What happens when we fall off of the saddle path?
A Calibrated Simulation with Forecasts Using @RISK and Evolver
“Well it’s a matter of continuity. Most people’s lives have ups and downs that are gradual, a
sinuous curve with first derivatives at every point. They’re the one who never get struck by
lightning. No real idea of cataclysm at all. But the ones who do get hit experience a singular
point, a discontinuity in the curve of life – do you know what the time rate of change is at the
cusp? Infinity, that’s what! And right across the point it’s minus infinity! How’s that for sudden
change, eh?” Thomas Pynchon, Gravity’s Rainbow, 1973.
“Linearity is a trap. The behavior of linear equations – like that of choirboys – is far from
typical. But if you decide that only linear equations are worth thinking about, self-censorship
sets in. Your textbooks fill with triumphs of linear analysis, its failure buried so deep that the
graves go unmarked and the existence of the graves goes unremarked. As the 18th century
believed in a clockwork world, so did the 20th in a linear one.” Ian Steward, Does God Play
Dice? The Mathematics of Chaos, 1989.
Dr. William Strauss, FutureMetrics
Capitalism - An economic system in which the means of production
and distribution are privately or corporately owned and development
is proportionate to the accumulation and reinvestment of profits
gained in a free market.
Inside job - A crime perpetrated by, or with the help of, a person
working for or trusted by the victim.
(Both from the American Heritage Dictionary, 2008)
What if we were to have 100 years of no growth?
What if conditions are such that there is no future
scenario under which growth will ever occur again?
We might characterize this as impossible, as a vision
that violates the outcome that we must realize as
innovating people.
In the presentation that follows I will show you our
world as it must be sometime in the future.
We will look briefly at the foundations of economic
growth theory and then we will use a simulation to
look at what the future might be.
Importantly (and unique to this research), this story will
be told from within the boundaries of modern economic
growth theory.
That is, rather than follow an ecological and/or
geographical path to explore limits to growth, this
research is an “inside job” that shows that when modern
growth theories are decoupled from assumptions that
have no basis in how the real world is developing but are,
for the most part, mathematical conveniences applied for
the sake of “stability”, then the long-run economic
outcome is not what it is supposed to be!
Many academics have accepted the significance of the reality
of dynamic discontinuity.
But dominant decision makers, connected to the politics of a
system that relies on increasingly complex systems to
maintain the orthodox expectation for an endlessly growing
standard of living, continue to assume that changes today
will lead to predictable and/or reversible outcomes.
This is a myth.
When reasons are based upon a flawed foundation,
bad choices can appear reasonable.
Perceptions, the conduit for knowledge, can be
denied or altered by dependence. Since capitalism
depends upon growth, this leads to a form of
addiction. One of the undisputed facts about
addiction is that it is a source of perceptual distortion.
Our growth imperative is thus complicated by a feedback loop that not only compounds the problem but
also provides all of us with an invisible veil of denial
that allows us to delink action and consequence.
Ponzi Scheme?
“In the eighty years or so after 1780 the population of Britain nearly tripled… the
average income of the population more than doubled… So strange were these events
that before they happened they were not anticipated, and while they were happening
they were not comprehended. “
Donald McCloskey, “The Industrial Revolution 1780-1860: A Survey”, 1981
Economic growth has
two pillars: population
growth and improving
productivity/technology.
Source: Clark, Gregory, A
Farewell to Alms: A Brief
Economic History of the World,
Princeton University Press,
2007.
Population growth must eventually stabilize… Japan is a current example
Source: Population in Japan; Statistics Bureau, Japan,
2007
So technology will keep us growing…
At the limit, what is the logical outcome?
Exponential growth!
x = 0.02
α = 0.33
n = 0.01
s = 0.1
Per Capita
y
c
x is technological
progress
δ = 0.05
70
60
α is capital intensity
50
40
n is population growth
30
20
s is savings
10
0
0
50
100
150
200
250
δ is depreciation
Units of Time
At what level of per capita consumption are we satiated?
!
Consumption (per person!) that is maybe a
bit more that 500 times greater than today?
x=0
α = 0.33
n= 0
s = 0.1
Per Capita
y
If growth flattens it is not
good news for capitalism.
c
δ = 0.05
700
At zero growth the return on
investment goes to zero!
600
500
400
ROI
300
ROI
9.00%
200
8.00%
100
7.00%
0
6.00%
0
50
100
150
Units of Time
200
250
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
0
50
100
150
200
250
The stock of trademarks is increasing exponentially. The fitted time series
forecast shows that by 2020 the stock of active trademarks could be more
than 3 times greater than in 2000. This suggests an exponentially increasing
array of goods that has helped support the growth of the general economy.
This non linearity is not considered in endogenous growth theory.
200,000
2,500,000
180,000
R² = 0.9976
Registered - Left Scale
Renewed
160,000
2,000,000
Stock - Right Scale
140,000
Time Series Forecast
120,000
1,500,000
100,000
80,000
1,000,000
60,000
40,000
500,000
20,000
2015
2010
2005
2000
1995
1990
1985
1980
1975
1970
1965
1960
1955
1950
1945
1940
1935
1930
1925
1920
1915
1910
1905
1900
1895
1890
1885
1880
1875
0
1870
0
Source: For period 1891 to 1970 the data on registered trademarks is taken from Historical Statistics of the United States: Colonial Times to 1970 (Series W 107 and W
108). These series are updated using data from the United States Patent and Trademark Office, US Department of Commerce, Annual Reports. The stock of trademarks
is computed based on methodology used by Greenwood and Uysal (2005).
Arthur C. Clarke’s third law of technology: “Any
sufficiently advanced technology is
indistinguishable from magic” (based on his work
from 1962).
As it turns out, that is prophetic with respect to
how growth theory has used technology to
provide for unending growth in per capita
measures of well-being.
What about BETTER QUALITY goods through innovation?
Increasing cost of innovation!
Ratio of R&D to Output for all Industries
6.00
R² = 0.9725
5.00
4.00
3.00
2.00
1.00
0.00
Source: BEA R&D satellite accounts, 2008
And decreasing “bang for the buck”! Another non linear input that really
messes up the mathematics of equilibrium and saddle path stability.
Source: BEA R&D satellite accounts, 2008
Using @RISK’s distribution fitting feature, we see that the first distribution
is normal and 80.0% of the data showed a net positive relationship between
the growth of R&D and the growth of output (that is, a given growth in R&D
generated a larger growth in output; a ratio of less than one). The second
distribution shows that R&D to Output is now greater than one.
What I have done so far is give a very brief overview on how assumptions
that are made in the name of making models congruent with the so-called
stylized facts of economic growth (those assumptions also make them
mathematically stable) are often very much ad-hoc or, worse, fly in the face of
recent data.
The second part of this presentation uses a computer simulation that is
completely self generating (completely endogenous). It is built to replicate
history using a complex non-linear set of relationships that do not depend on a
priori assumptions. In some forward looking cases key parameters are allowed
to vary using @RISK to drive that part of the simulation. Later, Evolver is used
to fit the future onto a sustainable path!
That simulation shows how the current path has no happy ending unless critical
assumptions that we now regard as axiomatic are altered.
But in no way will the future be what we think it will be.
“It’s tough to make predictions, especially about the future.”
Yogi Berra (maybe he got it from Niels Bohr).
Can we keep doing what we are doing
and keep getting what we got?
In general, logic that leads to uncomfortable conclusions
regarding the future is shunted or is nudged back onto a nice
saddle path to a well defined outcome that fits into the
paradigm of endless growth.
What follows is a simulation in which there are no
exogenous inputs and therefore is decoupled from the
assumptions of mainstream growth theory.
But the model is also derived from actual historical data.
Thus imbedded in the simulation are the axiomatic
assumptions that we don’t even think about.
The purpose of the simulation is to look forward into the
world that would be ours if there were no ecological
constraints.
We are going to look at the logical outcomes of
policy choices that are based on the foundations of
capitalism.
If one looks at the history of growth, a key role in explaining growth is the increasing
energy flows that are used to achieve the substitution of machines for animals and
people (machines that consume fossil fuel we should note). However it is also
important to account for the actual services that are performed by the energy. That is,
how efficiently is energy converted to useful work?
That is the heart of what technological progress is.
Energy Services per unit of raw
Technological Efficiency of Energy Conversion to Work
30.0%
25.0%
y=
-3E-07x3 +
energy input.
6E-05x2 - 0.0003x + 0.0468
R² = 0.9944
20.0%
15.0%
Energy Intensity of Output
10.0%
1900 = 1
1.2
5.0%
1
0.0%
1900
1904
1908
1912
1916
1920
1924
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
2004
2008
0.8
0.6
0.4
0.2
2008
2004
2000
1996
1992
1988
1984
1980
1976
1972
1968
1964
1960
1956
1952
1948
1944
1940
1936
1932
1928
1924
1920
1916
1912
1908
1904
0
1900
The quantity used and the
efficiency of the
conversion of natural
resources into output has
changed over time. This
suggests a further
shortcoming in the use of
production functions with
fixed factor intensities.
Although impossible to
solve mathematically, time
varying factor intensities
can be built into a
numerical simulation
model.
The “Production Function”
here at the heart of the simulation
ICT = Information and
Communications Technology
Not shown in the schematic are the feedback loops from GDP growth to
investment decisions and energy demand and many other sub feedback loops.
Calibration… MAPE’s are
very small.
Capital
20
15
10
5
0
1900
1912
1924
Capital : Calibration
Capital : MacroEnergyData6
1936
1948 1960
Time (Year)
1972
1984
1996
Labor
2008
4
3
2
1
0
1900
1912
1924
Labor : CalibrationGDP4
Labor : MacroEnergyData2
1936
1948 1960
Time (Year)
1972
1984
1996
2008
Primary Energy Intensity of Output
2
1.5
1
0.5
0
1900
1912
1924
1936
1948 1960
Time (Year)
1972
1984
1996
2008
Primary Energy Intensity of Output : Calibration
Primary Energy Intensity of Output : MacroEnergyData6
Efficiency of Primary Energy Conversion
0.2
0.15
0.1
0.05
0
1900
1912
1924
1936
1948 1960
Time (Year)
Efficiency of Primary Energy Conversion : Calibration
Efficiency of Primary Energy Conversion : MacroEnergyData6
1972
1984
1996
2008
The chart shows a sort of energy services Kuznets curve based on the efficiency of
the use of energy services to produce output. The economy reached a turning point
between 1970 and 1980.
Primary intensity has fallen steadily as more efficient manufacturing processes and
prime movers have been invented. But that in turn caused industrialization to grow
and that effect, in the first 3/4 of the century was more positive than the increase in
efficiency was negative.
However, since the mid-1970s the value added in the US has been shifting from
manufacturing to service .
Energy Service Intensity of Output
4
Primary Energy Intensity of Output
2
1.5
3
1
2
0.5
0
1900
1912
1924
1936
Primary Energy Intensity of Output : Calibration
Primary Energy Intensity of Output : MacroEnergyData6
1948 1960
Time (Year)
1972
1984
1996
2008
1
0
1900
1912
1924
1936
Energy Service Intensity of Output : Calibration
Energy Service Intensity of Output : MacroEnergyData6
1948 1960
Time (Year)
1972
1984
1996
2008
Gross Domestic Product
40
The calibrated output of the
simulation.
30
20
10
0
1900
1912
1924
1936
1948 1960
Time (Year)
1972
1984
1996
2008
Monetary Value of Output
Gross Domestic Product : Calibration
Gross Domestic Product : MacroEnergyData6
10,000
billions 1992 $
7,500
5,000
2,500
0
1900
1912
1924
1936
1948 1960
Time (Year)
Monetary Value of Output : Calibration
Monetary Value of Output : MacroEnergyData7
1972
1984
1996
2008
Gross Domestic Product
The future…
200
150
100
50
0
1900
1920
1940
1960
1980
Time (Year)
2000
Gross Domestic Product : FutureNoChange
Gross Domestic Product : MacroEnergyData5
2020
2040
The GDP index rises from
27.36 in 2007 to 105.62 in
2050. That is nearly four
times larger is just 43 years.
FutureNoChange
MacroEnergyData5
50%
75%
95%
Labor Intensity of Output
1
0.75
This is a problem for the unskilled
in the US.
0.5
0.25
0
1900
1938
1975
Time (Year)
2013
2050
The model predicts several impossible outcomes if allowed to proceed
to the point at which the system explodes (nearly infinite GDP). This is
the “doing what you did” scenario. That is, we remain on the path that
has carried us to today for the last 109 years.
If we do that, in 2080 very strange things happen. Given the growth
rate of population follows forecasted trends, the labor intensity of
output goes to 0.0034 (not much labor needed) and the exponential
growth of normalized output takes it to 2,487 (from 26.71 in 2007).
Consumption per labor hour goes from 7.526 (that is 7.5 times higher
than in 1900) in 2007 to 289.39; a 38.45 fold increase.
So whereas we have seen an increase in our standard of
living by a factor of about 7.5 in the last 107 years, we can
expect to see it increase by another 38.45 times in the next
70 or so years.
What does that mean?
1000
Primary Energy Demand
283.16
100
Simulated Primary Energy Demand
: MacroEnergyData6
10
6.95
1900
1906
1912
1918
1924
1930
1936
1942
1948
1954
1960
1966
1972
1978
1984
1990
1996
2002
2008
2014
2020
2026
2032
2038
2044
2050
2056
2062
2068
2074
2080
1
In 2007 the US used 6.95 times more
primary energy than in 1900. In 2080 it will
require 283.16 times more than in 1900.
That is a 40.74 times increase from 2007.
The usual limits-to-growth analysis would
impose a finite limit on energy production
as well as consider other by-products of
growth to tell a story about exceeding the
carrying capacity of the planet. A part of
that story is about waste. In our model so
far, waste is created but, in this scenario,
efficiency is increasing so that at the limit
(2083 or so when the system explodes),
there is no waste!
We again see that if the future is to be what is assumed in both economic
theory and in the minds of business and political decision makers there are
outcomes that violate possibility.
Suppose that there is a slowing growth in technological improvement but population
growth continues at the recent trend. The timing of the changes and the rates of change
are modeled using an @RISK simulation with each year’s parameters allowed to vary
around probability distributions. As the charts below show these changes bring
challenging times.
Gross Domestic Product
40
This is a BIG Problem
Actual GDP
35
Simulated GDP
5% - 95%
30
+/- 1 Std. Dev.
Output per Unit of Labor
25
8
20
7
Simulated Data
6
15
5% - 95%
+/- 1 Std. Dev.
5
10
4
The good times last
until about 2030
5
0
3
2
1
2100
0
-1
1900
1904
1908
1912
1916
1920
1924
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
2004
2008
2012
2016
2020
2024
2028
2032
2036
2040
2044
2048
2052
2056
2060
2064
2068
2072
2076
2080
2084
2088
2092
2096
2100
2095
2090
2085
2080
2075
2070
2065
2060
2055
2050
2045
2040
2035
2030
2025
2020
2015
2010
2005
2000
1995
1990
1985
1980
1975
1970
1965
1960
1955
1950
1945
1940
1935
1930
1925
1920
1915
1910
1905
1900
-5
To see if there is a way forward that does not end in a future in which we regress to
output per person levels seen in the 1960’s the tactic used is straightforward. I take
the original simulation conditions that showed infinite output and perform constrained
optimization (using Evolver to find the best set of parameters) on an idealized future
path for GDP. The result is to “re-hardwire” the production function by altering its
parameters with the objective of a sustainable future in terms of output per unit of
labor.
The choice of an objective is entirely arbitrary. The chart shows the objective as a level
of GDP about 100 times greater than in 1900 and about 3.71 times greater than in 2008
by 2200.
Ideal GDP Path
120
100
80
60
40
20
1900
1909
1918
1927
1936
1945
1954
1963
1972
1981
1990
1999
2008
2017
2026
2035
2044
2053
2062
2071
2080
2089
2098
2107
2116
2125
2134
2143
2152
2161
2170
2179
2188
2197
2206
0
The specification for the parameters in the production function and the other
relationships in the model are empirically determined from the data over the past 107
years. In other words, although the simulation model is built from data with no
assumptions regarding a theoretical economic growth model, embedded in the data are
the patterns that built the 20th century.
So what drives capitalism drives this model. This model therefore contains the
axiomatic assumptions that govern economic (and policy) decision making. Constrained
optimization on the production function parameters could only fit a portion of the ideal
path. Furthermore, the system is very sensitive to small parameter changes.
Here we see the system “falling off of the saddle path!”
Gross Domestic Product
Gross Domestic Product
200
200
150
150
100
100
50
50
0
1900
1944
1988
Gross Domestic Product : FutureOptimizedStableGrowth1
Gross Domestic Product : MacroEnergyDataFuture
2032
2076
Time (Year)
2120
2164
2208
0
1900
1944
1988
Gross Domestic Product : FutureOptimizedStableGrowth1
Gross Domestic Product : MacroEnergyDataFuture
2032
2076
Time (Year)
2120
2164
2208
If capital and labor and energy services continue on paths as
they have, the production function is invalid for a sustainable
future; and stable outcomes in the future with this production
function will become increasingly less likely.
It is like the knife edge on the saddle path is getting
thinner until there is no edge and we fall off.
This is not supposed to happen!
Gross Domestic Product
200
150
100
50
0
1900
1944
1988
Gross Domestic Product : FutureOptimizedStableGrowth1
Gross Domestic Product : MacroEnergyDataFuture
2032
2076
Time (Year)
2120
2164
2208
Can new relationships be imposed on the model that can
lead to a reasonable outcome?
Next the optimization was extended to the investment (savings) and depreciation
parameters, the growth and decay rates of the labor supply, and the parameters
controlling innovation (the efficiency of the conversion of primary energy to
energy services).
Gross Domestic Product
200
150
100
50
0
1900
1944
1988
Gross Domestic Product : FutureOptimizedStableGrowth1
Gross Domestic Product : MacroEnergyDataFuture
2032
2076
Time (Year)
2120
2164
2208
The optimized capital, labor, and the growth and
decay of innovation lead to a simulation that
follows the ideal path farther along; until 2096 and
then it collapses.
What throws the economy off the path is an increasing sensitivity to very
small changes in the continued growth in the ability to get useful work out
of primary energy (innovation). As the chart shows, the model continues to
extract more and more useful work out of a given quantity of primary
energy. This is sensible to a point. But eventually the marginal increase in
useful work is insufficient to power growth.
Energy Intensity of Output
1.2
Actual Energy Intensity of Output
1
Simulated Energy Intensity of Output
0.8
5% - 95%
+/- 1 Std. Dev.
0.6
Note that the @RISK simulation
allows the Energy Intensity of
Output to go below zero.
That is not possible.
0.4
However it does show the
possibility of a path that
leads to mathematical and
social chaos!
0.2
0
1900
1904
1908
1912
1916
1920
1924
1928
1932
1936
1940
1944
1948
1952
1956
1960
1964
1968
1972
1976
1980
1984
1988
1992
1996
2000
2004
2008
2012
2016
2020
2024
2028
2032
2036
2040
2044
2048
2052
2056
2060
2064
2068
2072
2076
2080
2084
2088
2092
2096
2100
-0.2
The solution to this simulation challenge lies in maintaining an optimal marginal
productivity of energy services (and therefore labor). Although there are no limits to the
accumulated stock of energy used in this simulation (that is, no limits to growth from
energy resources), there is no such thing as work without waste (infinite efficiency!).
Primary Energy Intensity of Output
 This has to be asymptotic to a
value that yields a stable GDP rather
than converge to zero.
2
1.5
1
Output per Unit of Labor
0.5
20
0
1900
15
1944
1988
2032
2076
Time (Year)
2120
2164
2208
Primary Energy Intensity of Output : FutureOptimizedStableGrowth1
Primary Energy Intensity of Output : MacroEnergyDataFuture
10
5
Gross Domestic Product
200
0
1900
150
1944
1988
2032
2076
Time (Year)
2120
2164
Output per Unit of Labor : FutureOptimizedStableGrowth1
Output per Unit of Labor : MacroEnergyDataFuture
100
And labor has to have a role
in producing the output.
50
0
1900
1944
1988
Gross Domestic Product : FutureOptimizedStableGrowth1
Gross Domestic Product : MacroEnergyDataFuture
2032
2076
Time (Year)
2120
2164
2208
2208
We asked earlier, “Can we keep doing what we are doing and keep getting
what we got?” The model says no.
I have shown that the foundations upon which we the last century’s growth
were based cannot work for another century. And that is notwithstanding
the limits to natural resources and the ability of the planet to process or
absorb waste.
When this research began the expectation was that the ecological
constraints that have been essentially excluded from this analysis would
motivate change before the issues of exponential growth.
But this work suggests that insustainablity
may be closer than we think.
To follow the ideal path just shown will require a fundamental change
literally in how we do business.
Bypassing concerns for ecological sustainability, and accepting that
exponential growth in impossible to sustain, and, even more in the
present, accepting that the paradigm that sustained the 20th century
will not work going forward for much longer, we are faced with a need
to radically overhaul the fundamental motives for business. More is not
going to better for long.
This is not only an overhaul of how we do
business but is also a shift in what living and
working is all about.
It is very hard to imagine such a future. And perhaps
that reality will never happen as we try to balance on
the ever narrowing saddle path by doing what we did.
But imagine it we must if we are to see a
future in which there are semblances of
the comforts that we have defined as
good and necessary…
“Heavier than air flying machines are impossible.”
Lord Kelvin, Royal Society, 1895
“There is no likelihood that man can ever tap the power of the atom.”
Robert Millikan, Nobile Laureate, Physics, 1923
"The concept is interesting and well-formed, but in order to earn better than a 'C,'
the idea must be feasible."
A Yale University management professor in response to Fred Smith's paper
proposing reliable overnight delivery service [Smith went on to found Federal Express Corp.]
"Everything that can be invented has been invented."
Charles H. Duell, Commissioner, U.S. Office of Patents, 1899
Brief Description of
the Production
Function
RETURN