Why Discrete Marginal Cost Matters
A firm’s cost schedule may report total cost at several output levels without giving a continuous cost function. To understand the cost of expanding production, compare the total cost at the two output levels. The key is to account for how much output changed as well as how much cost changed.
For example, if total cost rises by $120 while output increases by 10 units, the cost change is $120, but the marginal cost over that interval is $12 per additional unit. Dividing by the output increase puts comparisons between different-sized production changes on a common per-unit basis.
Reading the Cost Schedule Correctly
Use the total-cost values associated with the two output levels being compared. If the schedule reports fixed cost and variable cost separately, total cost is their sum at each output level. If it already reports total cost, do not add fixed cost a second time.
The standard short-run comparison assumes the production conditions and relevant input prices are unchanged between observations. In particular, the fixed cost of operating the chosen capacity is held constant. If fixed cost is constant, it appears in both total-cost observations and cancels when the difference is calculated. If fixed cost changes between observations, the total-cost difference includes that change and does not isolate the cost of additional output at unchanged capacity.
A schedule with output increases of one unit makes the calculation especially direct: the total-cost difference is already the cost per additional unit for that step. With larger output intervals, division is essential. If output changes from 30 to 50 units, for instance, the denominator is 20 units, not 50 units.
Worked Example: One-Unit Output Changes
A hypothetical workshop has the following short-run total-cost schedule. Its operating capacity and input prices remain unchanged across the listed output levels. Find marginal cost for each one-unit increase after output begins.
| Output, \(Q\) (units) | Total cost, \(TC(Q)\) (dollars) |
|---|---|
| 0 | 120 |
| 1 | 156 |
| 2 | 199 |
| 3 | 251 |
Method. For each adjacent pair, subtract the earlier total cost from the later total cost, then divide by the increase in output. Each output increase here is one unit.
From 0 to 1 unit, \(\Delta TC=156-120=36\) dollars and \(\Delta Q=1-0=1\) unit. Therefore, \(MC_{0\text{ to }1}=36/1=\$36\) per unit.
From 1 to 2 units, \(\Delta TC=199-156=43\) dollars and \(\Delta Q=2-1=1\) unit. Therefore, \(MC_{1\text{ to }2}=43/1=\$43\) per unit.
From 2 to 3 units, \(\Delta TC=251-199=52\) dollars and \(\Delta Q=3-2=1\) unit. Therefore, \(MC_{2\text{ to }3}=52/1=\$52\) per unit.
Check. The three cost increases add to \(36+43+52=\$131\). The total cost rise from 0 to 3 units is \(251-120=\$131\), so the interval calculations reconcile with the schedule.
Common mistake avoided: Reporting the three cost differences without stating that they are per unit. Here the differences do give the per-unit marginal costs because each output step is exactly one unit; that shortcut would not work for a larger step.
When Output Changes by More Than One Unit
For a larger interval, discrete marginal cost is the total cost increase divided by the number of additional units produced. The result is an interval rate. It does not reveal how the cost increase is distributed among the individual units unless further information about the cost schedule is available.
The schedule may have unequal output increments. Calculate each interval separately using its own change in output. Do not assume that every row represents the same number of additional units.
Worked Example: Unequal Output Intervals
A hypothetical plant records total cost at three production levels. Assume its operating capacity and input prices are unchanged. Calculate marginal cost from 20 to 30 units and from 30 to 50 units, then compare the results.
| Output, \(Q\) (units) | Total cost, \(TC(Q)\) (dollars) |
|---|---|
| 20 | 840 |
| 30 | 990 |
| 50 | 1,340 |
From 20 to 30 units: The cost change is \(990-840=\$150\). The output change is \(30-20=10\) units. Thus,
From 30 to 50 units: The cost change is \(1{,}340-990=\$350\). The output change is \(50-30=20\) units. Thus,
The second interval has a higher marginal cost by \(\$17.50-\$15=\$2.50\) per unit. This comparison describes the average cost per added unit in each interval; the data do not establish the marginal cost of every individual unit inside either interval.
Check. The increase from 20 to 50 units is \(1{,}340-840=\$500\), and the output increase is \(50-20=30\) units. The combined interval rate is \(\$500/30=\$16.67\) per unit to two decimal places. It lies between the two interval rates, as expected because the 20-to-50 interval combines 10 units at an interval rate of $15 and 20 units at an interval rate of $17.50.
Common mistake avoided: Dividing the second cost increase by 50, the final output level. The correct denominator is the additional output, \(50-30=20\), so the answer is $17.50 per unit, not $7 per unit.
Why the Formula Works
The calculation follows from what “cost per additional unit” means over an interval. The total cost change is the amount of extra spending associated with moving from the initial output to the final output. The output change counts how many extra units are produced. Dividing the former by the latter allocates the interval’s cost increase across its added output.
Subtract the initial total cost from the final total cost: \(\Delta TC=TC(Q_2)-TC(Q_1)\).
Subtract initial output from final output: \(\Delta Q=Q_2-Q_1\). For an increase, this quantity is positive.
By the definition of a per-unit rate, divide the added cost by the number of added units: \(MC=\Delta TC/\Delta Q\).
The same calculation can be viewed as the slope of a straight line joining the two observations on a graph with output on the horizontal axis and total cost on the vertical axis. Its rise is \(\Delta TC\), and its run is \(\Delta Q\). This is a secant slope across the observed interval. It is not automatically the slope at one endpoint or the exact cost of a particular unit.
If output rises by one unit, the interval contains one additional unit, so the ratio is that step’s total-cost increase. If output rises by several units, the ratio is the average cost increase per additional unit across the whole interval. The schedule alone does not tell us whether those added units each had exactly that cost.
Worked Example: Interpreting Marginal Cost Across Several Steps
A hypothetical packaging operation records the following costs under the same short-run capacity and input prices. Calculate the interval marginal cost for each listed increase, identify which interval has the greatest rate, and verify that the total change reconciles.
| Output, \(Q\) (packages) | Total cost, \(TC(Q)\) (dollars) |
|---|---|
| 0 | 500 |
| 4 | 620 |
| 9 | 815 |
| 15 | 1,115 |
From 0 to 4 packages: \(\Delta TC=620-500=\$120\) and \(\Delta Q=4-0=4\). Thus, \(MC_{0\text{ to }4}=\$120/4=\$30\) per package.
From 4 to 9 packages: \(\Delta TC=815-620=\$195\) and \(\Delta Q=9-4=5\). Thus, \(MC_{4\text{ to }9}=\$195/5=\$39\) per package.
From 9 to 15 packages: \(\Delta TC=1{,}115-815=\$300\) and \(\Delta Q=15-9=6\). Thus, \(MC_{9\text{ to }15}=\$300/6=\$50\) per package.
The greatest interval marginal cost is $50 per package, for the increase from 9 to 15 packages. For the full change from 0 to 15 packages, the total cost rise is \(1{,}115-500=\$615\), and the output rise is 15 packages. The overall interval rate is \(\$615/15=\$41\) per package.
Check. Weighting each subinterval rate by its output increase gives \(4(\$30)+5(\$39)+6(\$50)=\$120+\$195+\$300=\$615\), exactly the full cost change. The overall rate is therefore \(\$615/15=\$41\).
Common mistake avoided: Comparing raw cost increases—$120, $195, and $300—as if they were per-package costs. The intervals have different sizes, so each increase must be divided by its own output change before comparing rates.
Assumptions, Units, and Interpretation
A cost schedule comparison is most informative when it holds relevant conditions fixed. If capacity, input prices, or the period being measured changes between observations, the cost difference may reflect more than the extra output. In particular, if a fixed cost changes, the formula still calculates the total-cost change per additional unit, but that figure mixes the output-related cost change with the fixed-cost change.
For example, suppose fixed cost rises by $60 between two observations while output rises by 6 units. The observed total-cost difference includes that $60. To isolate the variable-cost increase, subtract the fixed-cost change from the total-cost change before dividing by the output increase. This adjustment is justified only when the fixed-cost change is known and the aim is specifically to isolate variable cost; otherwise, report the schedule’s total-cost change per added unit and state what it includes.
Keep units visible during the calculation. If cost is stated in thousands of dollars and output in units, the result is thousands of dollars per unit unless converted. If output is measured in hundreds of items, an increase from 2 to 5 represents 3 hundreds, not 3 individual items. Consistent units prevent answers that are off by a factor of 100 or 1,000.
| Situation | What to calculate | Interpretation |
|---|---|---|
| Output increases by one unit | Total-cost difference divided by 1 | Cost increase for that one-unit step |
| Output increases by several units | Total-cost difference divided by the full output increase | Average cost increase per added unit over the interval |
| Fixed cost changes between observations | Use total-cost difference for observed total cost; adjust only if isolating variable cost | The unadjusted rate includes the fixed-cost change |
| Output or cost uses scaled units | Convert or retain the stated units consistently | State the resulting cost-per-output unit clearly |
Common Exam Traps
- Using the final output instead of the output change. For a move from 30 to 50 units, divide by 20, not by 50.
- Forgetting the output interval. State which pair of output levels produced the rate. A schedule’s interval result is not automatically a value “at” the endpoint.
- Comparing total cost changes across intervals of different sizes. Divide each change by its own output increase before comparing marginal costs.
- Mixing up units. A result based on dollars and dozens of units is dollars per dozen, not dollars per individual unit.
- Ignoring a changed fixed cost. A difference in total cost includes any change in fixed cost; it isolates the cost of extra output only under the relevant constant-cost conditions.
A reliable calculation therefore begins by identifying the two observations, writing both differences explicitly, and labeling the units. Then interpret the result as an interval rate and check that the cost and output changes match the schedule.
Check Your Understanding
For each question, use the stated output units and distinguish the cost change from the per-unit rate.
- A hypothetical print shop has total cost of $250 at zero orders, $286 at one order, and $331 at two orders. What is marginal cost from one to two orders? (Answer: $45 per order.)
- A workshop’s total cost rises from $460 at 12 units to $596 at 20 units, with capacity and input prices unchanged. Calculate marginal cost over the interval. (Answer: $17 per unit.)
- A hypothetical producer has total cost of $300 at zero units, $425 at five units, and $635 at eleven units. Calculate marginal cost for each listed interval and identify the higher rate. (Answer: $25 per unit from 0 to 5; $35 per unit from 5 to 11; the 5-to-11 interval is higher.)
- A schedule reports total cost of $960 at 40 items, $1,110 at 50 items, and $1,450 at 70 items. Calculate marginal cost for each adjacent interval. (Answer: $15 per item from 40 to 50; $17 per item from 50 to 70.)
- A firm’s total cost rises from $700 at 10 units to $940 at 16 units. During this change, fixed cost rises by $60. Find (a) the observed total-cost change per added unit and (b) the variable-cost change per added unit, assuming the entire $60 fixed-cost increase is known. (Answer: (a) $40 per unit; (b) $30 per unit.)