Why Average Product Matters
A production total tells a manager how much a workplace produced, but not how that output relates to the resources used. If one workshop makes 120 units with 8 workers and another makes 150 units with 15 workers, the second workshop has greater total output, but the first has greater output per worker. Average product expresses this output-to-input relationship as a ratio.
For an actuarial candidate, the key is to identify exactly which input is in the denominator and to use a consistent measure of that input. Output per worker, output per worker-hour, and output per machine are different measures. They can each be useful, but they answer different questions.
Calculating Average Product from a Production Schedule
A production schedule reports total output at different input quantities. To calculate average product, divide the output in the relevant row by the amount of the input used in that row. When a schedule is for short-run production, other inputs—such as the number of machines or the size of a facility—are held fixed. Each resulting ratio describes output per unit of the changing input under those conditions.
For example, if 6 workers produce 96 packages during one shift, average product of labor is \(96/6=16\) packages per worker per shift. This is an average across the workers represented in the production total; it does not assert that every worker individually produced exactly 16 packages.
Worked Example: Reading Average Product from a Schedule
A hypothetical packing room uses the same equipment and shift length in each observation. Its schedule reports the following total output. Calculate average product of labor when there are 4 workers and when there are 6 workers, then interpret each result.
| Workers, \(L\) | Total packages, \(Q\) |
|---|---|
| 2 | 30 |
| 4 | 68 |
| 6 | 96 |
| 8 | 112 |
Method. Use \(AP_L=Q/L\) separately for each requested row. The input is workers, not the change in workers between rows.
With 4 workers, total output is 68 packages:
With 6 workers, total output is 96 packages:
The first ratio means that the 68 packages correspond to an average of 17 packages per worker for that shift; the second means the 96 packages correspond to an average of 16 packages per worker. The equipment and shift conditions are specified as unchanged, so the schedule provides a consistent basis for these comparisons.
Check. Multiplying each average product by its worker count recovers the reported total: \(17(4)=68\) and \(16(6)=96\). A common mistake is to divide the change in output between the 4-worker and 6-worker rows by 6. That computes neither row’s average product: average product at 6 workers uses total output at 6 workers, \(96/6=16\).
Average Product from a Production Function
A production function gives output as a function of input quantities. In a short-run setting with capital fixed at a specified level \(K\), write output as \(Q=f(L,K)\). For any positive labor quantity, average product of labor is the function’s output at that labor quantity divided by the labor quantity. The other input level must be the one specified in the function or problem; changing it can change output and therefore the ratio.
The calculation does not require differentiating the production function. A derivative describes a rate of change in output with respect to an input, whereas average product is a level of output divided by a level of input. Those quantities have the same broad dimensions—output per input—but are defined differently.
Worked Example: Evaluating Average Product from a Function
A hypothetical workshop operates with its machine capacity fixed. For the relevant shift, its production function is \(Q=20\sqrt{L}\), where \(L\) is the number of workers and \(Q\) is completed components per shift. Calculate average product at 25 workers and at 64 workers.
Method. First evaluate the production function to find total output at each labor quantity. Then divide that total by the same labor quantity.
At \(L=25\), \(\sqrt{25}=5\), so total output is:
Therefore:
At \(L=64\), \(\sqrt{64}=8\), so total output is:
Therefore:
Check. The ratios recover the function’s outputs: \(4(25)=100\) and \(2.5(64)=160\). The common mistake is to report \(Q=100\) or \(Q=160\) as average product. Those are total outputs; average product requires division by the corresponding labor quantity. These results apply to the stated fixed machine capacity and shift definition.
Why the Ratio Is Output per Input
The formula for average product follows from what the measure is intended to express: the total output distributed as an average amount across the units of input. The same reasoning applies whether the input units are workers, worker-hours, or machines, provided the numerator and denominator describe the same production period and operation.
Let \(Q\) be the output produced using \(L>0\) units of labor during the stated period and under the stated production conditions.
If the average output associated with each labor unit is \(a\), then \(L\) labor units at that average correspond to \(La\) units of total output.
Set \(La=Q\). Since \(L>0\), divide both sides by \(L\) to obtain \(a=Q/L\). Thus average product of labor is \(AP_L=Q/L\).
This justification establishes an aggregate average, not a claim about how much any particular worker produced. Production may involve workers sharing tasks, equipment, and materials, so the total-output ratio need not equal any worker’s individual contribution.
Choosing the Input Unit
The denominator determines the meaning of the measure. If labor is counted as people, the result is output per worker. If labor is counted as time worked, the result is output per worker-hour. A comparison is informative only when the input measure and the relevant production conditions are clear.
Worked Example: Output per Worker and per Worker-Hour
Two hypothetical service centers process claims in one shift. Center A has 8 workers, each working 6 hours, and processes 240 claims. Center B has 10 workers, each working 5 hours, and processes 250 claims. Calculate output per worker and per worker-hour for each center. Explain why the two measures differ.
Method. For output per worker, divide claims by the number of workers. For output per worker-hour, first calculate total worker-hours, then divide claims by that total.
For Center A, total worker-hours are \(8(6)=48\). Its output per worker is:
Its output per worker-hour is:
For Center B, total worker-hours are \(10(5)=50\). Its output per worker is:
Its output per worker-hour is:
Center A has higher output per worker for the shift, 30 claims compared with 25, because each worker’s shift is longer. Both centers have the same output per worker-hour, 5 claims. The results are consistent: \(30/6=5\) for Center A and \(25/5=5\) for Center B. Check. Multiplying claims per worker-hour by total worker-hours gives \(5(48)=240\) and \(5(50)=250\). A common mistake is to compare the centers’ output per worker as though the workers had worked the same number of hours.
Limits and Common Traps
Average product is straightforward to calculate, but interpreting it requires attention to the data and the model. It is undefined when the input quantity in the denominator is zero. It also does not, by itself, identify how output would change if the input quantity changed. For that question, the relevant measure is marginal product, which concerns the additional output associated with a change in input.
A ratio can also conceal differences in operating conditions. Output per worker may be affected by hours worked, equipment, technology, or the type of output being produced. Before comparing average products, check whether the compared observations use the same output unit and time period and whether other relevant inputs are comparable or held fixed.
Check Your Understanding
For each problem, identify the input used in the denominator and include its units in the result.
- A hypothetical print shop produces 126 posters in one shift using 9 workers. Find average product of labor. (Answer: 14 posters per worker per shift)
- With its number of machines held fixed, a workshop produces 84 parts using 6 technicians in a day. Calculate average product of labor and state what the result means. (Answer: 14 parts per technician per day)
- A production function for a fixed facility is \(Q=20\sqrt{L}\) parts per day. Find total output and average product at \(L=100\). (Answer: 200 parts per day; 2 parts per worker per day)
- A hypothetical repair depot completes 360 jobs in a day. It has 12 technicians, each working 4 hours. Calculate output per technician per day and per technician-hour. (Answer: 30 jobs per technician per day; 7.5 jobs per technician-hour)
- A workshop produces 240 units using 8 workers and 6 machines in a shift. Calculate average product of labor and average product of capital, and give each result’s units. (Answer: 30 units per worker per shift; 40 units per machine per shift)