not identify the fourth worker’s individual contribution.)
A shop reports output of 50 units with 4 workers and 68 units with 5 workers, but it also added a machine when hiring the fifth worker. Can the output difference be reported as labor’s marginal product with capital fixed? Explain. (Answer: No. The 18-unit increase occurred while both labor and capital changed, so labor’s contribution alone cannot be isolated.)
With capital fixed, output is \(q(L)=18L+3L^2-0.5L^3\). Find marginal product at \(L=4\), and determine whether marginal product is rising or falling there. (Answer: 18 output units per unit of labor; it is falling because \(dMP_L/dL=6-3(4)=-6\).)

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