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Dynamics of Learning by Doing under C onstraints:
Analysis of the Tipping Point
(Extended A bstract)
J. Bradley Morrison
MIT Sloan School of Management
The central notion in learning curve theory is that accumulating experience leads to
improved performance, or “leaming by doing.” The concept occupies a central role in
many strands of strategy and organization theory and forms the basis for such ideas as the
specialization of labor, organizational learning, knowledge transfer, and core
competences of the firm. Traditional learning curve theory considers the productive
activity of interest, such as the manufacture of airframes, in isolation from other demands
for critical resource inputs, such as direct labor hours. In contrast, many learning
situations are characterized by a competition for the learner’ s time between a new skill to
be learned and an old, proven means of accomplishing tasks. The learner's time is a
limited resource. The learner faces the challenge of allocating this resource to meet the
demand for certain output objectives while simultaneously trying to learn how to do
things a new and possibly better way.
The purpose of this paper is to extend learning curve theory by embedding the leamer in
a context in which he or she must achieve certain output objectives. Building on the
basic notion that accumulating experience leads to improved productivity, I formulate
and analyze a two-loop system dynamics model that incorporates a constraint to achieve
a specified level of output as well as the forgetting or deterioration of knowledge. I use
simulation analysis to demonstrate a mode of behavior in which leaming begins and then
stalls and another mode in which learning dominates so that the new skill becomes the
preferred manner of doing. I characterize the tipping point that distinguishes these two
modes. The discussion highlights key implications for managing implementation efforts
and organizational change.
A Model of Learning by Doing under C onstraints
The leamer has two choices for how to do the work that will achieve the productive
output of interest - an old way and a new way. As the leamer uses the new method, the
experience accumulated leads to increases in the productivity of this new method.
However, the leamer has mastered the old way, so productivity using the old, proven
method is high. At the outset, working in the new way requires considerably more time
to accomplish the same quantity of output. (The present analysis assumes that both
methods lead to similar or acceptable quality.) Since the learner wants to learn the new
way, all else equal, he or she will choose the new method in order to gain experience.
There are two complications. First, the learner must achieve a set rate of output - so all
else is not equal. Second, experience accumulated using the new way has a limited
useful life. If the new skill is not used, the experience will atrophy.
Figure 1 shows a stock and flow diagram of a model of learning by doing. The time
spent with the new skill leads to learning that accumulates in a stock of experience. The
stock is increased by a flow of learning and decreased by forgetting. Accumulating
experience increases the relative productivity with the new skill which in tum leads to
more time spent with the new skill and thus more learning, forming the Learning by
Doing Loop, reinforcing loop R. Accumulated experience with the new skill also
atrophies, as shown in the Forgetting Loop, balancing loop B.
Figure 1: Feedback Structure of Learning by Doing
Initial
Relative Pdy
Strength of
Learning Effect
+ J, Initial
bo experience
sri Relative
Productivity Productivity
with Old Skill with New Skill
Indicated
aa Rate
Time Spent ad By
‘ Mt New Skill Doing Loop
Time
a
=e | Experience
Learning Qs with New Skill Forgating
AS
@ ‘Time to
Forget
Forgetting
Loop
The complete model structure is described here.
4
Expt = fle —Fr)dt + Expo
Ly =TSNS;
F, =Exp;/ TF
where
Exp = Experience with New Skill (hours)
L=Leaming (hours/week)
F = Forgetting (hours/week)
TSNS = Time Spent with New Skill (hours/week)
TF = Time to Forget (weeks)
wo
Experience with New Skill is the accumulation of Learning less Forgetting. Learning is
based on the Time Spent with New Skill and is represented as a separate variable here
only for convenience in the diagram. Experience accumulates as hours of time spent with
the new skill. Forgetting is modeled as a loss from the stock of experience at a constant
fractional rate, as given by the Time to Forget.
TSNS; =max(0, min(TA, ITSNS,)
TA«PdyOld —ICR
ITSNSt =
*~ PdyOld « (I RPdyNew, )
where
TTSNS = Indicated Time Spent with New Skill (hours/week)
TA=Time Available (hours/week)
PdyOld = Productivity with Old Skill (widgets/hour)
ICR = Indicated Completion Rate (widgets/week)
RPdyNew = Relative Productivity with New skill (dimensionless)
The Time Spent with the New Skill is the Indicated Time Spent with the New Skill,
constrained to be not less than zero or more than the total time available. The Indicated
Time Spent with the New Skill is based on the learner's resource allocation policy. The
learner chooses to use all available time for learning the new skill consistent with the
need to achieve a set output objective, given by the Indicated Completion Rate. The
learner's allocation of time to the new skill must satisfy two equations:
TSOS*PdyOld + ITSNS*PdyOld*RPdyNew =ICR
TSOS +TSNS =TA
The first equation forces the total output to equal the Indicated Completion Rate. Total
output is achieved as the sum of output from the old way and output from the new way.
Output from the old way is the product of time spent with the old skill (TSOS) and the
productivity of that time (PdyOld). Similarly, output from the new way is the product of
time spent with the new skill (ITSNS) and the productivity of that time
(PdyOld*RPdyNew). The second equation assures that time spent with the old skill and
time spent with the new skill sum to the total time available. Solving these two equations
for ITSNS yields the allocation policy for the learner's time, as shown above.
RPdyNew; =max(0, min(0.99, IRPdyNew;)
IRPdyNew, =RelPdyp + SLE*( Exp;/ Expo - 1)
where
IRPdyNew = Indicated Relative Productivity with New skill (dimensionless)
RelPdyo = Initial Relative Productivity with new skill (dimensionless)
SLE = Strength of Leaming Effect (dimensionless)
Exp = Initial Experience with New Skill (hours)
The Relative Productivity with the New Skill is the ratio of the productivity with the new
skill to the productivity with the old skill and increases with the leaming effect based on
accumulated experience. Here it is assumed that productivity of the new skill will be
greater than zero and less than (99% of) that of the old skill during the learning phase of
interest here. The Indicated Relative Productivity with New skill is a straight line with a
slope equal to SLE/Expp and an intercept equal to (RelPdyo- SLE). This linear
formulation may be interpreted as a first-order approximation of a small portion of a
learning curve formulated as a power function or an exponential function.
To fully specify the model, the following parameter values are used:
TF = 12 weeks
PdyOld = 1 widget/hour
ICR= 30 widgets/week
TA = (ICR/PdyOld)/(1+IFNS*(IRPdy-1))(hours/week)
RelPdyo = 0.5
SLE = 0.25
Expy = TF * TA* IF (hours)
IF = Initial Fraction of time spent with new skill = 0.3 (dimensionless)
Model Behavior:
In this section, I use simulation analysis to explore what it takes for the learner to develop
a lasting proficiency with the new skill.
The simulations begin with a learner that has a small amount of experience with the new
skill. The leamer is exactly accomplishing the indicated completion rate and is learning
at exactly the rate necessary to offset the forgetting that is occurring. That is, the leaner
is spending exactly the amount of time with the new skill needed to maintain a constant
level of experience, and thus constant productivity. To give the leamer an opportunity to
learn more, the first test introduces extra time available. The hope is that the additional
time available will set in motion the reinforcing Learning by Doing Loop (R in Figure 1).
If the learner is endowed with additional time available with no increase in the
throughput objective, he will have slack capacity. He should allocate more time to
working with the new skill, thus stimulating leaming, which in turn will build experience
and increase productivity, leading to further slack capacity and thus still more time
allocated to working with the new skill. This virtuous cycle of learning will fuel the
increase in experience needed to master the new skill.
Figure 2 shows the results of a test in which a pulse of extra time (120 hours) is added in
week 10. The amount of experience (and thus also productivity) increases immediately
Figure 2: Simulating the Learner's Response
Response to a Pulse of Extra Time Available
300
225
150
Experience
(hours) Why
0 20 40 60 80 100 120 140
Time (Week)
Experience with New Skil : pulsetime hours
Experience with New Skill : initialequil hours
as expected. However, after this initial increase, experience degrades slowly from the
peak achieved at the time of the pulse back to the original levels. The addition of extra
time had led to an improvement, but this improvement is only temporary.
As seen in Figure 2, experience with the new skill begins a decline from the original peak
at an increasing rate from week 10 until approximately week 46. During this period, the
reinforcing loop R is dominating the behavior, working as a vicious cycle against the goal
of increasing leaming. Then, from week 46 onwards, experience continues to decline
toward the original level but now at a decreasing rate of decline. During this period, the
Forgetting Loop, balancing loop B in Figure 1, is dominating the behavior, guiding the
system back to its original equilibrium conditions.
Figure 3 shows the results from introducing pulses of various sizes to the stock of
experience. This set of simulations shows that pulses of various sizes lead to three
different terminals values for the level of experience. First, for some pulse sizes such as
the lowest four simulation runs in Figure 3, experience temporarily improves but then
slowly decreases over time back to the original level, a pattern identical to the test done
in Figure 2. Second, for one unique pulse size (the blue line in Figure 3), the pulse
immediately sets experience to a new level that stays constant thereafter. Third, for some
higher pulse sizes such as the highest five shown in Figure 3, a third outcome is reached
in which the learner accumulates experience to higher levels, eventually reaching a level
which is maintained thereafter. In these simulation runs, the system has passed a critical
threshold and entered into a regime in which the new skill is sustained at a permanently
higher level. That threshold is known as a tipping point. The one unique pulse (in the
blue line) brings the stock of experience just exactly to the tipping point. With any
greater amount of experience, the feedback structure brings the system to a new, higher
Figure 3: Response to Various Pulses of Experience
Response to Pulses of Various Sizes
600
300 | 7 ; j
Experience
(hours)
0 20 40 60 80 100 120 140
Time (Week)
equilibrium level of experience. With any smaller amount of experience, the system
retums to its original level of experience.
Model Analysis:
Under what conditions will the learner transition to and sustain the higher level of
experience with the new skill? The learner will do so when the level of experience
becomes great enough so that the reinforcing Learning by Doing Loop is working in a
favorable direction and dominating the Forgetting Loop.
Figure 4 shows a rate-level plot constructed by examining how the rates of flow in the
model depend on the stock of Experience with the New Skill. The plot shows the flow of
Learning as a function of the level of Experience. Even at very low levels of experience,
learning is positive (because the learner’s throughput goals are low enough that he can
afford to spend some time learning even if productivity is very low). The plot for
learning shows that as experience increases, learning increases. Learning increases at an
increasing rate because there are two benefits to increasing experience. First, increasing
experience increases the productivity of doing the work with the new skill to achieve the
required throughput. Second, as productivity with the new skill increases, the
opportunity cost of using the new skill decreases. Learning eventually reaches a
maximum when all of the allowed time is allocated to using the new skill, as seen in the
flat portion of the curve. Figure 4 also shows the flow of Forgetting. There is no
Forgetting when there is no experience, so the curve starts at the origin. For any positive
quantity of experience, forgetting occurs at a constant fractional rate. The curve is thus a
Figure 4: Rate-Level Plot of Learning by Doing
40
Learning ~,
20
Net Change
In
Experience 9
Net Change 7
-20
Forgetting
-40
0 50 100 150 200 250 300 350 400 450 500
Experience
straight line with a downward slope equal to the reciprocal of the time constant, Time to
Forget.
The net change in the level of experience as a function of the level of experience is the
difference between the inflow Learning and the outflow Forgetting. As seen in Figure 4,
the Net Change in Experience curve is the sum of the other two curves.
Figure 5 reproduces the Net Change in Experience curve from Figure 4 on an expanded
scale. The Net Change curve crosses the zero line at three different points. Each of these
points represents a level of experience at which the inflow from learning is exactly equal
to the outflow from forgetting and are thus levels at which the stock of experience is in
equilibrium. The arrows show the trajectory of the system in disequilibrium conditions,
at all other levels of experience. The leftmost equilibrium in Figure 5 is the starting point
for the simulations. This equilibrium is formally denoted a stable equilibrium, meaning
that small perturbations from it are compensated for by the system’s dynamics. The
balancing forgetting loop dominates behavior in this region, bringing stability to the
system. If experience drops below the equilibrium level (a shift to the left in the figure),
forgetting slows a bit so the net change is positive and the equilibrium level is restored.
If experience increases a little (moving right in the figure), forgetting speeds up so net
change is negative, again returning the level of experience to its original equilibrium
value. This equilibrium occurs where the rate-level plot crosses the zero axis with a
downward slope, thus characterizing a stable equilibrium. Similarly, the rightmost black
dot in Figure 5 labels another stable equilibrium. (The downward sloping region in the
right portion of the curve arises from the fixed constraint on the maximum amount of
time available to learn. In this region, the learner is spending all allowable time using the
new skill. If this constraint were removed, the curve would continue unbounded up and
Figure 5: The Tipping Point on the Rate-Level Plot of Learning by Doing
10
Unstable
Equilibrium
\
5
Net Change
In
Experience °
Stable /
5 Equilibrium Cert ne
-10
0 50 100 «6150 §=6200 §=6250 »= 300 3S 350 2 400 )= 450 = 500
Experience
to the right.)
In contrast, the third equilibrium, designated with a green dot, is in a region where the
rate-level plot crosses the zero axis in an upward sloping direction. The reinforcing loop
dominates behavior in this region, so the equilibrium is unstable. As the arrows show,
small perturbations away from the equilibrium are amplified, sending the system off
towards one of the other two equilibria. The unstable equilibrium point here is also the
tipping point, and the rate-level plot explains why. At all levels of experience to the left
of the tipping point, the system will be drawn to the leftmost stable equilibrium. Once
experience accumulates enough to move just rightward of the tipping point, the system
transitions onto a path towards the rightmost equilibrium.
The rate-level plot can now be used to restate the leamer’s goal. For the leamer wishing
to develop a sustained proficiency with new skill, the goal is to build enough experience
to get just past the tipping point. Although stability is often a desirable characteristic of
systems, it is the inherent instability in a critical region of this system's state space that
creates the opportunity for enduring change. In the learner’s case, reaching the unstable
equilibrium is the key to the transition to a sustained level of higher proficiency with the
new skill.
How much experience is enough to cross the tipping point? In the stylized model
presented here where all parameter values are known, an analytical solution is possible.
The tipping point is an equilibrium, so it must satisfy the condition that the inflows to the
stock of experience equal the outflows. Setting the flow equation for learning equal to
the flow equation for forgetting, substituting to remove the auxiliary variables, and
solving for Experience with New Skill yields a quadratic equation. The two roots of this
equation are the leftmost stable equilibrium in Figure 5 and the unstable equilibrium. The
first is the equilibrium for the initial conditions of the simulations, and the second is the
tipping point. The tipping point occurs when:
ICR
4*|TA-_—* _
“elle ( |
SLE *TA«IF
Expt — Exp, * 5
For the parameter values used in the simulations shown here, the quantity in brackets is
equal to two. Thus, to reach the tipping point, the learner must accumulate enough time
spent with the new skill to double his experience.
The equation for the tipping point is useful to develop an analytical understanding of the
system’ s properties, but its usefulness in practice will be limited by the degree to which
the parameter values are known or can be estimated with reasonable accuracy.
Sensitivity analyses that examining the effect of changes in various parameters can help
build understanding of system performance. For example, Figure 6 displays the rate-
level plots obtained using five different values for the Strength of Learning Effect (the
slope of the effect of experience on productivity). With no learning (Strength = 0, as in
the green line), forgetting dominates, the rate-level curve is always downward sloping,
and there is no tipping point. For the strong learning curve (Strength = 0.5) shown in the
brown line, any increase from the initial equilibrium sends the system towards the higher
equilibrium point. For moderate learning effects as in the other lines, the tipping point
moves to the left as the Strength of Learning Effect increases. Efforts to increase the
strength of this learning effect will bring the tipping point closer to the initial level of
experience.
Figure 6: Sensitivity Analysis: Rate-Level Plots by Strength of Learning Effect
Rate Level Plots for Various Effect Strengths
20
5
-10
-25
-40
0 50 100 150 200 250 300 350 400 450 500
Experience
Net Change : base hours/Week
zero : base hours/W eek
Net Change : le0 hours/Week
Net Change : le15 hours/Week
Net Change : 1e30 hours/Week
Net Change : 1e50 hours/Week
Discussion:
The paper has developed a system dynamics model based on extensions of learning curve
theory that incorporates a learner’s need to accomplish ongoing work while also meeting
the challenge of learning new skills. Simulations demonstrated that some attempts to
learn will be short-lived, while others will move the learner into a regime of sustained
higher proficiency. By characterizing the tipping point that distinguishes these two
modes of behavior, the preceding analysis has added to our understanding of the
dynamics of change.
A more effective approach to managing performance in systems with such tipping
potential may be to identify symptoms of system behavior that signal being near or past
the tipping point. The nature of unstable equilibria is such that systems rarely, if ever,
operate at or even very close to them. Learning about how to effectively manage
performance will thus be quite challenging. Learners may come quite close to a tipping
point and never realize how close they were to “getting over the hump.” More
insidiously, in the region of experience just below the tipping point, the system behavior
is dominated by a reinforcing loop acting as a vicious cycle to keep the leaner away from
the tipping point. Operating in this region is likely to feel much like swimming upstream.
The learner who perseveres to get past the critical threshold will then gain the benefit of a
reinforcing loop acting in his favor, as though the current were almost pulling him a long.
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