A shift of the frontier, and a move along it

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 derive old and new single-good maxima from their resource stocks and production requirements, then compare the two linear frontiers. Distinguish expansion, contraction, unchanged boundaries and crossing frontiers; explain a selective-productivity pivot, and separate changes in capacity from movements along or toward an existing boundary.

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. A frontier belongs to a specified resource and technology model

A production frontier describes the efficient combinations possible with a stated stock of resources and a stated technology. Changing the chosen output mix does not by itself change that frontier. If a workshop moves workers from A to B while their total hours and productive requirements remain the same, it selects another point within the same opportunity set. A change in the opportunity set requires a change in the conditions defining what can be produced.

An increase in productive resources can expand the set of feasible combinations. More usable machine-hours, additional suitable equipment, or better-trained labor can make output plans possible that were previously outside the boundary. The effect depends on what the added resources can do. An extra machine useful only for A need not increase the maximum B output. State the resource change instead of drawing every improvement as the same parallel outward movement.

Technology concerns how resources become output. If A can be produced with fewer hours per unit while total hours remain fixed, the A maximum increases. The B maximum may remain unchanged if B's method is unaffected. This selective improvement can pivot the frontier outward around the B intercept in the simple linear model. The graph's shape reflects which productive relationship changed.

The lesson compares two explicitly stated constant-rate models. Each has a resource stock H, an hours-per-A requirement, and an hours-per-B requirement. We calculate both sets of single-good maxima and compare the resulting straight frontiers. This is a controlled comparison of assumptions, not a prediction that every real resource or innovation changes production by a known exact amount.

Term	What it means
Frontier expansion	An increase in production possibilities that preserves the old feasible set in the stated model.
Frontier contraction	A reduction in production possibilities that removes some formerly feasible plans.
Selective productivity change	A change in resource requirements affecting one activity differently from another.
Movement along a frontier	A reallocation within unchanged productive possibilities.
Crossing frontiers	Boundaries where one model permits more of one extreme output but less of the other, so neither uniformly contains the other.

Another way: More of the same productive resource can scale capacity

Suppose a workshop initially has twenty-four productive hours, with four hours required per A and two per B. Its single-good maxima are six A and twelve B. If available productive hours rise to thirty-six while both per-unit requirements stay unchanged, the new maxima are nine A and eighteen B. Every old feasible combination remains feasible, and additional combinations become possible under the same technology.

The opportunity-cost rate remains two B per A because four hours per A divided by two hours per B is unchanged. The frontier expands without changing this relative production trade-off. This separates two ideas that are often confused: capacity is larger, but the cost of one output in units of the other can stay the same. An outward movement does not automatically imply a flatter or steeper straight frontier.

The proportional scaling conclusion relies on the model's constant rates and on all added hours being equally usable. If extra hours require tired workers, scarce supervision, or equipment that cannot operate longer, the effect may differ. The exercise states productive hours after those issues are resolved. A real application would need evidence that the nominal increase in time translates into the assumed productive capacity.

Resource losses can reverse the comparison. If productive hours fall while requirements remain fixed, both maxima fall and some formerly feasible plans become unavailable. A natural disaster, maintenance shutdown, or loss of usable equipment could motivate such a model, but the graph alone does not identify the cause. Derive the consequences from the specified loss rather than inventing an explanation from the direction of a line.

Another way: A selective productivity improvement changes the mix of possibilities

Now hold productive hours at twenty-four and B's requirement at two hours per unit, but reduce A's requirement from four hours to three. Maximum A rises from six to eight while maximum B stays twelve. The new frontier expands the possibilities involving A without moving the all-B extreme. It is an outward pivot in this simple model, not a parallel expansion of both intercepts.

The relative cost also changes. An A now uses three hours, which could produce 1.5 B at two hours each. The old rate was two B per A. Improved A productivity lowers A's production opportunity cost in B units. The reverse cost rises: producing one B now forgoes more potential A than before. These directional statements are consistent because the rates are reciprocals under the same technology.

A technology improvement does not require that actual output immediately move to the new boundary. Producers may need to learn the method or may choose a different mix. A changed frontier describes what is possible under the assumed adopted technology; actual production is a separate observation. Do not claim that an innovation necessarily increases both observed outputs merely because the opportunity set expands.

If one good's productivity improves while the other's worsens, the two frontiers may cross. One model can permit more A at the all-A extreme but less B at the all-B extreme. Neither feasible set then contains the other throughout. Calling the whole change 'an outward shift' would conceal the lost possibilities. The appropriate description names both changes and compares the relevant output mixes separately.

Another way: Movement, recovery and expansion answer different questions

A movement along an unchanged frontier reallocates existing productive capacity between outputs. A movement from an interior point toward the frontier uses previously unexploited possibilities under the same model. A frontier expansion changes those possibilities themselves. These three cases can all involve higher observed A output, so an increase in A alone does not identify which process occurred.

Suppose a workshop initially produces inside its twenty-four-hour frontier because some machines are idle. Reorganizing the schedule can increase both A and B while retaining the same productive resource stock and methods. That movement reduces underuse; it need not represent economic growth in productive capacity. If a new machine later raises usable capacity, that is a different change and should be represented by a new boundary.

A policy or event may affect several components simultaneously. Training might temporarily reduce production hours while improving later productivity. Investment may use current resources to increase future capacity. To keep the comparison intelligible, specify the period and separate the current trade-off from the later frontier change. A single arrow cannot explain both without the assumptions connecting them.

This distinction also prevents misleading claims about incentives. A producer's changed preference for A can move its chosen point along a frontier, but preference alone does not alter the technological maximum outputs. Conversely, a better production method can expand the frontier even if the producer keeps making the same observed combination. Preferences help select a point; resources and technology define the production possibilities in this model.

Another way: Compare models systematically and state the scope

For each model, divide its resource stock by its per-unit requirements to obtain the two intercepts. Record the old and new maxima in the same output units and period. Comparing daily and weekly maxima without conversion would create an apparent change that is only a measurement difference. The same caution applies if the definition of one output unit changes between the tables.

In these nonnegative two-good linear models, if neither new intercept is lower and at least one is higher, the feasible set expands. If neither is higher and at least one is lower, it contracts. If both are unchanged, the boundary is unchanged. If one rises and the other falls, the straight frontiers cross. The response symbols in our tasks are E for expansion, C for contraction, S for same boundary, and X for crossing.

An unchanged pair of intercepts establishes the same straight boundary only because the task specifies linear constant-rate models. Two curved frontiers could share intercepts and differ between them. Do not export the intercept shortcut to every imaginable technology. A method is reliable when its assumptions are explicit and its conclusion stays within them.

Finally distinguish potential gains from their distribution or value. An expanded opportunity set can preserve old options and make new ones available, yet the actual gains depend on the chosen plan and who receives its outputs. The graph does not prove that everyone benefits automatically. A complete economic discussion can recognize increased productive possibilities while still asking about adjustment costs, preferences, institutions, and distribution that the simple model omits.

Another way: Separate current investment from later capacity

Building new equipment can require current resources that would otherwise produce the two consumption goods. The current plan may therefore give up output even when the investment is expected to expand a future frontier. Those are different periods and different comparisons. A diagram of the later expansion does not show that the equipment was free, nor does a current output reduction alone prove that future productive capacity has contracted. State when each resource account applies.

5. Comparing two changes at a community workshop

A fictional community workshop initially has twenty-four productive hours. A bench A needs four hours and a planter B two. The old straight-frontier intercepts are six benches and twelve planters. The workshop is considering two separate proposals, each evaluated relative to this original model rather than combined without notice.

Proposal one adds twelve equally productive hours while keeping both production requirements unchanged. The new capacity is thirty-six hours, giving maxima of nine benches and eighteen planters. This is an expansion of both intercepts, while the opportunity cost remains two planters per bench. The extra hours permit more output without changing the relative rate at which time becomes the two goods.

Proposal two leaves capacity at twenty-four hours but improves bench production to three hours per bench. The new maxima are eight benches and twelve planters. The frontier pivots outward at the bench end, and the cost of a bench becomes 1.5 planters. This is also an expansion, but its shape and relative-cost implications differ from the first proposal.

The manager should not conclude that both proposals necessarily produce the same actual output mix or benefit every participant equally. Each changes the menu of feasible plans under its assumptions. The workshop still needs preferences, project requirements, and evidence about whether the proposed productive improvements can be achieved. The numerical comparison establishes what the models imply and identifies which assumptions distinguish their consequences.

6. Check the tempting inference

A new chosen output mix is not automatically a new frontier. Recovery from underuse differs from expanded capacity. A selective improvement can pivot the boundary; opposing changes can make frontiers cross. Equal intercepts imply equal boundaries only under the stated linear model, and expanded possibilities do not guarantee an equal gain for every person.

7. More productive hours, same rates

8. Only one production method improves

9. Opposing changes create crossing frontiers

10. Finish a capacity comparison

11. Guided practice

Old model:H24 hours, A needs 4 and B needs 2 per unit. New model:H36, with the same requirements. Calculate new maxima and boundary comparison. Both models are linear with constant rates, divisible outputs and no other constraint. Use E=expansion, C=contraction, S=same boundary, X=crossing.

12. Guided practice

Old model:H24, A needs 4, B needs 2. NewH 36 with the same rates. Use E=expansion, C=contraction, S=same, X=crossing. Complete the comparison.

13. Guided practice

Old model:H24, A needs 4, B needs 2. New model:H24, A needs 3, B needs 2. Calculate new maxima and comparison. Both models are linear with constant rates, divisible outputs and no other constraint. Use E=expansion, C=contraction, S=same boundary, X=crossing.

14. Practice

Old model:H30, A needs 5, B needs 3. New model:H20, A needs 5, B needs 4. Give new maxima and comparison. Both models are linear with constant rates, divisible outputs and no other constraint. Use E=expansion, C=contraction, S=same boundary, X=crossing.

15. Practice

Old model:H24, A needs 4, B needs 2. New model:H24, A needs 3, B needs 3. Give new maxima and comparison. Both models are linear with constant rates, divisible outputs and no other constraint. Use E=expansion, C=contraction, S=same boundary, X=crossing.

16. Somewhere new

A fictional workshop's old model hasH 32 productive hours, withA needing 4 hours andB 2. A shutdown reducesH to 24 while methods stay unchanged. Give new maxima and the boundary comparison. Both models are linear with constant rates, divisible outputs and no other constraint. Use E=expansion, C=contraction, S=same boundary, X=crossing.

17. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

18. Test question

A new comparison has oldH 40, A requirement 5, B requirement 2. NewH 40, A requirement 4, B requirement 4. Calculate new maxima and classify the two linear boundaries. Both models are linear with constant rates, divisible outputs and no other constraint. Use E=expansion, C=contraction, S=same boundary, X=crossing.

19. What you can do now

Compare both intercepts and the assumptions behind them. Explain why higher observed output alone cannot identify a frontier expansion.

	Your result
New maximum A
New maximum B
Boundary comparison symbol