Stock Flow Consistent Modeling session at 12th International Conference

1. The roads not taken:
graph theory and
macroeconomic regimes
in stock-flow consistent modelling
Miguel Carrión Álvarez
Grupo Santander
Dirk Ehnts
Free University, Berlin
12th International
Post Keynesian Conference
UMKC
27th September 2014

2. The economy as a network
Axel Leijonhufvud (INET, 2012):
Let me start by asking, what is your first association
when somebody talks to you about ‘the economy’?
Is the image you get: factories working,
supermarkets full of people, busy Wall St., or, what?
For today's purpose, I would like you to think of the
economic system, first, as a web of contracts,
contracts and understandings among agents in the
economy.

3. SFC macroeconomic modelling
Stock-flow consistent macroeconomic models
(Copeland, Tobin, Godley & Lavoie) is a framework
for looking at the macroeconomy from a monetary
or financial point of view. The basic object of
interest is the flow of funds between different
sectors of the economy.
We argue SFC modelling can be fruitfully recast in
the language of graph theory, in terms of
compatible flows on a network.

4. Beware the ‘nonsense math effect’
We want to avoid what Eriksson (2012) calls the nonsense
math effect, so if what follows is over your head, complain!
Economics should be used to illuminate the mathematics, and
not conversely. Further, the use of similar mathematics in
other disciplines adds nothing to the understanding by
economists. The graph metaphor is not as a prerequisite for
doing economics, but a novel way to represent what
economists already know. Once economists are comfortable
with the new representation they can take advantage of tools
from the new domain, but always projecting (familiar)
economic language onto (unfamiliar) mathematical language
and not the other way around.

5. The SIMplest model
Cash-flow specification of Model SIM (G&L 2007, Ch.3)
Net savings are of a different character from the rest of the rows of the
cash-flow specification
Taxes are an actual cash flow not leading to a stock accumulation.
Net savings are not an actual but a notional cash flow related to the
accumulation of a stock of savings.

6. The SIMplest model
Cash-flow specification of Model SIM (G&L 2007, Ch.3)
Net Savings are a notional cash flow from Households to Government:
• The stock of savings is an asset of Households and a liability of Government
• An increase in any asset stock has a corresponding cash flow from holder to issuer
• Cash flows from changes in asset stocks are part of each sector’s cash-flow balance
In each accounting period, cash balance is maintained by accounting convention
• Sector-by-sector equality of inflows and outflows
• For the Household sector:
(net savings) = (wage bill) - (consumption) - (taxes net of transfers)
• Interpreting this as a equality of sector inflows and outflows:
w = s + c + t
Constant price of cash implies no capital gains on the stock of savings

7. The SIMplest model
Balance sheet
Cash balance equations
for each sector are
obtained by
• equating columnwise
sums of cash flows to
zero; or, equivalently,
• equating sum of
incoming cash flows
for a sector to sum of
Transaction flow matrix outgoing cash flows
Accounting relations need to be augmented with
a dynamic closure to complete the model

8. The SIMplest closure
(G&L, Ch. 3)
There has to be
a simpler way
The answer:
graph theory

9. Some graph concepts
• A graph is a collection of
directed edges between
nodes.
• A directed edge has a
source node and a target
node.
• Sometimes this is called a
quiver because it is a
collection of arrows.
• Both edges and nodes can
carry labels, numerical or
otherwise.
• In a graph model of an
economy:
• Nodes are sectors into
which the economy is
decomposed and can carry
balance-sheet data
• Economic relations
between economic units or
sectors are represented by
directed edges between the
corresponding nodes. Edges
can be labelled by cash
flows, or by financial assets

10. Macroeconomics and Graphs:
an informal correspondence

11. Model SIM as a graph

12. More graph concepts:
spanning trees and elementary loops
A spanning tree is a subgraph
including
• all the sectors (nodes) and
• one fewer economic
relations (edges) than
sectors (nodes),
and such that the subgraph is
• connected and
• has no loops.
Adding any one edge to a
spanning tree results in a
closed elementary loop.
In our setting, finding a
spanning tree means
• selecting one fewer cash
flow than the number of
sectors
in such a way that
• every sector has at least
one cash flow in or out of
it, and
• there are no closed loops.
Adding an additional cash flow
to the tree would close a loop.

13. Spanning trees and
accounting relations
Each choice of spanning tree
divides the cash flows into
• independent variables,
which are the cash flows
not on the spanning tree;
and
• dependent variables,
those on the spanning
tree.
Dependent variables are
completely determined, via
accounting relations at each
node, by independent ones.
Solid:
dependent
Dotted:
independent
A choice of spanning tree for model SIM
Corresponding accounting relations

14. Macroeconomics and Graphs:
first few insights
• Elementary loops:
beyond the quadruple entry principle
• Multiple spanning trees: what do they mean?
– Dependent vs. Independent variables
– Monetary and Real drivers
– The meaning of Ceteris Paribus

15. Elementary loops:
beyond the quadruple-entry principle
Some elementary loops of consist of
only two economic relations (edges)
linking two sectors (nodes), but it's
possible to have loops with a larger
number of economic relations and
sectors. What we're calling
elementary loops are in fact a
generalisation of the quadruple-entry
principle of Godley and Lavoie
(2007, p.47ff)
A concrete example of [Copeland's]
legacy is represented by the quadruple-entry
system, which is a cardinal feature
of today's SFC models: that since
someone's inflow is someone else's
outflow, the standard double-entry
system of accounting, in its social version,
is doubled in a quadruple-entry system.
(Caverzasi and Godin, 2013 p.5)
Elementary loops for
previous choice of spanning tree
Model SIM contains an irreducible
‘sextuple entry’ represented by a
triangular cash flow loop.

16. Beyond the quadruple-entry principle:
propagation of government
expenditure in model SIM
A perturbation
of model SIM
How the perturbation
is resolved

17. Multiple spanning trees:
what do they mean?
“Space invader”: eight spanning trees of Model SIM

18. Dependent vs. independent variables
Some variables are called independent because they are determined
by behavioral relations. Other variables must then be called
dependent because they are determined by accounting. The
categories are mutually exclusive, there is no third kind of variable.
For instance, if taxation is determined as a percentage of previous
period income, taxation is an independent variable even though it
changes "automatically" with income. It is called independent because
it is not determined by accounting relations. If, on the other hand,
taxes (or transfers) are adjusted in each period to bring the
government budget deficit to zero, then taxes become a dependent
variable because they are determined by accounting relations: given
the level of government spending and private consumption (which
together determine income), there is only one level of taxes (net of
transfers) that solves the model.

19. “Real economy” factors of Model SIM
The consumption/wage-bill pair
of cash flows represents the
monetary realization of “real
economy” flows, which are
converted into monetary flows
by means of factor and product
prices. If c (consumption) is an
independent flow variable,
consumer demand is a driver of
the economy and we label the
institutional structure of the
economy as demand-driven.
Similarly, if w (the wage bill) is
an independent flow variable,
employer demand for labour
is a driver of the economy.
We call this a supply-driven
economy. It is possible for
both consumption and the
wage bill to be independent
variables, and then we have
an economy with both
supply-driven and demand-driven
characteristics.

20. Purely monetary factors of model SIM
Flows in the other dimension, taxation and savings, are purely monetary.
It is possible that t (taxation) is
an independent variable,
which we term a
“redistribution” institutional
arrangement. The level of
taxation then acts as a brake
on consumption, thus
introducing a negative
feedback on economic
activity.
Alternatively, private demand for savings
ΔS could be a driver of the economy. A
higher demand for savings translates into
less consumption, again acting as a brake
on the economy. We call this a “rentier
economy”. Since savings can be used
alternatively to run up wealth balances or
run down debts, another interpretation
could be named the “deleveraging
economy”.
And, if both taxes and savings are independent variables, we call the setup a
“financialized economy”. Both the demand for savings from the rentiers and
the taxes on households are restricting demand and thus are able to impose
unemployment on the economy.

21. The G&L closure of model SIM:
consumption and taxes as drivers
The “functional finance” closure of Model SIM

22. The G&L closure of Model SIM:
functional finance
Godley and Lavoie (2007, ch.3) favour a regime where driving variables are
taxation t, consumption c and, by implication, government expenditure g.
Taxation and government spending are both independent so the
government's fiscal position is fully autonomous, determining private savings
ΔS. This does not mean that households' desire for savings is determined by
government, but that actually realised savings is. In addition, the 'real
economy' is demand-driven, with government expenditure and private
consumption determining the wage bill w. We call this 'functional finance
economy' after Lerner (1943). The function of government spending is to
change the real economy and should not be stopped by 'traditional doctrine
about what is sound or unsound' (ibid. p. 39).
A closure of this system will require behavioural relations determining g,
t,and c; that is, private consumption and the government's fiscal stance.

23. The G&L closure of Model SIM:
functional finance
A closure of this system will require behavioural relations determining g, t,and c; that
is, private consumption and the government's fiscal stance. The closure presented in
Godley and Lavoie (2007, ch.3) is
• household income w is subject to a tax rate θ which can be negative, presumably
to represent the possibility that transfers exceed taxes;
• consumption c depends on disposable income (w – t) and start-of period stock of
savings S0 through coefficients α (household propensity to consume out of
disposable income) and β (household propensity to consume out of savings);
• government expenditure g is a policy lever explicitly left unspecified.
Because model SIM is a pure service economy firms accumulate no inventory, and
demand (c + g) is always met (there is no rationing), resulting in a wage bill w = g + c.

24. The Colonial closure of Model SIM:
Taxes, savings and labour as drivers
The “colonial” closure of Model SIM

25. Mosler/Kaboub closure of Model SIM
Fadhel Kaboub (Denison University) has described a pedagogical
experiment in which he requires his students to hand in a number of
'Denison Volunteer Dollars' for class credit, which students can obtain
by volunteering at local charities. The economics department, posing
as government, both issues DVDs to charities to pay for the
volunteering and collects the DVD tax at the end of the course. This
arrangement is a lot like the pure-service economy of model SIM.
There is a fixed tax t, which is an independent quantity decided
independently by the government. The private sector can decide how
much they want to work (possibly in excess of the tax liability) at a
fixed wage rate, therefore the wage bill w is also an independent
driver. Net savings ΔS are also a driver as the private sector can decide
to save money for future taxes. In this way, consumption c is a
dependent variable, as is government expenditure g (money will be
issued on demand to pay for earned wages).

26. Mosler/Kaboub closure of Model SIM
“They came up with this brilliant idea. They told everybody there was
going to be a tax on their hut. It was called a hut tax. Everyone had to
pay, what, 10 Pounds a month, something. Tax. Or they would get their
house burned down by the British. What happened? Everybody said all
right, what do we have to do to get the money to pay the tax? Ah, if
you come to the coffee plantation we’ll pay you one Pound a day to
work. Sure enough, people starting coming over to work, to earn the
money, so they didn’t get their house burned down. The tax, the
monetary system, created the unemployment.
Then the British hired the people so they could get the money to pay
the tax so they didn’t have their house burned down. They would
spend the money first and pay people, and then collect the tax, right.
And they spent more than they collected because some people saved
them [Pounds] for paying taxes later.”
Warren Mosler (2014)

27. The meaning of ‘ceteris paribus’
The spanning tree formulation
gives a precise meaning to the
phrase ceteris paribus, and
that meaning depends on the
“closure” or “analysis
framework“ (otherwise
referred to as “policy regime”).
“All other things being equal”
must mean
“all other independent
variables being equal”
Under the Godley-Lavoie closure
“changes in government expenditure, all
other things being equal”
means
“changes in government
expenditure at constant taxation and
consumption”
but something completely opposite
under an alternative closure where
taxation and consumption are
dependent on the wage bill and demand
for savings.

28. Graph algorithms: no thinking required
Use your energy for the economics, not the algebra.

29. Future directions
• Software tools
• Fuller analysis of dynamical closures
• Models of macroeconomic regime change
• Introducing the real economy
• Empirical models
– National accounts
– Flow of funds
– Leontieff matrices

30. Thank you for your attention
Miguel Carrión Álvarez
mcarrion@gruposantander.com
Dirk Ehnts
dirk.ehnts@fu-berlin.de