Price one more unit of capacity
When does a local marginal value stop predicting a larger change?
Entering midway? A marginal value is output change per small input change, keeping the model fixed.
Start with a concrete question
Allocate capacity first to a four-credit product capped at three units, then a two-credit product capped at four.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Predict before calculating: When does a local marginal value stop predicting a larger change?
Change one thing. Inspect what happens.
- Calculate value at capacity 2 and 3.
- Cross the breakpoint at 3 with a larger addition.
- Inspect left and right changes at each kink.
- State the capacity range where a quoted marginal value is valid.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Optimal value versus available capacity
Horizontal: capacity units. Vertical: optimal credits.
- 1. value function (solid line)
- 2. before and after (points)
Axes use the displayed ranges.
Read plotted values as a table
| Series | capacity units | optimal credits |
|---|---|---|
| value function | 0 | 0 |
| value function | 3 | 12 |
| value function | 7 | 20 |
| value function | 12 | 20 |
| before and after | 3 | 12 |
| before and after | 4 | 14 |
Follow the calculation
- First product earns 4 credits/unit up to 3 units; second earns 2 credits/unit up to 4 units. Both use one capacity unit.
- At capacity 3, optimal value=12. Adding 1 gives 14; gain=2.
- Local right-hand rate=2. Its prediction 2 is valid only until the next capacity breakpoint.
- At capacity 3 the left rate is 4 but the right rate is 2; at 7 the right rate becomes zero.
A local sensitivity is not a promise for a large change. When the active constraint or available product changes, recompute the optimum.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Price one more unit of capacity
Allocate capacity first to a four-credit product capped at three units, then a two-credit product capped at four.
Concepts to recall
A marginal value is output change per small input change, keeping the model fixed.
Understand the mechanism
The first three units each earn four credits, so value reaches 12 at capacity 3.
The next four earn two each, so value reaches 20 at capacity 7. Further capacity has no value in this model.
This piecewise slope is a resource sensitivity; at a kink the left and right slopes differ. A shadow-price claim needs its valid range and assumptions.
Symbols and calculation
V(C)=4 min(3,C)+2 min(4,max(0,C−3)). V is optimal credits; C is nonnegative capacity.
Worked example
First product earns 4 credits/unit up to 3 units; second earns 2 credits/unit up to 4 units. Both use one capacity unit.
At capacity 3, optimal value=12. Adding 1 gives 14; gain=2.
Local right-hand rate=2. Its prediction 2 is valid only until the next capacity breakpoint.
At capacity 3 the left rate is 4 but the right rate is 2; at 7 the right rate becomes zero.
Your investigation
Calculate value at capacity 2 and 3.
Cross the breakpoint at 3 with a larger addition.
Inspect left and right changes at each kink.
State the capacity range where a quoted marginal value is valid.
Limits and interpretation
- These are bounded teaching models with explicit assumptions, not evidence of a real-world intervention.
Try a fresh case
The first 2 capacity units earn 5 credits each and the next 3 earn 2 each. What is the gain from increasing capacity from 1 to 4?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Explain a fresh small numerical case with a diagram or table and an argument.
College investigation
Implement the method, test a boundary or counterexample, and justify its assumptions.
Workplace investigation
Use a small decision scenario, state units and constraints, and test how the recommendation changes.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.