Cost Accounting (14th Edition)

360 CHAPTER 10 DETERMINING HOW COSTS BEHAVE
Incremental Unit-Time Learning Model
In the incremental unit-time learning model, incremental time needed to produce the last
unit declines by a constant percentage each time the cumulative quantity of units produced
doubles. Again, consider Rayburn Corporation and an 80% learning curve. The
80% here means that when the quantity of units produced is doubled from X to 2X, the
time needed to produce the last unit when 2X total units are produced is 80% of the time
needed to produce the last unit when X total units are produced. Exhibit 10-11 is an Excel
spreadsheet showing the calculations for the incremental unit-time learning model for
Rayburn Corporation based on an 80% learning curve. Note how when units produced
double from 2 to 4 in column A, the time to produce unit 4 (the last unit when 4 units are
produced) is 64 hours in column B, which is 80% of the 80 hours needed to produce unit 2
(the last unit when 2 units are produced). We obtain the cumulative total time in column
D by summing individual unit times in column B. For example, to produce 4 cumulative
units would require 314.21 labor-hours (100.00 + 80.00 + 70.21 + 64.00).
Exhibit 10-12 presents graphs using Excel for the cumulative average-time learning
model (using data from Exhibit 10-10) and the incremental unit-time learning model
(using data from Exhibit 10-11). Panel A graphically illustrates cumulative average time
per unit as a function of cumulative units produced for each model (column A in
Exhibit 10-10 or 10-11). The curve for the cumulative average-time learning model is
plotted using the data from Exhibit 10-10, column B, while the curve for the incremental
unit-time learning model is plotted using the data from Exhibit 10-11, column E. Panel B
graphically illustrates cumulative total labor-hours, again as a function of cumulative
units produced for each model. The curve for the cumulative average-time learning model
is plotted using the data from Exhibit 10-10, column D, while that for the incremental
unit-time learning model is plotted using the data from Exhibit 10-11, column D.
Exhibit 10-11
Incremental Unit-Time Learning Model for Rayburn Corporation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
A B C D E F G H I
Incremental Unit-Time Learning Model for Rayburn Corporation
80% Learning Curve
Cumulative Individual Unit Time Cumulative Cumulative
Number for Xth Unit (y )*: Total Time: Average Time
of Units (X ) Labor-Hours
Labor-Hours per Unit:
Labor-Hours
E = Col D ÷ Col A
1 100.00
100.00
100.00
2 80.00 = (100 × 0.8) 180.00 90.00
3 70.21
250.21
83.40
4 64.00 = (80 × 0.8) 314.21 78.55
5 59.56
373.77
74.75
6 56.17
429.94
71.66
7 53.45
483.39
69.06
8 51.20 = (64 × 0.8) 534.59 66.82
9 49.29
583.89
64.88
10 47.65
631.54
63.15
11 46.21
677.75
61.61
12 44.93
722.68
60.22
13 43.79
766.47
58.96
14 42.76
809.23
57.80
15 41.82
851.05
56.74
16 40.96 = (51.2 × 0.8) 892.01 55.75
D14 = D13 + B14
= 180.00 + 70.21
*The mathematical relationship underlying the incremental unit-time
learning model is as follows:
y = aX b
where y = Time (labor-hours) taken to produce the last single unit
X = Cumulative number of units produced
a = Time (labor-hours) required to produce the first unit
b = Factor used to calculate incremental unit time to produce units
ln (learning-curve % in decimal form)
=
ln2
For an 80% learning curve, b = ln 0.8 ÷ ln 2 = –0.2231 ÷ 0.6931 = –0.3219
For example, when X = 3, a = 100, b = –0.3219,
y = 100 × 3 –0.3219 = 70.21 labor-hours
The cumulative total time when X = 3 is 100 + 80 + 70.21 = 250.21 labor-hours.
Numbers in the table may not be exact because of rounding.