Marginal Cost and Average Cost: The Algebra of Averages

Why Compare Marginal Cost With an Average?

A firm’s average cost summarizes the cost per unit across all units produced so far. Marginal cost describes the cost of an additional unit, or the cost per unit of a specified output increase. Comparing the two answers a practical question: as output expands, does the new production pull the average cost down, push it up, or leave it unchanged?

The logic is the same as adding a new observation to an average. If the new observation is below the existing average, it lowers the combined average; if it is above, it raises the combined average. In cost analysis, the “new observation” is the marginal cost of the additional output. The comparison must be made with the particular average being studied: marginal cost may be above average variable cost but below average total cost at the same output.

Definition: Let $q>0$ be output, $FC$ fixed cost, $VC(q)$ variable cost, and $TC(q)=FC+VC(q)$ total cost. Average variable cost is $AVC(q)=VC(q)/q$, and average total cost is $ATC(q)=TC(q)/q$. For a discrete increase from $q$ to $q+\Delta q$, where $\Delta q>0$, marginal cost over that interval is $MC=\Delta TC/\Delta q$. In a differentiable cost model, marginal cost at output $q$ is $MC(q)=dTC(q)/dq$.

In the short run, fixed cost does not change as output changes, so $\Delta TC=\Delta VC$ and $dTC/dq=dVC/dq$. This means the same marginal cost can be compared with both average variable cost and average total cost. The averages differ because total cost includes fixed cost, while variable cost does not.

The Discrete Algebra of Averages

For a discrete output change, the old average cost is based on $q$ units. The additional $\Delta q$ units have an incremental cost of $MC\Delta q$. Total cost after the increase is therefore the old total cost plus the cost of the added output. Dividing by the new output gives the new average.

ATC(q+\Delta q) =\frac{q\,ATC(q)+\Delta q\,MC}{q+\Delta q}

This is a weighted average: the old average is weighted by the number of units already produced, and the marginal cost is weighted by the number of additional units. Subtracting the old average from the new one gives the exact change.

ATC(q+\Delta q)-ATC(q) =\frac{\Delta q}{q+\Delta q}\bigl(MC-ATC(q)\bigr)

Because $\Delta q/(q+\Delta q)$ is positive, the sign of the change depends only on $MC-ATC(q)$. If marginal cost is below the old average total cost, the difference is negative and average total cost falls. If marginal cost is above the old average, the difference is positive and average total cost rises. If they are equal, the average is unchanged by that output increase.

The same reasoning applies to average variable cost when fixed cost is unchanged. Replace total cost with variable cost in the calculation: the additional variable cost is $MC\Delta q$, so the new average variable cost is a weighted average of the old average variable cost and marginal cost.

AVC(q+\Delta q)-AVC(q) =\frac{\Delta q}{q+\Delta q}\bigl(MC-AVC(q)\bigr)

Theorem: For a discrete output increase with unchanged fixed cost, $MC<AVC(q)$ implies that $AVC$ falls, and $MC>AVC(q)$ implies that $AVC$ rises. Likewise, $MC<ATC(q)$ implies that $ATC$ falls, and $MC>ATC(q)$ implies that $ATC$ rises. Equality with the relevant average leaves that average unchanged.

Worked Example: One Marginal Cost, Opposite Effects on Two Averages

A hypothetical workshop produces 40 units. Its fixed cost is $240 and its variable cost is $960. Producing 10 additional units raises total cost by $280. Find the marginal cost over the increase, then determine how average variable cost and average total cost change.

Method. First calculate each old average. Then calculate the marginal cost per added unit and compare it separately with each average. The common mistake avoided here is to assume that marginal cost must move both averages in the same direction.

The old total cost is $TC=240+960=\$1{,}200$. Thus the initial averages are:

AVC=\frac{960}{40}=\$24,\qquad ATC=\frac{1{,}200}{40}=\$30

The output increase is $\Delta q=10$, and the increase in total cost is $280. Therefore:

MC=\frac{280}{10}=\$28\text{ per unit}

Since fixed cost is unchanged, the $280 increase is also the increase in variable cost. New variable cost is $960+280=\$1{,}240$; new total cost is $1{,}200+280=\$1{,}480$; and new output is $40+10=50$. The new averages are:

AVC_{\text{new}}=\frac{1{,}240}{50}=\$24.80,\qquad ATC_{\text{new}}=\frac{1{,}480}{50}=\$29.60

Marginal cost of $28 exceeds the old $AVC$ of $24, so average variable cost rises to $24.80. But $28 is below the old $ATC$ of $30, so average total cost falls to $29.60. The calculation checks both comparisons: the marginal cost lies between the two starting averages, and the resulting averages move in opposite directions.

The Continuous Version

If total cost is differentiable as a function of output, the relationship can be expressed using derivatives. Assume $TC(q)=FC+VC(q)$, fixed cost is constant, and $q>0$. By definition, $ATC(q)=TC(q)/q$. Applying the quotient rule yields:

\frac{dATC}{dq} =\frac{q\,\frac{dTC}{dq}-TC(q)}{q^2} =\frac{MC(q)-ATC(q)}{q}

The denominator $q$ is positive, so the sign of the derivative is the sign of $MC(q)-ATC(q)$. Thus, at an output where marginal cost is below average total cost, a small increase in output reduces average total cost; where marginal cost is above it, a small increase raises average total cost. For variable cost, the same derivation applied to $AVC(q)=VC(q)/q$ gives:

\frac{dAVC}{dq} =\frac{MC(q)-AVC(q)}{q}

These are local results: they describe the direction in which the average changes at the output being examined. They do not, on their own, say whether marginal cost is rising or falling, or establish the overall shape of a cost curve.

Worked Example: Using the Derivative Rule for Average Variable Cost

A hypothetical producer has the differentiable short-run cost function $TC(q)=180+12q+0.2q^2$, where $q>0$. At $q=20$, determine marginal cost, average variable cost, and average total cost. Then use the derivative rule to identify which averages rise or fall as output increases slightly.

Method. Separate fixed and variable cost, calculate the two averages at the stated output, and compare each with marginal cost. This avoids treating the comparison with $ATC$ as if it automatically determined what happens to $AVC$.

The fixed cost is $FC=180$, and variable cost is $VC(q)=12q+0.2q^2$. Marginal cost is the derivative of total cost:

MC(q)=\frac{d}{dq}(180+12q+0.2q^2)=12+0.4q

At $q=20$, $MC=12+0.4(20)=\$20$ per unit. Variable cost at that output is $12(20)+0.2(20^2)=240+80=\$320$. Total cost is $180+320=\$500$. Therefore:

AVC(20)=\frac{320}{20}=\$16,\qquad ATC(20)=\frac{500}{20}=\$25

Marginal cost is above average variable cost but below average total cost. The derivative rules give:

\left.\frac{dAVC}{dq}\right|_{q=20} =\frac{20-16}{20}=0.20

\left.\frac{dATC}{dq}\right|_{q=20} =\frac{20-25}{20}=-0.25

At this output, average variable cost is increasing and average total cost is decreasing as output increases slightly. The derivative values are changes in dollars per unit of output; they are not the new average costs themselves. This is a local conclusion, not a claim that either average must keep moving in that direction at every output.

Reading the Comparison Correctly

To use the rule reliably, identify the average that the question asks about, and compare marginal cost with that average at the starting output. In a discrete problem, make sure marginal cost is calculated for the stated output interval. In a continuous problem, use marginal cost at the stated output. A comparison with average total cost does not determine the direction of average variable cost, or vice versa.

Identify the cost measure.
Use $AVC$ when the question concerns variable cost per unit, and $ATC$ when it concerns total cost per unit.

Find the starting average and marginal cost.
For a discrete change, divide the cost increase by the output increase. For a differentiable model, differentiate the cost function.

Compare and state the direction.
Marginal cost below the relevant average pulls it down; marginal cost above it pushes it up; equality leaves it unchanged for that comparison.

A frequent exam trap is to compare marginal cost only with average total cost and then use that conclusion for both averages. For example, suppose $AVC=\$24$, $ATC=\$30$, and $MC=\$28$. The incorrect shortcut is: “$MC<ATC$, so both averages fall.” That conclusion is wrong for average variable cost. The correct comparisons are $28>24$, so $AVC$ rises, and $28<30$, so $ATC$ falls. Compare each marginal cost with the particular average in question.

The algebra also explains why fixed cost matters differently for the two comparisons. With positive output, $ATC=AVC+FC/q$, so $ATC$ is at least as large as $AVC$ when fixed cost is nonnegative. Consequently, marginal cost can be above $AVC$ yet below $ATC$, as in the worked examples. The fact that fixed cost per unit changes as output changes is already reflected in the total-cost comparison; it does not mean fixed cost itself has become marginal cost.

Key takeaway: An average moves toward the value being added. In cost analysis, compare marginal cost with the relevant starting average: below pulls the average down, above pushes it up, and equality leaves it unchanged. Apply the comparison separately to average variable cost and average total cost.

Check Your Understanding

For each problem, calculate or compare marginal cost with the specified average before deciding the direction of change.

A repair shop has output $q=30$ and average variable cost of $18 per job. Producing 10 more jobs adds $220 of variable cost, with fixed cost unchanged. Find marginal cost and the new average variable cost. (Answer: $MC=\$22$ per job; new $AVC=\$19$ per job, so $AVC$ rises.)
A hypothetical plant has $q=50$ and total cost of $1,750. The next 5 units increase total cost by $150. Find marginal cost and the new average total cost. (Answer: $MC=\$30$ per unit; new $ATC=\$1{,}900/55\approx \$34.55$, down from $35.)
A firm has $q=20$, fixed cost of $100, and variable cost of $300. The next 4 units add $80 to variable cost. Find marginal cost and both new averages. (Answer: $MC=\$20$; new $AVC=\$380/24\approx \$15.83$, while new $ATC=\$480/24=\$20$; $AVC$ rises and $ATC$ is unchanged.)
A producer has $q=40$, $AVC=\$12$, and $ATC=\$19$. An increase of 8 units adds $112 to total cost, and fixed cost is unchanged. Find the new averages and their directions of change. (Answer: $MC=\$14$; new $AVC=\$592/48\approx \$12.33$, up; new $ATC=\$872/48\approx \$18.17$, down.)
A hypothetical cost function is $TC(q)=240+10q+0.25q^2$. At $q=20$, find marginal cost, $AVC$, and $ATC$, then state how each average changes for a small increase in output. (Answer: $MC=\$20$, $AVC=\$15$, $ATC=\$27$; $AVC$ rises and $ATC$ falls.)

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