Why Marginal Product Patterns Matter
A production schedule records how much output a firm can produce with different quantities of an input, holding the production setting fixed. Comparing output levels alone does not always reveal how productive each additional unit is. Marginal product makes that comparison: it measures the extra output associated with an increase in an input. Whether marginal product is rising, falling but positive, or negative tells us how total output changes as the input increases.
For this tutorial, consider a short-run setting in which labor varies and the firm’s other inputs, such as equipment and workspace, are held fixed. The same reasoning applies to another variable input. The classification describes the production data or function being analyzed; it does not claim that every firm must follow one particular pattern.
The sign and direction of marginal product answer different questions. Its sign tells whether output increases or decreases when labor increases. Whether marginal product itself is rising or falling tells whether the output added by successive increases is getting larger or smaller. In the falling-but-positive case, output still increases; it simply increases by less at the margin.
Three Patterns and Their Output Meaning
| Marginal product pattern | What happens to marginal product? | Effect on total output as labor increases |
|---|---|---|
| Increasing marginal product | It rises across the range being considered. | Output increases, with successive labor increases adding more output. |
| Diminishing but positive marginal product | It falls but remains above zero. | Output continues to increase, but successive labor increases add less output. |
| Zero marginal product | It equals zero at the point or interval considered. | An increase in labor adds no output at the margin. |
| Negative marginal product | It is below zero. | Output decreases as labor increases over the relevant range. |
These descriptions depend on comparing marginal products, not merely comparing total output levels. For instance, output can be higher with six workers than with five while the sixth worker’s marginal product is lower than the fifth worker’s. That is diminishing marginal product, not negative marginal product.
This relationship explains the connection to total output. Starting from an output level \(Q_1\), an input increase of \(\Delta L\) changes output by \(\Delta Q\), so the new output is \(Q_2=Q_1+\Delta Q\). If \(MP_L>0\) and \(\Delta L>0\), then \(\Delta Q>0\) and total output rises. If marginal product is positive but smaller than before, the increase in total output is smaller than the previous increase. If \(MP_L<0\), then \(\Delta Q<0\), so total output falls.
Worked Example: Classifying a One-Worker Production Schedule
A hypothetical workshop records output for different numbers of workers while keeping its machines and floor space unchanged. Classify the marginal product across the observed intervals and describe the change in total output.
| Workers \(L\) | Total output \(Q\) |
|---|---|
| 0 | 0 |
| 1 | 10 |
| 2 | 24 |
| 3 | 39 |
| 4 | 51 |
| 5 | 59 |
| 6 | 63 |
| 7 | 62 |
Method. Since labor rises by one worker at each step, calculate each marginal product as the difference between consecutive output values.
From 0 to 1 worker, \(MP_L=(10-0)/(1-0)=10\). From 1 to 2 workers, \(MP_L=(24-10)/(2-1)=14\). From 2 to 3 workers, \(MP_L=(39-24)/(3-2)=15\). Marginal product rises from 10 to 14 to 15, so it is increasing over these successive intervals. Total output rises by progressively larger amounts: 10, 14, and 15.
From 3 to 4 workers, \(MP_L=(51-39)/(4-3)=12\). From 4 to 5 workers, \(MP_L=(59-51)/(5-4)=8\). From 5 to 6 workers, \(MP_L=(63-59)/(6-5)=4\). Each marginal product is positive, but the sequence falls from 15 to 12 to 8 to 4. This is diminishing, positive marginal product. Total output still rises, from 39 to 63, but the successive additions shrink.
From 6 to 7 workers, \(MP_L=(62-63)/(7-6)=-1\). This is negative marginal product: output falls by one unit when the seventh worker is added. The table confirms the classification because output decreases from 63 to 62.
Check. The marginal products sum to the total output change across the full schedule: \(10+14+15+12+8+4-1=62\), equal to \(62-0\). The common mistake is to call every falling marginal product negative. Here marginal product falls from 15 to 4 but remains positive, so total output continues to rise over those intervals.
Using a Production Function
A production function can describe marginal product at input levels not listed in a table. The calculation has two parts: first obtain marginal product as the derivative of output with respect to the variable input; then use its values or sign to interpret output changes. This tutorial uses the resulting marginal product to classify ranges rather than to develop general derivative tests for production functions.
Worked Example: Finding Three Ranges From a Production Function
A hypothetical production process has output, in units per day, given by \(Q(L)=12L+2L^2-\frac{1}{3}L^3\), where \(L\) is the number of workers. The function is considered for \(L\geq 0\), with other inputs fixed. Identify where marginal product is increasing, where it is falling but positive, and where it is negative.
Step 1: Calculate marginal product. Differentiate output with respect to labor:
Step 2: Locate where marginal product changes direction. The expression can be written \(MP_L=16-(L-2)^2\). It rises as \(L\) increases up to \(L=2\), reaches 16 at \(L=2\), and falls after \(L=2\). Thus marginal product is increasing for \(0\leq L<2\) and falling for \(L>2\).
Step 3: Find where marginal product is positive or negative. Set it equal to zero:
The roots are \(L=6\) and \(L=-2\). Since the relevant input is \(L\geq0\), marginal product is positive for \(0\leq L<6\), zero at \(L=6\), and negative for \(L>6\). Combining this with its direction of change, marginal product is increasing for \(0\leq L<2\), falling but positive for \(2<L<6\), and negative for \(L>6\). At \(L=2\), it is at its highest value in the stated range; at \(L=6\), it is zero.
Step 4: Check the output interpretation. At \(L=2\), \(Q=12(2)+2(2^2)-\frac{1}{3}(2^3)=24+8-\frac{8}{3}=\frac{88}{3}\), or about 29.33 units. At \(L=6\), \(Q=72+72-72=72\). At \(L=7\), \(Q=84+98-\frac{343}{3}=\frac{203}{3}\), or about 67.67. Output is higher at 6 than at 2, then lower at 7 than at 6, consistent with positive marginal product before 6 and negative marginal product after 6.
Check. The derivative gives \(MP_L(2)=12+8-4=16\), and \(MP_L(6)=12+24-36=0\). A common mistake is to treat the point where marginal product starts falling as the point where output starts falling. In this example marginal product starts falling after 2, but remains positive until 6; output therefore continues to rise over that interval.
Discrete Data With Unequal Input Increases
Production schedules do not always increase labor by one unit at a time. In that case, compare output per unit of input using \(\Delta Q/\Delta L\), rather than comparing raw output differences alone. This distinction matters because an output increase of 24 units over two workers represents a different marginal product from an increase of 24 units over three workers.
Worked Example: Adjusting for the Size of Each Input Change
A hypothetical packing operation records daily output at labor levels of 0, 2, 4, 6, 8, and 10 workers. Output at those levels is 0, 28, 64, 90, 104, and 100 packages, respectively. Calculate the marginal product over each interval and classify the patterns.
Method. For each interval, divide the change in output by the change in workers. Each input change is two workers.
From 0 to 2 workers, \(\Delta Q=28-0=28\), so \(MP_L=28/2=14\) packages per worker. From 2 to 4 workers, \(\Delta Q=64-28=36\), so \(MP_L=36/2=18\). Marginal product rises from 14 to 18, and total output rises by a larger amount in the second interval.
From 4 to 6 workers, \(\Delta Q=90-64=26\), so \(MP_L=26/2=13\). From 6 to 8 workers, \(\Delta Q=104-90=14\), so \(MP_L=14/2=7\). Marginal product is positive but falls from 18 to 13 to 7. Total output continues to rise, from 64 to 104 packages, though by smaller increments per worker.
From 8 to 10 workers, \(\Delta Q=100-104=-4\), so \(MP_L=-4/2=-2\) packages per worker. The negative marginal product corresponds to a fall in total output from 104 to 100.
Check. Multiplying each interval’s marginal product by its two-worker increase gives output changes of \(28\), \(36\), \(26\), \(14\), and \(-4\). Their sum is \(100\), which matches the total change \(100-0\). The common mistake is to compare the raw differences without accounting for the two-worker intervals. Here the intervals happen to have equal size, but dividing by two is still necessary to report marginal product per worker.
A Short Derivation: Why Marginal Product Predicts Output Change
For a discrete production schedule, marginal product is defined as the output change divided by the input change. Rearranging that definition gives the exact relationship between the two changes:
If \(L_2>L_1\), then \(L_2-L_1>0\). Therefore, the sign of \(Q_2-Q_1\) is the sign of marginal product over that interval: positive marginal product means output rises, zero means there is no output change, and negative marginal product means output falls. If the marginal product for a later interval is smaller but still positive than for an earlier interval of the same input size, the later output increase is smaller. This is why “diminishing” does not mean “decreasing output.”
For a differentiable production function, the derivative \(dQ/dL\) is the instantaneous rate of output change with respect to labor. A positive derivative indicates that a small increase in labor locally raises output; a negative derivative indicates that it locally lowers output. The derivative describes a rate at a point, whereas a finite difference describes an average rate across an interval. Do not treat these as identical when interpreting a large discrete change.
Common Traps and Practical Interpretation
- Confusing a falling marginal product with negative marginal product. A positive value can fall and remain above zero. Total output then rises by smaller increments.
- Confusing total output with marginal product. Total output is the accumulated amount produced at an input level; marginal product is the additional output associated with an input increase.
- Ignoring the input interval. When labor changes by more than one unit, calculate \(\Delta Q/\Delta L\). A raw output difference is not output per worker unless \(\Delta L=1\).
- Assuming every increase in input must increase output. That is not true when the observed marginal product is negative. In that range, adding the input is associated with lower total output, holding the stated setting fixed.
- Using a classification outside the given range. A production function may have different marginal-product patterns at different labor levels. State the range supported by the data or calculation.
In applied economic analysis, these distinctions help describe what happens when a firm changes a variable input while holding its production setting fixed. They also keep productivity statements precise: “output grew more slowly” means output still increased, but by less; “output fell” requires negative marginal product over the relevant change.
Check Your Understanding
For each problem, calculate or interpret marginal product before classifying the output pattern. Assume other inputs remain fixed as stated.
- A greenhouse’s output is 0, 9, 21, 30, and 36 trays when labor is 0, 1, 2, 3, and 4 workers, respectively. Calculate marginal product across each interval and identify where it is increasing and where it is diminishing but positive. (Answer: Marginal products are 9, 12, 9, and 6 trays per worker; it is increasing from 0 to 2 workers, then diminishing but positive from 2 to 4 workers.)
- A hypothetical repair shop produces 0 units with 0 workers, 39 with 3, 72 with 6, and 90 with 9. Calculate marginal product per worker in each interval. Does output fall in any interval? (Answer: Marginal products are 13, 11, and 6 units per worker; output rises in every interval, with diminishing positive marginal product.)
- A workshop’s output at 0, 2, 4, 6, and 8 workers is 0, 28, 64, 90, and 104 units. Calculate marginal product per worker for each interval and classify the change from 2 to 8 workers. (Answer: Marginal products are 14, 18, 13, and 7 units per worker; from 2 to 8 workers marginal product is falling but positive, so total output rises by 76 units.)
- A production function is \(Q(L)=10L+2L^2-\frac{1}{3}L^3\) for \(L\geq0\). Find the marginal product at \(L=1\) and \(L=5\), then state whether a small increase in labor at each level raises or lowers output. (Answer: \(MP_L=10+4L-L^2\); \(MP_L(1)=13\), so output rises locally, and \(MP_L(5)=5\), so output also rises locally.)
- In a hypothetical facility, output is 120 packages with 8 workers and 112 packages with 10 workers. Calculate marginal product per added worker over the interval and explain the direction of total output. (Answer: \(MP_L=(112-120)/(10-8)=-4\) packages per worker; marginal product is negative over the interval and total output falls by 8 packages.)