Changes produce marginal values

Subtract adjacent totals and divide by the change in quantity.

MC=Δ TC/Δ Q, MR=Δ TR/Δ Q, MP_L=Δ Q/Δ L, and marginal utility is Δ TU/Δ Q. If quantity changes by more than one, divide by that change. Write which unit is being added. The change from four to five workers gives the fifth worker’s MP. It is not output per five workers.

The profit rule MR=MC uses two changes to select output. It does not calculate profit. Profit is TR-TC or (P-ATC)Q. A firm can satisfy MR=MC while earning profit, breaking even, or minimizing loss.

A marginal value answers a forward-looking question: what did the additional unit contribute? Begin with two adjacent total observations, subtract the earlier total from the later total, and divide by the number of units added. The order matters. If total cost rises from $410 at 20 units to $446 at 23 units, then MC=(446-410)/(23-20)=$12 per unit. Reporting $36 describes the total added cost of three units, not marginal cost per unit over the interval.

Tables often tempt students to divide a total by the new quantity. That produces an average. If five workers make 140 units and six make 164, the sixth worker’s marginal product is 24 units. Average product with six workers is 164/6. Both are legitimate numbers, but they answer different questions.

The MR=MC rule is a boundary condition, not a promise that a discrete schedule will contain an equality. Evaluate each successive block’s contribution to profit. In the example, unit four adds $15 of revenue at a $13 cost, whereas unit five adds $15 at an $18 cost. Output should end at four. Only a separate total or per-unit calculation reveals the resulting profit.

Marginal thinking also applies outside firms. A consumer buys another slice if marginal benefit covers price. A government expands pollution reduction while marginal social benefit exceeds marginal social cost. A student studies another hour if the expected score benefit outweighs the best alternative use of that hour. The labels change, but the incremental comparison remains.

Use three checks. Identify the exact unit added, preserve the direction of subtraction, and attach units. A negative marginal product can occur if added labor reduces total output, but a negative MC in a standard production problem is suspicious. Finally, do not average marginal values across unequal intervals unless the question asks for an interval average. A table may conceal changing increments.

A final exam habit is to annotate each table interval before touching the options. Write “unit 6” beside the change from five to six, and circle any interval where quantity rises by more than one. Then estimate the direction of the total: positive MC means TC must rise, positive MR means TR rises, and diminishing MP means successive output additions shrink. These annotations expose distractors built from reversed subtraction, skipped rows, or a missing divisor. They also make the explanation reproducible instead of dependent on recognizing a familiar number.