Reverse-Engineering the
Business Cycle with Petri Nets
Johnnie B. Linn III
Concord University
Athens, WV
Special Thanks for
• Online input-output tables for the U.S.
economy, 1998-2010.
– U.S. Bureau of Economic Analysis, Department of
Commerce, http://www.bea.gov.
• Online eigenvalue/eigenvector calculator for a
32 x 32 matrix.
– Bluebit Software, http://www.bluebit.gr.
The Problem
• “I do not think that the currently popular
DSGE models pass the smell test.”
– Robert Solow Professor Emeritus, MIT
• Prepared Statement, House Committee on Science and
Technology Subcommittee on Investigations and
Oversight. 1
• So what is DSGE?
Dynamic Stochastic General
Equilibrium (DSGE)
• Macroeconomic models founded on
microeconomic principles.2
• But DSGE failed to foresee the financial crisis.
Could agent-based modeling do better?
– The Economist, Jul 22nd 2010
Agent-Based Modeling
• “[C]omputational models for simulating the
actions and interactions of autonomous
agents (both individual or collective entities
such as organizations or groups) with a view to
assessing their effects on the system as a
whole.”3
So Where am I Going with This?
• The aggregate data we have are the products
of agents. Can we find all possible sets of
actions of agents that could have produced
our results?
• Then we can link each set of agents and their
actions with a particular set of behavioral
assumptions.
Petri Nets
• Bipartite Directed Graphs
• Used to Model Interconnected Causal Systems
Petri Nets and Their Firings
Petri Nets and Their Firings
Petri Nets and Their Firings
Petri Nets and Their Firings
Here, Causal Relationships of
Quantities
Disposable Income
Consumption
Matrix Representation for a Set of
Firings
𝑡
𝑀𝑡 = 𝑀0 + 𝐴′
𝑈𝑖
𝑖=1
An Adjacency Matrix and Firings
𝐴=
−1
1
1
−1
0 1 −1 1
=
+
1 0
1 −1
1 0 −1 1
=
+
0 1
1 −1
1
0
0
1
Decomposition of the Adjacency
Matrix
−1 1
𝐴=
1 −1
−1
𝑁=
0
0
𝐵=
1
0
−1
1
0
Key Assumption
𝑡
𝑀0 = −𝑁 ′
𝑈𝑖
𝑖=1
• All transitions are enabled, and can be fired in
any order.
• Makes internal cycles possible.
• Models independent actions by agents.
The Business Cycle, Starting Model
Value
Added
Gross Domestic
Product
NIPA Use Table, No Intermediate Detail
The Use of Commodities by Industries before Redefinitions (Producers' Prices)
(Millions of dollars)
Bureau of Economic Analysis
2010
IOCode
Commodities/Industries
Name
Total Intermediate
Total Value Added
Total Industry Output
Total Intermediate
Total Final Uses (GDP) Total Commodity Output
11284836 ...
...
...
14526547 ...
...
...
25811383
Creation/Destruction Cycle
Destructive Processes
Intermediate
Commodities
Gross Domestic Product/
Value Added
Creative Processes
Adjacency Matrix for the
Creation/Destruction Cycle
Int. GDP/VA Total
−1 −1
𝐴=
1
1
1
−1
• Creation Process is derived from the transpose
of the use table.
• Destruction Process is derived from the
untransposed use table.
Hard-Wiring the Technical
Transformation Processes
Int.
−11284836
𝐴=
11284836
GDP/VA
−14526547
14526547
Total
25811383
−25811383
• A token is denominated in dollars.
• Arc values can be scaled to best fit the
eigenvalue/eigenvector calculating algorithm.
Reverse-Engineering the Cycle
• Each firing delivers at least one new token to
the cumulative marking. It will be used to fill
the element of U that represents that firing.
• Working backwards…
𝑈𝑡 = 𝐵(𝑀𝑡 − 𝑀𝑡−1 ) + 𝐶𝑡
where Ct is a vector of constants that ensure that no
other elements of U are changed. Continuing..
𝑈𝑡−1 = 𝐵(𝑀𝑡−1 − 𝑀𝑡−2 ) + 𝐶𝑡−1
…
𝑈1 = 𝐵(𝑀1 − 𝑀0 ) + 𝐶1
So
𝑡
𝑈𝑖 = 𝐵(𝑀𝑡 − 𝑀0 ) + 𝐶
𝑖=1
but
𝑡
𝑀𝑡 − 𝑀0 = 𝐴′
𝑈𝑖
𝑖=1
so
𝑡
𝑡
𝑈𝑖 = 𝐵𝐴′
𝑖=1
𝑈𝑖 + 𝐶
𝑖=1
or
𝑡
(𝐼 − 𝐵𝐴′ )
𝑈𝑖 = 𝐶
𝑖=1
• In a cyclical process, U is a set of initial
processes C that have replicated themselves.
• So let
𝑡
𝐶=ν
𝑈𝑖
𝑖=1
then
𝑡
[ 1 − ν 𝐼 − 𝐵𝐴′ ]
𝑈𝑖 = 0
𝑖=1
• The matrix BA’ is a linear transformation of
the vector U with eigenvalues (1-νi).
Eigenvalues are normally expressed as λ , so
we have
𝑡
(𝜆𝐼 − 𝐵𝐴′ )
𝑈𝑖 = 0
𝑖=1
where
λ = 1-ν
If the Arcs are Hard-Wired to Data:
• The number of tokens accounted for by each
element of C will vary according to its process’
contribution to volume.
• Likewise the magnitudes of the elements of U
will vary according to their processes’
contributions to volume.
• The components of U remain orthogonal.
The Data
• NIPA Use Tables, 1998-2010
• Two sets
– Before redefinitions
– After redefinitions
• Sector Level
Part of a Use Table
The Use of Commodities by Industries before Redefinitions (Producers' Prices)
(Millions of dollars)
Bureau of Economic Analysis
2010
Commodities/Industries
IOCode
11
21
22
23
31G
42
44RT
48TW
51
FIRE
Name
Agriculture, forestry, fishing, and hunting
Mining
Utilities
Construction
Manufacturing
Wholesale trade
Retail trade
Transportation and warehousing
Information
Finance, insurance, real estate, rental, and
leasing
11
21
Agriculture,
forestry,
fishing, and
hunting
Mining
64636
0
816
33655
5642
8435
1747
10821
80294
50315
20092
6334
796
767
7486
9615
248
905
27369
20145
22
23
31G
42
Construc Manufact Wholesal
tion
uring
e trade
1
1186 220039
475
62057
8901 427580
321
193
3419
73548
6664
5714
875
17495
2218
4107 246347 1531048
74274
573
25774 220880
45147
38
36296
7570
760
14625
14011 108001
41648
510
6427
20439
11148
Utilities
4079
31524
115949
67088
The Computations
• 15 creative processes produce 15
commodities plus scrap and noncomparable
imports.
• 17 destructive processes reduce 15
commodities plus scrap and noncomparable
imports to intermediate or final uses.
– Scrap and noncomparable imports do not have
final uses.
The Computations (2)
• There are 32 processes for 18 quantities, so 14
processes are redundant.
• (λ I – BA’) is a 32 x 32 matrix.
• The matrix has 32 eigenvalues and 32
corresponding eigenvectors.
• We seek complex eigenvalues.
Summary of Results
• Complex eigenvalues were found.
• Average annual angle of rotation (alibi) about
24 degrees.
– Corresponds to cycle length of 15 years.
– Velocity of rotation increased significantly after
2007.
• Phase angles between creation process and
corresponding destruction process for each
commodity appear to have significance.
Alibi Results, 1998-2010
1.2
1
0.8
Alibi Before
0.6
Alibi After
0.4
0.2
0
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
What the Algorithm Did
• Algorithm had 14 degrees of freedom.
• Aligned the phase angles of all 15 creative
processes.
• Also, rotated each eigenvector to guarantee
one process to have a phase angle of zero
degrees, or sometimes 180 degrees.
• Not always the same process selected.
Raw Phase Angles, Before
Redefinitions
4
Creative
3
Agriculture, forestry, fishing, and hunting
2
Mining
Utilities
1
Construction
0
1996
-1
1998
2000
2002
2004
2006
2008
2010
2012
Manufacturing
Wholesale trade
Retail trade
-2
Transportation and warehousing
-3
-4
Information
Raw Phase Angles, After Redefinitions
2
Creative
Agriculture, forestry, fishing, and hunting
1
Mining
0
1996
1998
2000
2002
2004
2006
2008
2010
2012
Utilities
Construction
-1
Manufacturing
Wholesale trade
-2
Retail trade
Transportation and warehousing
-3
Information
-4
Phase Angle from Creative Datum,
Before Redefinitions
8
Creative
6
Agriculture, forestry, fishing, and hunting
Mining
4
Utilities
2
Construction
0
1996
Manufacturing
1998
2000
2002
2004
2006
2008
2010
2012
Wholesale trade
-2
Retail trade
-4
Transportation and warehousing
-6
Information
-8
Finance, insurance, real estate, rental, and
leasing
Phase Angle from Creative Datum,
After Redefinitions
5
Creative
4
Agriculture, forestry, fishing, and hunting
3
Mining
2
Utilities
1
Construction
0
1996
Manufacturing
1998
2000
2002
2004
2006
2008
2010
2012
Wholesale trade
-1
Retail trade
-2
-3
-4
Transportation and warehousing
Information
Agriculture, forestry, fishing, and
hunting
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Mining
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Utilities
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Construction
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Manufacturing
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Wholesale Trade
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Retail Trade
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Transportation and Warehousing
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Information
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Finance, insurance, real estate, rental,
and leasing
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Professional and business services
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Educational services, health care, and
social assistance
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Arts, entertainment, recreation,
accommodation, and food services
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Other services, except government
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Government
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Scrap, used and secondhand goods
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Noncomparable imports and rest-ofthe-world adjustment
Before Redefinitions
After Redefinitions
1.5
1.5
1
1
0.5
0.5
0
-1.5
-1
-0.5
0
0
0.5
1
1.5
-1.5
-1
-0.5
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0.5
1
1.5
Conclusions
• Close Trackers of Creative Processes
–
–
–
–
Utilities
Wholesale trade
Educational services, health care, and social assistance
Construction (added to fixed investment at time of outlays)
• Laggers, 1 to 6 months
– Agriculture, forestry, fishing, and hunting
– Mining
– Arts, entertainment, recreation, accommodation, and food
services
– Government
Conclusions (2)
• Laggers, 6 to 12 months
– Manufacturing
– Information
– Finance, insurance, real estate, rental, and leasing
– Other services, except government
Conclusions (3)
• More Problematic
– Retail trade (two year lag?)
– Professional and business services
– Transportation and warehousing
• Different subsectors in these may come to the
fore at different stages in the business cycle.
Future Work
• Structural Equations
• Sub-sector (summary) analysis.
References
1. Prepared Statement of Robert Solow, Professor
Emeritus, MIT, to the House Committee on
Science and Technology, Subcommittee on
Investigations and Oversight: “Building a Science
of Economics for the Real World,” July 20, 2010
2. http://en.wikipedia.org/wiki/Dynamic_stochasti
c_general_equilibrium.
3. http://en.wikipedia.org/wiki/Agentbased_model#cite_note-0