Stock-flow consistent models (SFC) are a family of macroeconomic models based on a rigorous accounting framework, that seeks to guarantee a correct and comprehensive integration of all the flows and the stocks of an economy. These models were first developed in the mid-20th century but have recently become popular, particularly within the post-Keynesian school of thought.^[1][2] Stock-flow consistent models are in contrast to dynamic stochastic general equilibrium models, which are used in mainstream economics.

The ideas for an accounting approach to macroeconomics go back to Knut Wicksell,^[3] John Maynard Keynes (1936)^[4] and Michał Kalecki.^[5][6] The accounting framework behind stock-flow consistent macroeconomic modelling can be traced back to Morris Copeland's development of flow of funds analysis back in 1949.^[1][7] Copeland wanted to understand where the money to finance increases in Gross National Product came from, and what happened to unspent money if GNP declined. He developed a set of tables to show the relationship between flows of income and expenditure and changes to the stocks of outstanding debt and financial assets held in the US economy.^[1][8]

James Tobin and his collaborators used features of stock-flow consistent modelling including the social accounting matrix and discrete time to develop a macroeconomic model that integrated financial and non-financial variables.^[1] He outlined the following distinguishing features of his approach in his Nobel lecture^[9]

1. Modelling changes between discrete short-run time periods rather than a long run equilibrium
2. Tracking changes in stocks of assets held by different groups
3. Multiple assets with different rates of return,
4. Modelling of monetary policy operations
5. Subjecting the demand functions to "adding up constraints"

Also Robert Clower based his Keynesian price and business cycle theories on stock-flow relations.^[10][11][12] A similar approach was developed in Germany by Wolfgang Stützel as Balances Mechanics.^[13]

The current SFC models mainly emerged from the separate economic tradition of the Post Keynesians, Wynne Godley being the most famous contributor in this regard.^[1] Godley argued in favour of wider adoption of stock-flow consistent methods, expressing the view that they would improve the transparency and logical coherence of most macro models.^[14] The Post Keynesians aimed at developing a macroeconomic theory that rejects the classical dichotomy, the neutrality of money and general equilibrium theory. Instead, they wanted to model the financial stocks and flows and their relations, the sectoral balances.^{[2][15][16]: 18 [17]} From some models of "monetary circuit theory", far-reaching consequences were derived, such as the thesis of a "monetary growth imperative", which, however, could be explained by inconsistent accounting.^[18][19] By respecting accounting constraints, "black holes" have to be avoided, where money vanishes without an offsetting entry in the balance sheet.^[20][21]

The models gained popularity at the beginning of the 21st century and especially after the beginning of the financial crisis of 2007–08,^[1] as some authors had foreseen the critical developments with accounting models.^[22][23] Wynne Godley, one of the pioneers of the SFC approach since the 1970s,^[22][24] had warned since 2000 in publications^[25][26] that the US housing market would weaken and cause a recession.^[22] In DSGE models, which dominate macroeconomics, crises usually cannot arise because of behavioural assumptions such as rational expectations and intertemporal optimisation. Although they treat stock and flow variables consistently, they usually model only individual stock variables such as physical capital, while monetary variables such as credit relations and debt are neglected.^[23][27] Therefore, attempts are made to analyse financial crises using stock-flow consistent models based on the accounting approach.^[28][29][30]

While ecological aspects were not considered by post-Keynesian authors like Godley or Lavoie, SFC models are now widely used within Ecological Macroeconomics.^[5] In addition to cash flows, resource or energy flows and stocks or an ecosystem with renewable resources are modeled.^{[31][32][33][34]} References are made in particular to the flow-fund models of Nicholas Georgescu-Roegen.^[32][34] Other approaches^[1] integrate concepts of SFC models into agent-based modeling (ABM)^[5][35][36] or input–output models.^[31][37]

Current researchers in the SFC approach to macroeconomic modelling are based in University of Limerick, Levy Economics Institute and University of Oxford.

SFC models usually consist of two main components: an accounting part and a set of equations describing the laws of motion of the system. The consistency of the accounting is ensured by the use of three matrices: i) the aggregate balance sheets, with all the initial stocks, ii) the transaction flow, recording all the transactions taking places in the economy (e.g. consumption, interests payments); iii) the stock revaluation matrix, showing the changes in the stocks resulting from the transactions (the transaction flow and the stock revaluation matrix are often merged in the full integration matrix). The matrices are built respecting intuitive principles. Someone's asset is someone else's liability and someone's inflow is someone else's outflows. Furthermore, each sector and the economy as a whole must respect their budget constraint. No fund can come from (or end up) nowhere.

The second component of SFC models, the behavioural equations, include the main theoretical assumption of the model. Most of the papers in the existing literature are based on post-Keynesian theory. However, the behavioural equations are not restricted to a single school of thought.^{[nb 1]}

Most SFC models are formulated in discrete time,^[1] but can also be formulated in continuous time as differential equations or differential-algebraic equations.^[20][39]

Simple models can be solved analytically and investigated by means of concepts of dynamical system theory such as bifurcation analysis.^[1][31][18] More complex models must be numerically simulated.^[1]

The comprehensive accounting framework has several advantages. Tracking all the monetary flows taking place in an economy and the way they accumulate, allows for a consistent integration of the real and the financial side of the economy (for a detailed discussion see Godley and Lavoie, 2007). Furthermore, as balance sheets are updated in any period, SFC models can be used to identify unsustainable processes, for example a prolonged deficit of a sector will result in an unsustainable stock of debt. These models were used by Wynne Godley in forecasting, showing promising results.^[40] Moreover, from a modelling perspective, the consistent accounting framework prevents the modellers from leaving "black holes" i.e., unexplained parts of the model.

• Wynne Godley, Marc Lavoie: Monetary Economics. An Integrated Approach to Credit, Money, Income, Production and Wealth. Palgrave Macmillan, New York 2012, ISBN 978-0-230-30184-9.
• Dimitri B. Papadimitriou, Gennaro Zezza: Contributions in Stock-flow Modeling: Essays in Honor of Wynne Godley. Palgrave Macmillan, London 2012, ISBN 978-1-349-33340-0, doi:10.1057/9780230367357.
• Eugenio Caverzasi, Antoine Godin: Post-Keynesian stock-flow-consistent modelling: a survey. In: Cambridge Journal of Economics 39(1), 2015, pp. 157–187, doi:10.1093/cje/beu021.
• Michalis Nikiforos, Gennaro Zezza: Stock-Flow Consistent Macroeconomics Models: A Survey. In: Journal of Economic Surveys 31(5), 2017, pp. 1204–1239, doi:10.1111/joes.12221.
• Emilio Carnevali, Matteo Deleidi, Riccardo Pariboni, Marco Veronese Passarella: Stock-Flow Consistent Dynamic Models: Features, Limitations and Developments. In: Philip Arestis, Malcolm Sawyer (eds.): Frontiers of Heterodox Macroeconomics, Palgrave Macmillan, Cham 2019, pp. 223–276. doi:10.1007/978-3-030-23929-9 6.
• Oliver Richters, Erhard Glötzl: Modeling economic forces, power relations, and stock-flow consistency: a general constrained dynamics approach. In: Journal of Post Keynesian Economics, 2020, doi:10.1080/01603477.2020.1713008