Productivity Changes and Short-Run Cost Curves

Why Productivity Changes Matter for Short-Run Costs

A firm can produce the same quantity using fewer units of a variable input when productivity improves. If the input is labor and the wage is unchanged, fewer labor units mean lower variable cost at that output. The change does not, by itself, alter the firm's fixed cost: in the short run, fixed inputs and their costs remain fixed.

To trace the effect correctly, compare the labor required at the same output level before and after the productivity change. Then use the wage to calculate variable cost and derive the other short-run cost measures. A productivity gain need not reduce the labor required by the same proportion at every output level, so marginal and average costs should be calculated from the relevant schedule rather than assumed to move uniformly.

Definition: A productivity improvement means that a given quantity of inputs can produce more output, or equivalently, that less of a variable input is required to produce a given output. In a short-run labor model, let $L(q)$ denote the labor required to produce output $q$. With a constant wage $w$ per labor unit and fixed cost $FC$, variable cost is $VC(q)=wL(q)$, and total cost is $TC(q)=FC+VC(q)$. These relationships assume labor is the only variable input and its wage does not change.

Average variable cost is $AVC=VC/q$, average fixed cost is $AFC=FC/q$, and average total cost is $ATC=TC/q$, for $q>0$. For a discrete increase in output, marginal cost is the change in total cost divided by the change in output. Because fixed cost is unchanged in the short run, that change in total cost is the change in variable cost.

AVC(q)=\frac{wL(q)}{q}, \qquad ATC(q)=\frac{FC+wL(q)}{q}, \qquad MC=\frac{\Delta TC}{\Delta q}=\frac{w\Delta L}{\Delta q}

The last expression applies to a discrete output interval when the wage is constant and labor is the only variable input. If other variable inputs are used, their changing costs must also be included. If the wage changes at the same time as productivity, the cost comparison combines two effects and cannot be attributed to productivity alone.

From a Production Schedule to Cost Measures

A production schedule reports output associated with input use. To make a short-run cost comparison, read or construct the labor requirement for each output level in each productivity situation. Calculate variable cost by multiplying labor by the wage; add the same fixed cost to obtain total cost. The following sequence keeps the comparison consistent.

Hold output fixed for the comparison.
Compare the labor required to produce the same $q$ before and after the productivity change.

Calculate variable cost.
Multiply each labor requirement by the wage, provided the wage is unchanged.

Calculate total and average costs.
Add fixed cost to variable cost for total cost; divide by output for the relevant average.

Calculate marginal cost by interval.
Divide the change in total cost by the change in output. With fixed cost unchanged, this is wage times the additional labor per additional output unit.

Worked Example: Comparing Complete Cost Schedules

A workshop has fixed cost of $FC=\$120$ per production period. Labor costs $w=\$12$ per labor unit. The table gives labor requirements before and after a productivity improvement. Find variable cost, total cost, average variable cost, average total cost, and marginal cost for each output interval. Assume the wage and fixed cost are unchanged.

Output $q$	Original labor $L$	Improved labor $L$
0	0	0
4	2	1
8	5	3
12	9	6

Method. Apply $VC=wL$ separately to each schedule, add $\$120$ for total cost, and divide by output for the average measures. For marginal cost, use the change in cost divided by the change in output in each interval.

At output $q=4$, original variable cost is $12(2)=\$24$, and improved variable cost is $12(1)=\$12$. Total costs are $120+24=\$144$ and $120+12=\$132$. The average measures are $AVC=\$24/4=\$6$ and $ATC=\$144/4=\$36$ originally; after the improvement they are $AVC=\$12/4=\$3$ and $ATC=\$132/4=\$33$.

At output $q=8$, original variable cost is $12(5)=\$60$, and improved variable cost is $12(3)=\$36$. Thus total costs are $\$180$ and $\$156$. The averages are $AVC=\$60/8=\$7.50$ and $ATC=\$180/8=\$22.50$ originally; they are $AVC=\$36/8=\$4.50$ and $ATC=\$156/8=\$19.50$ after the improvement.

At output $q=12$, original variable cost is $12(9)=\$108$, while improved variable cost is $12(6)=\$72$. Total costs are $\$228$ and $\$192$. The averages are $AVC=\$108/12=\$9$ and $ATC=\$228/12=\$19$ originally; after the improvement they are $AVC=\$72/12=\$6$ and $ATC=\$192/12=\$16$.

For the interval from 0 to 4, original marginal cost is $(144-120)/4=\$6$ per output unit, and improved marginal cost is $(132-120)/4=\$3$. From 4 to 8, the respective calculations are $(180-144)/4=\$9$ and $(156-132)/4=\$6$. From 8 to 12, they are $(228-180)/4=\$12$ and $(192-156)/4=\$9$.

Check. At each positive output, improved labor is lower, so variable cost and total cost are lower. Fixed cost remains $\$120$ in both cases. The marginal-cost calculations also agree with the labor increments: for example, from 8 to 12 original labor rises by 4, so $12(4)/4=\$12$ per output unit; improved labor rises by 3, so $12(3)/4=\$9$. The common mistake avoided here is mixing input requirements from one schedule with costs calculated from another.

Why Lower Labor Requirements Reduce Cost

The cost result follows directly from the labor requirement and the unchanged wage. At any given output, if the improved process requires less labor, its variable cost is lower by the wage multiplied by the labor saved. Since fixed cost is the same in both situations, total cost falls by exactly that amount as well. Dividing by the same positive output gives lower average variable and average total cost at that output.

L_{\text{improved}}(q)<L_{\text{original}}(q) \quad\Longrightarrow\quad VC_{\text{improved}}(q)<VC_{\text{original}}(q) $$ $$ VC_{\text{original}}(q)-VC_{\text{improved}}(q) =w\left[L_{\text{original}}(q)-L_{\text{improved}}(q)\right]

This comparison is point by point: it applies at an output level where the improved process actually uses less labor. A productivity improvement over some range does not guarantee lower labor requirements at every possible output, nor does a lower average labor requirement automatically imply a lower marginal cost in every interval. Marginal cost depends on the additional labor required between two output levels.

The connection to marginal product helps explain the direction of the result. Over a discrete interval, marginal product of labor is the additional output divided by the additional labor, $\Delta q/\Delta L$. Under the same constant wage assumption, marginal cost is $w\Delta L/\Delta q$. Thus, when the interval's marginal product is higher, less labor is needed per additional output unit and marginal cost is lower. This reciprocal relationship is a local comparison; average productivity and marginal productivity are not interchangeable.

Worked Example: A Gain That Varies Across Output Levels

A small packaging plant has fixed cost $FC=\$240$ and pays a constant wage of $w=\$15$ per labor unit. Before and after a process upgrade, labor requirements are as follows. Calculate cost measures at outputs 6, 12, and 18, then compare marginal costs over each six-unit interval.

Output $q$	Original labor	Improved labor
0	0	0
6	3	2
12	7	5
18	12	9

At $q=6$, variable costs are $15(3)=\$45$ and $15(2)=\$30$; total costs are $\$285$ and $\$270$. Therefore, original $AVC=\$45/6=\$7.50$ and $ATC=\$285/6=\$47.50$. Improved $AVC=\$30/6=\$5$ and $ATC=\$270/6=\$45$.

At $q=12$, variable costs are $15(7)=\$105$ and $15(5)=\$75$; total costs are $\$345$ and $\$315$. The average costs are $AVC=\$105/12=\$8.75$ and $ATC=\$345/12=\$28.75$ originally, compared with $AVC=\$75/12=\$6.25$ and $ATC=\$315/12=\$26.25$ after the upgrade.

At $q=18$, variable costs are $15(12)=\$180$ and $15(9)=\$135$, giving total costs of $\$420$ and $\$375$. The averages are $AVC=\$180/18=\$10$ and $ATC=\$420/18=\$23.33$ originally, compared with $AVC=\$135/18=\$7.50$ and $ATC=\$375/18=\$20.83$ after the upgrade. Average total costs are rounded to the nearest cent.

For the interval from 0 to 6, marginal costs are $15(3)/6=\$7.50$ and $15(2)/6=\$5$. From 6 to 12, labor rises by 4 originally and 3 after the upgrade, so marginal costs are $15(4)/6=\$10$ and $15(3)/6=\$7.50$. From 12 to 18, the labor increases are 5 and 4, giving marginal costs of $15(5)/6=\$12.50$ and $15(4)/6=\$10$.

Check. At each listed output, the improved process uses less labor and has lower variable and total cost. The interval calculations separately confirm lower marginal cost in this particular schedule. Do not assume that outcome without checking the labor increments: a productivity improvement can vary by output range.

Productivity Can Improve Unevenly

An upgrade may leave labor requirements unchanged at low output but reduce them at higher output. In that case, costs at the unchanged output level are also unchanged, while costs at the affected levels fall. Even when labor requirements fall at every listed output, the labor saved need not be the same in each interval. That is why it is useful to distinguish three statements:

Variable cost: falls at an output level if fewer labor units are needed there and the wage is unchanged.
Average cost: falls at that output level if total cost falls while output and fixed cost are unchanged.
Marginal cost: falls over an interval only if the additional labor per additional output unit is lower over that interval.

Worked Example: Identifying an Uneven Productivity Effect

A finishing shop has fixed cost $FC=\$90$, and labor costs $w=\$10$ per unit. Original labor requirements at outputs 0, 5, 10, 15, and 20 are 0, 2, 5, 9, and 14. After a process change, the requirements are 0, 2, 4, 7, and 11. Find variable cost at output 20, the variable-cost saving there, average variable and total cost at that output, and marginal cost from 15 to 20. Also express the savings as percentages of original variable and total cost at output 20.

At $q=20$, original variable cost is $10(14)=\$140$ and improved variable cost is $10(11)=\$110$. The saving is $\$140-\$110=\$30$. Original total cost is $90+140=\$230$; improved total cost is $90+110=\$200$.

Original $AVC=\$140/20=\$7$ and $ATC=\$230/20=\$11.50$. Improved $AVC=\$110/20=\$5.50$ and $ATC=\$200/20=\$10$.

From output 15 to 20, original labor increases from 9 to 14, or by 5 units. Original marginal cost is $10(5)/5=\$10$ per output unit. Improved labor increases from 7 to 11, or by 4 units, so improved marginal cost is $10(4)/5=\$8$.

The variable-cost saving as a share of original variable cost is $\$30/\$140=0.214285\ldots$, or 21.43%. The total-cost saving as a share of original total cost is $\$30/\$230=0.130434\ldots$, or 13.04%. The total-cost percentage is smaller because fixed cost does not fall. These percentages are rounded to two decimal places.

Check. The dollar saving is $\$30$ for both variable cost and total cost, but the percentage denominators differ. Using total cost as the denominator for the variable-cost percentage, or treating fixed cost as a saving, would give the wrong comparison.

Common Exam Trap: Reversing Marginal Cost

Marginal cost is cost change divided by output change, not output change divided by cost change. The units reveal the error. If cost is measured in dollars and output in units, marginal cost has units of dollars per output unit. Its reciprocal has units of output units per dollar.

For example, in the first worked example's original schedule, variable cost rises from $\$24$ at output 4 to $\$60$ at output 8. The incorrect calculation $4/36$ reverses the ratio and gives $0.1111$ output units per dollar. The correct calculation is $(60-24)/(8-4)=36/4=\$9$ per output unit. Since fixed cost is unchanged, using the change in total cost would give the same result: $(180-144)/4=\$9$.

Another frequent error is to assume that improved productivity reduces fixed cost. It changes how effectively variable inputs produce output; it does not automatically change rent, equipment charges, or other fixed costs in the short run. The comparison also requires the same output level and unchanged input prices if the goal is to isolate the productivity effect. This treatment of productivity and short-run cost is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Key takeaway: At a fixed output, compare input requirements first. With an unchanged wage, fewer labor units mean lower variable cost; unchanged fixed cost means the same dollar reduction in total cost. For marginal cost, compare the additional labor required per additional output unit in each interval.

Check Your Understanding

Use the schedules and assumptions stated in each question. Give monetary answers in dollars.

A shop has fixed cost $\$90$ and pays $\$18$ per labor unit. At output 5 it requires 2 labor units with the old process and 1 with a new process; at output 10 it requires 5 and 3, respectively. At output 10, find each process's $AVC$ and $ATC$, and find each process's marginal cost from output 5 to 10. (Answer: Old $AVC=\$9$, $ATC=\$18$, and $MC=\$10.80$; new $AVC=\$5.40$, $ATC=\$14.40$, and $MC=\$7.20$.)
A producer has $FC=\$64$, a wage of $\$16$, and labor requirements at outputs 0, 4, and 8 of 0, 1, and 3 under the old method and 0, 0.5, and 2 under the improved method. At output 8, calculate each method's $AVC$ and $ATC$, then compare marginal cost from output 4 to 8. (Answer: Old $AVC=\$6$, $ATC=\$14$, and $MC=\$8$; improved $AVC=\$4$, $ATC=\$12$, and $MC=\$6$.)
A plant has fixed cost $\$200$ and a wage of $\$25$. At outputs 10 and 20, old labor requirements are 4 and 9, while improved requirements are 3 and 7. Find variable-cost savings, $AVC$, and $ATC$ at output 20, and marginal cost from 10 to 20 under both methods. (Answer: Variable-cost saving is $\$50$; old $AVC=\$11.25$, $ATC=\$21.25$, and $MC=\$12.50$; improved $AVC=\$8.75$, $ATC=\$18.75$, and $MC=\$10$.)
Over the relevant range, an old process produces 4 units per labor unit, while an improved process produces 5 units per labor unit. If the wage is $\$12$, fixed cost is $\$100$, and the target is output 20, calculate labor requirements, variable costs, total costs, and the percentage reductions in variable and total cost. (Answer: Labor requirements are 5 and 4; variable costs are $\$60$ and $\$48$; total costs are $\$160$ and $\$148$; reductions are 20% in variable cost and 7.5% in total cost.)
A firm has wage $\$20$. At outputs 0, 5, 10, and 15, old labor requirements are 0, 2, 4, and 7; improved requirements are 0, 2, 3, and 6. Calculate marginal cost in each five-unit interval for both methods. Identify where marginal cost is lower after improvement, and find the variable-cost saving at output 15. (Answer: Old marginal costs are $\$8,\$8,\$12$; improved marginal costs are $\$8,\$4,\$12$. Marginal cost is lower only from 5 to 10, and variable-cost saving at output 15 is $\$20$.)

Output \(q\)	Original labor \(L\)	Improved labor \(L\)
0	0	0
4	2	1
8	5	3
12	9	6

What You'll Learn