A boat builder in Warri wins an order for sixteen identical fishing boats. The first boat takes the crew 800 hours. The tenth does not take 800 hours: the crew has learned where the delays are, jigs have been made, and the work has settled into a rhythm.
A budget that assumes 800 hours a boat would overstate labour cost by thousands of hours and a price quoted on that basis would lose the contract. The learning effect is the improvement in efficiency that comes with repetition of a task, and the learning curve is the mathematical model that describes it.
This topic explains the effect, the learning rate, the tabular approach that works for doublings of output and its limitation, the learning curve formula y = axb that works for any number of units, how to find the learning rate from data, and how to calculate the time for any single unit and the cost of a batch. It builds on the earlier topic on correlation, regression and time series.
The next topic covers the learning curve in budgeting: steady state and limitations.
In 30 seconds
When a skilled, repetitive task starts, workers learn and the time per unit falls; eventually the effect wears out and the time becomes constant (a steady state). An x% learning curve means that every time cumulative output doubles, the cumulative average time per unit falls to x% of its earlier value.
So with an 80% curve, the average time for the first 2 units is 80% of the time for the first unit, for the first 4 units 80% of the average for 2, and so on. The tabular approach shows cumulative average, total time and incremental time for 1, 2, 4, 8, 16 units but cannot give the time for a single unit such as the third.
The formula y = axb can: y = cumulative average time per unit, x = cumulative units, a = time for the first unit and b = log (learning rate) ÷ log 2 (a negative number; use 0.8 for 80%). Total time for x units = y × x. Time for the nth unit = total for n units − total for n − 1 units. The learning rate is found from (cumulative average ÷ a) = ratenumber of doublings.
Learning affects labour and labour-driven overhead, not materials. Full marks come from correct application of the formula and a sensible comment.
The learning effect and the learning curve
The effect
When a team starts a skilled task for the first time and the task becomes repetitive, the workers find quicker ways, make fewer mistakes and gain knowledge and confidence, so the time to do the work falls. This improvement is the learning effect. It was first noticed in the United States in the 1940s in aircraft manufacture, a highly skilled activity where labour time is a large part of cost.
It is strongest in labour-intensive work and weakest where machines set the pace. After a time the effect is exhausted and the time per unit stabilises. The effect matters for costs, for budgeting and forecasting of labour and production, and for pricing, because a quotation can allow for the savings that will be made.
The learning rate
The effect is measured as a percentage, the learning rate: for example 80% or 90%. An x% curve means that when cumulative output doubles, the cumulative average time per unit becomes x% of what it was. The lower the percentage, the faster the learning. Note that it is the cumulative average time of all units made so far that falls to x%, not the time of the last unit.
The tabular approach
For doublings of output, a table can be built:
- cumulative units
- cumulative average time per unit (multiplying by the rate at each doubling)
- total time (units × cumulative average)
- incremental time for the extra units (the change in total)
- average time for the additional units (incremental time ÷ extra units)
Its weakness: it works only for doublings, and gives the time for a batch (for example units 3 and 4 together) but not for a single unit, since the later units in a batch take less time than the earlier ones. For that, use the formula.
The formula
y = axb, where y is the cumulative average time per unit for x units, a is the time for the first unit, x is the cumulative number of units and b = log of learning rate ÷ log 2. The rate is entered as a decimal. For an 80% curve b = log 0.8 ÷ log 2 = −0.32193; for 85%, −0.2345; for 90%, −0.1520.
A calculator with a log key does the work, and x raised to a negative power is the reciprocal of x raised to the positive power. Then:
- Cumulative average for x units = a × xb.
- Total time for x units = x × a × xb.
- Time for the nth unit = total for n units − total for (n − 1) units.
- Time for units m to n = total for n units − total for (m − 1) units.
Finding the learning rate
If the time for the first unit and the cumulative average for some later number of units are known, the learning rate is found by dividing the average by a and taking the root that matches the number of doublings. If 4 units (two doublings) average 405 hours when the first took 720, then 405 ÷ 720 = 0.5625, and the rate is √0.5625 = 0.75, a 75% curve. For 8 units there are three doublings and the cube root is taken.
For other numbers use logs: b = log (average ÷ a) ÷ log x, and the rate = 2b.
Using the curve for costing and budgeting
The learning effect reduces costs that vary with labour time: direct labour and overheads that are absorbed on labour hours. It does not normally reduce material cost, because the amount of material needed has nothing to do with the speed of work.
To use the curve in a cost estimate:
- calculate the total labour hours for the batch (or for the units still to be made) from the formula
- multiply by the labour rate (and the variable overhead rate per hour)
- add the materials, which do not fall
Pricing on a cost-plus basis can allow for the lower cost of later units, and the sales budget then reflects lower prices.
In budgetary control, actual hours should be compared with hours expected allowing for learning, not with the time of the first unit. The conditions for the curve to apply, the steady state and the practical limits are the subject of the next topic.
Layouts for the learning curve
- Tabular: units; cumulative average (× rate at each doubling); total; incremental; average of the additional units.
- Formula: b = log rate ÷ log 2; y = a xb; total = x y.
- Single unit: total(n) − total(n − 1).
- Rate: (average ÷ a) = ratedoublings.
- Cost: hours × labour (and variable overhead) rate.
Hours to one decimal place; money in naira; each answer is rounded once, at the end.
Worked examples
Example 1: the tabular approach. The first of a new boat takes 800 hours and there is an 85% learning curve. Required: the table for 1, 2, 4, 8 and 16 boats.
- Cumulative average. 1: 800.
2: 800 × 0.85 = 680.
4: 578.
8: 491.3.
16: 417.605. - Total time. 1: 800. 2: 1,360. 4: 2,312. 8: 3,930.4. 16: 6,681.68.
- Incremental time. Boat 2: 560. Boats 3 and 4: 952 (average 476). Boats 5 to 8: 1,618.4 (average 404.6). Boats 9 to 16: 2,751.28 (average 343.9).
- Check. Each doubling lowers the cumulative average by 15%, but the average of the additional boats falls faster, because later boats are quicker than earlier ones.
Example 2: the formula, the rate and the cost. Using the 85% curve and a = 800: (a) the time for the tenth boat; (b) the learning rate if the first 8 boats of a different model average 307.2 hours and the first took 600; (c) the labour cost of the first 8 boats at ₦1,500 an hour, compared with no learning.
- b.
log 0.85 ÷ log 2 = −0.070581 ÷ 0.30103 = −0.2345. - Tenth boat.
For 10 boats: y = 800 × 10−0.2345 = 466.26 hours; total 4,662.6.
For 9 boats: y = 800 × 9−0.2345 = 477.92; total 4,301.3.
The tenth boat took 4,662.6 − 4,301.3 = 361.3 hours, much less than the first. - Rate. 8 units is three doublings.
307.2 ÷ 600 = 0.512; the cube root is 0.8, so an 80% curve. - Cost.
Total time for 8 boats = 3,930.4 hours (table)
at ₦1,500 = ₦5,895,600.
With no learning: 8 × 800 = 6,400 hours × 1,500 = ₦9,600,000.
Learning saves ₦3,704,400 of labour, which a price quoted on the first-unit time would fail to pass on to the customer.
Expect variants: a requirement to find the time of a later unit, finding the rate when the 8th or 16th unit is given, learning applied to the labour cost of a contract with a given number of units, or a question on the conditions needed for learning.
Common mistakes
- Applying the rate to the time of the last unit instead of to the cumulative average.
- Using 80 instead of 0.8 in the log.
- Taking y as the total time when it is the cumulative average.
- Finding the nth unit by subtracting averages, not totals.
- Applying learning to materials.
- Using the tabular approach for a number of units that is not a doubling.
Quick revision
- x% curve: cumulative average time × x% each time output doubles.
- y = axb; b = log rate ÷ log 2; total = xy; nth unit = total(n) − total(n − 1).
- Rate: average ÷ a = ratedoublings.
- Affects labour and labour-driven overhead, not materials.
How the marks are earned
- Method: the correct formula and b.
- Calculation: totals and differences, in hours then naira.
- Comment: use in budgeting and pricing.
Practice questions
These are original questions, written in the style of the Skills examination. Work each one before you open the answer.
Recall check: multiple choice
Question 1. The learning effect is:
- A rise in wages
- A fall in prices
- The improvement in efficiency as a task is repeated
- A new machine
Show the answer
C. Time per unit falls with experience.
Question 2. An 80% learning curve means that each time cumulative output doubles:
- Total time falls to 80%
- The cumulative average time per unit falls to 80% of its earlier value
- The last unit takes 80% of the time
- Cost rises by 80%
Show the answer
B. It is the cumulative average that falls.
Question 3. The tabular approach can show only:
- Doublings of output
- Any number of units
- Single units
- Materials
Show the answer
A. Use the formula for other numbers.
Question 4. The learning curve formula is:
- y = a + bx
- y = ab
- y = x/a
- y = axb
Show the answer
D. y is the cumulative average time.
Question 5. The learning factor b equals:
- log (learning rate) ÷ log 2
- The learning rate
- Log 2
- The first unit time
Show the answer
A. It is negative for rates below 100%.
Question 6. In y = axb, y is:
- The time for the last unit
- The total time
- The cumulative average time per unit for x units
- The first unit time
Show the answer
C. Multiply by x for the total.
Question 7. The time for the nth unit is:
- y
- Total for n units less total for n − 1 units
- y ÷ n
- Total ÷ 2
Show the answer
B. Subtract totals, not averages.
Question 8. The learning effect mainly reduces:
- Materials
- Rent
- Depreciation
- Labour and labour-driven overhead
Show the answer
D. Materials are unaffected.
Question 9. In the log formula an 80% rate is entered as:
- 0.8
- 80
- 8
- 800
Show the answer
A. As a decimal.
Question 10. With a 90% curve and a first unit of 100 hours, the cumulative average for the first 2 units is:
- 100 hours
- 95 hours
- 90 hours
- 80 hours
Show the answer
C. 100 × 0.9.
Section B style: written questions
Question 11 (10 marks). (a) Explain the learning effect. (3 marks) (b) Explain what a 90% learning curve means. (3 marks) (c) State why the tabular approach is limited. (2 marks) (d) State which costs are affected. (2 marks)
Question 12 (15 marks). The first unit of a product takes 400 hours and there is a 90% learning curve. Labour costs ₦1,200 an hour.
(a) Prepare the tabular schedule for 1, 2, 4 and 8 units: cumulative average, total time and incremental time. (8 marks) (b) Calculate the labour cost of units 5 to 8. (3 marks) (c) State FOUR conditions for a learning curve to apply. (4 marks)
Question 13 (20 marks). The first unit of a machine took 1,000 hours and an 80% learning curve applies.
(a) Calculate b. (3 marks) (b) Calculate the cumulative average and total time for 12 units. (5 marks) (c) Calculate the time for the twelfth unit. (6 marks) (d) The first four units of a different job averaged 640 hours each against 1,000 for the first. Calculate the learning rate. (3 marks) (e) State TWO limitations of the learning curve. (3 marks)
Model answers
Show the model answer to Question 11
(a) When a skilled, repetitive task starts, workers become quicker and make fewer mistakes with experience, so the time per unit falls until a steady state is reached (3).
(b) Each time cumulative output doubles, the cumulative average time per unit becomes 90% of what it was (3).
(c) It works only for doublings of output and cannot give the time of a single unit (2).
(d) Labour and overheads that vary with labour time; not materials (2).
Show the model answer to Question 12
(a) Cumulative average: 400; 360; 324; 291.6. Total time: 400; 720; 1,296; 2,332.8. Incremental: unit 2 320; units 3 and 4 576; units 5 to 8 1,036.8 (8).
(b) 1,036.8 × 1,200 = ₦1,244,160 (3).
(c) Any four: stable conditions with low labour turnover; labour-intensive repetitive work; no change in production method; short gaps between units; motivated employees (4).
Show the model answer to Question 13
(a) b = log 0.8 ÷ log 2 = −0.3219 (3).
(b)
- y = 1,000 × 12−0.3219 = 449.35 hours
- total = 12 × 449.35 = 5,392.2 hours (5).
(c)
- For 11 units: y = 1,000 × 11−0.3219 = 462.11; total 5,083.2.
- The twelfth unit took 5,392.2 − 5,083.2 = 308.9 hours (6).
(d)
- Four units is two doublings.
640 ÷ 1,000 = 0.64 - √0.64 = 0.8, an 80% curve (3).
(e) Any two: hard to measure the rate and first unit time accurately; automation removes learning; changes in design restart it; high labour turnover destroys it (3).
Pathfinder on Learning curve rate and effect
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
Learning curve concept, learning rate from quarterly shut-down costs, forecast year costs, steady state.
Revise first: Layouts for the learning curve
Learning rate from batch times, time for 400 more units, minimum price with opportunity cost of scarce labour.
Revise first: Layouts for the learning curve
Source: ICAN Pathfinder, Performance Management. Diets covered: March/July Diet, 2020 to May Diet, 2026. All 13 diets in the Pathfinder are mapped.
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