Interpreting Short-Run Cost Curve Graphs

Why Reading the Graph Matters

A short-run cost graph compresses several kinds of information into one picture. At a chosen output, the height of a cost curve gives a cost level; the slope of the total-cost curve gives marginal cost; and the position of marginal cost relative to an average-cost curve shows whether that average is rising or falling. These readings are related, but they are not interchangeable.

For a firm operating with at least one fixed input, output is usually shown on the horizontal axis and cost on the vertical axis. Total costs are measured in dollars, while average and marginal costs are measured in dollars per unit of output. Always check the axis labels and the output level before interpreting a plotted point.

Definition: At output $q>0$, total cost $TC(q)$ is the firm's total short-run cost, fixed cost $FC$ does not vary with output, and variable cost is $VC(q)=TC(q)-FC$. Average fixed cost is $AFC(q)=FC/q$, average variable cost is $AVC(q)=VC(q)/q$, and average total cost is $ATC(q)=TC(q)/q=AFC(q)+AVC(q)$. If total cost is differentiable, marginal cost is $MC(q)=dTC(q)/dq$, the additional cost of a small increase in output at $q$.

What the Curves Show

A total-cost graph plots $TC(q)$ against output $q$. Its vertical intercept, at $q=0$, is fixed cost when there is no variable cost at zero output. The vertical distance between the total-cost curve and the fixed-cost level is variable cost. The total-cost curve's height is not the cost of producing one unit; it is the cost of producing all $q$ units.

An average-cost graph plots cost per unit against output. At a given $q$, read the height of the $ATC$ curve for total cost per unit, or the height of the $AVC$ curve for variable cost per unit. The identity $ATC=AFC+AVC$ means the vertical gap between $ATC$ and $AVC$ equals $AFC$. Since fixed cost is spread over more units as output increases, $AFC$ falls as $q$ rises.

The marginal-cost curve shows the cost of producing an additional unit at a particular output when costs are smooth. It is a separate curve from average cost: a curve's height at $q$ gives a cost per unit, but that height does not say how much the total cost of all units produced is. The shape of any of these curves depends on the firm's cost conditions; do not assume a curve is rising or falling merely because it is labelled “cost.”

Graph-reading rule: At output $q$, a curve's height gives the value of the quantity named by that curve. The slope of the total-cost curve gives $MC$ only when the curve is differentiable. For data or a straight segment between two plotted outputs, the segment's slope is the change in total cost divided by the change in output; it is an average marginal cost over that interval, not necessarily the marginal cost at either endpoint.

Why Marginal Cost Determines the Direction of Average Cost

A useful graph-reading result connects the height of $MC$ with the direction of the $ATC$ curve. Assume $TC(q)$ is differentiable and $q>0$. Then $ATC(q)=TC(q)/q$, and the derivative of an average is:

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

The second equality follows because $TC(q)/q=ATC(q)$. Since $q>0$, the denominator is positive: if $MC<ATC$, then $dATC/dq<0$, so average total cost is falling; if $MC>ATC$, then $dATC/dq>0$, so it is rising. At a smooth interior minimum of $ATC$, its derivative is zero, which requires $MC=ATC$. The same argument applies to $AVC$, using $VC(q)$ and its marginal cost, which equals $MC$ when fixed cost is constant.

This is a statement about the direction of an average, not about the average's level. For example, when $MC<ATC$, $ATC$ falls as output increases, even if $MC$ itself is rising. The comparison must be made at the same output.

Worked Example: Reading All Cost Measures at One Output

A hypothetical firm has the short-run total-cost curve $TC(q)=120+4q+q^2$, where output is measured in units and cost is in dollars. Read the cost measures at $q=10$, determine whether $ATC$ is rising or falling there, and locate where $MC$ and $ATC$ are equal.

Method. Read the total-cost level from the function, separate the fixed and variable components, divide by output for averages, and differentiate total cost for marginal cost.

At $q=10$, total cost is $TC(10)=120+4(10)+10^2=120+40+100=\$260$. Fixed cost is $FC=\$120$, so variable cost is $VC(10)=260-120=\$140$.

The average measures are $AFC(10)=120/10=\$12$ per unit, $AVC(10)=140/10=\$14$ per unit, and $ATC(10)=260/10=\$26$ per unit. They check: $12+14=26$.

Marginal cost is the derivative $MC(q)=4+2q$, so $MC(10)=4+20=\$24$ per unit. Since $24<26$, $ATC$ is falling at $q=10$.

For the intersection, set marginal cost equal to average total cost. Here $ATC(q)=120/q+4+q$, so $4+2q=120/q+4+q$. Subtracting $4+q$ gives $q=120/q$, and therefore $q^2=120$. The positive output is $q=\sqrt{120}$, approximately $10.954$. At this output $MC=ATC$; the common value is $4+2\sqrt{120}$, approximately $\$25.91$ per unit.

Check. At $q=10$, $MC$ is below $ATC$, consistent with $ATC$ falling before its minimum; at the calculated positive output, the two measures are equal. The common mistake avoided here is treating total cost, $\$260$, as though it were either average cost or marginal cost.

Following the Curves When Output Changes

To explain what happens as output changes, compare readings at two outputs rather than relying on a curve's name. Total cost changes by the vertical difference between its two points. Each average cost can move differently: $AFC$ falls as its fixed-cost numerator is divided across more units, while $AVC$ depends on variable cost per unit. The direction of $ATC$ can be checked by comparing $MC$ with $ATC$, or directly by comparing the two average-cost readings.

A change over a finite output interval also has a useful slope interpretation. If output rises from $q_1$ to $q_2$, the slope of the total-cost segment is $[TC(q_2)-TC(q_1)]/(q_2-q_1)$. This is the average additional cost per extra unit across that interval. It may differ from the marginal cost at the starting or ending output, especially when the total-cost curve is curved.

Worked Example: Comparing Cost Measures at Two Outputs

A hypothetical producer has $TC(q)=200+10q+q^2$. Compare the cost measures at $q=4$ and $q=8$, and describe what changes when output rises from 4 to 8 units.

Method. Evaluate total cost at both outputs, compute each average, and compare marginal cost with average total cost at each endpoint. The fixed cost is $\$200$; variable cost is $10q+q^2$.

At $q=4$, $VC(4)=10(4)+4^2=40+16=\$56$, and $TC(4)=200+56=\$256$. Thus $AFC(4)=200/4=\$50$, $AVC(4)=56/4=\$14$, and $ATC(4)=256/4=\$64$ per unit. Marginal cost is $MC(q)=10+2q$, so $MC(4)=10+8=\$18$ per unit.

At $q=8$, $VC(8)=10(8)+8^2=80+64=\$144$, and $TC(8)=200+144=\$344$. Thus $AFC(8)=200/8=\$25$, $AVC(8)=144/8=\$18$, and $ATC(8)=344/8=\$43$ per unit. Marginal cost is $MC(8)=10+16=\$26$ per unit.

Total cost rises by $344-256=\$88$ as output rises by 4 units. The total-cost segment's slope over this interval is $88/4=\$22$ per extra unit. Average fixed cost falls by $\$25$ per unit, while average variable cost rises by $\$4$ per unit, so average total cost falls by $25-4=\$21$, from $\$64$ to $\$43$. This agrees with the direct difference, $64-43=\$21$.

At both endpoints $MC<ATC$: $18<64$ and $26<43$. The average-cost direction rule is consistent with the falling $ATC$. The common mistake avoided is assuming that because marginal cost rises from $\$18$ to $\$26$, average total cost must also rise. Its level remains below $ATC$ at these outputs, and the computed average falls.

Read a Total-Cost Slope Carefully

A graph may provide only plotted points or straight segments rather than a differentiable formula. In that case, a secant slope between two points is the average increase in total cost per additional unit across that interval. A tangent slope at one point, when it exists, is the marginal cost at that output. Confusing these readings can produce an incorrect answer even when the arithmetic is accurate.

Worked Example: Tangent Slope Versus Interval Slope

A hypothetical firm's smooth total-cost curve is $TC(q)=100+20q+0.5q^2$. At $q=10$, find $ATC$ and point marginal cost. Then find the average increase in total cost per unit from $q=5$ to $q=10$, and explain why it need not equal point marginal cost.

Method. Use the curve's height divided by output for $ATC$, its derivative for point marginal cost, and the difference quotient for the interval slope.

At $q=10$, $TC(10)=100+20(10)+0.5(10^2)=100+200+50=\$350$. Therefore $ATC(10)=350/10=\$35$ per unit.

Differentiating gives $MC(q)=20+q$, so point marginal cost at $q=10$ is $MC(10)=20+10=\$30$ per unit.

At $q=5$, $TC(5)=100+20(5)+0.5(5^2)=100+100+12.5=\$212.50$. The total-cost increase from 5 to 10 is $350-212.50=\$137.50$, and output rises by $10-5=5$ units. The interval slope is therefore $137.50/5=\$27.50$ per unit.

The interval slope, $\$27.50$, is below the point marginal cost of $\$30$ at $q=10$ because the marginal-cost curve $20+q$ rises over the interval. The common mistake avoided is reporting the secant slope as marginal cost at the endpoint. They would agree if marginal cost were constant over the interval, but this curve is not linear in total cost.

Common Graph-Reading Traps

Confusing height with slope. The height of the total-cost curve is $TC$; its slope is $MC$. Read the axis and the quantity named by the curve.
Calling an interval slope point marginal cost. A change in total cost divided by a finite change in output is an average over that interval. Point marginal cost is a derivative, or the tangent slope when the graph is smooth.
Comparing curves at different outputs. To determine whether an average is rising at $q$, compare $MC(q)$ with that average at the same $q$.
Assuming all averages move together. $AFC$, $AVC$, and $ATC$ can move in different directions. Check their readings or the relevant marginal-versus-average comparison.
Forgetting the units. Total cost is in dollars; average and marginal costs are in dollars per unit. A graph may use different scales, so a visual height comparison across separate axes is not automatically meaningful.

Key takeaway: At a specified output, read cost levels from curve heights, read point marginal cost from the slope of total cost when it is smooth, and use $MC$ relative to an average to determine whether that average is rising or falling. For a finite interval, label the total-cost segment's slope as an interval average, not a point value.

Check Your Understanding

Work from the stated cost functions or graph readings, keeping point values distinct from interval slopes.

A hypothetical firm's total-cost curve is $TC(q)=60+6q+q^2$. At $q=6$, find $TC$, $AFC$, $AVC$, $ATC$, and $MC$. Is $ATC$ rising or falling there? (Answer: $TC=\$132$, $AFC=\$10$, $AVC=\$12$, $ATC=\$22$, and $MC=\$18$ per unit; $ATC$ is falling.)
A total-cost graph passes through $(q,TC)=(0,90),(4,150),(8,250)$, with cost in dollars. At $q=8$, find $AFC$, $AVC$, and $ATC$. Also find the average increase in total cost per unit from $q=4$ to $q=8$. (Answer: $AFC=\$11.25$, $AVC=\$20$, $ATC=\$31.25$ per unit, and the interval slope is $\$25$ per unit.)
A hypothetical firm's cost function is $TC(q)=40+12q+0.5q^2$. At $q=4$, calculate $TC$, all three average-cost measures, and $MC$. State the direction of $ATC$. (Answer: $TC=\$96$, $AFC=\$10$, $AVC=\$14$, $ATC=\$24$, and $MC=\$16$ per unit; $ATC$ is falling.)
A hypothetical firm's cost function is $TC(q)=160+8q+q^2$. Find the positive output where $MC=ATC$ and the common cost per unit, rounding both to two decimals. (Answer: $q=\sqrt{160}\approx12.65$ units, and $MC=ATC=8+2\sqrt{160}\approx\$33.30$ per unit.)
On a smooth total-cost graph, $TC(6)=\$310$, $TC(10)=\$430$, and the tangent slope at $q=10$ is $\$34$ per unit. Find $ATC$ and point $MC$ at $q=10$, then find the average increase in total cost per unit from 6 to 10. (Answer: $ATC=\$43$ per unit, point $MC=\$34$ per unit, and the interval slope is $(430-310)/(10-6)=\$30$ per unit.)

What You'll Learn