Reading Profit and Loss on the Cost-Curve Graph

Reading Profit and Loss on a Cost-Curve Graph

A cost-curve graph can show not only which output a competitive firm chooses, but also whether that output yields an economic profit or loss. The key is to read price and average total cost at the same output. Their vertical difference is profit or loss per unit; multiplying that difference by the output gives the firm’s total profit or loss.

This interpretation uses the standard short-run model of a price-taking firm: the market sets a price that the individual firm takes as given, and the firm chooses its output to maximize profit. The graph typically shows the horizontal market-price line alongside the firm’s marginal-cost (MC) and average-total-cost (ATC) curves. The focus here is how to read the profit or loss at a chosen positive output, not whether the firm should produce or shut down.

Definition: Let $q>0$ be a firm’s output, $P$ the market price per unit, $C(q)$ total economic cost, and $ATC(q)=C(q)/q$ average total cost. Economic profit is $\pi(q)=Pq-C(q)$. At a chosen output $q^*$, the per-unit profit or loss is $P-ATC(q^*)$, and total profit or loss is $[P-ATC(q^*)]q^*$.

The Price Line, ATC Curve, and Profit Rectangle

In the graph, output $q$ is on the horizontal axis and dollars per unit are on the vertical axis. Because a competitive firm is a price taker, its price line is horizontal at $P$: the price does not change as this one firm varies its output. The ATC curve, in contrast, can rise or fall with output. At the chosen output $q^*$, compare the height of the price line with the height of the ATC curve at $q^*$.

If $P>ATC(q^*)$, the firm earns positive economic profit. If $P<ATC(q^*)$, it incurs an economic loss. If the two heights are equal, it breaks even in economic terms. The comparison is made at $q^*$; comparing price with ATC at some other output does not give the firm’s profit at its chosen output.

The associated rectangle is an auxiliary rectangle: its width is $q^*$ and its height is the vertical gap $P-ATC(q^*)$. Its signed area equals total economic profit. For a loss, the height is negative; equivalently, the loss amount is a rectangle of width $q^*$ and height $ATC(q^*)-P$. The rectangle uses the ATC value at $q^*$ as a horizontal level across its width. It is not generally the region between the price line and the varying ATC curve over every output from zero to $q^*$.

\begin{aligned} \pi(q^*) &= Pq^* - C(q^*)\\ &= Pq^* - ATC(q^*)q^*\\ &= [P-ATC(q^*)]q^*. \end{aligned}

This identity gives the graph’s rectangle its meaning. The area $Pq^*$ represents revenue, while $ATC(q^*)q^*$ represents total cost at that output. Subtracting the second amount from the first leaves the signed profit. The identity does not say that the ATC curve has the same height at every output within the rectangle.

Why the Rectangle Formula Holds

The derivation depends on the definition of average total cost, not on a special shape for the cost curves. For positive output, average total cost is total cost divided by output. Rearranging gives total cost as average total cost multiplied by output. Revenue is price multiplied by output. Substituting both identities into the definition of profit yields the formula.

Write total revenue.
A price-taking firm sells $q^*$ units at the given price $P$, so total revenue is $Pq^*$.

Rewrite total cost.
By definition, $ATC(q^*)=C(q^*)/q^*$. Since $q^*>0$, multiplying by $q^*$ gives $C(q^*)=ATC(q^*)q^*$.

Subtract cost from revenue.
Thus $\pi(q^*)=Pq^*-C(q^*)=Pq^*-ATC(q^*)q^*=[P-ATC(q^*)]q^*$.

In the standard competitive-firm model, a positive, interior profit-maximizing output is identified where price equals marginal cost on the relevant rising portion of the MC curve. Once $q^*$ is identified, use the price and ATC at that output to read profit or loss. The MC condition identifies output; the price–ATC gap measures the resulting profit or loss.

Worked Graph Interpretations

Worked Example: Positive Economic Profit

A price-taking firm faces a market price of $28 per unit. Its short-run total cost is $C(q)=100+4q+q^2$, where $q$ is output in units. Find the positive interior profit-maximizing output, then calculate the per-unit margin and total profit. Interpret the graph without treating the ATC curve as horizontal.

Method: identify output from price and marginal cost, then use the price–ATC gap. Marginal cost is the derivative of total cost:

MC(q)=\frac{dC}{dq}=4+2q.

At an interior profit maximum, set $P=MC(q)$. Therefore $28=4+2q$, so $2q=24$ and $q^*=12$. Marginal cost is increasing because its slope is $2>0$, so this output satisfies the relevant second-order condition.

Now calculate average total cost at $q^*=12$. Total cost is $C(12)=100+4(12)+12^2=100+48+144=292$. Thus $ATC(12)=292/12=73/3=\$24.33$ per unit, rounded to cents. The exact per-unit margin is $28-73/3=11/3$ dollars, or approximately $3.67 per unit.

Total profit is the gap times output: $(11/3)(12)=\$44$. As a check, revenue is $28(12)=\$336$, and revenue minus total cost is $336-292=\$44$. The results agree.

Graph interpretation. At $q=12$, the price line is $28 per unit and the ATC curve is at $24.33 per unit, so the vertical gap is about $3.67. Multiplying that gap by the horizontal width of 12 units gives $44. The auxiliary rectangle uses the ATC height at $q=12$ across the width; it should not be described as lying above the ATC curve at every output from 0 to 12.

Common mistake avoided: Do not multiply the price–ATC gap by the height of the ATC curve at a different output. Both curves must be read at $q^*=12$.

Worked Example: Economic Loss at a Positive Output

A hypothetical firm sells a product at a market price of $18 per unit. Its total cost is $C(q)=64+6q+q^2$. Conditional on producing a positive interior output, find the profit-maximizing quantity and measure the loss shown by the graph. This calculation alone does not decide whether the firm should produce.

Method: find the conditional interior output, then calculate the price–ATC gap. Marginal cost is $MC(q)=6+2q$. Set marginal cost equal to price: $18=6+2q$, so $2q=12$ and $q^*=6$. Since marginal cost is increasing, this is the interior profit-maximizing quantity conditional on producing.

At this output, total cost is $C(6)=64+6(6)+6^2=64+36+36=\$136$. Average total cost is $136/6=68/3$, or approximately $22.67 per unit. The per-unit margin is $18-68/3=-14/3$, or approximately $-\$4.67$ per unit.

The signed rectangle area is $(-14/3)(6)=-\$28$, so the firm incurs a $28 economic loss at this output. Check using total revenue: $18(6)=\$108$, and $108-136=-\$28$. The sign and amount agree.

Graph interpretation. At $q=6$, ATC is above the price line by $14/3$ dollars per unit. The loss rectangle has width 6 and height $14/3$, for a positive loss amount of $(14/3)(6)=\$28$. It is constructed using ATC at $q=6$, not by tracing the ATC curve across the interval.

Common mistake avoided: A negative signed profit is not a negative dollar amount of loss. State the result as economic profit of $-\$28$, or equivalently an economic loss of $28.

Worked Example: Break-Even at the Profit-Maximizing Output

A competitive firm has total cost $C(q)=25+5q+q^2$ and faces a market price of $17 per unit. Find its positive interior profit-maximizing output and determine whether the graph shows profit, loss, or break-even.

Method: use marginal cost to identify output and compare price with ATC there. Marginal cost is $MC(q)=5+2q$. Set $17=5+2q$, giving $2q=12$ and $q^*=6$. Since MC is increasing, this is the relevant interior maximum.

Total cost at this output is $C(6)=25+5(6)+6^2=25+30+36=\$91$. Hence $ATC(6)=91/6$, approximately $15.17 per unit. The price–ATC gap is $17-91/6=(102-91)/6=11/6$, approximately $1.83 per unit.

The profit is $(11/6)(6)=\$11$. Check directly: revenue is $17(6)=\$102$, and $102-91=\$11$. The graph therefore shows positive economic profit, not break-even.

Graph interpretation. At $q=6$, the price line is above the ATC curve by $11/6$ dollars per unit. The auxiliary rectangle has width 6 and height $11/6$, so its area is $11. Break-even would require the price line and ATC curve to have equal height at the chosen output.

Common mistake avoided: The fact that price equals marginal cost at the chosen output does not imply break-even. Break-even is determined by comparing price with ATC at that output.

How to Read a Graph Reliably

When a graph gives curves rather than equations, use a consistent sequence. First identify the market-price line. Next identify the output being evaluated, typically the positive interior output where the price line meets the rising part of MC. At that horizontal coordinate, read the ATC height and compare it with price. Finally, multiply the vertical gap by the output width. If the graph is scaled, apply both axis scales before calculating area.

Key takeaway: At positive output $q^*$, total profit is the signed vertical gap $P-ATC(q^*)$ multiplied by $q^*$. A positive gap means profit, a negative gap means loss, and a zero gap means break-even. The rectangle is an algebraic area using the single ATC value at $q^*$; it is not generally the region between the price line and the varying ATC curve.

A common graphical error is to treat the ATC curve as if it were a horizontal line over the entire output interval. ATC is a value that can vary with output. The rectangle formula remains correct because total cost at $q^*$ is $ATC(q^*)q^*$, not because ATC has that value at every smaller quantity. Another error is to report only the per-unit gap as total profit; total profit also depends on the number of units produced.

This distinction matters whenever a graph is used to infer a firm’s dollar outcome. A gap of $2 per unit has different total implications at outputs of 5 and 50 units. Always report the vertical gap, the output, and their product, then check the result using revenue minus total cost if cost data are available.

Check Your Understanding

For each problem, identify the output if needed, compare price with ATC at that output, and use the correct area interpretation.

A firm has $C(q)=81+3q+q^2$ and faces $P=\$19$. Find its positive interior profit-maximizing output, ATC there, and total profit or loss. (Answer: $q^*=8$, $ATC=\$21.125$, economic loss $=\$17$.)
A firm has $C(q)=40+2q+q^2$ and faces $P=\$14$. Find the positive interior output and verify the signed profit using revenue minus cost. (Answer: $q^*=6$, $ATC=\$44/3$, signed profit $=-\$4$, or a $4 loss.)
A firm has $C(q)=72+4q+q^2$ and faces $P=\$24$. Calculate the profit-maximizing output, the per-unit margin, and total profit; state the revenue-minus-cost check. (Answer: $q^*=10$, per-unit margin $=\$2.80$, profit $=\$28$; revenue $=\$240$, cost $=\$212$.)
On a graph, a firm’s positive interior profit-maximizing output is 12 units. At that output, the market price is $30 per unit and ATC is $32.50 per unit. Find total profit or loss and explain which ATC value the auxiliary rectangle uses. (Answer: economic loss $=\$30$; the rectangle uses the ATC value of $32.50 at 12 units, not the varying ATC curve across all outputs.)

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