One more, rather than all or nothing

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

You will calculate each increment's benefit minus cost and compare cumulative results across every feasible whole quantity, including zero when allowed. Explain when stopping at the first negative increment is valid and when a later improvement requires evaluating a larger sequential plan. Distinguish a positive total from the highest total.

2. Starting point

Use the stated alternatives, quantities and units. Separate a model's assumptions from the conclusion derived from them.

3. Terms to use precisely

4. The decision can be one more rather than everything

Many choices concern how much of an activity to do, rather than whether to do all of it or none. A club can prepare one more display panel, a workshop can complete one more item, or a learner can spend one more fixed period on practice. Marginal analysis asks what changes when the quantity increases by one specified unit. It compares the additional benefit with the additional relevant cost of that increment.

The unit must be clear. One more hour differs from one more ten-minute period; one more panel differs from one more complete exhibition. A marginal benefit of eight units has no clear interpretation until we know the increment it belongs to. Our tables number successive increments and give comparable benefit and cost figures for each. Those figures are stipulated teaching units, not claims that every real experience can be measured exactly.

Total benefit describes all benefits from the selected quantity. Marginal benefit describes the extra benefit from adding the next unit. The same distinction applies to total and marginal cost. If two panels provide benefits of ten and seven, the total benefit of two is seventeen, while the marginal benefit of the second is seven. Confusing those quantities can make an unhelpful extra unit appear attractive merely because the entire activity still has a positive total.

A decision should compare changes caused by the proposed increment. A fixed cost already unavoidable under all current alternatives does not become an extra cost every time a unit is added. An additional setup charge triggered only by the next batch, however, can be relevant to that increment. The previous lesson's distinction between sunk and avoidable costs helps identify what belongs in the marginal comparison.

Term	What it means
Marginal benefit	The additional benefit of one specified increment.
Marginal cost	The additional relevant cost of that increment.
Marginal net benefit	Marginal benefit minus marginal cost.
Cumulative net benefit	The sum of marginal net benefits through a selected quantity.
Sequential increments	Units that must be taken in order, so quantity q includes the first q increments.

Another way: Calculate additional net benefit first

For each increment, subtract its marginal cost from its marginal benefit. If the benefit is nine and the cost is five, the increment adds four net units under the model. If the benefit is four and the cost is six, the increment reduces net benefit by two. Keep the signed result: negative two is not a quantity to round away simply because the benefit itself is positive.

A positive marginal benefit alone is insufficient to justify an increment. Almost any extra activity can offer something useful while using resources that have more valuable alternatives. The relevant comparison includes both sides. Similarly, a positive marginal cost does not imply that the increment is a mistake when its additional benefit is larger. The sign of the difference summarizes the supplied trade-off.

When marginal benefit equals marginal cost, the increment changes net benefit by zero. The model is indifferent between the adjacent quantities unless another consideration or a tie rule is supplied. Do not claim that equality establishes a unique best quantity. Our scored cases use a unique highest cumulative net result, so a single quantity can be constructed without guessing how a tie should be resolved.

The cost figures must include the relevant alternative use of the resources under the stated model. A free material handed to a club may still require time to process, and that time may have another use. The lesson does not automatically add unstated costs, but it teaches the habit of asking what the cost column represents. An exact calculation from an incomplete column remains conditional on that incomplete model.

Another way: Build cumulative results from the increments

To evaluate a quantity of two units, add the first two marginal benefits and the first two marginal costs, or equivalently add the first two marginal net benefits. Both routes should give the same cumulative net result. This is a useful arithmetic check because it connects the total and marginal views without treating them as separate theories.

Suppose marginal net benefits for four successive units are five, three, negative one, and negative four. The cumulative net values at quantities zero through four are zero, five, eight, seven, and three. The highest value is eight at quantity two. Quantity four still has a positive cumulative net value, but it is worse than quantity two. Looking only for a positive total would miss the better stopping point.

Include zero when it is feasible. If every available increment reduces net benefit and no prior commitment requires production, choosing zero can be best under the supplied objective. A table beginning at quantity one should not silently force the learner to select an activity that is inferior to the stated zero baseline. Conversely, if the task specifies a minimum requirement, apply that constraint before selecting among quantities.

Our basic tasks treat increments as sequential: choosing the third requires taking the first and second. This describes quantities of one activity, not unrelated projects that can be selected independently. Under this rule, a quantity q includes the first q rows. If projects can be selected separately, a different model is needed. Read that distinction before trying to skip an unfavorable row and collect only later favorable increments.

Another way: Know when the simple stopping rule works

When marginal benefits do not rise and marginal costs do not fall as quantity increases, marginal net benefits do not rise either. Under these conditions and the sequential model, once an increment has negative net benefit, later increments cannot restore a higher cumulative total. Stopping before the first negative increment then gives a highest net result, subject to ties and other constraints.

Without those conditions, stopping at the first negative increment may be wrong. A second unit might require temporary setup while a later unit provides a large additional benefit. For example, marginal net values of four, negative three, and six produce cumulative values zero, four, one, and seven. Quantity three is best among those feasible quantities, even though the second increment alone is unfavorable. The intermediate loss must be evaluated as part of the complete sequential plan.

This is why our general solution method compares all feasible cumulative quantities rather than blindly applying a slogan. In the regular diminishing-benefit cases, that comparison confirms the familiar stopping rule. In a case with a later improvement, it prevents premature stopping. State which assumptions justify a shortcut before using it.

A further distinction concerns discrete and continuous choices. This course uses whole increments such as panels or fixed time blocks. We compare neighboring quantities in a table. More advanced models can allow continuously divisible quantities and use equality conditions under additional mathematical assumptions. The introductory lesson should not pretend that a neat equality is guaranteed in a discrete table where marginal benefit can jump from above cost to below it.

Another way: The best quantity is conditional on the objective

Maximizing the supplied cumulative net benefit is a stated objective, not a claim that every real decision has one uncontested measure. A club may also face a spending limit, a deadline, a minimum service requirement, or a concern about who receives the benefits. If the problem supplies such a constraint, identify the feasible quantities first and maximize only over that set. A numerically attractive but impossible quantity is not a valid recommendation.

The marginal values themselves may be forecasts. A change in expected attendance or material requirements can change a row and therefore the best quantity. Recalculate the cumulative totals when the inputs change. Do not treat a result from yesterday's table as a permanent fact about the activity's ideal size. The model provides a transparent relationship between assumptions and conclusion.

A useful final explanation names the selected quantity, its cumulative net value, and the comparison with nearby or competing feasible quantities. 'Two units give eight, compared with five at one and seven at three' shows why two is preferred under the supplied objective. Saying only 'more is better' or 'stop when it becomes expensive' does not identify the relevant trade-off.

Marginal reasoning also helps evaluate a small proposed revision to an existing plan. If a club already intends two panels and someone proposes a third, focus on what that third panel adds and uses. Then check whether a larger change in the plan needs a broader cumulative comparison. The distinction keeps incremental questions precise without losing sight of the complete feasible alternatives.

Another way: Averages can hide the last increment

A project with total benefit twenty after four units has average benefit five per unit. That average does not tell us the fourth unit's additional benefit. The first three could contribute eighteen and the fourth only two. Compare the fourth unit's own added benefit with its added cost before extending the plan. An average summarizes the selected quantity; a marginal figure answers the question about changing it.

5. How many display panels should a club prepare?

A fictional science club can prepare up to four display panels before an exhibition. Each additional panel requires one more work block, and later panels require completing the earlier ones. The supplied marginal benefits are twelve, nine, six, and three teaching units. Marginal costs, including the stated alternative use of each work block, are five, five, seven, and seven. The club's bounded objective is to maximize cumulative net benefit.

The marginal net values are seven, four, negative one, and negative four. Starting from zero panels with zero net benefit, the cumulative values are zero, seven, eleven, ten, and six. Two panels therefore give the highest supplied result, eleven. Four panels still give a positive total, but preparing the last two reduces the total below what two panels achieve. A positive final total is not evidence that every increment was worthwhile.

The club can explain the proposed third panel precisely. It adds six benefit units and uses seven cost units, reducing net benefit by one. Under this table, declining the third panel preserves the higher total. Because benefits fall and costs do not fall, later increments do not create a recovery that would overturn the stopping point.

Now imagine that a new proposal changes the fourth panel's benefit substantially. The club should evaluate the revised cumulative table, not insist that the first negative row always ends the discussion. The sequential model might make taking an unfavorable intermediate step worthwhile as part of a better complete plan. The original answer remains correct for the original numbers; the revised proposal requires its own calculation and explanation.

6. Check the tempting inference

Do not compare a marginal cost with the entire activity's benefit. A positive total is not necessarily the highest total. Include zero if feasible and respect sequential requirements. Stopping at the first negative increment needs conditions that rule out a later recovery; otherwise compare all feasible cumulative plans.

7. One increment's net contribution

8. A positive total is not the maximum

9. A later improvement defeats an unsafe shortcut

10. Finish a cumulative comparison

11. Guided practice

Sequential increments have benefits 10,8,4,2 and costs 4,5,6,6. Zero is feasible. Give best whole quantity, highest cumulative net and cumulative net at all four increments.

12. Guided practice

Three sequential increments have benefits 7,5,2 and costs 3,4,5. Zero is feasible. Complete the cumulative comparison.

13. Guided practice

Benefits for three sequential increments are 9,5,2; costs 3,6,7. Zero is feasible. Give best quantity, its cumulative net and cumulative net at all three.

	Your result
Best whole quantity
Highest cumulative net benefit
Cumulative net if all increments taken