Short-Run Production and Cost Problem Workshop

From Inputs to Costs: A Consistent Workflow

A short-run production problem often gives information in different forms: a production schedule, input prices, a fixed cost, or selected points on a cost curve. The task is to translate these facts into a consistent set of cost measures. The central discipline is to keep the units and intervals clear: labor is an input, output is a quantity produced, and marginal cost compares a change in cost with the corresponding change in output.

Assume a firm operates over a short-run period in which at least one input is fixed. Let $q$ denote output, $F$ fixed cost, and $VC(q)$ variable cost. Total cost is the sum of fixed and variable cost. At positive output, average costs are cost per unit; marginal cost is the additional cost per additional unit, or, with discrete data, the cost change divided by the output change.

Definition: In the short run, total cost is $TC=F+VC$. For $q>0$, average fixed cost is $AFC=F/q$, average variable cost is $AVC=VC/q$, and average total cost is $ATC=TC/q=AFC+AVC$. Over an output interval from $q_1$ to $q_2$, with $q_2>q_1$, discrete marginal cost is $MC=\Delta TC/\Delta q=(TC_2-TC_1)/(q_2-q_1)$. Since fixed cost does not change with output in the short run, this also equals $\Delta VC/\Delta q$.

If the firm hires $L$ units of labor at wage $w$, labor cost is $wL$. When labor is the only variable input, $VC=wL$; if other inputs are also variable, their costs must be included as well. At zero output, fixed cost may still be positive, but average costs are undefined because division by zero is not possible.

A Practical Calculation Sequence

Work across the data in a fixed order. First identify which inputs are fixed and which vary, then use the production schedule to connect input quantities to output. Calculate variable cost from the prices and quantities of variable inputs, add fixed cost to obtain total cost, and only then calculate average and marginal measures. This order makes it easier to locate a mistaken input quantity, price, or output interval.

Map inputs to output.
Read the production schedule or use the stated production function to determine the output associated with each variable-input quantity.

Build variable cost.
Multiply each variable input quantity by its price and add the resulting costs. Include any other variable cost specified per unit of output.

Build cost measures.
Add fixed cost to variable cost, divide by output for averages, and use matching changes in cost and output for discrete marginal cost.

Check consistency.
Verify $ATC=AFC+AVC$, confirm marginal cost uses the stated interval, and compare the pattern of marginal product with the pattern of marginal cost when the assumptions permit.

Worked Production and Cost Schedules

The following examples use hypothetical firms. Monetary amounts are in dollars. Per-unit cost measures are rounded to two decimal places only at the final step.

Worked Example: Labor, Output, and a Complete Cost Schedule

A small workshop has fixed cost of $120 per period and hires labor at $24 per worker-period. Its short-run production schedule is shown below. Assume labor is the only variable input. Find variable cost, total cost, average variable cost, average total cost, and marginal cost at each positive-output observation, using the change from the preceding row for marginal cost.

Method and assumptions. The number of workers is the labor input $L$, each worker-period costs $24, and fixed cost remains $120 at every output level. Thus $VC=24L$ and $TC=120+VC$. Average costs require positive output, and discrete marginal cost is computed between adjacent schedule rows.

Labor $L$	Output $q$	Variable cost $VC$	Total cost $TC$	$AVC$	$ATC$	Interval $MC$
0	0	$0	$120	undefined	undefined	—
1	8	$24	$144	$3.00	$18.00	$3.00
2	18	$48	$168	$2.67	$9.33	$2.40
3	27	$72	$192	$2.67	$7.11	$2.67
4	33	$96	$216	$2.91	$6.55	$4.00

For $L=2$, variable cost is $24(2)=\$48$, and total cost is $120+48=\$168$. Therefore $AVC=48/18=\$2.666\ldots$, or $2.67, and $ATC=168/18=\$9.333\ldots$, or $9.33. The marginal cost from 8 to 18 units is $(168-144)/(18-8)=24/10=\$2.40$ per unit.

For the last interval, output rises from 27 to 33, while total cost rises from $192 to $216. Thus $MC=(216-192)/(33-27)=24/6=\$4.00$ per unit. The marginal product of the fourth worker is 6 units, lower than the preceding worker's 9 units. With a constant wage and labor as the only variable input, this lower output increment corresponds to higher marginal cost.

Check. At $L=3$, $AFC=120/27=\$4.44$ and $AVC=72/27=\$2.67$, so their sum is $7.11, matching $ATC=192/27=\$7.11$ after rounding. The common mistake to avoid is dividing a cost change by the number of workers rather than by the output change; marginal cost here is cost per unit of output.

Worked Example: A Production Function with a Per-Unit Materials Cost

A plant's short-run production function is $q=12\sqrt{L}$, where $L$ is labor. Labor costs $18 per unit, a variable material costs $3 for each unit of output, and fixed cost is $90. Find the cost function in terms of output and calculate the cost measures at $q=24$, including marginal cost over the interval from $q=12$ to $q=24$.

Method and assumptions. Capital and the $90 fixed cost are unchanged in the short run. Labor and materials are variable. Solve the production function for labor: $\sqrt{L}=q/12$, so $L=(q/12)^2=q^2/144$. Variable cost is labor cost plus materials cost.

VC(q)=18\left(\frac{q^2}{144}\right)+3q=\frac{q^2}{8}+3q, \qquad TC(q)=90+\frac{q^2}{8}+3q.

At $q=24$, required labor is $L=24^2/144=576/144=4$. Labor cost is $18(4)=\$72$, and material cost is $3(24)=\$72$. Therefore $VC=72+72=\$144$ and $TC=90+144=\$234$. The averages are $AVC=144/24=\$6.00$ and $ATC=234/24=\$9.75$.

At $q=12$, labor is $12^2/144=1$, so $VC=18(1)+3(12)=18+36=\$54$. The cost increase from 12 to 24 units is $144-54=\$90$, and the output increase is 12. Hence the interval marginal cost is $90/12=\$7.50$ per unit.

Check and common mistake. At $q=24$, fixed cost per unit is $90/24=\$3.75$, and $AFC+AVC=3.75+6.00=\$9.75=ATC$. Do not treat the materials cost as fixed: it rises with output, so omitting it would understate both variable cost and marginal cost.

Why Productivity and Marginal Cost Move Together

When labor is the only variable input and the wage is constant, the connection between productivity and marginal cost follows directly from the cost of hiring the additional labor. For a change from $L_1$ to $L_2$, let the additional labor be $\Delta L=L_2-L_1$, and let its output increment be $\Delta q=q_2-q_1>0$. The extra variable cost is $w\Delta L$, so dividing by the output increment gives the discrete relationship below.

MC=\frac{\Delta VC}{\Delta q} =\frac{w\Delta L}{\Delta q} =\frac{w}{\Delta q/\Delta L} =\frac{w}{MP_L},

Here $MP_L=\Delta q/\Delta L$ is the marginal product of labor over the same interval. Thus, if wage is constant and marginal product falls, marginal cost rises. This conclusion is conditional: it does not follow from diminishing marginal product alone if the wage changes, other variable inputs have costs that also change, or the intervals being compared do not match.

If an additional variable input costs $m$ per unit of output, its cost contributes $m\Delta q$ to the cost change. In that case, with labor as the other variable input, $MC=w/MP_L+m$. The next example checks this relationship directly from a schedule.

Worked Example: Separating Labor Cost from Materials Cost

A packaging firm has fixed cost of $80, pays $30 per unit of labor, and uses materials costing $2 per unit of output. Its production schedule is: $L=0,1,2,3,4$ produces $q=0,10,22,31,37$, respectively. Calculate variable and total cost, average variable and total cost, and marginal cost at $L=3$. Explain why marginal cost changes between the two intervals ending at $L=2$ and $L=3$.

Method and assumptions. Labor and materials are variable, labor is priced at $30 per unit, and materials cost $2 per unit of output. Therefore $VC=30L+2q$, while $TC=80+VC$.

At $L=3$, output is 31. Labor cost is $30(3)=\$90$, and material cost is $2(31)=\$62$. Thus $VC=90+62=\$152$, and $TC=80+152=\$232$. Average variable cost is $152/31=\$4.9032\ldots$, or $4.90 per unit. Average total cost is $232/31=\$7.4838\ldots$, or $7.48 per unit.

At $L=2$, output is 22, so variable cost is $30(2)+2(22)=60+44=\$104$, and total cost is $80+104=\$184$. From $L=2$ to $L=3$, output rises by $31-22=9$, while total cost rises by $232-184=\$48$. Therefore interval marginal cost is $48/9=\$5.333\ldots$, or $5.33 per unit.

The third worker adds 9 units of output, so labor cost per added output unit is $30/9=\$3.333\ldots$. Materials add $2 for each added output unit. Their sum is $3.333\ldots+2=\$5.333\ldots$, agreeing with the marginal-cost calculation. The preceding interval adds 12 units of output, giving $30/12+2=\$4.50$ marginal cost.

Check and common mistake. At $L=3$, $AFC=80/31=\$2.58$ and $AVC=\$4.90$; using unrounded values, their sum is $232/31=\$7.48$. Do not compute marginal cost as the change in labor cost alone: the added materials cost is part of the total variable-cost change.

Reading the Numbers as a Consistency Check

A cost schedule can be checked both horizontally and vertically. Horizontally, each row should satisfy $TC=F+VC$, and each average should equal its corresponding total divided by the row's output. Vertically, marginal cost should use cost and output from the same two rows. If fixed cost is truly constant, changes in total cost and variable cost over an interval must be equal.

These checks also help when plotting cost data. Each average cost is associated with a particular output, while a discrete marginal cost belongs to the interval between two output observations. Plotting an interval marginal cost as though it were a cost observed precisely at the endpoint can obscure that distinction. Where the production schedule shows smaller additional output from successive units of labor at an unchanged wage, the resulting labor cost per added unit of output increases; if other variable costs are present, include them before drawing a conclusion about total marginal cost.

Key takeaway: A reliable short-run cost solution keeps the production schedule, input prices, and output intervals aligned. Calculate costs from the inputs first, then derive averages and marginal measures, and verify each result using the cost identities and the relevant productivity relationship.

Check Your Understanding

For each problem, show the input-cost calculation before rounding any per-unit measure.

A repair shop has fixed cost of $70 and pays $21 per labor unit. One labor unit produces 7 units of output, and two labor units produce 17 units. Find variable cost, total cost, $AVC$, and $ATC$ at two labor units, and interval $MC$ from one to two labor units. (Answer: $VC=\$42$, $TC=\$112$, $AVC=\$2.47$, $ATC=\$6.59$, and interval $MC=\$2.10$ per unit.)
A short-run production function is $q=10\sqrt{L}$. Labor costs $16 per unit and fixed cost is $64. At output 30, find labor required, variable cost, total cost, $AVC$, and $ATC$; also calculate interval $MC$ from output 20 to 30. (Answer: $L=9$, $VC=\$144$, $TC=\$208$, $AVC=\$4.80$, $ATC=\$6.93$, and interval $MC=\$8.00$ per unit.)
A firm has fixed cost of $50, labor cost of $12 per unit, and material cost of $1 per output unit. Its schedule is $L=1,4,9$ producing $q=12,24,36$. At $L=9$, find $VC$, $TC$, $AVC$, and $ATC$. Then find marginal cost from $q=12$ to 24 and from $q=24$ to 36. (Answer: $VC=\$144$, $TC=\$194$, $AVC=\$4.00$, $ATC=\$5.39$; interval marginal costs are $\$4.00$ and $\$6.00$ per unit, respectively.)
A producer's fixed cost is $100. Variable cost is $60 at output 20 and $105 at output 30. Find $AVC$ and $ATC$ at output 30, then calculate marginal cost from 20 to 30. (Answer: $AVC=\$3.50$, $ATC=\$6.83$, and interval $MC=\$4.50$ per unit.)
With labor as the only variable input, wage $24, and marginal product of labor equal to 8 units over an interval, calculate marginal cost over that interval. If wage remains unchanged but marginal product falls to 6 units over the next comparable interval, calculate the new marginal cost and state the direction of change. (Answer: $MC=\$3.00$ per unit at marginal product 8, then $MC=\$4.00$ per unit at marginal product 6; marginal cost rises.)

What You'll Learn