Why Total Product Matters
A production schedule records how much output a firm produces at different quantities of an input. Turning that schedule into a production curve makes the input-output relationship visible: each point pairs a quantity of the input with the output associated with it. To interpret the curve correctly, you must know which input varies, which other conditions are being held fixed, and what the axes measure.
This tutorial focuses on total product: the output level associated with a specified quantity of a variable input under a particular production setup. A curve is not meaningful without that setup. For example, output at a given quantity of labor may differ if the firm has different equipment, technology, or access to materials.
The notation \(f(L;\bar K)\) emphasizes that the output level depends on the quantity of labor and on the fixed capital quantity. The semicolon separates the input that varies in the schedule from the condition being held fixed. It does not imply that capital is unimportant; it means only that the schedule describes one particular capital setup.
A production schedule may give output at only selected quantities of labor. Each row then supplies a coordinate for the graph: the labor quantity is the horizontal coordinate, and its corresponding total output is the vertical coordinate. The graph should be labelled with units, such as worker-hours per day and units produced per day. Without units and the fixed-input condition, a plotted curve can be ambiguous.
From a Schedule to a Production Curve
Suppose a workshop uses a fixed set of machines and varies the number of worker-hours assigned to a shift. The schedule below reports output per shift.
| Labor, \(L\) (worker-hours per shift) | Total product, \(Q\) (items per shift) |
|---|---|
| 0 | 0 |
| 2 | 12 |
| 4 | 21 |
| 6 | 27 |
| 8 | 30 |
To construct the curve, label the horizontal axis “Labor, worker-hours per shift” and the vertical axis “Total product, items per shift.” Plot the ordered pairs \((0,0)\), \((2,12)\), \((4,21)\), \((6,27)\), and \((8,30)\). The resulting points describe the schedule for this workshop with its machines and other production conditions fixed.
The point \((4,21)\), for example, means that four worker-hours per shift are associated with total output of 21 items per shift. It does not mean 21 items per worker-hour; that would be a different quantity. Reading the coordinates in their correct order is essential.
If the schedule provides values only at selected input quantities, the observed points are known, but the exact output between those points is not determined by the table alone. A line drawn between points can help display the relationship, but treating an intermediate value as exact requires an additional assumption about the production relationship. Do not silently substitute a guessed or interpolated output for a value absent from the schedule.
Worked Example: Plotting a Schedule and Reading Its Intercept
A packaging station has the same equipment and work area throughout a shift. Its schedule reports the following pairs: 0 workers produce 0 cartons, 1 worker produces 7 cartons, 2 workers produce 15 cartons, and 3 workers produce 20 cartons. Identify the axes, list the points to plot, and state the vertical intercept and output at 2 workers.
Method. Treat the schedule as a short-run production relationship with workers varying and equipment and work area fixed. The input quantity goes on the horizontal axis; total output goes on the vertical axis.
Horizontal axis. Label it “Workers per shift.” Vertical axis. Label it “Cartons per shift.” Since the schedule gives one output for each listed worker quantity, the plotted coordinates are \((0,0)\), \((1,7)\), \((2,15)\), and \((3,20)\).
The vertical intercept is the output where the horizontal coordinate is zero. The schedule gives \(Q=0\) when \(L=0\), so the curve intersects the vertical axis at \((0,0)\). At \(L=2\), the table gives \(Q=15\) cartons per shift.
Check. The requested output is read from the row for 2 workers, not from the row for 15 cartons. The answer is consistent with the plotted point \((2,15)\). This avoids the common mistake of reversing the axes or reporting an input quantity as output.
What the Intercept Says
The vertical intercept is the output level on the production curve when the variable input on the horizontal axis is zero. In a labor schedule, it is the output associated with zero labor, while the stated fixed inputs and technology remain as specified. If the schedule gives \(Q=0\) at \(L=0\), the curve passes through the origin. If the schedule instead reports positive output at \(L=0\), that positive value is the vertical intercept for that schedule.
Do not assume an intercept merely from the appearance of a graph or from what seems economically plausible. Use the given production relationship or schedule. A table that starts at \(L=1\) does not establish the output at \(L=0\); the intercept is then not determined by those observations alone.
The horizontal axis intercept is a different idea: it is the labor quantity at which output equals zero. When the schedule includes \((0,0)\), that point lies on both axes. In other schedules, the horizontal intercept may occur at a positive labor quantity, or the available information may not establish one. Always state which intercept is being requested.
Why the Graph Uses These Coordinates
The coordinate choice follows directly from what the schedule records. For a fixed technology and fixed quantities of other inputs, the production relationship assigns an output to each relevant quantity of the variable input. In symbols, \(Q=f(L;\bar K)\). A schedule row that states labor \(L_0\) produces output \(Q_0\) therefore records the ordered pair \((L_0,Q_0)\). Plotting all such pairs displays the same mapping as the schedule, without changing the underlying information.
This also explains why the fixed-input condition belongs in the description of the curve. If the fixed capital quantity changes from \(\bar K_1\) to \(\bar K_2\), the schedule represents \(f(L;\bar K_1)\) in one case and \(f(L;\bar K_2)\) in the other. A point on one schedule cannot automatically be combined with a point from the other to make a single curve for one unchanged production setup.
Identify the technology and the inputs held fixed, and name the input quantity that varies.
Put the variable input on the horizontal axis and total output on the vertical axis.
Write the input quantity first and its corresponding total output second.
Use an observed point when the input quantity appears in the schedule; do not claim an unsupported intermediate value.
For the vertical intercept, find the output at zero units of the horizontal-axis input.
Worked Example: Reading Output at Several Input Quantities
A print shop holds its press capacity and paper supply conditions fixed for a day. Its production schedule is: 0 operator-hours, 0 posters; 2 operator-hours, 18 posters; 4 operator-hours, 31 posters; 6 operator-hours, 40 posters; and 8 operator-hours, 46 posters. State the curve’s intercept, read output at 4 and 8 operator-hours, and explain whether the schedule determines output at 5 operator-hours.
Method. The variable input is operator-hours per day, so place it on the horizontal axis. Total posters per day belong on the vertical axis. The fixed press capacity and paper conditions identify which production schedule is being graphed.
The plotted points are \((0,0)\), \((2,18)\), \((4,31)\), \((6,40)\), and \((8,46)\). The schedule states that zero operator-hours correspond to zero posters, so the vertical intercept is \((0,0)\).
At 4 operator-hours, the schedule gives 31 posters per day. At 8 operator-hours, it gives 46 posters per day. These are direct readings of the relevant rows, with the input and output units retained.
The schedule has no row for 5 operator-hours. It therefore does not give an exact output at that input quantity. One could calculate a value between the observations only by adopting an extra interpolation rule or a more fully specified production function; neither is supplied here.
Check. The output levels at the listed quantities match their schedule entries, and the intercept comes from the zero-input row. The common mistake avoided is treating a graph drawn between observed points as proof of an exact intermediate output.
Comparing Curves Without Mixing Schedules
A production curve describes one production setup. If a schedule gives output under two different fixed quantities of capital, plot and label the two schedules separately. The horizontal and vertical axes may have the same units, but each curve still corresponds to its own fixed-input condition.
Worked Example: Selecting the Correct Fixed-Input Schedule
A small fabrication business records daily output for two fixed machine setups. With Setup A, the schedule is: 0 technician-hours, 0 components; 1 hour, 6 components; 2 hours, 11 components; and 3 hours, 15 components. With Setup B, it is: 0 technician-hours, 0 components; 1 hour, 9 components; 2 hours, 17 components; and 3 hours, 23 components. Construct the coordinates for Setup B, identify its intercept, and give output at 2 technician-hours. Explain why the two schedules should not be combined into one curve.
Method. Select the requested fixed-input condition first. For Setup B, technician-hours per day are the horizontal coordinate and components per day are the vertical coordinate. The Setup B coordinate pairs are \((0,0)\), \((1,9)\), \((2,17)\), and \((3,23)\).
The zero-hour row gives the vertical intercept \((0,0)\). At 2 technician-hours per day, Setup B produces 17 components per day according to its schedule.
Setup A and Setup B have different fixed machine setups. Their points describe different production relationships, so one curve should not combine coordinates from both. For example, the Setup A point at 1 hour is \((1,6)\), whereas Setup B’s point at 1 hour is \((1,9)\); each belongs to its labelled setup.
Check. Every listed Setup B coordinate uses the same input-output ordering and the same machine condition. The answer at 2 hours is 17, not Setup A’s 11. The common mistake avoided is silently switching fixed-input conditions while reading a single curve.
Common Interpretation Traps
- Reversing the coordinates. The variable input is the horizontal coordinate and output is the vertical coordinate. A schedule entry of 3 worker-hours and 20 items becomes \((3,20)\), not \((20,3)\).
- Leaving off units. “Output is 20” is incomplete if the schedule measures items per shift. Preserve the units when reading a level.
- Assuming the intercept. The zero-input output must come from the schedule or a stated production relationship. If no zero-input information is supplied, the intercept may be unknown.
- Mixing production setups. Keep separate any schedules with different fixed inputs or technologies, even if they use the same axes.
- Inventing an intermediate value. A schedule gives exact values at its listed input quantities. An unlisted quantity requires an additional assumption before an exact output can be calculated.
Production curves provide a compact way to read the level of output associated with a quantity of an input, while keeping the production conditions visible. The key discipline is to identify the setup, label the axes correctly, plot the schedule’s ordered pairs, and read only what the stated information supports. Those steps make the curve a reliable representation of total product rather than a source of avoidable misinterpretation.
Check Your Understanding
For each question, keep the input-output ordering and the stated fixed-input conditions explicit.
- A greenhouse holds its heating system fixed and records 0, 1, 2, and 3 gardener-hours with outputs of 0, 4, 9, and 13 trays of seedlings. What are the axis labels and the plotted coordinate at 2 gardener-hours? (Answer: Horizontal axis: gardener-hours; vertical axis: trays of seedlings; coordinate \((2,9)\).)
- A workshop schedule lists 0 machine-hours with output 3 batches, 2 machine-hours with output 10 batches, and 4 machine-hours with output 16 batches. What is the vertical intercept, and what does it mean for this schedule? (Answer: \((0,3)\); under the stated schedule’s fixed conditions, output is 3 batches at zero machine-hours.)
- A food processor’s schedule, with its equipment fixed, reports 1, 3, and 5 operator-hours producing 8, 19, and 27 cases. What output is specified at 3 operator-hours, and is output at 4 hours determined exactly by these entries alone? (Answer: 19 cases; no, the schedule alone does not specify the exact output at 4 hours.)
- Two laboratory setups have outputs at 2 technician-hours of 12 and 15 test kits, respectively. The first schedule holds one instrument configuration fixed and the second holds a different configuration fixed. What should be plotted for each setup at 2 hours, and may these be treated as two points from one unchanged production curve? (Answer: Plot \((2,12)\) for the first setup and \((2,15)\) for the second; no, they belong to different fixed-input conditions.)
- A repair facility records 0, 2, and 4 mechanic-hours per shift producing 0, 11, and 19 repaired units. A candidate reports the point at 4 hours as \((19,4)\) and calls 11 the vertical intercept. Identify both errors and give the correct answers. (Answer: The coordinates are reversed and the intercept was read from the wrong row; the correct point is \((4,19)\), and the vertical intercept is \((0,0)\).)