Management And Accounting Web
Provided by James R. Martin, Ph.D., CMA
Professor Emeritus, University of South Florida
Learning or Experience Curve Main Page
Contents
Introduction
Wright's Cumulative Average Model
Crawford's Incremental Unit Time (or Cost) Model
Using Wright's Model to find the Learning Rate
Using Crawford's Model to find the Learning Rate
Comparing the Learning Rates
Concluding Comment
Note and References
The theory of the learning curve or experience1 curve is basedon the simple idea that the time required to perform a task decreases as aworker gains experience. The basic concept is that the time, or cost, ofperforming a task (e.g., producing a unit of output) decreases at a constantrate as cumulative output doubles. Learning curves are useful for preparing costestimates, bidding on special orders, setting labor standards, scheduling laborrequirements, evaluating labor performance, and setting incentive wage rates.
There are two different learning curve models. The originalmodel was developed by T. P. Wright in 1936 and is referred to as the CumulativeAverage Model or Wright's Model. A second model was developed later by a team ofresearchers at Stanford. Their approach is referred to as the Incremental UnitTime (or Cost) Model or Crawford's Model. Simple learning curve problems aremore easily introduced with Wright's model, although Crawford's model is widelyused in practice. Thus, we will examine Wright's model first and Crawford'ssomewhat more involved approach second.
Wright's Cumulative Average Model
In Wright's Model, the learning curve function is defined as follows:
Y = aXb
where:
Y = the cumulative average time (or cost) per unit.
X = the cumulative number of units produced.
a = time (or cost) required to produce the first unit.
b = slope of the function when plotted on log-log paper.
= log of the learning rate/log of 2.
For an 80% learning curve b = log .8/log 2 = -.09691/.301 = -.32196
If the first unit required 100 hours, the equation would be:
Y = 100X-.322
The equation for cumulative total hours (or cost) is found bymultiplying both sides of the cumulative average equation by X.
Since X times Xb = X1+b, the equationis:XY = aX1+b
Thus, the equation for cumulative total labor hours is,
XY = 100X1-.322 = 100X.678
An 80 percent learning curve means that the cumulative averagetime (and cost) will decrease by 20 percent each time output doubles. In otherwords, the new cumulative average for the doubled quantity will be 80% of theprevious cumulative average before output is doubled. For example, assume thatdirect labor cost $20 per hour in the problem above. The cumulative averagehours and cost as well as cumulative total hours and cost are provided below fordoubled quantities 1 through 8.
| Table 1:Example of Wright's Model With an 80% Learning Curve | ||||
| 1 Cumulative Output X | 2 Cumulative Total Labor Hours XYh | 3 Cumulative Average Labor Hours Yh | 4 Cumulative Total Labor Cost XYc | 5 Cumulative Average Labor Cost Yc |
1 2 4 8 | 100 160 256 409.6 | 100 80 64 51.2 | $2,000 3,200 5,120 8,192 | $2,000 1,600 1,280 1,024 |
Note that the cumulative average columns, 3 and 5 decrease by20% as output is doubled, or the new cumulative average is 80% of the previouscumulative average. The cumulative total columns 2 and 4 increase at a rateequal to twice the learning rate, or 160% in this case. Since these rates ofchange remain constant, tables for doubled quantities can be developed easily.However, for quantities in between the doubled quantities, the equations arerequired. For example, assume the firm has produced eight units as indicated inthe table. How much will it cost to produce ten additional units? Any of theequations for Yh, Yc, XYh or XYc maybe used to solve the problem. However, working with the equation for cumulativetotal cost is the fastest way to obtain the solution. The answer is found bysubtracting the cost of the first 8 from the cost of producing the first 18.Using the equation for cumulative total cost generates the answer in two steps as follows:
Cost of first 18 = XYc= $2,000(18).678 = $14,194
Less cost of first8 -8,192
Cost of 10 additionalunits $6,002
Thus, producing 10 additional units will require approximately$6,002 of additional direct labor cost.
Crawford's Incremental Unit Time (or Cost) Model
The equation used in Crawford's model is as follows:
Y = aKb
where: Y = the incremental unit time (or cost) of the lotmidpoint unit.K= the algebraic midpoint of a specific production batch or lot.
X (i.e., the cumulative number of units produced) can be used in the equation instead of K to find the unit cost of any particular unit, but determining the unit cost of the last unit producedis not useful in determining the cost of a batch of units. The unit cost of eachunit in the batch would have to be determined separately. This is obviously nota practical way to solve for the cost of a batch that may involve hundreds, oreven thousands of units. A practical approach involves calculating the midpointof the lot. The unit cost of the midpoint unit is the average unit cost for thelot. Thus, the cost of the lot is found by calculating the cost of the midpointunit and then multiplying by the number of units in the lot.
Since the relationships are non linear, the algebraic midpointrequires solving the following equation:
K = [L(1+b)/(N21+b - N11+b)]-1/bwhere: K = the algebraic midpoint of the lot.
L= the number of units in the lot.
b= log of learning rate / log of 2
N1= the first unit in the lot minus 1/2.
N2= the last unit in the lot plus 1/2.
Once Yc is determined for the algebraic midpoint ofa lot, then the cost of the entire lot is found by multiplying Yc bythe number of units in the lot as indicated above.
An example of an 80 percent learning curve based on Crawford'sunit time (or cost) model can be developed in much the same way we developedTable 1, except that the unit values for the doubled quantities decrease by 20%rather than the cumulative average quantities.
| Table 2: Example of Crawford's Model with an 80% Learning Curve | ||||
| 1 Cumulative Output X | 2 Incremental Unit Labor Hours Yh | 3 Cumulative Total Labor Hours Kh (Yh) | 4 Incremental Unit Labor Cost Yc | 5 Cumulative Total Labor Cost Kc(Yc) |
1 2 4 8 | 100 80 64 51.2 | 100 180 314.2 534.6 | $2,000 1,600 6,284 1,024 | $2,000 3,600 6,284 10,692 |
Notice from Table 2 that the unit labor hours (column 2) andunit labor cost (column 4) decrease by 20% each time the cumulative output isdoubled. However, the cumulative total labor hours (column 3) and cumulativetotal labor cost (column 5) increase by a variable rate. This means that columns3 and 5 are much more difficult to develop. It also means that the cumulativetotal hours and cost generated by the two models are not compatible when basedon the same learning rate. For example, compare column 2 in Table 1 with column3 in Table 2. The cumulative total hours for 8 units is 409.6 based on Wright'smodel and 534.6 based on Crawford's model. Another difference is that thecumulative average hours and cost decrease by a variable rate in Crawford'smodel. This does not present a problem when using Crawford's model because thecumulative averages are not required for predicting cost.
To illustrate the use of the algebraic midpoint equation andCrawford's approach, assume that the firm in the example above has produced 2units and wants to determine the cost of producing 4 additional units. One wayto find the answer is to calculate the unit cost for each unit 3 through 6 andthen sum those values. That works reasonably well for a lot of 4 units, butwould not be a practical way to determine the cost of 40, 400, or 4,000additional units. The midpoint of the lot is:
K = [L(1+b)/(N21+b - N11+b)]-1/b= [4(.678)/(6.5.678 - 2.5.678)]1/.322
= [2.712/(3.55758 - 1.86124)]3.10559 = 4.2938
The cost of the mid point unit is:
Yc = $2,000(4.29385)-.322 = $1,250.99
and the total cost for the lot of 4 = 4(1,250.99) = $5,005
An alternative is to use the equation for hours as follows:
Yh = 100(4.29385)-.322 = 62.5494 hours
Then the total cost for the lot of 4 is 4(62.5494)($20) = $5,004.
Finding the Learning Rate When Doubled Quantities are not available
The equations provided above show how to use the learningcurve to predict the time and cost of a specific quantity of units assuming thatwe know the learning rate. An important question, ignored to this point, is howdo we find the learning rate in the first place? If we have data for two lots ofunits we can find the learning rate by using simultaneous equations. Forexample, assume two lots have been produced, one lot contained 2 units and asecond lot contained 4 more units.
| Lot | Number of Units in the Lot | Cumulative Units | Labor Hours for the Lot | Cumulative Labor Hours |
| 1 2 | 2 4 | 2 6 | 72 111 | 72 183 |
We can solve for the learning rate using either Wrights modelor Crawford's model, but the procedures and learning rates are different.
Using Wright's Model to find the Learning Rate:
The equations for the 2 lots are:
XY = aX1+b
72 = a(2)1+b
183 = a(6)1+b
Converting these to the log forms we have:
log 72 = log a + (1 + b)(log 2)
log 183 = log a + (1 + b)(log 6)
Calculating the log values indicated we have:
1.8575 = log a + (1 + b)(.301)
2.2625 = log a + (1 + b)(.7782)
1.8575 = log a + .301 + .301b
2.2625 = log a + .7782 + .7782b
Subtracting the first equation from the second equationprovides the following equation which can easily be solved for b.
.405 = .4772 + .4772b
b = -.151299
Substituting b into either of the original equations, a = 40.
Then the learning rate is found by using the equation for b,i.e.,
b = log of learning rate / log of 2
-.151 = Log of learning rate / .301
log of learning rate = -.151(.301) = -.04545
The learning rate = the antilog = 10-.04545 = .90
Thus, the equation for cumulative average hours is:
Y = 40X-.151
and the equation for cumulative total hours is:
XY = 40X.849
Using Crawford's Model to find the Learning Rate:
To find the learning rate using Crawford's model, we must findthe algebraic midpoint for each lot which is needed in the equations that mustbe solved simultaneously. We can't use the formula for K because it includes thevalue of b which is unknown. Thus, we must use the alternative midpoint formulasdescribed by Liao [see p. 309].
The midpoint of the first lot is:
A = [(L + 1)/3] + .5 = (2+1)/3 + .5 = 1.5
The midpoint of subsequent lots is:
A = (L/2) + total units in all preceding lots = 4/2 + 2 = 4
After finding approximate midpoints we can develop twoequations, one for each lot as follows:
Find the average hours for the midpoint units:
72/2 = 36 for the midpoint in lot 1.
(183 - 72)/4 = 27.75 for the midpoint in lot 2.
Then the equations are:
36 = a(1.5)b
27.75 = a(4)b
Converting to the log forms:
log 36 = log a + b(log 1.5)
log 27.75 = log a + b(log 4)
1.5563 = log a + b(.17609)
1.44326 = log a + b(.603)
Changing the signs in equation 2 and then adding the twoequations provides:
.11304 = -.4259b
b = - .2654
Then a is determined:
36 = a(1.5)-.2654
a = 40.09
The equation for incremental unit time is:
Y = 40.09X-.2654
The learning rate is found by using the equation for b asindicated above in the example for Wright's model.
b = log of learning rate/log of 2
-.2654 = log of learning rate/.301
Log of learning rate = (.301)(-.2654) = .079885
The learning rate = antilog .079885 = .83198.
Comparing the two learning rates we have .90 for Wright'smodel and .832 for Crawford's model. This reinforces the fact that the twomodels are not compatible when the same learning rate is used. In other words,the same set of data will always generate two different learning rates under thetwo separate models because unit time and cumulative average time do notdecrease at the same rate. The best model is the one that generates time andcost estimates that are closest to the actual results.
Learning curves range from around 70% to 100%. A learningcurve below 70% is rare. A 100% learning curve indicates no learning at all. Onthe other hand, a 50% learning curve would indicate that no additional time orcost would be required for additional units beyond the first unit, since thecumulative average time, (in Wright's model) or the incremental unit time (inCrawford's model) would decrease by 50% each time output doubled. This meansthat the cumulative total time would not increase because it would equal 100% ofthe previous cumulative total time.
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1 The term experience curve is more of a macro concept, while the term learning curve is a micro concept. Theterm experience curve relates to the total production, or the total output ofany function such as manufacturing, marketing, or distribution. The developmentof experience curves is attributed to the work of Bruce Henderson of the BostonConsulting Group around 1960.
Liao,S. S. 1988. The learning curve: Wright's model vs. Crawford's model. Issues in Accounting Education (Fall): 302-315.
Morse, W. J. 1972. Reportingproduction costs that follow the learning curve phenomenon. TheAccounting Review (October): 761-773.
* For more information on the learning curve models, see the LearningCurve Bibliography.
