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The roads not taken: 
graph theory and 
macroeconomic regimes 
in stock-flow consistent modelling 
Miguel Carrión Álvarez ...
The economy as a network 
Axel Leijonhufvud (INET, 2012): 
Let me start by asking, what is your first association 
when so...
SFC macroeconomic modelling 
Stock-flow consistent macroeconomic models 
(Copeland, Tobin, Godley & Lavoie) is a framework...
Beware the ‘nonsense math effect’ 
We want to avoid what Eriksson (2012) calls the nonsense 
math effect, so if what follo...
The SIMplest model 
Cash-flow specification of Model SIM (G&L 2007, Ch.3) 
Net savings are of a different character from t...
The SIMplest model 
Cash-flow specification of Model SIM (G&L 2007, Ch.3) 
Net Savings are a notional cash flow from House...
The SIMplest model 
Balance sheet 
Cash balance equations 
for each sector are 
obtained by 
• equating columnwise 
sums o...
The SIMplest closure 
(G&L, Ch. 3) 
There has to be 
a simpler way 
The answer: 
graph theory
Some graph concepts 
• A graph is a collection of 
directed edges between 
nodes. 
• A directed edge has a 
source node an...
Macroeconomics and Graphs: 
an informal correspondence
Model SIM as a graph
More graph concepts: 
spanning trees and elementary loops 
A spanning tree is a subgraph 
including 
• all the sectors (no...
Spanning trees and 
accounting relations 
Each choice of spanning tree 
divides the cash flows into 
• independent variabl...
Macroeconomics and Graphs: 
first few insights 
• Elementary loops: 
beyond the quadruple entry principle 
• Multiple span...
Elementary loops: 
beyond the quadruple-entry principle 
Some elementary loops of consist of 
only two economic relations ...
Beyond the quadruple-entry principle: 
propagation of government 
expenditure in model SIM 
A perturbation 
of model SIM 
...
Multiple spanning trees: 
what do they mean? 
“Space invader”: eight spanning trees of Model SIM
Dependent vs. independent variables 
Some variables are called independent because they are determined 
by behavioral rela...
“Real economy” factors of Model SIM 
The consumption/wage-bill pair 
of cash flows represents the 
monetary realization of...
Purely monetary factors of model SIM 
Flows in the other dimension, taxation and savings, are purely monetary. 
It is poss...
The G&L closure of model SIM: 
consumption and taxes as drivers 
The “functional finance” closure of Model SIM
The G&L closure of Model SIM: 
functional finance 
Godley and Lavoie (2007, ch.3) favour a regime where driving variables ...
The G&L closure of Model SIM: 
functional finance 
A closure of this system will require behavioural relations determining...
The Colonial closure of Model SIM: 
Taxes, savings and labour as drivers 
The “colonial” closure of Model SIM
Mosler/Kaboub closure of Model SIM 
Fadhel Kaboub (Denison University) has described a pedagogical 
experiment in which he...
Mosler/Kaboub closure of Model SIM 
“They came up with this brilliant idea. They told everybody there was 
going to be a t...
The meaning of ‘ceteris paribus’ 
The spanning tree formulation 
gives a precise meaning to the 
phrase ceteris paribus, a...
Graph algorithms: no thinking required 
Use your energy for the economics, not the algebra.
Future directions 
• Software tools 
• Fuller analysis of dynamical closures 
• Models of macroeconomic regime change 
• I...
Thank you for your attention 
Miguel Carrión Álvarez 
mcarrion@gruposantander.com 
Dirk Ehnts 
dirk.ehnts@fu-berlin.de
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The Roads Not Taken: Graph Theory and Macroeconomic Regimes in Stock-Flow Consistent Modeling

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Stock Flow Consistent Modeling session at 12th International Conference

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The Roads Not Taken: Graph Theory and Macroeconomic Regimes in Stock-Flow Consistent Modeling

  1. 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. 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. 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. 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. 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. 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. 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. 8. The SIMplest closure (G&L, Ch. 3) There has to be a simpler way The answer: graph theory
  9. 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. 10. Macroeconomics and Graphs: an informal correspondence
  11. 11. Model SIM as a graph
  12. 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. 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. 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. 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. 16. Beyond the quadruple-entry principle: propagation of government expenditure in model SIM A perturbation of model SIM How the perturbation is resolved
  17. 17. Multiple spanning trees: what do they mean? “Space invader”: eight spanning trees of Model SIM
  18. 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. 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. 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. 21. The G&L closure of model SIM: consumption and taxes as drivers The “functional finance” closure of Model SIM
  22. 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. 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. 24. The Colonial closure of Model SIM: Taxes, savings and labour as drivers The “colonial” closure of Model SIM
  25. 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. 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. 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. 28. Graph algorithms: no thinking required Use your energy for the economics, not the algebra.
  29. 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. 30. Thank you for your attention Miguel Carrión Álvarez mcarrion@gruposantander.com Dirk Ehnts dirk.ehnts@fu-berlin.de

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