From Production Data to Cost Data
A production schedule tells a firm how much output it can produce with specified quantities of inputs. To convert that information into costs, attach a price to each input: multiply each variable input quantity by its unit price, add those expenditures, and then include fixed cost when calculating total cost. The result is a cost schedule: each listed output level is paired with its corresponding cost.
The key is to keep the production and cost information aligned. If a row says that three labor units produce 32 units of output, the cost assigned to that output must use the cost of those three labor units, along with any other variable inputs required by that row. A cost schedule does not come from output alone; it depends on the input quantities used and their prices.
For one variable input, such as labor, let \(L\) denote the quantity of labor and \(w\) its price per unit. If fixed cost is \(F\), then the variable and total costs at a scheduled output level are calculated from the labor required for that output:
With several variable inputs, add the cost of each one. If input \(j\) has quantity \(x_j\) and price \(p_j\), then \(VC=\sum_j p_jx_j\), and \(TC=F+\sum_j p_jx_j\). These expressions assume that the listed input quantities are the quantities used for that production plan, input prices are constant over the schedule, and there are no additional variable costs omitted from the data. If any of those assumptions changes, the cost calculation must reflect the changed information.
Why Input Quantities Determine Cost
The mapping from production to cost follows directly from expenditure. Buying \(L\) units of labor at \(w\) dollars per unit costs \(wL\) dollars. With multiple inputs, the firm pays the price of each input for every unit purchased, so the total variable expenditure is the sum of those input expenditures. Fixed cost is then added because it is part of total cost even when it does not vary with output.
This reasoning also explains why a production schedule must identify input use, not just output. The same output level could, in principle, be produced with different input combinations. If their input prices differ, the cost of the production plans can differ as well. The schedule and prices together specify the cost mapping.
Record the output level and the quantity of every variable input used to produce it.
Multiply each input quantity by its unit price, using consistent units such as dollars per labor hour and labor hours.
The sum is variable cost for that production plan.
This gives total cost at that output level for the stated short-run period.
Worked Schedule With One Variable Input
Worked Example: Labor Cost at Each Scheduled Output
A small workshop has fixed equipment with a short-run cost of $240 per week. Labor is the only variable input, and one labor unit costs $18 per week. The production schedule lists the following output for each labor quantity. Assume the equipment and its cost remain fixed, labor is priced at $18 for every row, and there are no other variable costs. Construct the variable-cost and total-cost schedules.
| Labor units | Output units |
|---|---|
| 0 | 0 |
| 1 | 8 |
| 2 | 19 |
| 3 | 32 |
| 4 | 43 |
| 5 | 51 |
| 6 | 56 |
Method. For each row, multiply labor units by $18 to obtain variable cost. Then add the weekly fixed cost of $240 to obtain total cost.
At zero labor, \(VC=0\times \$18=\$0\), so \(TC=\$240+\$0=\$240\). At one labor unit, \(VC=1\times \$18=\$18\), so \(TC=\$240+\$18=\$258\). Continue in the same way: at two labor units, \(VC=2\times \$18=\$36\) and \(TC=\$276\); at three, \(VC=3\times \$18=\$54\) and \(TC=\$294\); at four, \(VC=4\times \$18=\$72\) and \(TC=\$312\); at five, \(VC=5\times \$18=\$90\) and \(TC=\$330\); at six, \(VC=6\times \$18=\$108\) and \(TC=\$348\).
| Output units | Labor units | Variable cost | Total cost |
|---|---|---|---|
| 0 | 0 | $0 | $240 |
| 8 | 1 | $18 | $258 |
| 19 | 2 | $36 | $276 |
| 32 | 3 | $54 | $294 |
| 43 | 4 | $72 | $312 |
| 51 | 5 | $90 | $330 |
| 56 | 6 | $108 | $348 |
Check. Every total-cost entry is exactly $240 greater than the corresponding variable-cost entry. At output 32, for example, the schedule uses three labor units, so variable cost is $54 and total cost is \( \$240+\$54=\$294 \). This verifies that the input quantity, price, output, and cost columns remain aligned.
When More Than One Variable Input Is Used
A production plan may use multiple variable inputs. The same procedure still works, but each input must be priced separately before expenditures are added. Do not multiply one input price by a combined quantity that mixes different units: labor hours and fuel units, for example, cannot be added as if they were the same input.
Worked Example: Combining Labor and Materials Costs
A hypothetical plant has weekly fixed cost of $175. For each scheduled output, it uses the listed labor units and material units. Labor costs $30 per unit, and materials cost $4 per unit. Assume these prices are constant, the schedule gives all variable inputs used, and there are no other variable costs. Find variable and total cost at each output.
| Output units | Labor units | Material units |
|---|---|---|
| 0 | 0 | 0 |
| 10 | 1 | 3 |
| 24 | 2 | 7 |
| 39 | 3 | 12 |
| 51 | 4 | 18 |
Method. Calculate labor expenditure as labor units times $30 and material expenditure as material units times $4. Add those two amounts to obtain variable cost; then add $175 to obtain total cost.
At output 0, both inputs are zero, so \(VC=\$0+\$0=\$0\) and \(TC=\$175\). At output 10, labor costs \(1\times \$30=\$30\), materials cost \(3\times \$4=\$12\), so \(VC=\$30+\$12=\$42\) and \(TC=\$175+\$42=\$217\).
At output 24, labor costs \(2\times \$30=\$60\), materials cost \(7\times \$4=\$28\), so \(VC=\$60+\$28=\$88\) and \(TC=\$175+\$88=\$263\). At output 39, labor costs \(3\times \$30=\$90\), materials cost \(12\times \$4=\$48\), so \(VC=\$138\) and \(TC=\$175+\$138=\$313\). At output 51, labor costs \(4\times \$30=\$120\), materials cost \(18\times \$4=\$72\), so \(VC=\$192\) and \(TC=\$175+\$192=\$367\).
| Output units | Labor cost | Materials cost | Variable cost | Total cost |
|---|---|---|---|---|
| 0 | $0 | $0 | $0 | $175 |
| 10 | $30 | $12 | $42 | $217 |
| 24 | $60 | $28 | $88 | $263 |
| 39 | $90 | $48 | $138 | $313 |
| 51 | $120 | $72 | $192 | $367 |
Check. At output 39, the $138 variable cost is the sum of $90 in labor and $48 in materials; adding fixed cost gives $313. The calculation keeps the input categories separate until both have been converted to dollars.
Discrete Output Levels and Feasible Production Plans
A production schedule often lists only particular input quantities and the resulting output levels. The corresponding cost schedule therefore gives cost points for those listed output levels. It does not, by itself, establish the input requirement or cost at an output between two rows. Interpolating between rows requires an additional assumption about how input use changes between the listed plans.
A related distinction arises when a question asks for enough production to meet a target rather than the cost at a particular scheduled output. If labor is available only in whole units, a target that lies between two listed output levels may require choosing the next feasible production plan. The cost belongs to the inputs used by that plan. If those inputs produce more than the target, do not silently relabel their scheduled output as the target.
Worked Example: Meeting a Minimum Output Target
A workshop has fixed weekly cost of $310. Its production schedule shows output of 0, 7, 17, 29, and 38 units with, respectively, 0, 1, 2, 3, and 4 labor units. A labor unit costs $95 per week, and labor can be hired only in whole units. Assume the fixed input and its cost do not change, and there are no other variable costs. What are the costs for the scheduled production plans, and what is the least-cost listed plan that can meet a target of at least 26 units?
Method. First map each scheduled labor quantity to costs using \(VC=95L\) and \(TC=310+VC\). Then identify the smallest scheduled output that is at least 26.
At output 0, \(VC=0\times \$95=\$0\), and \(TC=\$310\). At output 7, \(VC=1\times \$95=\$95\), and \(TC=\$310+\$95=\$405\). At output 17, \(VC=2\times \$95=\$190\), and \(TC=\$500\). At output 29, \(VC=3\times \$95=\$285\), and \(TC=\$595\). At output 38, \(VC=4\times \$95=\$380\), and \(TC=\$690\).
The scheduled output of 17 is below the target of 26, while 29 meets it. With whole labor units and the listed plans, the least-cost feasible plan meeting the target uses three labor units. It produces 29 units, with variable cost $285 and total cost $595. The cost is attached to the scheduled plan producing 29 units; the schedule does not establish a separate cost for producing exactly 26 units.
Check. The plan with two labor units costs \( \$310+\$190=\$500 \), but produces only 17 units, so it fails the target. The next listed plan costs $595 and produces 29, confirming the target choice.
Common Mistakes in the Mapping
A frequent error is to use output as though it were the quantity of the variable input. If labor costs $18 per labor unit, multiplying output by $18 is not justified unless the problem specifically says each output unit requires one labor unit. The production schedule is what links output to labor; the input price then converts labor into expenditure.
For example, at output 32 in the first worked example, the wrong calculation would be \(32\times \$18=\$576\), treating 32 output units as 32 labor units. The schedule instead assigns three labor units to output 32. The correct calculation is \(3\times \$18=\$54\) of variable cost, followed by \( \$240+\$54=\$294 \) of total cost. The wrong method overstates variable cost by $522 and total cost by the same amount.
- Omitting fixed cost from total cost. Variable cost at zero output can be zero, but total cost is still the stated fixed cost.
- Adding input quantities before pricing them. Price each input in its own units, then add the dollar expenditures.
- Using the wrong row. Match each output level to the exact input quantities associated with it in the production schedule.
- Inventing an unlisted cost point. A discrete schedule does not imply linear input use or cost between rows.
- Confusing a target with the scheduled output. When a feasible plan exceeds a minimum target, report both the plan's actual scheduled output and its cost.
Check Your Understanding
For each problem, use only the production plans and cost assumptions stated.
- A shop has fixed cost of $90 and uses one variable input costing $16 per unit. Its schedule lists output 0, 6, 15, and 27 for input quantities 0, 1, 2, and 3. Find variable cost and total cost at output 15. (Answer: variable cost $32; total cost $122)
- A producer has fixed cost of $140. At output 18, it uses two labor units costing $25 each and five material units costing $3 each. Assume these are its only variable inputs. Find variable cost and total cost. (Answer: variable cost $65; total cost $205)
- A plant has fixed cost of $450 and a weekly labor price of $72 per labor unit. The schedule lists output 0, 11, 25, 42, and 54 for labor quantities 0, 1, 2, 3, and 4. Find variable and total cost at output 42. (Answer: variable cost $216; total cost $666)
- A workshop has fixed cost of $200 and labor costs $64 per unit. The listed plans produce 0, 12, 25, and 39 units using 0, 1, 2, and 3 labor units. If the requirement is at least 31 units and labor must be whole units, identify the least-cost listed plan that meets the requirement and its costs. (Answer: 3 labor units produce 39 units; variable cost $192; total cost $392)
- A firm has fixed cost of $120. Its schedule lists output 8, 20, and 35 using 1, 2, and 3 labor units, respectively. Labor costs $25 per unit, and packaging costs $2 per output unit. Assume these are the only variable costs. Find total cost at outputs 20 and 35, and the increase in total cost between those scheduled output levels. (Answer: total cost $210 at output 20; $265 at output 35; increase $55)