Primer: Understanding Stock – Flow Norms

A stock–flow norm is a relationship between stock and flow variables within an economic model, which we assume also holds true for the real economy. Normally, we do not expect the holdings of an asset to become "too large" relative to incomes. They are core behavioral relationships within stock-flow consistent (SFC) models. Within this article, I explain how these stock – flow norms work within these models. I will discuss the implications of stock-flow norms in later articles, but I note that they are incredibly important for the analysis of government debt dynamics.



The models I present here are based on those within the text Monetary Economics by Godley and Lavoie, from Chapter 2. I have simplified the notation. It should be noted that these concepts are similar to those found within earlier Keynesian models. Since I want to focus on how stock-flow norms work within these models, I do not want to get distracted with the history of the concept. The text discusses the history as well as giving further references.

What Are Stocks And Flows?

A stock variable is an economic variable that is a quantity (and not a parameter) that is measured at the end of an accounting period. For example, variables that would appear on the balance sheet, such as money holdings or inventory at the end of the month.

A flow variable is somewhat harder to define. For variables associated with money, flows are transactions that change of a monetary stock variable during accounting period. For example, variables that would appear on income statements. For the other side of transactions (since money is exchanged for goods or services), things are more complex. Flows for goods would be a change of status of the number of goods over a period, such as a the number of cars sold (which affects the stock of inventory). However, there are no stocks associated with transactions in services.

(A fuller definition is given on Wikipedia. Note that the previous paragraph was corrected thanks to the input of Ramanan in the comments.)

For bond market economics, the classic stock variable is the amount of government debt outstanding, and the corresponding flow variable is the government deficit over a time period. Also, GDP is a flow variable. Therefore, a debt-to-GDP ratio is the ratio of a stock to a flow variable. Many economists warn that you should not mix stock and flow variables. Correspondingly, a debt-to-GDP ratio looks like a mix of concepts. However, it is necessary to tie the relationships between flow variables to the relationships between stock variables, and therefore economic models have to include something that tells us about those ratios.

Inventory Ratios

Stock-flow norms can be expressed in real terms. An important example is the inventory-sales ratio. The chart above shows the inventory/sales ratio of motor vehicles and parts inventories (at the wholesale level) in the United States. As an example, if the ratio is 1.4, that means it would take 1.4 months of sales at that current rate to clear that current level of inventory.

Production decisions by businesses are not modelled very well in mainstream economic models (that is, Dynamic Stochastic General Equilibrium – DSGE – models). Those models focus on household behaviour as the driver for the economy. Within those models, businesses adjust prices to prevent a rise in inventories (which are often assumed to be zero). Although this may happen in some industries, such as the commodity industry, this does not appear to reflect reality very well, as shown above. The spectacular rise in auto inventories during the last recession was not the result of deliberate choice by automakers.

Stock-flow consistent models are more complete, as they attempt to create behavioural rules for every sector of the economy. Within my models (which are fairly basic adaptations of the models in the Godley and Lavoie textbook) my model business sectors adjust production — hence employment — based on the amount of inventory they hold relative to sales. This is a thus stock-flow norm embedded in the behaviour of businesses.

Household Consumption

The most visible application of stock–flow norms is in the area of modelling the consumption function of the household sector. This is a large area of research and economic squabbling.

A typical consumption function for the household sector is given by (using my non-standard notation):

C = (1-s)*I + d*W;

where:
C = consumption,
s = savings rate,
I = total income = wage income + dividend income + interest income,
d = wealth drawdown,
W = wealth.

The fact that I include interest income within total income income is a modelling choice that affects the dynamics. If it is not included in the income measure, household asset/income ratios will become dependent upon the rate of interest.

This consumption function is commonly called the Modigliani consumption function.

For simplicity, let's assume that households only hold bank deposits as financial assets. Since I have included interest as part of income, the change in financial assets (bank deposits) is given by:

ΔW = I - C.

If we substitute the consumption equation into previous, we get:

ΔW = s*I - d*W.

We can then recast this to:

ΔW =I*d*(s/d - W/I).

The ratio W/I is the wealth to income ratio. We can interpret this last equation as an adjustment of the wealth to income ratio towards a target ratio which is given by s/d. If the wealth to income ratio already equals the target, and income is not changing, wealth will not change. If the wealth to income ratio is below target, there will be positive net savings which will push the wealth to income ratio towards target.

The linearity of the consumption function creates a duality between two possible models of household behaviour. We could either specify consumption directly, or we can specify household behaviour in terms of attempting to reach a target level of wealth. If we make our behavioural relationships more complex (nonlinear) the duality will probably break down. For example, if we have a more complex consumption function, the wealth to income ratio may not tend towards a constant value.

(As an aside, these dynamics are first order linear system, which also describes the standard Adaptive Expectations model, as discussed in this primer.)

Example Simulation Results

Within a full stock-flow consistent model, the results are complicated by the impact of the different sectors upon each other. I will give an example of how household savings react to a step change in income, but I assume that this steady income can be maintained despite the interaction of the household with the other sectors within the economy. The parameters for the household sector behaviour are:

1. The household saves 20% of income, putting the savings into money holdings.
2. It draws down (spends) 10% of its money holdings.
3. Money does not pay interest (for simplicity).
4. Income before t=0 was $100/period, which then jumps to $110 at t=0.

(Note: these parameters are not meant to be realistic estimates.)

The chart above shows the simulation results. The ratio of the savings rate to the drawdown rate was 2:1, which is why the initial stock of savings was $200. At t=0, the amount of spending rose above the amount of money drawn down, and so the stock of money saved rises. The stock of money asymptotically converges towards $220 (=2×$110).

Impact Of Income Growth

One key property of these models is that the ratio of asset holdings to income drops as the rate of growth of nominal income rises. In the chart below, I show what happens if the initially static income of the household (growth rate of 0%) rises to 5% per period (keeping other parameters constant).

Although the stock of savings rises, the stock-flow ratio drops from an initial 200% to converge towards a new steady state of 140% of income. This behaviour means that if you want to specify the consumer behaviour in terms of seeking a target assets-to-income ratio within a linear model, the parameters are dependent upon the assumed nominal income growth.

(UPDATE) It is possible to define SFC models where the steady state debt-to-GDP ratio is stable, even if nominal GDP growth rates change. This comes from an assumption that households attempt to stabilise their real assets versus real income. As Ramanan noted in the comments, there is such a model in Chapter 11 of the Godley and Lavoie text. However, the dynamics of household behaviour in that model are more complex than what I want to discuss within this article.

In practice, the dependence of the assets-to-income ratio to nominal growth rates is strong. In my view, it is no accident that the government debt-to-GDP ratio moved inversely to nominal GDP growth rates throughout the developed world during the post-WWII era, as government debt is an important component of private sector assets. I will cover this topic more fully in later articles.

Comparison To Mainstream Models

In modern mainstream models (DSGE models), household savings behaviour is specified in terms of the solution of an optimization problem. Households are assumed to optimize utility over time. The simple behavioural rules used above look primitive relative to an optimizing framework.

In practice, the difference is smaller than might appear. If we created two parallel models, the consumption function used above would probably be a very good first order approximation of the behaviour of an optimizing household over a reasonable time interval. Additionally, even optimising models need to take into account that there are wide divergences between savings patterns between the different age cohorts, and so aggregate behaviour will deviate from what looks optimal for an individual household.

We would need fairly extreme changes in economic behaviour to create significant deviations in the behaviour between the models. I suspect that the differences would be larger in an inflationary environment. But if we were calibrating to real-world data at present, it would be hard to distinguish the model classes.

On the other hand, SFC models can be solved without relying on fairly dubious assumptions and linearisations. Correspondingly, if you want to actually get a model that you can use, the best line of attack is to look into stock-flow consistent modelling.

See Also:

• The text Monetary Economics by Wynne Godley and Marc Lavoie offers an excellent introduction to SFC models. (Amazon affiliate link.)
• My "theme article" with links to my main articles on SFC models.
• sfc-models.net

 (c) Brian Romanchuk 2014