P2 – Advanced Management Accounting CH4 – Learning curves

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Chapter 4 Learning curves Chapter learning objectives:

Lead Component Indicative syllabus content

A.1 Evaluate techniques for analysing and managing costs for competitive advantage.

(d) Apply learning curves to estimate time and cost for activities, products and services.

• Learning curves and their use in predicting product/service costs, including derivation of the learning rate and the learning index.

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1. Introduction to learning curves A learning curve is the mathematical expression of the commonly observed effect that, as complex and labour-intensive procedures are repeated, unit labour times tend to decrease.

• The learning process starts from the point when the first unit comes off the production line.

• From then on, each time cumulative production is doubled, the cumulative average time per unit is a fixed percentage of its previous level.

Wrights Law Wrights law states that, as cumulative output doubles, the cumulative average time per unit falls to a fixed percentage (the learning rate) of the previous average time.

Illustration 1 A 90% learning curve means that each time cumulative output doubles, the cumulative average time per unit falls to 90% of its previous value.

The concept of logarithms • We need to understand the concept of logarithms because they appear in the

definition of “b”, i.e. “the learning coefficient”.

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• Logarithms are related to powers, i.e. 103 = 1,000

103 = 10 × 10 × 10

If we need to know n given that 10n = 1,000, we know x = n = 3

Let’s try something more difficult:

What if the equation were 10n = 2.5?

The log function can be used to rearrange an equation containing powers.

10n = 2.5 is equivalent to n = log102.5 = 0.39794

This is read as “log 2.5 to base 10”.

Note:

A calculator will provide the logarithm of any number if the button marked “log” or “log 10x” is pressed.

For example, using a calculator, the log of 72 equals 1.8573, which means 101.8573 = 72.

• Please remember the following rules:

o log(x*y) = log x + log y

o log xy = y log x

• Logarithms are useful when you need to derive non-linear functions y = axn.

• In costing and learning curve theory, this will be useful when we look at deriving the learning rate.

• The logarithms of y and the logarithms of axn must be the same and so y = log a + n log x. This gives a linear function similar to y = a + nx, the only difference being that in place of x and y, the logarithms of x and y must be used.

• Values for n and for log a can be deduced using simultaneous equations. The value of a log a can be translated into a useful number using “antilogs”, i.e. the 10x button on the calculator.

Illustration 2 Suppose the relationship between x and y can be described by the function y = axn. The following applies:

If x = 500, y = 50,250

If x = 1,000, y = 42,750

Substitute these values into log y = log a + n log x and we find that:

Equation 1: 50,250 = a × 500n

log 52250 = log a + n log 500

4.7011 = log a + 2.69897n

Equation 2: 42,750 = a × 1000n

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log 42750 = log a + n log 1000

4.6309 = log a + 3n

Equation 1 – Equation2

0.0702 = -0.30103n

n = -0.2332

Replacing n with -0.2332 in Equation 1 we find:

4.7011 = log a + 2.69897 × (-0.2332)

log a = 5.3305

a = 105.3305

a = 214,042 and our function is y = 214,042x-0.2332

Test Your Understanding 1 - Logarithms It will take 100 hours to make the first unit of a new product. A 90% learning curve applies.

Complete the following table:

Cumulative Incremental

Units Average time per unit Total time Units Total time Average time

per unit

Test Your Understanding 2 – 80% learning curve The first unit of a product is expected to take 100 hours to make. An 80% learning curve is expected to apply.

a) Calculate the average time per unit for the first 18 units. b) Calculate the time it takes to make the 20th unit.

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2. The steady rate Eventually, the learning curve effect will cease, and the time to make each successive unit will stabilise at a constant time per unit. This is because there is a limit to manual dexterity and/or other limiting factors come into play such as a limit on how quickly materials can be supplied.

3. Learning curve and pricing The initial cost estimate for a new product may be very high, but if costs fall along a learning curve, the company may be able to sell at a lower and hence more competitive price.

Test Your Understanding 3 – Average cost per unit It is estimated that the cost of the first unit of a new product is $850, but an 80% learning curve is expected to apply. It is estimated that the company will make and sell 2,000 units during the first year.

Calculate the average cost per unit for the first 2,000 units.

4. General conditions for a learning curve to apply The following conditions are pre-requisites for a learning curve to apply:

• The activity should be labour-intensive.

• The process should be repetitive for each unit.

• Labour turnover should be low, and there should be no prolonged breaks in production.

5. Experience curves • These are similar to learning curves.

• They cover all costs, not just labour costs.

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Experience curves cover the following Material costs

• Material costs may decrease slightly with quantity discounts, but will not decrease by large amounts.

Variable overheads

• Variable overheads often follow the pattern of direct labour and so may decrease in a similar way

Fixed overheads

• Fixed overheads will decrease per unit as more units are made.

6. Solutions to Test Your Understanding questions

Test Your Understanding 1 - Logarithms Cumulative Incremental

Units Average time per unit Units Average time per unit Units Average time per unit

1 100 100 1 100 100

2 90 180 1 80 80

4 81 324 2 144 72

8 72.9 583.2 4 259.2 64.8

16 65.62 1049.92 8 466.72 58.34

Test Your Understanding 2 – 80% learning curve a) Solution

a = 100

b = !"# !"#$%&'( !"#$% !"#$!"# !

b = !"# !.!!"# !

b = !!.!"#"$!.!"#"!

b = - 0.3219

x =18

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y = 100. 18-0.3219

y = 100 × 0.39436

y = 39.43 hours b) Solution

We will find the value for the 20th unit by finding the total time for 20 units and then subtracting the time for the first 19 units.

X = 20

Y = 100. 20-0.3219

Y = 38.12 hours

Total time for 20 units = 38.12 hours × 20 units

Total time for 20 units = 762.48 hours

X = 19

Y = 100. 19-0.3219

Y = 38.76 hours

Total time for 19 units = 38.76 hours × 19 units

Total time for 19 units = 736.44 hours

Time it takes to make the 20th unit = 762.48 hours – 736.44 hours

Time it takes to make the 20th unit = 26.04 hours

Test Your Understanding 3 – Average cost per unit The average cost per unit for the first 2,000 units can be calculated as follows:

For an 80% learning curve = -0.3219 which can be achieved by (!"# !.!!"# !

)

Y = $850 × 2,000-0.3219

Y = $73.59 per unit This means that the selling price can be set at far below the cost of the first unit, making the company more competitive.

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7. Chapter summary