PPF Practice Problems: Points, Costs, and Shifts
At 8 bicycles a week, the fictional economy below can produce 48 crates of food. Raising bicycle output to 12 costs 12 crates: 3 crates per additional bicycle. These PPF practice problems ask you to separate the total sacrifice from the cost per unit.
The same table also lets you check whether a point is attainable, reverse the direction of a tradeoff, and decide whether the frontier has shifted. Try the six exercises before reading their worked answers.
Read the frontier before calculating
A production possibilities frontier (PPF) shows the limit of what an economy can produce with its resources and technology. In the two-good model used here, a point on the frontier is productively efficient: producing more of either good requires giving up some of the other. An interior point is attainable but productively inefficient; a point outside is currently unattainable. OpenStax explains these distinctions.
Being on the frontier doesn't establish allocative efficiency: whether the output mix matches society's preferences. A graph of production limits can't tell us whether people would prefer more bicycles or more food.
For every exercise, use this original teaching model:
- Output is measured per week. Put bicycles on the horizontal axis and crates of food on the vertical axis.
- The table lists points on the frontier. Join adjacent points with straight line segments, allowing fractional output. Don't draw a smooth curve through them.
- Every nonnegative combination on or below those segments is attainable. Nothing beyond 20 bicycles or above the frontier is attainable.
- Resources and technology stay fixed unless an exercise explicitly changes them. Ignore trade, inventories, and production from other periods.
| Point | Bicycles per week | Maximum crates of food per week |
|---|---|---|
| A | 0 | 60 |
| B | 4 | 56 |
| C | 8 | 48 |
| D | 12 | 36 |
| E | 16 | 20 |
| F | 20 | 0 |
This is a piecewise-linear frontier. The tradeoff stays constant within each segment and changes at the listed corners.
For example, an output of 2 bicycles falls halfway between A and B. The food maximum there is 58 crates, halfway between 60 and 56. The combination (2, 58) is on the frontier; (2, 57) is inside.
Keep the cost's units attached
When moving along the frontier toward more bicycles:
Opportunity cost per bicycle = crates of food given up / bicycles gained
Report the cost as a positive quantity: how much food is given up. The graph's slope is negative because food output falls as bicycle output rises. Its absolute value gives the cost per bicycle on that segment.
Resources suited to one activity may be less suited to another, as Federal Reserve Education's increasing-opportunity-cost lesson explains. Moving more resources into bicycle production can therefore require giving up progressively more food. Our table represents that increasing cost with successively steeper segments.
For comparisons between producers, try the separate comparative advantage practice problems.
Six production possibilities frontier practice problems
For each calculation, write both the number and its units. Coordinates below always mean (bicycles, crates of food).
1. Classify four production points
Classify P (10, 42), Q (10, 40), R (10, 44), and A (0, 60) as on, inside, or outside the frontier. Which are productively efficient? Can the frontier alone tell you which attainable point society prefers?
2. Find the five segment costs
For each adjacent pair from A → B through E → F, calculate food sacrificed and cost per additional bicycle. Is opportunity cost constant or increasing as bicycle output rises?
3. Cross more than one segment
Find the total food sacrifice and average cost per additional bicycle for C → E. Compare that average with the cost per bicycle on D → E. What does the first additional bicycle starting at C cost?
4. Reverse the direction
For D → C, find bicycles sacrificed, food gained, and the opportunity cost of one additional crate of food. Relate it to the C → D cost per bicycle.
5. Bring idle capacity back into use
The economy starts at S (6, 40). Resources that were idle return to use, allowing output to reach C (8, 48). The economy's stock of resources and its technology stay the same. Has the frontier shifted? Does food output fall to make room for the extra bicycles in this move? Then classify a subsequent move from C to D.
6. Change the technology
A new food-production technique doubles the maximum food output at every fixed bicycle quantity. The maximum bicycle output remains 20. The new frontier has twice the original food coordinate at every bicycle quantity, with all nonnegative points on or below it attainable.
Find the new food maximum at 12 bicycles. Classify T (12, 60) relative to both frontiers. Is the change a movement along the old frontier, a parallel outward shift, or an outward change with one unchanged intercept? Does the new frontier tell you that the economy will actually produce at T?
Worked answers
1. Two efficient points, one inside, one outside
At 10 bicycles, use segment C → D. Ten is halfway between 8 and 12, so maximum food is:
48 − (10 − 8) × 3 = 42 crates
| Point | Position | Interpretation under this model |
|---|---|---|
P (10, 42) |
On | Attainable and productively efficient |
Q (10, 40) |
Inside | Attainable and productively inefficient |
R (10, 44) |
Outside | Currently unattainable |
A (0, 60) |
On | Attainable and productively efficient |
An endpoint can be efficient: starting at A, producing any bicycles requires sacrificing food. But productive efficiency doesn't rank the attainable points by what society wants. That requires information about preferences, which this table doesn't supply.
2. One, two, three, four, then five crates
| Move | Food given up | Bicycles gained | Opportunity cost per bicycle |
|---|---|---|---|
| A → B | 4 crates | 4 | 4 / 4 = 1 crate |
| B → C | 8 crates | 4 | 8 / 4 = 2 crates |
| C → D | 12 crates | 4 | 12 / 4 = 3 crates |
| D → E | 16 crates | 4 | 16 / 4 = 4 crates |
| E → F | 20 crates | 4 | 20 / 4 = 5 crates |
Cost rises from one segment to the next and stays constant within each. On C → D, the whole four-bicycle increase requires giving up 12 crates. Divide by the four bicycles gained to get 3 crates per bicycle.
3. The average is 3.5 crates per bicycle
C → E gains 16 − 8 = 8 bicycles and sacrifices 48 − 20 = 28 crates:
Average cost = 28 crates / 8 bicycles = 3.5 crates per bicycle
D → E costs 4 crates per bicycle. Starting at C, the first additional bicycle takes production to (9, 45), so it costs 3 crates. The average over the eight-bicycle increase is between the two segment rates. Use the segment you actually enter to find the cost of the next unit.
4. One-third of a bicycle per crate
D → C sacrifices 12 − 8 = 4 bicycles to gain 48 − 36 = 12 crates:
Cost per crate = 4 bicycles / 12 crates = 1/3 bicycle per crate
On C → D, the cost was 3 crates per bicycle. Reversing direction on that same segment gives its reciprocal, 1/3 bicycle per crate. The units reverse along with the fraction; a result of “3 bicycles per crate” would describe a different tradeoff.
5. Production recovers without a frontier shift
At 6 bicycles, the original frontier allows 56 − (6 − 4) × 2 = 52 crates, so S (6, 40) is inside. Moving from S to C adds 2 bicycles and 8 crates of food. Both outputs rise as unused capacity returns to production. There is no reduction in food output along this move.
The frontier stays fixed because the economy's resources and technology haven't changed. Production has moved from inside the frontier to its boundary. The later C → D move is along the frontier: once production is efficient, gaining those 4 bicycles requires giving up 12 crates.
6. Capacity expands, with the bicycle intercept unchanged
At 12 bicycles, the new food maximum is 2 × 36 = 72 crates. T (12, 60) was outside the old frontier because 60 > 36; it is inside the new one because 60 < 72.
The food-axis intercept rises from 60 to 120 crates, while the bicycle-axis intercept F (20, 0) stays fixed. Every point on the original frontier with positive food output now has a higher food maximum at the same bicycle quantity. The frontier expands unevenly, so this is an outward change with one unchanged intercept. OpenStax discusses technology as a source of expanded production possibilities.
This is a change in production capacity, not a movement along the old frontier. T is now attainable but productively inefficient: at 12 bicycles, the economy could produce another 12 crates of food. The expanded frontier doesn't tell us which point will be chosen or how society's welfare will change.
Revisit the decision you missed
Turn one mistake into a recall prompt: “Why can both outputs rise from an inside point?” Answer it before checking the explanation, then try again later. The flashcard review guide explains how to space those attempts.
Keep practicing full calculations: find the segment, check feasibility, and divide with units.
Production note: AI tools substantially assisted with research, writing, and exercise construction. All production quantities and scenarios are fictional. The worked-answer arithmetic was checked with an exact-fraction script. These exercises have not been tested in a classroom or used to measure learning outcomes. The illustration is AI-generated.