GDP is a flow and is the sum of the all expenditure flows over a given period

I have two days of teaching left in Helsinki and my next stop on Friday is Dublin where I will be discussing unification and exit. Should be a fun topic. Its Wednesday back home already and today I consider a matter that came up in one of my classes that I am taking in macroeconomics at the moment at the University of Helsinki. Students really struggle when first introduced to the idea of a stock and a flow. They can easily be led into defining a flow as a stock. Getting this absolutely right is one of the key building blocks in understanding basic macroeconomics and the links between the expenditure system and financial accumulation. Modern Monetary Theory (MMT) builds heavily on the difference between stocks and flows and is also what we call stock-flow consistent. So all flows that inform stocks are accounted for in a consistent way. So, for example, we know that when households save, which is the residual of disposable income that is not consumed and a flow, this accumulates into a stock of financial wealth. Today, I am seeking to clarify the issue in my class that we did not have sufficient time to deal with in detail last week. And after that, some music to restore sanity.

GDP is a flow

In my classes last week at Helsinki University we discussed stocks and flows and how they fed into understanding national accounting and financial balances (flow of funds).

I was asked by a student to comment on the so-called ‘bath tub’ analogy that was often used in mainstream macroeconomics textbooks and is still in vogue in some circles.

Within the core MMT group this analogy came up some years ago and my comment then was that it was not a valid representation of the concepts being discussed because it confused stocks with flows.

It also cannot reasonably be used to establish definitive causalities between injections into and leakages from the spending stream, because of the interdependence between the two types of flows.

Here is why.

It was first proposed in Kenneth Boulding’s 1946 book – The Economics of Peace (Prentice-Hall) – where he outlined the basic Keynesian model in the context of the Post WW2 reconstruction requirements in Europe and Asia, which he described as the challenge of “capital rebuilding”.

Capital was defined as:

… the sum total of the valuable things possessed by the individuals of a society, excluding ‘claims’, that is, mere titles to property. The word is used to mean both the inventory of these valuable things – the houses, factories, machines, livestock, stocks of raw materials, and goods in all stages of production – and also to meant he sum of the values of these things.

So Kenneth Boulding was really talking about what economists call ‘stocks’.

It gets tricky here.

He says that:

If, for instance, production in a community is worth 1,000,000 dollars a year, and consumption amounts to 900,00 dollar’ worth a year, there will be a net increase of 100,000 dollars in the value of the stock of capital. Thus a community can increase its rate of accumulation of capital only by increasing production, by decreasing consumption, or by some combination of both.

He says that “this difference is sometimes called ‘saving’, but since this word is ambiguous, it should be avoided as far as possible.

He later noted that ‘saving’ resulted from “the difference between earnings and spendings … an increase in the holdings of money. There is no point in an individual merely accumulating money, that is, cash, beyond a certain point”.

He then says that individauls will make portfolio choices and perhaps “buy securities of some kind”.

So implicit in this construction is the notion that income flows (more about which later) that are not fully recycled back into the spending stream and thus constitute what we consider to be a flow of saving in each period, form the basis of wealth accumulation (a stock of financial assets).

So in this case: income minus consumption equals saving (all flows).

Saving then accumulates to wealth (a stock), which can be held in different forms.

So you can see that Kenneth Boulding is jumping from a difference between flows to an accumulation of a stock, which is fine in the context he is working in – a concentration on wealth construction.

In Chapter 7 Unemployment: The Problem, Kenneth Boulding rehearses the basic Keynesian model that was becoming mainstream in the academy by this point.

This is where he introduces his “Bathtub Theorem”.

He writes in terms of quantities:

The rate of accumulation of the stockpile of goods is equal to the rate of production less the rate of consumption … just as the rate at which water accumulates in a bathtub is the difference between the rate at which it runs in from the faucet and the rate at which it runs out through the drain, so the rate of accumulation of goods is equal to the difference betwen the rate at which goods are added to the stock and the rate at which goods are drained away … If production (the flow from the faucet) is greater than the consumption (the flow down the drain) it is clear that the water in the economic bathtub (the total stock of goods) must rise.

He then applies the ‘Theorem’ to the monetary system – where “expenditure is income”. He rehearses the argument that what applies to the individual “are not true of the system as a whole” – this is the standard Keynesian argument about the fallacy of composition.

He writes to illustrate the point:

To an individual, his money income flows into his pocket, again rather as water flows into a bathtub, and his money expenditure flows out of his pocket just as water disappears down the drain The amount in his pocket (or his bank balance) obviously depends on the relative size of income and expenditure; when he is getting money faster than he is spending it, the amount of money in his pockets increases, just as a bathtub will fill up if water is running in faster than it runs out ….

When we look at our whole society, however, and add up all incomes of all people, and add up all the expenditures of all people, it is clear … that not only must these totals be equal, but that they are simply different ways of looking at exactly the same thing!

We don’t see much more of the ‘bathtub’ after this.

The point is that the individual with a net inflow of income (a flow) can save (a flow) and accumulate those savings in the form of a bank balance (a stock).

The ‘bathtub’ analogy was meant to describe the initial income flows (in and out) and the way in which the difference between the two flows would accumulate as wealth (or a water storage) in the bath.

Using it that way is fine but somewhat confusing as I will explain.

But it should be clear that Kenneth Boulding was not thinking of the bathtub level as national income. He sort of fudges a step in the process of accumulation and many people after him didn’t seem to get the point that he was talking about a stock of wealth rather than a flow of income.

Trying to conceive of the bathtub water level as GDP is therefore not valid.

In macroeconomics, we define a ‘steady-state’ or ‘equilibrium’ as a situation where there are no forces present to change the current flow of output and income being produced.

This concept of ‘equilibrium’ is not defined as a ‘market clearing’ state, which means it does not necessarily mean that all the productive resources are being employed.

An economy can come to rest in the macro sense and be plagued by elevated levels of persistent unemployment.

This is why Keynes advocated ‘exogenous’ injections of government spending and/or reductions in tax rates in order to provide an external stimulus to the economy when it is stuck in a undesirable ‘steady state’.

The basic macroeconomic relations tell us that total income (GDP) = total spending = total output.

If spending changes so does output and income in the same direction.

Which should disabuse anyone of the IMF type claims during the GFC that a nation could enjoy a fiscal contraction expansion. That was claptrap.

So equilibrium means GDP remains constant and we can define this in terms of the condition that total injections into the spending stream equal total leakages from the spending stream.

The idea is that in any period, spending is creating income which then leads to household saving (leakage), government taxation rising (leakage) and import spending (leakage).

If there was no offsetting injections into the expenditure stream next period, these leakages would mean that the initial income created would not fully cycle back into the spending stream.

So, to balance the leakages, spending injections have to come from outside the cycle in the form of government spending, business investment and export revenue.

When the leakages equal the injections then the system comes to rest.

Note also that these leakages are themselves dependent on the level of income (and hence spending).

Further, we consider that Gross Domestic Product (GDP) to be the sum of the expenditure coming from Household consumption, Private investment (capital formation), Government spending (recurrent and capital) and Exports minus Imports.

The national account expenditure components (consumption, investment, government spending, exports minus imports) are flows and sum to GDP, which is also a flow.

GDP – that is the market value of all final goods and services produced in a period – is a flow of dollars (or whatever currency that is relevant).

The leakages from the income-expenditure system are also flows – saving from disposable income, taxation revenue to government and import expenditure that flows to the rest of the world.

A flow is measured as a quantity over time, not at a point in time.

A stock is a level measured at a point in time.

So your weekly income is a flow, while your bank balance (if you are lucky enough to have one) is a stock.

So why is the ‘bath tub’ analogy problematic.

The water in a bath or a reservoir is a stock not a flow.

Mountain streams are channelled into dams (reservoirs) to accumulate water. The streams are flows, the accumulation is a stock.

The bath tub analogy constructs the spending injections as water flows into the bath coming from the taps.

The leakages are characterised in the analogy as water draining from the bath via the drain hole.

And, according to the analogy, if the flows in from the tap equals the flows out into the drain, then GDP will be stable.

Which is where the analogy fails.

The failure is because it characterises GDP as a stock – a reservoir of water in the bath.

Considering GDP to be a stock is a basic error that students often make.

Clearly, the ‘bath tub’ analogy is not a good pedagogic device because it perpetuates the confusion that students have in terms of stocks and flows.

On the one hand we go to great length to teach students that GDP is a flow of spending, income and output, but then they come across the ‘bath tub’ analogy which tells them that GDP is the water level in a bath, which is a stock.

A better (but still not perfect) analogy is that is that the expenditure-income process is like a river flowing endlessly into some unspecified horizon.

Accordingly, the injections are the flows into the river from its source (or tributaries) and the leakages are the outflows from the river to other places.

The level of the river will rise and fall according to the inflows and outflows but it will still be flowing.

It also allows us to understand dynamic shifts in equilibrium as the volume of water moving down the river increases and decreases according to inflow shocks and the adjustments that follow via the leakages.

Further, trying to isolate causality between the injections (say, government spending) and leakages (say, tax revenue) is impossible in this type of framework.

That is because there are three injections in the standard framework pushing water into the GDP flow which generates income, which triggers the three leakages.

Which spending source caused the income that allows taxes to be paid, for example? Taxes flow from the flow of income.


I didn’t get time in the class last week to clarify these details.

I hope that sets the record straight.

Boulding’s bathtub theorem is about accumulation not income determination.

European and UK Teaching and Speaking Tour, February 2020

  • Monday, February 03, 2020 – Meetings in Helsinki.
  • Tuesday, February 04, 2020 – Teaching, University of Helsinki – 16.15-17.45 Porthania P674 – all lectures are public.
  • Wednesday, February 05, 2020 – Teaching, University of Helsinki – 10.15-11.45, Language Centre in Fabianinkatu room 207.
  • Thursday, February 06, 2020 – Teaching, University of Helsinki – 10.15-11.45 Main building, Hall 16.
  • Friday, February 07, 2020 – Presentation at Italian Parliament, Rome – details to follow.
  • Saturday, February 08, 2020 – Events in Rome with activists – details to follow.
  • Sunday, February 09, 2020 – Travel.
  • Monday, February 10, 2020 – Breakfast presentation – ‘The future of monetary policies’ – Nordic West Group, Helsinki (closed event). Later in the day: Speaking on ‘What is the meaning of political economy today?’ at Think Corner, Helsinki – 17:00 to 19:00 (public event).
  • Tuesday, February 11, 2020 – Teaching, University of Helsinki – 16.15-17.45, Porthania room 723.
  • Wednesday, February 12, 2020 – Teaching, University of Helsinki – 10.15-11.45, Language Centre in Fabianinkatu room 207.
  • Thursday, February 13, 2020 – Teaching, University of Helsinki – 10.15-11.45 Main building, Hall 16.
  • Friday, February 14, 2020 – Meetings, Dublin.
  • Saturday, February 15, 2020 – Speaking at public event at Wynn’s Hotel, 35-39 Abbey Street Lower, North City, Dublin – 14:00 to 17:00. All Welcome.
  • Sunday, February 16, 2020 – Athens.
  • Monday, February 17, 2020 – Meetings and Presentations, Athens.
  • Tuesday, February 18, 2020 – Paris, Reception, French Senat, Palace of Luxembourg – 18:00.
  • Wednesday, February 19, 2020 – Presentation to the French Treasury, Closed event. 9:00 to 12:00. Afternoon – press interviews.
  • Thursday, February 20, 2020 – Paris, Presentation to French Senate Commission, Palace of Luxembourg – 8:30-10:30.
  • Thursday, February 20, 2020 – London, GIMMS presentation, MMT education – afternoon – Details.
  • Friday, February 21, 2020 – Manchester, GIMMS presentation, The Harwood Room in the Barnes Wallis Building, University of Manchester, Details.
  • Saturday, February 22, 2020 – MMTed Masterclass Workshop, London, for Details and Tickets. Limited spaces available.
  • Sunday, February 23, 2020 – Amsterdam – private meetings.

GIMMS Events, London and Manchester, February

I encourage you to support these public events in the UK:

1. February 20, 2020 – I will speaking in London about the recent political events in the UK and how an understanding of MMT is essential to rebuild a progressive political force in Britain. Criminologist Steve Hall will also talk and will focus on the current rise of populism in the West.

The event will be at the Unite the Union (Diskus Theatre) in central London and will run from 13:30 to 17:00.

For – Details.

2. February 21, 2020 – The same show moves to Manchester.

The event will be at the Barnes Wallis Building (The Harwood Room) at the University of Manchester and will run from 13:30 to 16:30.

For – Details.

3. February 22, 2020 – MMTed – with help from – GIMMS – will hold a three-hour MMT Masterclass in London between 14:00 and 17:00.

This is a teaching seminar exclusively and will suit those who want to build their understanding of macroeconomics from an MMT perspective.

For more details – MMT Masterclass, London.

There are still vacancies available.

Keef Hartley Band

I felt like listening to some organ and guitar this morning and what better album but – Halfbreed.

I purchased my original copy in 1970 just days after it arrived via import in Australia (it was released in the UK in 1969).

The song is Born to Die.

The band is the Keef Hartley Band (named after its drummer and leader).

They were one of the few British bands to appear at the first Woodstock festival.

Keef Hartley was also the only other musician on John Mayall’s first released the Blues Alone (the first album I ever purchased as a teenager).

This was their first album and was probably their best.

They had lots of changes in line-up which Keef Hartley justified as being like a ‘jazz outfit’, where musicians can freewheel in and out and improvise.

I liked that about them and formed a penchant in the early 1970s in bands I played in for free form.

The lead guitar player on this song and album was Ian Cruickshank, one of the more competent British guitar players who few have ever heard about. He died in 2017.

After starting out in this sort of blues outfit he became a Django Reinhardt devotee and played in his band – Ian Cruickshank’s Gypsy Jazz. He also wrote several books about Django.

This song features beautifully phrased guitar (Gibson SG) and a perfect blend of Hammond B-3 through some Leslie Cabinets.

What a combination!

That is enough for today!

(c) Copyright 2020 William Mitchell. All Rights Reserved.

This Post Has 35 Comments

  1. I don’t understand this. Are you only talking about flows of money, or savings in money, or does real goods get involved? I assume any valuable real thing is a stock. Is it possible to outline a simpler situation where say I buy a stock of wood, and then add my labor and turn it into a stock called a ‘chair’ and then sell it to you?

  2. Way back in the late 60s, in Engineering college the Profs made a big deal out of being *sure* to keep the units straight.

    If you do this then you know how to make the equation work.

    Here in economics, flows are always something / a=time-period.

    So, GDP is $$ / year.
    My income is $$ / month.
    My bank balance is so many $$ on a certain day.

    Maybe Economics can do what Engineering does\did. That is always insist on all the units being stated.

    So a graph of GDP of the US over a decade would be sure to say “GDP/yr” and never say “GDP”.

    OTOH, remember that MS economics is not about finding and communicating the truth, rather it is *all* about hiding the truth from the students. And indoctrinating them with a false dogma.

    But we need to r

  3. Steve,
    Me too. If people kept the dimensional analysis straight, how could there be a problem? I admit that there is a problem, only how could there be? I don’t understand those people.
    I’m not sure about attaching dimensions to nouns. Nouns sort of mean what they are, and I think GDP is intrinsically, essentially, a flow. It has to be because so many of the goods included in GDP are consumables. Otherwise, if we found out that people in the U.S. had eaten a trillion tons of food since 1776, that could be a valid statistic, but what could we *do* with it?
    So GDP/year already has a meaning. It means an acceleration in our use of stuff.

  4. Another view of stock & flows. Anyone who’s ever run a (productive) business knows that the earnings statement (the measure of flow) is the important management control, It gives you a real-time day-by-day financial view of your business operations. You have certain expectations on your income & expenses & when you see deviations in those expectations, you drill down into your earnings accounts to see where the deviations are coming from & assess your options for correcting them. The balance sheet (the measure of stock) is of much less importance in managing your business – it’s more a measure of the safety margin (or available growth opportunity) for your business – something you keep a general eye on, but certainly not day by day.

    That’s why I see GDP (& GDI) so important. I see it as the earnings statement for the economy albeit, being macro, necessarily showing zero earnings. But there are expectations on those GDP/GDI accounts & when they deviate from those expectations it would seem that proper management would dictate drilling down into those accounts to see where the deviations are coming from to more quickly & certainly assess the options for correcting them. I see the MMT view as opening up this possibility more clearly.

  5. Jerry,

    ” Is it possible to outline a simpler situation where say I buy a stock of wood, and then add my labor and turn it into a stock called a ‘chair’ and then sell it to you?”

    This is how I would look at it.

    Purchase wood, flow – increase stock of wood

    Add labour, flow – produce chair, increase stock of chairs, reduce stock of wood

    Sell chair, flow (of income) – increase stock of money – reduce stock of chairs .

    Is that what you meant?

  6. Ed Zimmer,

    The income statement is an integral part of the balance sheet, in effect.

    It is part of the retained earnings item in the equity component of the liabilities side of the balance sheet.

    The income statement (and flow of funds statement for that matter) derive from changes in the balance sheet from one time period to the next..

  7. you can sort of use the bathtub analogy, by comparing it to how high the tap is turned on.

    A tap on “full blast” corresponds to a high gdp; a dripping tap is low gdp; turning the tap up or down is like gdp growth or contraction.

    Seems to work ok to me.

  8. “A tap on “full blast” corresponds to a high gdp”

    But this only mentions the pipes. The level of water in the tub ought to represent a stock of something, but a stock of what? Private sector savings? Accumulated reserves of your currency in foreign countries? Government savings because we haven’t convinced them that they don’t need to do that?
    Plus we can and should differentiate the flows into and the leakages out of GDP. Treating the GDP as one undistinguished flow denies us that.

    Apparently, as a technical point, the U.S. Treasury does keep accounts with private banks. It’s not so that Treasury will have money to spend later. It’s to keep from denuding the private banks of reserves after the quarterly tax payments come in. It’s another way of loaning reserve funds to the banks for the sake of smooth running.

  9. “The income statement (and flow of funds statement for that matter) derive from changes in the balance sheet from one time period to the next.” – Henry Rech.

    Henry, I find that to be incorrect & very dangerous thinking, Income & expense is what’s being measured (& in reality, all that is measurable). Thinking of those variables as the difference between accumulated income & expense just adds uncertainty (& manipulability) into the measurements. We have more than enough of that in valuing RGDP over NGDP. (Yes, RGDP allows GDP time-series estimates, but they’re “estimates” – not “measurements”.) That’s my fear of the Fed’s FOFA program – that NGDP will degenerate into accumulation estimates that will be treated as measurements.

    Look at it another way. Fiat currency has only perceptive value, ie, it’s best thought of as tokens or coupons. One can count coupons & have confidence in that count. But what confidence can one have in accumulating the count of coupons (whose value can change from day to day).

  10. Ed,

    “I find that to be incorrect & very dangerous thinking, Income & expense is what’s being measured (& in reality, all that is measurable).”

    It’s simple, basic accounting. Nothing dangerous about it at all.

    Do you not believe income and expenses have no impact on the balance sheet?

    What do you think retained earnings in the equity component on the liability side of the balance sheet are?

  11. Should have said:

    “Do you not believe income and expenses have an impact on the balance sheet?”

  12. Despite all the effort people are still confused. “I have found out what economics is; it is the science of confusing stocks with flows” (Michał Kalecki)
    I might sound boring but we have to define stock and flows in a way consistently used across empirical sciences and in engineering. A chemist, electronic engineer or physicist won’t have any problems identifying what’s a stock and what’s a flow.

    Please look up the Wikipedia article about “Mass flow rate”

    A stock is a measure of something for example the number of electrons or monetary value.
    A flow is a measure of how much of something passes an imaginary boundary in a unit of time. For example how much money is deposited onto my account from all sources of income. Or how many electrons divided by 1.602176634×10−19 pass though an imaginary boundary over one second. (this is the measure of electrical current denominated in amperes, let’s not forget that the direction of flow of electrical current is opposite to the direction of flow of electrons)
    We can talk about an average value of flow ΔM/Δt or an abstraction of a value of flow at a time t dM/dt where “d/dt” stands for a derivative. In economics people tend to use a discrete time abstraction so they talk about ΔM/Δt.

    The original bathtub analogy is misleading for the reasons explained by Bill

    In my opinion the statement “Purchase wood, flow – increase stock of wood Add labour, flow – produce chair, increase stock of chairs, reduce stock of wood Sell chair, flow (of income) – increase stock of money – reduce stock of chairs .” – is imprecise and prone to misinterpretation, a flow is how much wood is purchased per unit of time. The same about flow of labour (has to be defined per unit of time) and flow of money (which has to be defined per unit of time)

    (Δ pieces of wood) is not the same as (Δ pieces of wood)/Δt

    Going further the following statement is also prone to misinterpretation:
    “The income statement (and flow of funds statement for that matter) derive from changes in the balance sheet from one time period to the next.

    it is only true that net inflow is equal to the change in the value of a balance sheet entry in a unit of time. But net inflow is equal to the difference between the inflow and the outflow. We have to separately count all the income/revenue transactions and divide their value by the time we have accrued them to get the average value of the income/revenue flow. The same about the expenses/expenditures. A flow is an amount of something passing a boundary not a change in value of something in a unit of time (or a derivative of a stock in a continuous time abstraction). Only if there is one inflow and nothing else, we can say that the flow it is equal to Δstock/Δt.

  13. what makes things more confusing is that we use symbol Δ to think about a “change in the value of something” and also to think about “the count or value of something passing an imaginary boundary”

  14. Adam,

    Strictly speaking you are correct, however, I think you are nit picking.

    The delta t is implied.

    Accountants don’t describe income as $earned/unit of time. The time period is implicit. When an accountant describes profit it is specified as a $ amount for such and such a time period (e.g. the year ending 30/6/2020).

    “(Δ pieces of wood) is not the same as (Δ pieces of wood)/Δt” – it is if the time period is implicit.

  15. Adam,

    “”The income statement (and flow of funds statement for that matter) derive from changes in the balance sheet from one time period to the next.”

    This is strictly correct, the only thing I have not done is specify the time period, which isn’t necessary.

    “…all the income/revenue transactions and divide their value by the time we have accrued them to get the average value of the income/revenue flow. ”

    Accountants don’t deal in average values – their mission is precision to the cent.

    “A flow is an amount of something passing a boundary not a change in value of something in a unit of time (or a derivative of a stock in a continuous time abstraction).”

    Now your showing your confusion and self contradiction.

    “A flow is an amount of something passing a boundary not a change in value of something in a unit of time”

    You are getting yourself tangled up in comparing a physical model with a monetary one and more self contradiction.

  16. Adam,

    “Purchase wood, flow – increase stock of wood”

    If you want to be entirely correct and pick me up on every detail not only should the delta t be specified but also the the time at which the stock is measured should be specified.

    However, as I have said above, given the non specific/general nature of my explanation to Jerry above, these are both implicit and it was not necessary to specify either.

  17. Dear Henry Rech,

    The core problem experienced by people studying economics without having prior experience with “hard” science or engineering is exposed by this exchange of comments. I am not “nitpicking”. I am not questioning your personal level of understanding of what is taught as “economics”. I am questioning the paradigm employed in economics by the majority of economists, excluding ones who started as engineers and some people who were curious and intelligent enough to figure out things on their own. Michał Kalecki and Bill Phillips belong to the category of engineers and I claim this was the reason they were among the few who actually contributed something in the 20th century.

    “When an accountant describes profit it is specified as a $ amount for such and such a time period (e.g. the year ending 30/6/2020).” “You are getting yourself tangled up in comparing a physical model with a monetary one”.

    No, the views expressed above are objectively incorrect. One cannot specify the value of a flow without explicitly specifying the time period used in the measurement of the quantity/volume of something passing the boundary. Example: How much do I pay for renting a property? $950. Is this per week or per month? I know that the meaning of the term “flow” is defined differently by some economists and possibly accountants than by physicists or electronic engineers. It is not my fault.

    I know that accountants produce “accurate statements”. The value of profit in the year ending 30/6/2020 may not be the same as the value of the flow of profits measured for example in January 2020 and expressed in annual form. These are two separate things. I was referring to estimating a value of a flow for modelling purposes. My own estimation of disposable income is not based on the statement from a month when I got an extra bonus. I need to average. I don’t want to get into the details of averaging etc. This will be too boring and too confusing.

    The use of “slightly modified” language and dodgy 19th century methodology (e.g. “short-run” vs “long-run”) is precisely the reason why economics is called “dismal science” and why there is a “replication crisis”. We can add widespread abuse of statistical methods on top of this.

    A causal model of an electronic circuit, chemical reactions in a vessel or a biological system is usually expressed in continuous time as a system of explicit ordinary differential equations or (in some cases) has to be expressed as a system of implicit differential-algebraic equations. What is the reason a “monetary model” needs to be different?

    I suggest having a look at “Principles of Object-Oriented Modeling and Simulation with Modelica 3.3: A Cyber-Physical Approach” by Peter Fritzson to see the context of what I am talking about.

    I can provide more references if required.

    We can define a macroeconomic model in discrete time, what started with Tinbergen and was later further developed by Godley into “stock-flow consistent” modelling methodology. More developed models are expressed as systems of implicit difference-algebraic equations. They cannot be easily implemented using System Dynamics framework which only supports ordinary differential equations. (We can re-implement a discrete time model in continuous time, it will be integrated in discrete time anyway). Let’s leave this topic as it is not for blog comments.

    Please take a model of the economy defined by an equation X_t+1=F(X_t,X_t+1) where X is a vector of variables, for example GROWTH model from 11th Chapter of Godley and Lavoie “Monetary Economics” (2007) and change sampling period from one year to one quarter or one month. If you attempt doing this (and I did), this is exactly where sampling period is relevant. It cannot be “assumed away”.

    I don’t want to chuck the whole model here because Bill would object. Some of this stuff will be published at some point of time.

    this is from the actual Python code, implementing a version of GROWTH as a discrete-time model with a variable time step


    The equation describing inventories

    in_ = ( c.y – c.s – c.NPL / c.UC ) * self.dT + p.in_

    You see I have to multiply the value of the flow (c.y is the current real GDP, c.s is the current real sales) by the value of the time time step to calculate the value of the inventories in the next time step. It is using forward Euler method. The value of flow itself is invariant to the length of the time step (obviously I am not talking about the changes in the value). Regardless how I measure it (every month or every year) the flow e.g. real GDP has to be expressed in [units/year] if I express time step in [years]. For example units can be US dollars from 2012.

    We can produce whatever economic theories we want and make any statements we like. It is nice to have a debate etc. I suggest building a numerical model and running it. If it produces something what resembles the real world, chances are we know what we are doing. But it is extremely likely that if we do something wrong the simulation will fail.

  18. Adam,

    Firstly, just so you know, I was on my way to an engineering degree but at the last moment I decided I wanted to fix the problems of the world so I took economics. (You can see what a great job I’ve done!)

    Anyway, you have made a mountain out of a mole hill. You have traversed the field from delta t to time derivatives to, now, integration. You have raised System Dynamics and Python based modelling, etc., etc., etc..

    Funnily enough, I cannot recall seeing any of these elements the last time I opened a serious book on financial accounting. I can only wonder at the monstrous aberrations in income and asset measurement perpetrated by the accounting profession, giving no due regard to the matters you have raised. I can only wonder at the millions of auditors reports which have failed to raise the question of time derivatives etc. in verifying asset statements in balance sheets. They have clearly managed to hoodwink authorities since time immemorial.

    I have no doubt, should you raise these matters in the proper academic circles, you will be well on your way to the next Nobel Prize in economics.

    Sorry (well not really) for the silly diatribe, but you have gone way over the top. You have made a relatively simple subject completely impenetrable.

  19. Adam,

    And, again, just so that you know that there is nothing personal in my comments above, there have been many comments in Bill’s blog written by you which I have enjoyed and much admired.

  20. I do thank everyone who has tried to improve my understanding and lessen my confusion. If anyone has some energy left- what is a common problem we might have of with an economic idea that wasn’t really stock-flow consistent with the accounting? How much better is this type of accounting able to help us understand the economy in terms of real production and consumption of things? Or does it also suffer from assuming money exchanges as a measure of economic well-being? Is it better able to represent that money is not usually a fixed supply? Or that changes in the value of money verses real goods happens?

    So because it isn’t fair to ask Bill to explain everything I don’t understand, I have done some investigating myself. So I watched 2 and a half hours of Marc Lavoie explaining STC to UMKC econ students. And read a few other shorter explanations. But mostly I come away with this as the highlight- Stock-Flow Consistent models “integrate money seen as a flow with money seen as a stock”. But the value of that effort is what is escaping me. And I might be missing something very, very simple- I sure wasn’t trying to ask a difficult question but it looks like maybe I did.

  21. Why is a claim that your model isn’t stock-flow consistent any stronger than that your model doesn’t acknowledge the government budget constraint? Maybe that is really at the heart of my question.

  22. Jerry,

    “Why is a claim that your model isn’t stock-flow consistent any stronger …”

    Doesn’t MMT say it is stock flow consistent?

  23. Dear Jerry Brown (at 2020/02/14 at 4:54 pm)

    You are confusing yourself.

    The point of the post is that we should use correct pedagogy to illustrate key concepts in economics. Otherwise, students get confused and make errors in their reasoning.

    The ‘bathtub’ example is a case in point.

    It was taken from Kenneth Boulding who used it to illustrate how stocks of capital accumulate after a series of flows (where saving occurs). He made the pedagogical error, I believe, in leaving out the saving bit and expressing the accumulation as the difference between two flows. But in the context he was working it was easy to fill in the gap.

    Subsequently, economists of all persuasions have used the example, probably without even knowing its provenance. And they have used it in the context of summarising the national account flows.

    The level of the bath is claimed to be GDP and the inflows and outflows the injections into and leakages from the spending stream. So while Boulding knew the bath level was a reasonable analogy for capital stock he would have shuddered to see how his example is used to represent GDP which is a flow!

    So in the context of my current teaching commitments I was trying to get students to clearly understand the difference between a stock and a flow, and somehow, the bathtub story entered the fray.

    I never use it and eschew its use.

    One of the reasons why it is important to understand this is because people do not realise the fiscal deficits represent the difference between two flows – here today, gone tomorrow. But public debt is a stock that accumulates in a stock-flow consistent manner as a result of matching debt issuance to the flow difference.

    Please don’t seek more from my post than was intended.

    best wishes

  24. Hi mel

    The water in the tub would be national wealth/assets that are due to accumulated output over time.
    (I find it interesting that there isn’t a single headline figure for national wealth like there is for gdp).
    You could think of the outflow through the hole in the bathtub as a source of depreciation of assets, or something like that.

    Sure, we want to break gdp down, but the introductory premise for measuring gdp that I’ve heard is “imagine a nation only produced one good, and we’ll call that good output”. Using water instead of output seems ok to me. Given that gdp is reported as a single number, not too mislead to use a single quantity of something.

    If we want to look at more complexity, the analogy also needs more complexity – for example we could have multiple taps going into the bath to represent different types of flows. Or it could by multiple baths that are connected in a network. Such as the pipe into “bath B” coming from the hole out of “bath A”. If we give “bath A” a second hole, that can represent a leakage that doesn’t get to “bath B”.

    I think the term “savings” is not right to use as a stock of wealth for macro, because it conflicts with the ‘S’ component of GDP. So “savings” as defined in macro must also be a flow as well.

  25. Stock-Flow Consistency

    I don’t know examples from macro. I sort or wish I did, a little.

    In my imagination, a stock/flow INconsistent model would be the one that works with Debt/GDP ratios without considering the units for that quantity.
    People see a Debt/GDP ratio of maybe 2.7 (really 2.7*year) and run screaming “Ahh! We’re broke. BROKE! We owe twice as much as we’re worth! More even!”
    Although in the golden years of western prosperity after WWII, families took on debt/income ratios of 3 as a matter of course (actually 3*year), and lived well, and paid off their 30-year mortgages by the end of the 70s. Because the debt was a 1-time stock level, and the income was a recurring flow, and it would eventually chip the debt down to zero. Stable jobs and reliable income were vital in making that work.
    (It was in the late 70s that the Powell memo had been circulating, and the War on Payroll heated up, and our present shit started off toward the fan.)

  26. Thank you Bill. It is true I am good at confusing myself by adding unnecessary complications on occasion. Maybe I was overthinking this one- thought I was missing a very important point about the post and I was anxious to find out what it might be.

  27. An example of a stock/flow inconsistent model is according to Godley and Shaikh, the “standard neoclassical model”. The paper is available on Anwar Shaikh’s web site. “An important inconsistency at the heart of the standard macroeconomic model”. Obviously there are ways around the inconsistency and the New Keynesian model of the monetary economy built without money by Woodford (2003, “Interest and Prices”) seems to be stock-flow consistent.

    The problem with capital accumulation in a bathtub is that GDP is a nominal measure of the value of goods and services produced for the market in a unit of time (usually one year). What is accumulated is physical capital and we can calculate its historical value by the simple accrual of investment minus depreciation but its current value depends on the expected value of future profits. This was demonstrated by Sraffa and hidden again under the carpet by Samuelson. (in Cambridge capital controversy). That’s why it is difficult to model the process of accumulation in the real economy.

    There is nothing trivial in the stock-flow consistency topic, it actually goes to the heart of what’s fundamentally wrong with mainstream economics and I disagree with the statement that I have made “a mountain out of a mole hill”. Yes it is true that “cash flow” is defined in accounting as as a ΔM in a stated period. This definition is a correct instrument to capture information required to build a picture of a company’s (or household’s) finances within the fully consistent accounting framework. I am not questioning this. But it belongs to the realm of accounting not economics.

    I will still argue that a “flow” in macroeconomics has to be defined as ΔM/Δt, in the same way a flow of water is dimensioned in the hydraulic model of the economy built by Phillips in 1949 (Monetary National Income Analogue Computer).

    Let me explain what in my opinion is wrong with economics. It is quite simple. By using incorrect analogies and dodgy models, students who attempt to learn about the economy get confused. It is easy to get confused by thinking about flows dimensioned in [$] instead of [$/year]. And then they have to learn intermediate macro with all the intimidating algebra. Then to study statistics which is quite perplexing on its own. When they graduate they continue confusing others. The ruling corporate oligarchy pays a lot of money to sustain this process. But the actual entrepreneurs and bankers know very well how things really work. Please watch Alan Greenspan: “The United States can pay any debt it has because we can always print money to do that”

    We should apply the methodology from empirical sciences and engineering and simply forget about the “short-run” and “long-run” analysis in the context of Walrasian equilibrium. The whole marginal theory was made up to discredit classical political economy developed by Marx. But Marx was right. Prices of commodities are cost-prices, they have nothing to do with marginal utility. His solution to value transformation problem was correct. I may write about this later.

    Marginal utility only exists in the world inhabited by 19th century cyborgs, it has nothing to do with the humans. This is where the rot started. By putting the research of Dr Victor Frankenstein at the heart of the scientific paradigm. The whole ninety trillion-dollar global economy still revolves around this creation. Like in the original novel, the creation has escaped its creators.

  28. Adam,

    “We should apply the methodology from empirical sciences and engineering……..”

    Unfortunately the empirical sciences and engineering do not quite understand the fallacy of composition the way Keynesian macroeconomists do.

  29. BTW, would it improve the bathtub analogy to make it very plain that we start each “year” with an empty tub?
    Now, all rises of the water level come from the flows in vs out more plainly.
    And the level of the water is just for the year.

  30. “we start each “year” with an empty tub”

    So who empties the tub, and why? We’re in real danger of “assume a can opener” here. If we drop a bad analogy, then we spare ourselves the effort of explaining to people why they have to ignore all the things that are wrong with it.

  31. Dear Henry Rech,

    Exactly the opposite is true regarding the fallacy of composition. It is defined as “something is true of the whole because it is true of part of the whole”.

    The most obvious example in economics is the “paradox of thrift”. It is only a paradox in an invalid neoclassical framework where aggregate income is independent of aggregate demand. If you look at SFC models introduced in “Monetary Economics” by Godley and Lavoie (the closest we have got so far in Post Keynesian economics to so-called Dynamic Systems methodology), Section 7.4.2 analyses this paradox in great detail.

    It is easy to demonstrate this phenomenon – and much more – using properly designed dynamic SFC models. These are “emergent phenomena” which can be seen as results of the interaction of relatively simple components of the system, yielding results which seem to be contradictory to the intuition about individual components, because of the presence of feedback loops. We can also reproduce a pseudo-Goodwin cycle or the Gibson paradox. This stuff will be published at some point of time.

    But wait, there is more. We know how to study emergent phenomena of different kind in empirical sciences. When I build an object detector by training a neural network, I am employing complexity science. I would claim that only in this type of framework we can properly link what is considered “microeconomics” but should be rather named behavioural economics, and macroeconomics. We should actually investigate phenomena such as bubbles, unstable inflation, etc in this type of framework.

  32. “… If we drop a bad analogy, then we spare ourselves the effort of explaining to people why they have to ignore all the things that are wrong with it.”

    Perhaps analogies ought to be left to poets (and poetic prose-writers, provided they’re good enough writers)?

    They’re the people best-equipped – actually the only people equipped – to use them properly. The rest of us had better use them only with the greatest caution, and drop them the moment they start to come apart in our hands.

    (Damn it! – I’ve just ignored my own warning!)

  33. Dear Robert H

    There is nothing wrong in seeking analogies if they use is restricted to the area they work. As we can build (an obviously imperfect and limited) model of the economy using a system of ordinary differential equations, we need to look at physical phenomena described by similar equations. This is where the hydraulic analogy is coming from. Please look for MONIAC (an analogue computer built by Bill Phillips in 1949). There was nothing unusual about building analogue computers at that time.

    We can find more advanced papers written by Bill Phillips about employing optimal control theory in calibrating fiscal policy. “Stabilisation Policy in a Closed Economy” is an article worth reading even now. The author clearly states that “aggregate demand [is measured in] in units per year”, that’s why I was ranting so loudly about this issue. Special care was given to the presence of multiple lags in the productive system. (In modern times optimal control approach has been superseded by robust control that is control in the presence of uncertainty, this may be relevant to the main topic).

    About analogue computers. This is a real story. I was visiting Intrepid Aircraft Carrier museum in NYC a few years ago. There is a guided missile submarine Growler anchored next to Intrepid. The tour guide was a navy veteran. He showed us the missile control room. I asked him – “what’s this? it is looking like an analogue computer used to precalculate the trajectory of the missile”. The guide was stunned. Apparently I had been the first non-military visitor who had an idea about the contraption. He asked me where I came from. Then I felt I was in hot water. How could I explain that I studied analogue electronic design in a former Eastern Bloc country? Obviously we had access to normal digital computers (in the late 1980s). But we studied so much of analogue signal processing that it would have been totally impossible for me not to recognise the dials and small screens used to control the Regulus missile.

    People don’t have to think about operational amplifiers and feedback loops while thinking about the flow of saving and consumption. (I had to memorise the schematics of µA741 for one of the exams maybe this screwed up my brain completely). The hydraulic analogy is good enough for normal people.

    Bill Phillips is only remembered for his econometric research into the relationship between the money wage changes and unemployment. He made much more contribution than this for example he introduced the idea of adaptive expectations. It was a great success of neoclassical economists that the contribution of Bill Phillips has been effectively erased when monetary policy became a panacea for all the macroeconomic issues and fiscal policy has been effectively demoted.

    I am not advocating borrowing the analogue missile controller from Growler to calibrate the fiscal stimulus here in Australia but we need to restore the sanity in thinking about how the economy actually works. We can still learn a lot from Bill Phillips and Michał Kalecki.

  34. “(Damn it! – I’ve just ignored my own warning!)”

    Exactly! You got it! We can’t really converse without analogies. We need them to tie new ideas to old, known, ideas so as to demonstrate the new meanings.
    From what I’m reading lately, the Vienna Circle in the 1930s tried to establish a new logical language for expressing scientific ideas unambiguously. Later on, Bertrand Russel judged that in his Tractatus, Wittgenstein had created a logical language, and also demonstrated how little we have when we have one of those.

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