Why Put Production and Cost Measures in One Table?
A production schedule records how output changes as a firm uses more of a variable input. A cost schedule translates those input choices into dollar amounts. Combining both schedules makes the links between productivity and cost visible: the output added by an extra unit of labor affects the cost of producing additional output, while fixed cost is spread across the units produced.
The short run is a period in which at least one input is fixed. In the examples below, capital is fixed and labor is variable. The table is a set of discrete observations, so marginal measures describe changes between listed rows. They do not necessarily describe a smooth curve or a derivative.
These formulas assume that labor is the only variable input, its price per unit is constant, and fixed cost is unchanged across the rows. If there are other variable inputs, their costs must also be included in variable cost. If labor prices vary with the quantity hired, variable cost need not equal a constant wage multiplied by labor input.
Worked Example: Building a Complete Table
Worked Example: Building a Complete Table
A workshop has fixed capital during the period. Its production schedule lists labor input \(L\) and output \(Q\) as follows: \(L=0,1,2,3,4,5\) and \(Q=0,10,24,36,46,54\). Each labor unit costs \(w=\$12\) for the period, and fixed cost is \(FC=\$36\). Construct the production and cost measures for every row.
Method and assumptions. Use the change from the previous row for marginal measures. Labor rises by one unit each time, so \(MP_L=\Delta Q\). Variable cost is \(\$12L\), and total cost adds the fixed \(\$36\). Ratios per unit of output are undefined at \(Q=0\).
Productivity. At \(L=1\), \(MP_L=(10-0)/(1-0)=10\), and \(AP_L=10/1=10\). At \(L=2\), \(MP_L=(24-10)/(2-1)=14\), while \(AP_L=24/2=12\). At \(L=3\), \(MP_L=(36-24)/(3-2)=12\), and \(AP_L=36/3=12\). At \(L=4\), \(MP_L=(46-36)/(4-3)=10\), and \(AP_L=46/4=11.5\). At \(L=5\), \(MP_L=(54-46)/(5-4)=8\), and \(AP_L=54/5=10.8\).
Costs. Variable costs are \(12L\): respectively \(\$0,\$12,\$24,\$36,\$48,\$60\). Adding \(\$36\) of fixed cost gives total costs of \(\$36,\$48,\$60,\$72,\$84,\$96\). For example, at \(L=3\), \(AFC=36/36=\$1.00\), \(AVC=36/36=\$1.00\), and \(ATC=72/36=\$2.00\) per unit of output. Marginal cost from \(L=2\) to \(L=3\) is \((72-60)/(36-24)=\$1.00\) per additional output unit.
For each later row, calculate the average measures by dividing the cost by that row’s output. Calculate marginal cost by dividing the change in total cost by the change in output. The full table is:
| \(L\) | \(Q\) | \(MP_L\) | \(AP_L\) | \(FC\) | \(VC\) | \(TC\) | \(AFC\) | \(AVC\) | \(ATC\) | \(MC\) |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | — | — | $36 | $0 | $36 | — | — | — | — |
| 1 | 10 | 10 | 10 | $36 | $12 | $48 | $3.60 | $1.20 | $4.80 | $1.20 |
| 2 | 24 | 14 | 12 | $36 | $24 | $60 | $1.50 | $1.00 | $2.50 | $0.86 |
| 3 | 36 | 12 | 12 | $36 | $36 | $72 | $1.00 | $1.00 | $2.00 | $1.00 |
| 4 | 46 | 10 | 11.5 | $36 | $48 | $84 | $0.78 | $1.04 | $1.83 | $1.20 |
| 5 | 54 | 8 | 10.8 | $36 | $60 | $96 | $0.67 | $1.11 | $1.78 | $1.50 |
Dollar amounts per unit of output in the table are rounded to the nearest cent; productivity values are shown without rounding. For example, at \(L=4\), \(AFC=36/46=\$0.7826\ldots\), \(AVC=48/46=\$1.0435\ldots\), and \(ATC=84/46=\$1.8261\ldots\), giving the displayed values. At \(L=5\), marginal cost is \((96-84)/(54-46)=12/8=\$1.50\) per additional output unit.
Check. Each row satisfies \(TC=FC+VC\). At \(L=4\), for example, \(\$84=\$36+\$48\). For positive output, \(ATC=AFC+AVC\); at \(L=4\), \(\$1.83\) is the rounded sum of \(\$0.78+\$1.04\), with the one-cent difference arising because the components were rounded separately. Using unrounded values gives \(84/46=(36/46)+(48/46)\). The table’s marginal costs also equal \(\Delta VC/\Delta Q\), since fixed cost does not change.
Why Marginal Cost Is Linked to Marginal Product
With one variable input and a constant input price, the relationship between the two marginal measures follows directly from their definitions. Between adjacent rows, the change in total cost is the change in variable cost because fixed cost is unchanged. If labor costs \(w\) per unit, that change in variable cost is \(w\Delta L\).
This calculation applies when \(\Delta Q>0\), labor is the only variable input, and the wage \(w\) is constant over the interval. It explains why, under these assumptions, a larger marginal product means a lower marginal cost for that interval. In the first example, from \(L=1\) to \(L=2\), marginal product is 14 and marginal cost is \(12/14=\$0.8571\ldots\), displayed as \(\$0.86\). From \(L=4\) to \(L=5\), marginal product falls to 8 and marginal cost rises to \(12/8=\$1.50\).
The identity does not say that every row’s average cost equals wage divided by average product. In fact, under the same assumptions, \(AVC=VC/Q=wL/Q=w/AP_L\) for \(L>0\) and \(Q>0\). This is an average relationship; it is not the marginal-cost formula.
Worked Example: Labor Increases in Larger Steps
Worked Example: Labor Increases in Larger Steps
A small processing operation uses a fixed facility. Its data show \(L=0,2,4\) labor units and output \(Q=0,18,30\) batches per period. The wage is \(w=\$15\) per labor unit and fixed cost is \(FC=\$90\). Calculate the productivity and cost measures, paying particular attention to the size of each labor increment.
Method. Marginal product is output change divided by labor change, not simply the raw output difference when labor changes by more than one unit. Variable cost is \(15L\). At \(L=2\), \(MP_L=(18-0)/(2-0)=9\) batches per labor unit, and \(AP_L=18/2=9\). Costs are \(VC=15(2)=\$30\) and \(TC=90+30=\$120\). Thus \(AFC=90/18=\$5.00\), \(AVC=30/18=\$1.6667\ldots\), and \(ATC=120/18=\$6.6667\ldots\). Marginal cost over this interval is \((120-90)/(18-0)=\$30/18=\$1.6667\ldots\) per batch.
At \(L=4\), \(MP_L=(30-18)/(4-2)=12/2=6\) batches per labor unit, and \(AP_L=30/4=7.5\). Variable cost is \(15(4)=\$60\), so total cost is \(\$90+\$60=\$150\). Then \(AFC=90/30=\$3.00\), \(AVC=60/30=\$2.00\), and \(ATC=150/30=\$5.00\). Marginal cost from \(L=2\) to \(L=4\) is \((150-120)/(30-18)=30/12=\$2.50\) per batch. This agrees with \(w/MP_L=15/6=\$2.50\). Average cost measures are rounded to cents here; the unrounded values at \(L=2\) are each \(1.6667\ldots\) where applicable.
Check. At \(L=4\), total cost is \(\$150=\$90+\$60\), and average total cost is \(ATC=AFC+AVC=\$3.00+\$2.00=\$5.00\). The method avoids treating the 12-batch output increase as the product of one labor unit: two units of labor were added.
Worked Example: Recovering Input Cost from Total Cost
Worked Example: Recovering Input Cost from Total Cost
A firm’s fixed-capital production schedule is \(L=0,1,2,3\) and \(Q=0,8,18,26\). Its observed total costs are \(\$50,\$68,\$86,\$104\), respectively. Assume one variable input, labor, is used at a constant price. Find fixed cost and the labor price, then complete the measures for \(L=3\).
Method. At \(L=0\), variable cost is zero, so \(TC=FC=\$50\). Each additional labor unit increases total cost by \(\$68-\$50=\$18\), \(\$86-\$68=\$18\), and \(\$104-\$86=\$18\). The repeated increment is consistent with a constant wage of \(w=\$18\) per labor unit. Therefore, at \(L=3\), \(VC=18(3)=\$54\) and \(TC=50+54=\$104\), matching the given total cost.
From \(L=2\) to \(L=3\), output rises from 18 to 26, so \(MP_L=(26-18)/(3-2)=8\). Average product at \(L=3\) is \(AP_L=26/3=8.6667\ldots\). The cost measures at \(Q=26\) are \(AFC=50/26=\$1.9231\ldots\), \(AVC=54/26=\$2.0769\ldots\), and \(ATC=104/26=\$4.00\). Marginal cost for the last interval is \((104-86)/(26-18)=18/8=\$2.25\) per unit of output. It also equals \(w/MP_L=18/8=\$2.25\).
Check. The average-cost components sum: \(50/26+54/26=104/26=4\). Total cost is consistent both with the observed schedule and with \(FC+VC=50+54=104\). This method avoids assuming fixed cost is zero merely because it is not shown as a separate column.
Common Mistakes and Table Checks
A complete table is useful only if each column is calculated from the correct base. Several errors are especially likely when the data are discrete:
- Using a level instead of a change for marginal values. Marginal product uses \(\Delta Q/\Delta L\), and marginal cost uses \(\Delta TC/\Delta Q\), between the same two adjacent rows.
- Dividing by the wrong increment. If labor rises from 2 to 4 units, divide the output change by 2 to find marginal product per labor unit. Do not report the entire output change as per-unit marginal product.
- Confusing marginal cost with average total cost. At \(L=3\) in the first example, \(MC=(72-60)/(36-24)=\$1.00\); \(ATC=72/36=\$2.00\). They answer different questions.
- Including fixed cost in variable cost. Fixed cost belongs in total cost and average fixed cost, but does not change between rows under the short-run assumptions used here.
- Calculating an average at zero output. At \(Q=0\), cost-per-output ratios require division by zero, so \(AFC\), \(AVC\), and \(ATC\) are undefined. Likewise, \(AP_L\) is undefined at \(L=0\).
- Rounding too early. Keep ratios unrounded until the final presentation. Separately rounded \(AFC\) and \(AVC\) may not add exactly to displayed \(ATC\).
A reliable final check is to verify \(TC=FC+VC\) in every row, \(ATC=AFC+AVC\) wherever output is positive, and \(MC=\Delta TC/\Delta Q\) for each interval with positive output change. Under the constant-wage, one-variable-input assumptions, also check \(MC=w/MP_L\).
Check Your Understanding
For each problem, use the stated short-run assumptions and keep marginal measures tied to the change between adjacent rows.
- A workshop has \(L=0,1,2\), output \(Q=0,7,17\), wage \(w=\$10\) per labor unit, and fixed cost \(\$30\). At \(L=2\), find marginal product, average product, total cost, average total cost, and marginal cost for the last interval. (Answer: \(MP_L=10\), \(AP_L=8.5\), \(TC=\$50\), \(ATC=\$2.94\) per output unit, and last-interval \(MC=\$1.00\) per output unit.)
- A firm uses \(L=0,2,4\) and produces \(Q=0,16,28\). Labor costs \(\$24\) per unit and fixed cost is \(\$72\). At \(L=4\), find marginal product for the last interval, average product, total cost, average total cost, and last-interval marginal cost. (Answer: \(MP_L=6\), \(AP_L=7\), \(TC=\$168\), \(ATC=\$6.00\) per output unit, and \(MC=\$4.00\) per output unit.)
- With fixed cost \(\$42\), wage \(\$14\), and \(L=0,1,2,3\), a production schedule gives \(Q=0,12,22,29\). At \(L=3\), calculate marginal product, average product, variable cost, total cost, average total cost, and marginal cost for the last interval. Check the wage-to-marginal-product relationship. (Answer: \(MP_L=7\), \(AP_L=29/3=9.6667\), \(VC=\$42\), \(TC=\$84\), \(ATC=\$2.90\) per output unit, and \(MC=\$2.00=w/MP_L\) per output unit.)
- A firm has \(FC=\$60\), \(w=\$16\), and output \(Q=0,8,20,29\) at \(L=0,1,2,3\). At \(L=3\), calculate marginal product, average product, variable cost, total cost, average fixed cost, average variable cost, average total cost, and marginal cost. Round per-unit dollar amounts to cents. (Answer: \(MP_L=9\), \(AP_L=29/3=9.6667\), \(VC=\$48\), \(TC=\$108\), \(AFC=\$2.07\), \(AVC=\$1.66\), \(ATC=\$3.72\), and last-interval \(MC=\$16/9=\$1.78\), each per output unit where applicable.)
- A production schedule has \(L=0,1,2\) and \(Q=0,10,25\). Fixed cost is \(\$40\), labor costs \(\$12\) per unit, and a proposed total-cost column lists \(\$40,\$52,\$68\). Identify any incorrect entry. Then find marginal product, average product, variable cost, corrected total cost, average total cost, and marginal cost at \(L=2\). (Answer: the \(L=2\) total cost should be \(\$64\), not \(\$68\); \(MP_L=15\), \(AP_L=12.5\), \(VC=\$24\), \(ATC=\$2.56\), and last-interval \(MC=\$12/15=\$0.80\), per output unit where applicable.)