And other observations from an explorable Stock-Flow Consistent model

"Whether the policy debate is health care, infrastructure, education, or climate change, the same question inevitably arises: But how are you going to pay for it?" Stephanie Kelton, *The deficit myth*, 2020.

This is a popular analogy: the finances of a sovereign state's government are like the finances of a household. A government receives tax revenues and spends on expenditure programs. It runs a fiscal deficit when its expenditures exceed its revenues, and this deficit needs to be covered by some form of borrowing. New borrowings increase future expenditures and can trigger unsustainable spirals of deficits and borrowings. Responsible governments should therefore always aim to produce balanced budgets. Whenever possible, they should even aim to produce fiscal surpluses and pay back their existing debts.

In her bestselling book *The Deficit Myth* (2020), Stephanie Kelton calls this Government-Household analogy "undoubtly the most pernicious" of all the myths surrounding fiscal deficits. This analogy is wrong, she tells us, because households are currency users while the government is the currency issuer: it spends its currency into existence. Of course, the fact that the government can manufacture its own currency does not mean that there are no limits to its spending; but deficits are not evidence of overspending, inflation is. Once we fully aknowledge the role of the government as the currency issuer, it becomes clear that the sum of all past fiscal deficits—what we call the "National Debt"—really isn't a debt in the usual sense of the term. It is "nothing more than a footprint from the past", a statement of the total amount of currency issued by the government to date.

The distinction between currency user and issuer is an important starting point, but to really understand the difference between a government's finances and those of an individual household, it is necessary to zoom out and look at the economy from the macro level. Stephanie Kelton brings us there by introducing the Stock-Flow Consistent framework pioneered by British economist Wynne Godley. This framework starts from a simple accounting axiom: "everything comes from somewhere and everything goes somewhere". Seen from the perspective of a government's finances for instance, this leads to what Godley and Kelton playfully call the "one-equation model of the world":

Government financial balance $+$ Nongovernment financial balance $=$ Zero.

This equation may seem trivial, but it fundamentally contradicts what the Government-Household analogy suggests: "instead of making us poorer, fiscal deficits increase our wealth and collective savings". To see it clearly, think in terms of "buckets" or *stocks*, and consider the *flows* of funds between them. Whenever the government bucket runs a fiscal deficit, the nongovernment bucket—that is, the rest of the economy including firms and households—receives a surplus of funds:*the set of all households*.

The reverse is, of course, also true: whenever the government runs a fiscal surplus, the rest of the economy must be in deficit. And in practice, it is not obvious what determines the government's financial balance. For instance, the government might try to reduce its debt by decreasing spending and increasing taxes. But the rest of the economy might refuse—or simply be unable—to go into deficit and react by also cutting spending: wages for the firms and consumption for the households. What should the government do when this happens? Double down on debt reduction at the risk of triggering an austerity-driven recession?

The aim of this blog post is to make this type of macroeconomic reasoning more intuitive. It goes beyond the one-equation model of the world and introduces a Stock-Flow Consistent model with six equations and six variable—a sightly modified variant of *Model SIM: The Simplest Model with Government Money* in Wynne Godley's and Marc Lavoie's textbook on *Monetary Economics* (2006). My main contribution is to make this model explorable: you will be able to control various behavioural parameters using sliders, and observe the stock and flow variables evolve to new states on a dynamic representation of the economic system. Hopefully, this will help you refine your own understanding of simple macroeconomic phenomena, and contribute to putting the tired Government-Household analogy to rest.

Let's consider an economy made of three entities: the Government, a set of Firms and a set of Households. These entities own stocks and regularly exchange flows of funds over discrete economic cycles $0, 1, \ldots, i$. The model we are interested in rigorously accounts for the variation of the stocks and flows over time through *behavioural equations* describing the spending behaviour of each economic entity, and through *accounting identities* keeping track of each stock's inputs and outputs.

Let's start by familiarizing ourselves with the notations and the graphical representation of the model. For clarity, we exclusively use bold capital letters to refer to *stocks*, and bold small letters to refer to *flows*.

The Government

• It controls a flow of *expenditures* ${\color{g} \boldsymbol{g}}$ directed to the firms

• It controls a *tax rate* ${\color{theta} \theta}$ applied directly on the wages

• It deducts a flow of *tax revenues* ${\color{t} \boldsymbol{t}_i}$ from the households

• It owns a negative stock of *debt* ${\color{G} \boldsymbol{G}_i}$

The Firms

• They control a *level of spending on income* $\color{alpha} \alpha_1$

• They control a *level of spending on savings* $\color{alpha} \alpha_2$

• They pay *wages* ${\color{w} \boldsymbol{w}_i}$ directed to the households

• They own a positive stock of *savings* ${\color{F} \boldsymbol{F}_i}$

The Households

• They control a *level of spending on income* $\color{beta} \beta_1$

• They control a *level of spending on savings* $\color{beta} \beta_2$

• They spend on *consumption* ${\color{c} \boldsymbol{c}_i}$ directed to the firms

• They own a positive stock of *savings* ${\color{H} \boldsymbol{H}_i}$

At this stage it is important to distinguish between the *model parameters* $\; \color{g} \boldsymbol{g}$ $\; \color{theta} \theta$ $\; \color{alpha} \alpha_1$ $\; \color{alpha} \alpha_2$ $\; \color{beta} \beta_1$ $\; \color{beta} \beta_2 \;$ and the *model variables* $\; {\color{t} \boldsymbol{t}_i}$ $\; {\color{G} \boldsymbol{G}_i}$ $\; {\color{w} \boldsymbol{w}_i}$ $\; {\color{F} \boldsymbol{F}_i}$ $\; {\color{c} \boldsymbol{c}_i}$ $\; {\color{H} \boldsymbol{H}_i}$. The parameters are directly controllable by each economic entity; their values represent policy decisions (by the government) or individual choices (by firms and households). The variables are not directly controllable; their values are functions of the parameters. Since there are six unknown variables in the present case, fully determining the model will require six independent equations. Running a simulation will then consist in fixing the parameters and observing how the variables evolve over multiple economic cycles.

Already, we can make two important observations.

*1) Government expenditures $\color{g} \boldsymbol{g}$ are a parameter.*
As the currency issuer, the government controls its level of spending $\color{g} \boldsymbol{g}$ directly in an unconstrained way.

*2) The tax revenues ${\color{t} \boldsymbol{t}_i}$ are a variable.*
The government has direct control over the tax rate $\color{theta} \theta$ but this can translate into different levels of tax revenues ${\color{t} \boldsymbol{t}_i}$ depending on the size of the wages ${\color{w} \boldsymbol{w}_i}$.

* As we will see in the following, increasing government expenditures tends to result in higher tax revenues even when the tax rate remains fixed: government expenditures tend to finance themselves.*

Let's now describe the spending behaviour of the three economic entities in more details. For a start, we assume that the firms and households spend according to their disposable incomes and savings:

spending $=$ level of spending on income $\times$ disposable income + level of spending on savings $\times$ disposable savings.

The firms spend on wages ${\color{w} \boldsymbol{w}_i}$ controlled by a level of spending on income $\color{alpha} \alpha_1$ and a level of spending on savings $\color{alpha} \alpha_2$. Their disposable income are the government expenditures ${\color{g} \boldsymbol{g}}$ plus the consumption ${\color{c} \boldsymbol{c}_i}$ and their disposable savings are the previous stock ${\color{F} \boldsymbol{F}_{i-1}}$. Hence we can write: $${\color{w} \boldsymbol{w}_i} = {\color{alpha} \alpha_1} \, ({\color{g} \boldsymbol{g}} + {\color{c} \boldsymbol{c}_i}) + {\color{alpha} \alpha_2} \, {\color{F} \boldsymbol{F}_{i-1}} . \tag{1} $$ The households spend on consumption ${\color{c} \boldsymbol{c}_i}$ controlled by a level of spending on income $\color{beta} \beta_1$ and a level of spending on savings $\color{beta} \beta_2$. Their disposable income are the wages ${\color{w} \boldsymbol{w}_i}$ minus the taxes paid ${\color{t} \boldsymbol{t}_i}$ and their disposable savings are the previous stock ${\color{H} \boldsymbol{H}_{i-1}}$. Hence we can write: $${\color{H} \boldsymbol{c}_i} = {\color{beta} \beta_1} \, ({\color{w} \boldsymbol{w}_i} - {\color{t} \boldsymbol{t}_i}) + {\color{beta} \beta_2} \, {\color{H} \boldsymbol{H}_{i-1}} . \tag{2}$$ For the government, the level of expenditures $\color{g} \boldsymbol{g}$ is a parameter controlled directly. The tax revenues ${\color{t} \boldsymbol{t}_i}$ are extracted from the wages ${\color{w} \boldsymbol{w}_i}$ with a tax rate ${\color{theta} \theta}$ such that: $${\color{t} \boldsymbol{t}_i} = {\color{theta} \theta} \, {\color{w} \boldsymbol{w}_i} . \tag{3}$$ We now have one equation per flow variable. We focus next on the stock variables.

Here, the idea is to apply a mass balance approach to our economic system and explicitly keep track of each stocks' inputs and outputs. This is done through the simple accounting identity:

current stock $=$ previous stock $-$ spending $+$ revenues.

The firms spend on wages ${\color{w} \boldsymbol{w}_i}$ and receive expenditures ${\color{g} \boldsymbol{g}}$ and the consumption ${\color{c} \boldsymbol{c}_i}$. Hence we can write: $${\color{F} \boldsymbol{F}_i} = {\color{F} \boldsymbol{F}_{i-1}} - {\color{w} \boldsymbol{w}_i} + {\color{g} \boldsymbol{g}} + {\color{c} \boldsymbol{c}_i} . \tag{4}$$ The households spend on consumption ${\color{c} \boldsymbol{c}_i}$ and taxes ${\color{t} \boldsymbol{t}_i}$ and receive wages ${\color{w} \boldsymbol{w}_i}$. Hence we can write: $${\color{H} \boldsymbol{H}_i} = {\color{H} \boldsymbol{H}_{i-1}} - {\color{c} \boldsymbol{c}_i} - {\color{t} \boldsymbol{t}_i} + {\color{w} \boldsymbol{w}_i} . \tag{5}$$ The government spends on expenditures ${\color{g} \boldsymbol{g}}$ and receives tax revenues ${\color{t} \boldsymbol{t}_i}$. Hence we can write: $$ {\color{G} \boldsymbol{G}_i} = {\color{G} \boldsymbol{G}_{i-1}} - {\color{g} \boldsymbol{g}} + {\color{t} \boldsymbol{t}_i} . \tag{6} $$ We now have a system of six equations and six variables and the model is almost ready to be simulated.

One particularity of the system of equations defined above is that it is recursive, meaning that the state of the model for an economic cycle $i$ depends on its state for the previous economic cycle $i-1$, which in turn depends on its state for previous the economic cycle $i-2$, etc. In practice, this means that initial stock values ${\color{F} \boldsymbol{F}_0} \;\; {\color{H} \boldsymbol{H}_0} \;\; {\color{G} \boldsymbol{G}_0}$ need to be defined, from which all the successive states of the model will result. For simplicity, the initial stocks are assumed to be null:
$${\color{F} \boldsymbol{F}_0} = {\color{H} \boldsymbol{H}_0} = {\color{G} \boldsymbol{G}_0} = 0 . \tag{0}$$
A couple of observations follow directly. First, *the sum of stocks remains null at all economic cycles $i$*. Indeed, summing equations $(4)+(5)+(6)$ shows that the sum of stocks remains constant over time:
$$ {\color{F} \boldsymbol{F}_i} + {\color{H} \boldsymbol{H}_i} + {\color{G} \boldsymbol{G}_i} = {\color{F} \boldsymbol{F}_{i-1}} + {\color{H} \boldsymbol{H}_{i-1}} + {\color{G} \boldsymbol{G}_{i-1}} . $$
This is just another formulation of Wynne Godley's "one-equation model of the world": *in a closed economy, a sector's deficit is necessarily another sector's surplus.* And this is true in particular for the initial state:
$$ {\color{F} \boldsymbol{F}_i} + {\color{H} \boldsymbol{H}_i} + {\color{G} \boldsymbol{G}_i} = {\color{F} \boldsymbol{F}_0} + {\color{H} \boldsymbol{H}_0} + {\color{G} \boldsymbol{G}_0} = 0 .$$
Second, *the government debt is the negative of the firms' and households' savings*. This is simply obtained by rearranging the equation above:
$$ {\color{G} \boldsymbol{G}_i} = - {\color{F} \boldsymbol{F}_i} - {\color{H} \boldsymbol{H}_i} .$$
A corrolary to which we'll come back later is that *a high government debt is not necessarily the sign of excessive government spending, but can also be the sign of excessive saving by other economic entities.*

Our model is a system of six recursive equations with a null initial state, and we can compute the values of the six variables $\; {\color{t} \boldsymbol{t}_i}$ $\; {\color{G} \boldsymbol{G}_i}$ $\; {\color{w} \boldsymbol{w}_i}$ $\; {\color{F} \boldsymbol{F}_i}$ $\; {\color{c} \boldsymbol{c}_i}$ $\; {\color{H} \boldsymbol{H}_i}$ at any economic cycle $i$ given predefined values of the behavioural parameters $\; \color{g} \boldsymbol{g}$ $\; \color{theta} \theta$ $\; \color{alpha} \alpha_1$ $\; \color{alpha} \alpha_2$ $\; \color{beta} \beta_1$ $\; \color{beta} \beta_2$. This is what the interface below lets you explore in real time.

The graphical representation on the leftat the top is dynamically adjusted to show the current state of each variable. The stocks are represented by vertical bars of varying heights and the flows are represented by arrows of varying widths. The variables are also plotted as functions of time on the rightbelow.

The parameters are directly controllable using sliders, and are initialized with reasonable values. The government spending is average, the tax rate is low, the firms and households spend most of their incomes and a lower proportion of their savings: $$\quad \begin{array}{ccc} {\color{g} \boldsymbol{g} = 20} \quad & {\color{alpha} \alpha_1 = 0.8} \quad & {\color{beta} \beta_1 = 0.7}\\ {\color{theta} \theta = 0.2} \quad & {\color{alpha} \alpha_2 = 0.4} \quad & {\color{beta} \beta_2 = 0.2} \end{array}\quad.$$

Go ahead and explore the model yourself! You can start by clicking on the "reset stock" button to reset the simulation and observe the model converge to its equilibrium state. Or you can increase government expenditures by moving the slider ${\color{g} \boldsymbol{g}}$ to the right and observe the model converge to a new equilibrium state, with larger flows and stocks. If you're not sure what to try, a guided tour is presented in the next section.

**An empty economy.**

${\color{g} \boldsymbol{g} = 0}$

Without government expenditures $\color{g} \boldsymbol{g}$, there is no economy.

**The government starts spending.**

${\color{g} \boldsymbol{g} = 20}$

Government expenditures $\color{g} \boldsymbol{g}$ turn into wages $\color{w} \boldsymbol{w}_i$, which then turn into tax revenues $\color{t} \boldsymbol{t}_i$: spending enables taxation.

**The government reduces the tax rate.**

${\color{theta} \theta = 0.2}$

Part of the wages $\color{w} \boldsymbol{w}_i$ is now used for consumption $\color{c} \boldsymbol{c}_i$ which then turns into more wages $\color{w} \boldsymbol{w}_i$ triggering a positive feedback loop.

**Firms start saving up.**

${\color{alpha} \alpha_1 = 0.8 \quad \alpha_2 = 0.4}$

Part of the firms' income ${\color{g} \boldsymbol{g}} + {\color{c} \boldsymbol{c}_i}$ starts feeding into the firms' stock $\color{F} \boldsymbol{F}_i$, temporarily decreasing the wages $\color{w} \boldsymbol{w}_i$ and the tax revenues $\color{t} \boldsymbol{t}_i$, producing government debt $\color{G} \boldsymbol{G}_i$.

**Households start saving up.**

${\color{beta} \beta_1 = 0.7 \quad \beta_2 = 0.2}$

Part of the households' income ${\color{w} \boldsymbol{w}_i} - {\color{t} \boldsymbol{t}_i}$ starts feeding into the households' stock $\color{H} \boldsymbol{H}_i$, temporarily decreasing the consumption $\color{c} \boldsymbol{c}_i$, the wages $\color{w} \boldsymbol{w}_i$ and the tax revenues $\color{t} \boldsymbol{t}_i$, producing more government debt $\color{G} \boldsymbol{G}_i$.

**Firms and Households save excessively.**

${\color{alpha} \alpha_1 = \alpha_2 = 0.1} \quad {\color{beta} \beta_1 = \beta_2 = 0.1}$

Most of the firms' and households' incomes feed into their respective stocks $\color{F} \boldsymbol{F}_i$ and $\color{H} \boldsymbol{H}_i$, drying up the wages $\color{w} \boldsymbol{w}_i$, the consumption $\color{c} \boldsymbol{c}_i$ and the tax revenues $\color{t} \boldsymbol{t}_i$, leading to an ever increasing government debt $\color{G} \boldsymbol{G}_i$.

After exploring the model for a while, it becomes clear that the variables converge to an equilibrium state for most combinations of parameter values when given enough time. By definition, such an equilibrium state is reached when all the variables remain constant over time, satisfying: \begin{equation*} \begin{cases} {\color{t} \boldsymbol{t}_{i-1}}\\ {\color{w} \boldsymbol{w}_{i-1}}\\ {\color{c} \boldsymbol{c}_{i-1}}\\ {\color{F} \boldsymbol{F}_{i-1}}\\ {\color{H} \boldsymbol{H}_{i-1}}\\ {\color{G} \boldsymbol{G}_{i-1}} \end{cases} = \begin{cases} {\color{t} \boldsymbol{t}_i}\\ {\color{w} \boldsymbol{w}_i}\\ {\color{c} \boldsymbol{c}_i}\\ {\color{F} \boldsymbol{F}_i}\\ {\color{H} \boldsymbol{H}_i}\\ {\color{G} \boldsymbol{G}_i} \end{cases} = \begin{cases} {\color{t} \boldsymbol{t}_*}\\ {\color{w} \boldsymbol{w}_*}\\ {\color{c} \boldsymbol{c}_*}\\ {\color{F} \boldsymbol{F}_*}\\ {\color{H} \boldsymbol{H}_*}\\ {\color{G} \boldsymbol{G}_*} \end{cases} . \tag{$\ast$} \end{equation*} The equilibrium variables can then be expressed solely as functions of the parameters by substituting the set of equalities $(\ast)$ into equations $(0$-$6)$. For instance, equation $(6)$ becomes: $${\color{G} \boldsymbol{G}_*} = {\color{G} \boldsymbol{G}_*} - {\color{g} \boldsymbol{g}} + {\color{t} \boldsymbol{t}_*} \quad \Rightarrow \quad {\color{t} \boldsymbol{t}_*} = {\color{g} \boldsymbol{g}} \,.$$ The full equilibrium state can be expressed as follow: $$\bbox[#fafafa, 15px, border:2px solid #e6e6e6]{ \begin{aligned} {\color{t} \boldsymbol{t}_*} &= {\color{darkG} \boldsymbol{g}} \phantom{\frac{\beta_1}{\beta_1}}\\ {\color{w} \boldsymbol{w}_*} &= {\color{darkG} \frac{\boldsymbol{g}}{\theta}} \phantom{\frac{\beta_1}{\beta_1}}\\ {\color{c} \boldsymbol{c}_*} &= {\color{darkG} (1-\theta) \, \frac{\boldsymbol{g}}{\theta}} \phantom{\frac{\beta_1}{\beta_1}}\\ {\color{F} \boldsymbol{F}_*} &= {\color{darkF} \frac{1-\alpha_1}{\alpha_2}} \, {\color{darkG} \frac{\boldsymbol{g}}{\theta}} \phantom{\frac{\beta_1}{\beta_1}}\\ {\color{H} \boldsymbol{H}_*} &= {\color{darkH} \frac{1-\beta_1}{\beta_2}} \, {\color{darkG} (1-\theta) \, \frac{\boldsymbol{g}}{\theta}} \phantom{\frac{\beta_1}{\beta_1}}\\ {\color{G} \boldsymbol{G}_*} &= - \left[{\color{darkF} \frac{1-\alpha_1}{\alpha_2}} + {\color{darkH} \frac{1-\beta_1}{\beta_2}} \, {\color{darkG} (1-\theta)}\right] \, {\color{darkG} \frac{\boldsymbol{g}}{\theta}} \end{aligned}}$$

And a number of important observations can be made about this equilibrium state.

• *Government expenditures finance themselves.*

In a closed economy at equilibrium, ${\color{t} \boldsymbol{t}_*} = {\color{darkG} \boldsymbol{g}}$. All government expenditures eventually come back in the form of tax revenues, irrespectively of the tax rate. Higher government expenditures simply generate higher tax revenues. In the long term, the tax rate does not control the amount of funding that the government is able to raise, but rather the speed with which a given expenditure comes back in the form of tax revenues.

• *The government controls the size of the economy.*

All the variables other than the tax revenues are proportional to the ratio $\displaystyle {\color{darkG} \frac{\boldsymbol{g}}{\theta}}$, which is directly controlled by the government. The wages, consumption and all the stocks grow when government expenditures increase or the tax rate decreases (stimulus), and they all shrink when government expenditures decrease or the tax rate increases (austerity).

• *The government debt depends on the saving behaviour of firms and households.*

The government debt ${\color{G} \boldsymbol{G}_*}$ depends on the coefficients ${\color{darkF} \frac{1-\alpha_1}{\alpha_2}}$ and ${\color{darkH} \frac{1-\beta_1}{\beta_2}}$, which are controlled by firms and households respectively. Ultimately, the government debt would vanish entirely if firms and households stopped saving: setting ${\color{alpha} \alpha_1} = {\color{beta} \beta_1} = 1$ results in ${\color{G} \boldsymbol{G}_*} = 0$.

• *Taxes and spending are necessary to balance the economy.*

The equilibrium state is undefined if $\,{\color{theta} \theta} = 0\,$ or $\,{\color{alpha} \alpha_2} = 0\,$ or $\,{\color{beta} \beta_2} = 0$ (these values result in a division by zero). Without taxes, the government debt increases indefinitely. The same happens when Firms and Household stop spending. One simple policy reform that would guarantee that a small proportion of savings is always spent is to implement a wealth tax.

The model presented in this blog post is, of course, extremely simplistic. It was designed as a starting point, a student-friendly introduction to fully coherent stock-flow macroeconomics. In the words of the authors themselves, "very strong simplifying assumptions will have to be made initially and the reader is asked to suspend disbelief until more realistic systems are introduced". These more realistic systems introduced later include a banking sector with private money, the consideration and adequate accounting of price inflation, as well as the consideration of open economies. The textbook is relatively technical, but I strongly encourage interested readers to explore it further!