# Graph representations of economic models

# A *SIM*ple economic model

Model *SIM*, the *simplest model with government money*, from Chapter 3 of
Godley and Lavoie (2007) is a
Stock-Flow Consistent model
with 11 equations.

The first four equalise demands and supplies of household consumption, government expenditure, taxes and labour. The equation numbers match those from Godley and Lavoie (2007):

Disposable income, , is household wage income less taxes:

where taxes are levied as a fixed proportion, , of money income:

and are the two policy variables in the model.

Household consumption is a function of current disposable income, , and past accumulated wealth, :

Changes in government debt, in the form of cash money, arise from differences between government expenditure () and tax revenue ():

Households hold that debt as a vehicle for savings:

In a closed economy with no investment, the national income identity is simply:

Alternatively, in income terms:

which re-arranges to express labour demand as:

# Models as graphs

A convenient way to think about a model like the one above is as a directed graph where the:

- nodes (vertices) correspond to the variables in the model
- edges link variables such that an edge that runs
*from**to*denotes as an endogenous (left-hand side) variable and as (one of) its exogenous (right-hand side) variables. may itself be an endogenous variable if it appears on the left-hand side of another equation

Below is a visualisation of Model *SIM* as a directed graph.

This way of depicting a model isn’t new. See, for example, the application of
path analysis to
structural equation modelling. A
recent application in economics is by Fennell *et al* (2014). This also uses a
model from Godley and Lavoie (2007).

What makes this representation useful is that it allows us to see the directions of causation in a model and to use techniques from graph analysis to further understand the relationships that underpin that model. Click on the nodes to see measures (calculated using NetworkX) of:

- in-degree and out-degree: the number of dependent variables (if any) and the number of variables for which the current node is a dependent variable, respectively
- betweenness centrality: the proportion of shortest paths between any two pairs (not including pairs that include the current node) that pass through this node
- closeness centrality: the reciprocal of the sum of the shortest path distances from the current node to each of the other nodes (normalised by the sum of the minimum possible distances)
- PageRank: a variant of eigenvector centrality that gauges the ‘importance’ of a variable in terms of the importance of the variables that it depends on

The ‘Endogenous’ button in the bottom-right corner of the visualisation highlights endogenous variables i.e. those with in-degree greater than zero. The ‘SCC’ button highlights the strongly connected component of the graph. This identifies a cycle or feedback mechanism in the model.

For further information on graph analysis as applied to economic modelling, see
Fennell *et al* (2014). Antoine Godin, one of the authors of that paper, shows
here how such analysis can inform the order
of a model’s equations for more efficient solution.

# Sources

Fennell, P., O’Sullivan, D., Godin, A., Kinsella, S. (2014)
‘Visualising Stock Flow Consistent Models as Directed Acyclic Graphs’
(05/09/2014),
*Social Science Research Network*

http://dx.doi.org/10.2139/ssrn.2492242

[Accessed 08/12/2015]

Godley, W., Lavoie, M. (2007),
*Monetary economics: An integrated approach to
credit, money, income, production and wealth*,
Palgrave Macmillan