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Wednesday, September 10, 2014

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

Chart: U.S. Auto Inventory-Sales Ratio

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;
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

Chart: Simulation Of The Impact Of An Increase In Income

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).
Chart: Impact Of Increases Income Growth On The Asset-Income Ratio

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:

 (c) Brian Romanchuk 2014


  1. "A stock variable is an economic variable that is a quantity (and not a parameter) that is measured at the end of a 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 an economic variable that is the change of a stock variable during accounting period. For example, variables that would appear on income statements."

    That's an incomplete description of flows because not all flows are changes in stocks, although flows affect changes in (other) stocks. For example, consider consumption: it is not a change in the stock of anything. Also GDP itself: it is not the change in the stock of something.

    To see this consider a pure service economy. The output is not a change of the stock of something, although the output is a flow.

    1. Thanks, I will take a look at re-writing that. My thinking was t focus on the monetary side of the transaction, the flows would result in a change of balance sheet entries. The other side of the transaction would be harder to characterise. I did not want to write a 300 word definition of what stocks and flows are within a 1000 word description of stock-flow norms. I'll probably link to a better description,

  2. Also about your main point on stock-flow norms, can you see Sec 11.7.3 of Godley/Lavoie's book and Fig 11.3C? There the government debt to GDP doesn't seem to change in the long run. (Although the expression at the end of the chapter has the norm depending on the growth rate).

    1. Thanks, it was awhile since I read the text, and had not noted that example. When I look at it, it appears the mechanism would be the fact that behaviour is driven by real parameters, and not parameters on nominal quantities, such as I used. I did note that a more complex specification could lead to a stable debt-to-GDP ratio, but I will add the reference. However, my view is that the inverse dependence upon nominal growth rates fits the data better, wven if it does not match the priors of economists.

  3. (Sorry if this appears multiple times)

    Also don't forget that in open economies, the point on stock-flow norms can work the reverse. It is quite possible that output rises fast but debt/gdp keeps rising.

    So there is no general principle about stock-flow norms and growth.

    1. Even in a closed economy, the government debt-to-GDP ratio will change if there is a change in the distribution of income. In fact, that's what my next SFC article will be about. This article is meant as a basic explanation of the dynamics of one sub-sector of the economy, without worrying about the interaction with other sectors. My comments on the government debt-to-GDP ratio were more an indication of why this matters, without getting too far into the details.

    2. Im sorry for my question but my mind got so blurry.
      It is possible that stock over flow ratios keep rising altough the flow variable rise fast. So wouldnt it be meaningless to interpret this ratios? or would it be a trouble to adobt this ratios in an empirical model?

    3. The ratios are hard to interpret, but I guess they can be useful under some circumstances. I do not think you can conclude too much about them, without adding other qualifiers. For example, debt-to-income (with GDP being national income) ratios are affected by growth rates; the ratios tend to be higher when nominal growth rates are low.

  4. "However, there are no stocks associated with transactions in services."

    (I think this is a late addition as a result of the Ramanan comment.)

    This wording creates a worry for me because the stock of money does still change for each party to the transaction. I think you intend that the money stock change would be an unstated and automatic assumption

    This is an important point for me BECAUSE I believe that new government debt first appears as a change to GDP AND TO THE MONEY SUPPLY when government first pays for labor (a service). If this is correct, then the stock of money increases when time (the essence of service) is traded for new money.

    I would agree that if a person already has money, the trade of money for service is a trade of money stock for time but no new money nor average stock measure is changed.

    1. I may have to look at the wording again, but the idea is that a transaction in a monetary economy consists of
      (a) Person A gives $100 to Person B [monetary side]
      (b) Person B provides a service to Person A. [real economy side]

      The (a) side is associated with a change in monetary stocks. But on the (b) side, there is no stock of "services". If Person B provided something tangible like a television, then the number of televisions in a stock variable - inventory - would change.

      As for your example, a government purchase will add to GDP, and it will also result in a change in government liabilities. These transactions will show up in two sets of related but distinct accounting frameworks - the GDP accounts, and the Flow of Funds (financial flows).

      However, not all government spending will directly add to GDP, and so there are limits to the mechanical link between government spending and GDP. For example, a transfer payment to a retired person does not add to GDP, but it would expand government liabilities. But if that person then spends the money, that spending would add to GDP (personal consumption), but even that would be reduced if the spending reduced inventories (which is a negative investment).

  5. "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)."

    I am wondering if there is a typo here. When I multiply the top equation by 1/d*I and rearrange, I come up with

    ΔW =I*d*(s/d - W/I).

    Thanks for the thought-provoking blog post.

    1. You are correct, I updated the text. I was spending so much time deciding how to format the equation in a text editor that apparently I didn't bother double-checking what I wrote.


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