Why Compare Average and Marginal Product?
A production report may show that output per worker is rising even as each newly added worker contributes less output than the workers already employed. There is no contradiction: output per worker is an average, while the contribution of an additional worker is marginal. Reading both measures together helps explain what a productivity table says—and what it does not say.
This comparison is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. The central task is to keep the measures and their units distinct, then use their relationship to interpret observed production data.
The word “marginal” means an incremental change, not the total output level. For a discrete table, calculate marginal product between two observations. If the input rises by more than one unit, divide the output change by the input change; the result is output per added input unit over that interval. For a differentiable production function \(Q=f(L)\), the point marginal product is \(MP_L=f'(L)\), the rate of change of output with respect to the input at a specified input level.
The units are a useful check. If output is measured in packages per shift and labor in workers, both average and marginal product are packages per worker per shift. If labor is measured in hours, the unit is packages per labor-hour. The numerical values can differ because the measures answer different questions: average product asks how much output corresponds to each input unit overall; marginal product asks how much output changes for an additional input unit.
Reading Both Measures in a Production Table
For every row with positive input, average product is calculated from total output divided by the input at that row. Marginal product is calculated from the change in output divided by the change in input between rows. Do not calculate a marginal product by dividing a row’s total output by its input—that repeats the average-product calculation.
Worked Example: Comparing Productivity Across a Staffing Schedule
A hypothetical workshop records output per shift at several staffing levels. Other production conditions are held fixed. Calculate average product at each positive staffing level and marginal product between successive observations. Identify where average product rises and where it falls.
| Workers, \(L\) | Output per shift, \(Q\) |
|---|---|
| 2 | 22 |
| 4 | 48 |
| 6 | 78 |
| 8 | 100 |
Method. Divide each output level by its corresponding number of workers for average product. For marginal product, take the output difference and divide by the two-worker increase. This avoids confusing output per worker with the output associated with added workers.
At \(L=2\), average product is \(22/2=11\) output units per worker per shift. At \(L=4\), it is \(48/4=12\). At \(L=6\), it is \(78/6=13\). At \(L=8\), it is \(100/8=12.5\).
From 2 to 4 workers, marginal product is \((48-22)/(4-2)=26/2=13\) output units per added worker per shift. From 4 to 6, it is \((78-48)/(6-4)=30/2=15\). From 6 to 8, it is \((100-78)/(8-6)=22/2=11\).
The first two marginal products, 13 and 15, exceed the average products at the starting levels, 11 and 12. Average product rises over both intervals, from 11 to 12 and then from 12 to 13. In the last interval, marginal product is 11, below the starting average product of 13, and average product falls to 12.5. The calculations are consistent: an increment above the current average pulls the new average up, while an increment below it pulls the new average down.
Check. The output changes are 26, 30, and 22, and each interval adds two workers; dividing gives marginal products of 13, 15, and 11. Each average-product value also reproduces its row’s total output when multiplied by the worker count.
Why Marginal Product Changes the Average
The relationship can be derived directly from the definitions. Suppose input rises from \(L_0>0\) to \(L_1=L_0+\Delta L\), where \(\Delta L>0\). Let the marginal product over this interval be \(MP_L=\Delta Q/\Delta L\), so the new output is \(Q_1=Q_0+MP_L\Delta L\). The new average product is therefore
Subtract the starting average product \(AP_{L_0}=Q_0/L_0\). Putting the two terms over a common denominator gives
Because \(\Delta L/(L_0+\Delta L)\) is positive, the sign of the change in average product is the sign of \(MP_L-AP_{L_0}\). Thus, over this interval, average product rises if marginal product exceeds the starting average product, falls if marginal product is below it, and stays unchanged if they are equal. This result applies to a positive starting input and a positive input increment; it compares the interval’s marginal product with the starting average, not automatically with the ending average.
For a smooth production function, the corresponding local result follows from \(AP_L=f(L)/L\). Differentiating for \(L>0\) gives
Since \(L\) is positive, the direction in which average product changes locally depends on whether point marginal product is above or below average product at that input level. The discrete formula and the derivative formula express the same average-versus-increment idea, but they use different definitions of marginal product: an interval rate in a table versus a point rate for a differentiable function.
From a Productivity Report to a Careful Interpretation
A productivity report may use output per worker as its average measure and compare it across staffing levels. To infer marginal product from such data, the report must also provide total output and input levels, or otherwise provide the change in each. Average product at one staffing level alone does not reveal the output change caused by adding a worker.
In addition, interpreting a difference as the effect of extra labor requires care about what else changed. The definition of labor’s marginal product in a multi-input production setting holds other inputs fixed. If a report compares periods in which staffing, equipment, hours, or production conditions all changed, the observed output difference is not by itself the marginal product of labor. It is still possible to calculate a descriptive output change per worker added, but that ratio should not be interpreted as isolating labor’s contribution unless the comparison supports that interpretation.
Worked Example: A One-Worker Increase in a Productivity Report
A hypothetical print shop reports output per shift at three staffing levels, with its other production conditions unchanged. Calculate average product and marginal product, then explain whether average product rises at each addition.
| Workers, \(L\) | Output per shift, \(Q\) |
|---|---|
| 20 | 200 |
| 21 | 214 |
| 22 | 226 |
| 23 | 235 |
Method. Calculate row averages as \(Q/L\), then calculate each one-worker marginal product as the change in output. The common error to avoid is using the new row’s output per worker as the output added by the new worker.
At 20 workers, average product is \(200/20=10\). At 21 workers, it is \(214/21=10.190476\ldots\), or 10.19 to two decimal places. At 22 workers, it is \(226/22=10.272727\ldots\), or 10.27 to two decimal places. At 23 workers, it is \(235/23=10.217391\ldots\), or 10.22 to two decimal places.
The marginal product from 20 to 21 workers is \(214-200=14\) output units. From 21 to 22, it is \(226-214=12\). From 22 to 23, it is \(235-226=9\). Each interval adds one worker, so these output differences are also output units per added worker per shift.
Average product rises from 10 to 10.19 and then to 10.27 because the marginal products 14 and 12 exceed the respective starting averages, 10 and 10.19. It then falls to 10.22 because the next marginal product, 9, is below the starting average of about 10.27.
Common mistake and correction. A mistaken calculation might call \(214/21\approx10.19\) the marginal product from 20 to 21 workers. That is the average product at 21 workers. The marginal product is \((214-200)/(21-20)=14\). As a check, \(14\) exceeds the starting average of 10, and the average does rise to about 10.19.
Marginal Product in a Production Function
When output is specified as a differentiable function of an input, average product is the function’s output divided by the input, while point marginal product is its derivative. A derivative describes the local rate of change; it need not equal the output difference for a whole additional worker. If the question asks for a discrete one-worker change, calculate that finite difference rather than substituting the derivative without comment.
Worked Example: Comparing a Point Marginal Product with a One-Worker Change
A hypothetical production process has output per shift \(Q=f(L)=18L-0.5L^2\), for labor levels in the range being considered. Find average product and point marginal product at 10 workers. Then calculate the exact output change from 10 to 11 workers and compare it with the point marginal product at 10.
Method. Use \(Q/L\) for average product and differentiate \(f(L)\) for point marginal product. Separately evaluate the function at both integer staffing levels to obtain the discrete change. This avoids treating a local rate as automatically identical to a finite-interval change.
At \(L=10\), output is \(f(10)=18(10)-0.5(10^2)=180-50=130\). Average product is \(130/10=13\) output units per worker per shift.
The derivative is \(f'(L)=18-L\), so point marginal product at 10 is \(f'(10)=18-10=8\) output units per worker per shift.
At 11 workers, output is \(f(11)=18(11)-0.5(11^2)=198-60.5=137.5\). The exact one-worker output change is \(137.5-130=7.5\) output units. It is less than the point marginal product of 8 at \(L=10\), because the derivative declines as labor increases across the interval.
Check. At 10 workers the marginal product, 8, is below average product, 13; the average product at 11 workers is \(137.5/11=12.5\), lower than 13. The function-based calculation agrees with the average-versus-marginal relationship.
Common Traps and a Practical Check
The most reliable interpretation starts by identifying the denominator and the interval for each measure. Average product belongs to a particular input level. A discrete marginal product belongs to a change between two input levels. A point marginal product belongs to a particular input level in a differentiable model.
- Using total output as marginal product. \(Q/L\) is average product. Marginal product requires an output difference divided by an input difference, or a derivative in the smooth model.
- Ignoring the size of an input step. If labor rises by three units, divide the output change by three to obtain output per added unit over that interval.
- Comparing the wrong quantities. To explain whether an interval raises or lowers average product, compare its marginal product with average product at the interval’s starting input level.
- Mixing units or conditions. Check that output and input units are consistent and that other inputs are held fixed before interpreting a ratio as a marginal product of one input.
- Equating an interval change with a derivative. For a discrete increment, evaluate the finite change. Use a derivative for a local rate, unless the question specifically asks for an approximation.
A quick check is to ask whether the observed average moved in the direction predicted by the marginal product. If marginal product is above the starting average, the new average should be higher; if it is below, the new average should be lower. A mismatch often signals an arithmetic error, an incorrect interval, or a comparison that does not hold other relevant conditions fixed.
Check Your Understanding
For each problem, distinguish the average measure at an input level from the marginal measure over an input change. Give units where relevant.
- A greenhouse produces 84 trays with 7 workers and 108 trays with 9 workers per shift, with other conditions unchanged. Calculate average product at both staffing levels and marginal product over the interval. (Answer: Average products are 12 and 12 trays per worker per shift; marginal product is 12 trays per added worker per shift.)
- A repair team’s output rises from 150 completed jobs with 15 technicians to 168 jobs with 16 technicians. Calculate the new average product and the one-technician marginal product, then state whether average product rises. (Answer: New average product is 10.5 jobs per technician; marginal product is 18 jobs per added technician; average product rises from 10 to 10.5.)
- A production schedule has \(Q=36\) when \(L=3\) and \(Q=60\) when \(L=6\). Calculate average product at each input level and marginal product over the interval. Is average product higher or lower at 6? (Answer: Average products are 12 and 10 output units per input unit; marginal product is 8 output units per added input unit; average product is lower at 6.)
- For \(Q=20L-0.5L^2\), calculate average product and point marginal product at \(L=8\), then calculate the exact output change from \(L=8\) to \(L=9\). (Answer: Average product is 16, point marginal product is 12, and the exact one-unit output change is 11.5.)
- A mill’s output changes from 320 units at 40 labor-hours to 350 units at 45 labor-hours. Calculate average product at each observation and marginal product per added labor-hour. Does the average rise or fall, and is that consistent with the marginal comparison? (Answer: Average products are 8 and about 7.7778 units per labor-hour; marginal product is 6 units per added labor-hour; average product falls, consistently because 6 is below the starting average of 8.)