Reading a Short-Run Production Schedule
A production schedule organizes the relationship between an input and output. In the short run, a firm can vary some inputs while at least one other input remains fixed. A schedule can then show how output changes as the firm adds labor to a fixed amount of plant, equipment, or other capacity. Reading it correctly requires separating three measures: total product, marginal product, and average product.
This tutorial focuses on completing and interpreting a schedule, not on deriving a firm's costs. The calculations are aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
A schedule normally assumes that the production method and all other inputs remain unchanged while the listed labor amounts vary. This fixed-input condition matters: if the firm adds equipment at the same time as workers, the change in output cannot be attributed to labor alone. The schedule describes output at the stated input levels; it does not, by itself, explain every cause of productivity.
At zero labor, total product may be zero, but average product is undefined because division by zero is not possible. For positive labor, average product is calculated separately at each labor level. Marginal product instead compares output across two labor levels. Keep track of which measure is being requested and, for marginal product, which interval is being compared.
A Method for Completing the Schedule
For each row with a positive labor amount, divide total product by labor to find average product. For each change from one labor level to the next, subtract the earlier total product from the later total product and divide by the increase in labor. If labor increases one worker at a time, the denominator is one and the marginal product equals the change in output.
Confirm that the schedule varies labor while the other production conditions are unchanged. Otherwise, the calculated output change is not attributable to labor alone.
Use the given output at each labor amount. Total product is a level of output, not output per worker or the change from the prior row.
Take the change in total product and divide by the change in labor. Label the interval when labor does not increase by one unit.
Divide that row's total product by its labor input. Do not divide by the change in labor.
A rising marginal product means the next labor addition produces more output than the preceding addition; a diminishing marginal product means it produces less.
Worked Schedules
In the examples, the output figures are hypothetical. Unless noted otherwise, labor rises one worker at a time and other inputs are held fixed. Product measures are in units of output per worker or per interval, as appropriate.
Worked Example: Completing a One-Worker-at-a-Time Schedule
A hypothetical print shop keeps its presses and work area fixed and records output as it adds workers. Complete the schedule, then identify where marginal product rises and where it diminishes.
| Workers \(L\) | Total product \(Q\) | Marginal product \(MP_L\) | Average product \(AP_L\) |
|---|---|---|---|
| 0 | 0 | — | Undefined |
| 1 | 12 | 12 | 12 |
| 2 | 30 | 18 | 15 |
| 3 | 54 | 24 | 18 |
| 4 | 72 | 18 | 18 |
| 5 | 84 | 12 | 16.8 |
| 6 | 90 | 6 | 15 |
Method. With one additional worker per row, subtract the previous total product to find marginal product. Divide each positive-labor row's total product by its worker count to find average product.
Marginal product calculations. At one worker, \(12-0=12\). At two workers, \(30-12=18\). At three workers, \(54-30=24\). At four workers, \(72-54=18\). At five workers, \(84-72=12\). At six workers, \(90-84=6\).
Average product calculations. At one worker, \(12/1=12\). At two, \(30/2=15\). At three, \(54/3=18\). At four, \(72/4=18\). At five, \(84/5=16.8\). At six, \(90/6=15\).
Marginal product rises over the first three additions: \(12\), then \(18\), then \(24\). It diminishes from the fourth addition onward: \(18\), then \(12\), then \(6\). Total product continues to rise at every listed labor level, because each marginal product is positive. Average product is unchanged between three and four workers: the fourth worker's marginal product of \(18\) equals the prior average product of \(18\).
Check. The marginal products sum to \(12+18+24+18+12+6=90\), matching the schedule's total product at six workers. The method avoids confusing a worker's marginal contribution with the average output per worker.
Worked Example: Uneven Labor Intervals
A hypothetical packing operation reports output at three workforce levels, with several workers added between recorded rows. Calculate marginal product per added worker for each interval, calculate average product at the reported workforce levels, and state what can and cannot be concluded about the intermediate additions.
| Workers \(L\) | Total product \(Q\) | Marginal product over interval | Average product \(AP_L\) |
|---|---|---|---|
| 0 | 0 | — | Undefined |
| 2 | 22 | 11, from 0 to 2 | 11 |
| 5 | 49 | 9, from 2 to 5 | 9.8 |
| 9 | 73 | 6, from 5 to 9 | 8.111… |
Marginal product calculations. From zero to two workers, output rises by \(22-0=22\) while labor rises by \(2-0=2\), so marginal product over that interval is \(22/2=11\) units per worker. From two to five workers, output rises by \(49-22=27\) and labor by \(5-2=3\), so the interval marginal product is \(27/3=9\). From five to nine workers, output rises by \(73-49=24\) and labor by \(9-5=4\), so the interval marginal product is \(24/4=6\).
Average product calculations. At two workers, \(22/2=11\). At five workers, \(49/5=9.8\). At nine workers, \(73/9=8.111\ldots\), or approximately \(8.11\) units per worker if rounded to two decimals.
The reported interval marginal products decline from \(11\) to \(9\) to \(6\), so the average output gain per added worker is lower in each successive interval. However, the schedule does not show output at every individual labor level. It cannot establish the exact marginal product of each unreported worker or prove that marginal product fell at every step within an interval.
Check. The interval gains reconcile: \(22+27+24=73\). The method avoids treating \(27\), the total gain from two to five workers, as marginal product per worker; that gain must be divided by the three added workers.
Why Marginal and Average Product Move Together
The relationship between marginal and average product follows directly from how an average is formed. Suppose output with \(L-1\) workers is \(Q_{L-1}\), and output with \(L\) workers is \(Q_L\). With one worker added, the marginal product of that addition is \(MP_L=Q_L-Q_{L-1}\), while the prior average product is \(AP_{L-1}=Q_{L-1}/(L-1)\).
Since \(Q_{L-1}=(L-1)AP_{L-1}\), dividing the new total product by the new workforce gives:
Subtracting the previous average product produces:
Because \(L\) is positive, the sign of the change in average product is determined by whether the new worker's marginal product is above, equal to, or below the previous average product. If \(MP_L>AP_{L-1}\), average product rises; if \(MP_L=AP_{L-1}\), it is unchanged; if \(MP_L<AP_{L-1}\), it falls. This conclusion compares a marginal value with the prior average, not automatically with the new average.
Worked Example: Diminishing Marginal Product and Falling Total Product
A hypothetical repair facility holds its tools and floor space fixed. Its total product is \(0, 14, 32, 47, 57,\) and \(55\) jobs with \(0, 1, 2, 3, 4,\) and \(5\) workers, respectively. Calculate the missing product measures and distinguish diminishing marginal product from a fall in total product.
Marginal products. The successive output changes are \(14-0=14\), \(32-14=18\), \(47-32=15\), \(57-47=10\), and \(55-57=-2\). Thus marginal product is \(14, 18, 15, 10,\) and \(-2\) jobs per added worker.
Average products. At one worker, \(14/1=14\). At two, \(32/2=16\). At three, \(47/3=15.666\ldots\), approximately \(15.67\). At four, \(57/4=14.25\). At five, \(55/5=11\).
Marginal product rises from \(14\) to \(18\) with the second worker, then diminishes: \(15<18\), \(10<15\), and \(-2<10\). Total product falls only with the fifth worker, from \(57\) jobs to \(55\), because that addition has negative marginal product. Earlier marginal products diminish but remain positive, so total product still rises through four workers.
Check. The five marginal products sum to \(14+18+15+10-2=55\), the total product at five workers. The method avoids calling every diminishing marginal product a decrease in total output: a positive but smaller increment still increases the total.
Interpreting the Patterns and Avoiding Traps
A rising marginal product means successive additions of labor are associated with larger output increments. A diminishing marginal product means those increments become smaller. These statements concern comparisons of marginal product, not whether total product is rising or falling. If marginal product is positive, total product rises; if it is zero, total product is unchanged over that addition; if it is negative, total product falls.
Diminishing marginal product is interpreted under the fixed-input condition: labor changes while other inputs and production conditions remain fixed. It does not mean that labor itself has declined, or that total output must already have declined. A schedule with only a few reported rows also supports conclusions only about the changes shown. When rows skip labor levels, interval marginal product is an average over the workers added in that interval; it does not reveal each worker's exact contribution.
- Wrong calculation: With output rising from \(22\) to \(49\) as labor rises from \(2\) to \(5\), report marginal product as \(49-22=27\). Correction: \(27\) is the total output change over the interval; per added worker, marginal product is \(27/(5-2)=9\).
- Wrong interpretation: Marginal product falls from \(18\) to \(15\), so total product falls. Correction: The marginal product is still positive, so total product rises, but by a smaller amount.
- Wrong average: Use the increase in output divided by the number of workers to calculate average product. Correction: Average product is total product divided by the entire labor input at that row; the change-in-output calculation is for marginal product.
- Wrong comparison: Conclude that average product rises whenever marginal product is positive. Correction: Average product rises when the new marginal product exceeds the prior average product; a positive marginal product can still be below the average.
Check Your Understanding
For each problem, calculate the requested schedule measures before checking the final answer.
- A hypothetical greenhouse holds its growing space and equipment fixed. Total product at labor levels \(0,1,2,3,4\) is \(0,9,23,39,51\) trays. Find the marginal product at each addition and average product for positive labor levels. Identify where marginal product rises and diminishes. (Answer: Marginal products are \(9,14,16,12\); average products are \(9,11.5,13,12.75\); marginal product rises through the third addition and diminishes at the fourth.)
- A hypothetical sorting line produces \(0,33,65,80\) parcels with \(0,3,7,10\) workers. Find interval marginal product per added worker and average product at each positive labor level. What does the schedule say about the intervals? (Answer: Interval marginal products are \(11,8,5\); average products are \(11,65/7\approx9.29,\) and \(8\); average interval marginal product declines across the three reported intervals.)
- A hypothetical workshop reports total product \(0,10,24,39,51,57\) with \(0,1,2,3,4,5\) workers. Calculate successive marginal and average products, and identify why average product first falls. (Answer: Marginal products are \(10,14,15,12,6\); average products are \(10,12,13,12.75,11.4\); average product first falls at four workers because that worker's marginal product \(12\) is below the prior average product \(13\).)
- A hypothetical facility records total product \(0,16,35,50,62,60\) with \(0,1,2,3,4,5\) workers. Find all marginal products and average product at five workers. State which addition causes total product to fall. (Answer: Marginal products are \(16,19,15,12,-2\); average product at five workers is \(60/5=12\); total product falls with the fifth addition, when marginal product is \(-2\).)
- A hypothetical production unit reports output \(0,26,66,84\) at labor levels \(0,2,6,9\). Calculate marginal product per added worker for each interval and average product at the three positive labor levels. Explain the limitation of these figures. (Answer: Interval marginal products are \(13,10,6\); average products are \(13,11,\) and \(84/9\approx9.33\); they show declining average marginal product across reported intervals but do not establish each individual worker's marginal product.)