Calculate a marginal value from a table

Marginal means a change between neighboring totals.

The word marginal means additional: how much a total changes when the activity increases by one unit. A table does not usually print that change directly. It prints totals at successive quantities, and you recover the marginal value by comparing neighboring rows. If total cost is $146 at seven units and $163 at eight, the eighth unit adds $17 to cost. The calculation is MC₈=(TC₈-TC₇)/(8-7)=($163-$146)/(1)=$17 per unit. The subscript matters: the difference between rows seven and eight belongs to the eighth unit, because that is the unit added. It does not describe the seventh unit and it is not the average of the two rows.

The general rule is marginal value=(Δtotal value)/(Δquantity). A typical economics question normally supplies enough information to calculate marginal values with discrete differences. Introductory test questions do not require derivatives. If a table advances one row at a time, compare the neighboring totals. If it skips quantities, divide the total change by the number of units in the interval. When quantity rises by exactly one, the denominator is one and the marginal value is simply the difference between totals. When quantity jumps by more than one, divide by the size of the jump. If total cost rises from $210 at ten units to $246 at thirteen units, the table supports an average marginal cost over those three units of ($246-$210)/(13-10)=$12 per unit. Without the missing rows, it does not reveal the separate cost of the eleventh, twelfth, and thirteenth units.

The same neighboring-row method applies across the course. Marginal product is the change in total product divided by the change in an input. Marginal revenue is the change in total revenue divided by the change in output. Marginal utility is the change in total utility divided by the change in consumption. The economic interpretation changes, but the mathematics does not.

Read one row slowly. Hiring the fifth worker raises total output from 92 to 108, so the fifth worker’s marginal product is 108-92=16 units. With five workers, average product is total output divided by the number of workers: 108/5=21.6 units per worker. If each extra unit of output adds $6 to revenue in a competitive product market, the fifth worker’s marginal revenue product is 16×$6=$96. The same row therefore contains a total, a marginal value, an average, and a dollar value. They answer different decisions.

An average is a level divided by a count. Average product is total output per worker. Average total cost is total cost per unit. The marginal-average rule provides a useful check. A marginal value above the current average pulls the average upward. A marginal value below it pulls the average downward. This is why marginal cost crosses average variable cost and average total cost at their minimum points.

The rule is easier to understand with grades. If your current average is 82 and the next score is 91, the new average must rise. If the next score is 74, it must fall. The new score is marginal to the collection, while the existing average summarizes all earlier scores. In the table, the third worker’s marginal product of 27 exceeds the two-worker average product of 21, so average product rises. The fifth worker’s marginal product of 16 is below the four-worker average of 23, so average product falls. Marginal product crosses average product near average product’s maximum for the same mathematical reason that marginal cost crosses average cost near average cost’s minimum.

Several tempting calculations answer the wrong question. Dividing $163 by eight gives average total cost, not marginal cost. Subtracting 7 from 8 gives the change in quantity, not the change in cost. Dividing the total change by the final quantity mixes a change in the numerator with a level in the denominator. Using nonadjacent rows without dividing by their quantity difference exaggerates the per-unit change. A good scratch-work line always shows both differences and the unit.

One schedule, three calculations

A workshop’s total output rises from 72 to 88 when a fifth worker is hired. The fifth worker’s marginal product is 16. Average product with five workers is 88/5=17.6. If each additional unit sells for $6 in a competitive output market, the worker’s marginal revenue product is 16× $6=$96.

On the exam, finish the calculation with a decision. A marginal cost of $17 means producing the eighth unit adds $17 to total cost. It does not by itself say whether the unit should be produced. Compare it with the marginal revenue or marginal benefit of that unit. A marginal product of 16 tells how much output the fifth worker adds. Compare its revenue value with the worker’s marginal resource cost to decide whether hiring is profitable. The number is evidence. The economic rule turns it into an answer.

Write the unit before choosing an option

Cost per additional unit, output per additional worker, revenue per additional unit, and utility per additional item are not interchangeable. If the stem asks for the marginal product of labor, an answer measured in dollars is either marginal revenue product or an unrelated value.