Ponzi Maths–Part 1

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This is an unplanned post that partly pre-empts what I’ll be writing in the February Debtwatch Report, where I will explain in full my theory of money creation in a pure credit economy. So this is somewhat out of sequence, and will undoubtedly be badly explained compared to what I put together for February. 

I will also have to finish this in a later post–probably in the first couple of days of the New Year–because Sydney’s fireworks beckon, and we have to be on board the cruiser we’re watching them from at 7pm.  But what is here is part of a long-promised explanation of my model of money creation. In a couple of days I’ll publish the punch line, which is a newly developed model of a Ponzi Scheme.

To begin at the beginning: in the last year, I have developed a method of building dynamic models of financial processes using a table layout that is closely related to the accounting methodology of “double-entry book-keeping”. The table represents the flows in and out of bank accounts:

Type Asset (1)  Liability(-1)
Account Loans or Reserves Deposit Accounts
Activity Flow in/out Flow in/out

As I illustrate below, a dynamic model of a financial system can easily be derived from this basic schema, since each row represents a specific relationship between accounts, and the entries in a column represent all the action for that account. Simply add up the entries in each column, and you have a system (of differential equations) that represents the relevant model of the financial system.

I originally developed this as a means to communicate my dynamic modelling to other economists, since the vast majority of them have never studied differential equations–let alone systems dynamics. When I presented a model as a system of ODEs (“Ordinary Differential Equations”), they frequently failed to grasp the logic–and colleagues who are systems engineers, such as Trond Andresen or Mike Radzicki, found a similar response to their sophisticated flowchart models. Economists, even non-orthodox ones, simply aren’t accustomed to thinking dynamically, and normally lack any exposure to the sophisticated tools that engineers in particular use routinely today.

This applied in spades when I first presented my model of the endogenous creation of money to the bi-annual Post Keynesian Economics Conference in Kansas City in 2006. A room packed with about 100 conference participants broke out in a vigorous debate, with many criticising my analysis because “You must have made mistakes in your double-entry book-keeping.” 

I knew the model was accurate, so the thought occurred to me that, if so, it should be possible to present the model in double-entry book-keeping format. Sure enough, when I presented exactly the same model to the Society of Heterodox Economists conference later that year (with several people from the previous conference in attendance), the reaction was far better.

One person even commented that he was a bit disappointed because my presentation was less mathematical than usual! I then informed him that he’d actually seen a presentation involving a six-dimensional dynamic system.

Since then, I have found that this method was not merely a presentation tool, but also a very useful development tool for building dynamic systems. I’ve built models of non-bank lending, a credit crunch, etc., all of which will turn up in (non-neoclassical!) economics journals at some stage, and in my forthcoming books.

Since Bernie Madoff’s spectacular collapse, I’ve wanted to add an extension to model Ponzi finance, and at 11.30am Sydney time on December 30th, I’ve cracked the basic mathematics of a Ponzi Scheme. I can’t resist posting (part of!) this before the New Year, as a “Happy New Year” present to my loyal band of bloggers.

But first things first: the mathematics of an absolutely minimalist pure credit economy. In February I’ll explain why I believe that the endogenous expansion of credit money–and not the “money multiplier”–is the key driver still today in the growth of the financial system (and also why I differ from Austrians like Peter Schiff in my analysis of the economy, and how it might be reformed to avoid asset bubbles in future). For now, just take this on board as a thought experiment: IF there was no Central Bank, and no Government, how would money be created in a pure credit economy?

The answer, in a nutshell, is “by the banking system creating matching deposits when it issues loans”. In this model, loans by the banks create deposits, which then finance economic activity. Economic activity consists of firms that own factories hiring workers to work in them to produce goods for sale, using borrowed money to finance their wage payments (and their own consumption and inter-firm purchases as well), and banks making profits on the spread between loan and deposit rates of interest.

The basic mechanics of this system, which can function indefinitely at a constant level of production with a single loan injection, are shown in the following table, which has 4 accounts: a Firm Sector Loan, Firm Sector Deposit, Bank Sector Deposit (strictly the Banksector’s  profit and loss account), and Workers Deposit.

For now I’ll use a simple capital letter to indicate each flow–in every case this will be replaced by some appropriate product (for example, “A” below will be replaced by the interest rate on loans times the current outstanding loan balance for the firm sector).

The six processes in this model are:

  1. Interest accrues on the outstanding loan at the rate +A;
  2. The bank pays the firm interest on the balance in its deposit account by a transfer B from its deposit account to the firm’s deposit account;
  3. The firm sector transfers the sum C from its deposit account to the bank’s deposit account to pay the interest on the outstanding debt. The bank is then obliged to record that the outstanding debt has been reduced by that amount–hence the -C entry in the Firm Loan account;
  4. The firm hire workers to produce output–a flow D goes from the Firm’s Account to the Workers’;
  5. The bank is obliged to pay interest to the workers on the balance in their accounts; the flow E goes from the bank’s deposit account to the workers;
  6. Finally, the bank and the workers consume some of the output of the firm sector and pay for this with transfers from their accounts of F and G.







Firm Loan (FL)

Firm Deposit (FD)

Bank Deposit (BD)

Worker Deposit (WD)

Interest on Loan





Interest on Deposit





Pay Interest on Loan





Pay Wages





Interest on Deposit     -E +E






This simple system describes a self-sustaining economy which could function indefinitely at a constant level.  Simulating this system with equilibrium values yields a model in which bank accounts remain at a constant level and finance a constant level of income for all classes over time. The first graph below indicates the equilibrium values of bank accounts, and the second shows the equilibrium annual incomes that result (I don’t want to scare off non-mathematical readers with mathematical symbols right now, so anyone who wants to see what A to G are, please scroll to the bottom of this blog entry). 

Notice that incomes are substantially greater than the size of the initial loan. A lot of people who have attempted to build a monetary model have made the mistake of believing that the spending the loan can finance is identical to the amount of the loan itself. This ignores the fact that the loan finances a turnover of economic activity–in effect, it ignores what economists call the velocity of circulation. The interest bill is also effectively paid out of the “small change” from the profits capitalists make–whereas a lot of analysts have presumed that the interest couldn’t be paid at all. I’ll go into what was wrong with that perspective in more detail in the February Debtwatch Report; for now take it from me–and from the simulations–that paying interest on debt is a breeze for capitalists in a productive economy in which there is no unproductive debt.

The impact of unproductive debt–money borrowed simply to speculate on rising asset prices–is the focus of this post, but unfortunately time has got away from me and I have to get off the blog and on to the boat.

Thanks to all readers, and especially to the blog participants. Most people will finish 2008 in a state of absolute bewilderment. Members of the debtdeflation blog will only feel that way tonight if they overdo the alcohol consumption!

So here’s a toast to you all. Happy New Year, and I look forward to corresponding with you all in 2009.

Steve Keen 


Flow Letter Symbolic Value Explanation
 A  rL × FL  Loan interest rate times outstanding loan
 B  rD × FD  Deposit interest rate times deposit bal
 C  rL × FL  Loan interest rate times outstanding loa
 (1-s)/tF  Workers share of surplus divided by time lag in production
 E  rD × WD   Deposit interest rate times deposit balance
 F  BD/tB  Account balance divided by time lag in consumption
 G WD/tW   Account balance divided by time lag in consumption



Parameter Values

These are just for illustrative purposes–they are not derived from any fit with empirical data–but they are reasonable values nonetheless for this toy monetary economy model.

Parameter Meaning Value
rL Interest rate on loans 5%
rD Interest rate on deposits 1%
s Capitalists share of surplus from production 33%
tF Delay between financing production and selling output 2 months=1/6th Year
tB Consumption time lag for bankers 1 year
tW Consumption time lag for workers 2 weeks=1/26th Year

About Steve Keen

I am Professor of Economics and Head of Economics, History and Politics at Kingston University London, and a long time critic of conventional economic thought. As well as attacking mainstream thought in Debunking Economics, I am also developing an alternative dynamic approach to economic modelling. The key issue I am tackling here is the prospect for a debt-deflation on the back of the enormous private debts accumulated globally, and our very low rate of inflation.
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13 Responses to Ponzi Maths–Part 1

  1. Effit says:

    Hi Steve
    Hope you enjoyed your river cruise and the fireworks. The fireworks looked wonderful on TV – yes, a terrible waste of money, but bring joy to many.

    I’ve read your Ponzi Maths 1 a couple of times…Phew…not sure if someone like me with Grade 3 Arithmetic and a poor grasp of Economics 101 will ever catch on. I think I get the gist of it, but I may have to read it again and again.

  2. tommyt says:

    Hi Steve, thanks for your greetings. I could not resisit sending you in a new ‘brickbats’ post from …yes Craig James, perhaps he is a nostradamus figure, anyway in today’s ‘smh’,finance section,page 27,”…fear of recession wiped 43% from the market, or 2699points…it closed at 3722points…” later in the piece Mt Craig is quoted “Mr James was optimistic about the market reaching 4700 points by the end of the year.” We shall see, we shall see!

  3. Bullturnedbear says:

    Good model Steve. I think the steady state model is a good model for Mankind.

    The problem is that one can’t take Mankind out of the model. The psychology of mankind has dictated that when people are feeling bullish the trend is for growth and risk rising over time. In history this has been the predominant trend by far. Once that goes too far though people switch to risk aversion and go the other way. Regardless of Government intervention or “rules”, when Mankind is bullish they seem to ignore the “rules” and risk and growth take over.

    I know you believe this too as I have read what you have written on Minsky in the past.

    Therefore, I believe at present that systems and models can sometimes be good at predicting or explaining outcomes but not so good at causing outcomes. I get excited when you talk about solutions to the problems. But I get sceptical when I read the solutions presented by yourself and others.

    Mankind and history seem to keep repeating themselves.

    An example of this may be the belief that a gold standard would prevent a bubble. If Mankind wanted an investment and debt boom, “to get more done”. They would either invent a new way to create debt within a gold standard. Or they would convince the politicians to scrap the standard. When Mankind becomes bullish, we find a way to run with the bulls. The older the bull market gets the more crazy it becomes. The bull market that finished in 2007 was very very old.

  4. Bullturnedbear says:

    Hi Tommyt,

    If Craig James was true to himself and his readers. He should have published his prediction for the market at the end of 2008 that was made in 2007. I checked back. Not that hard to find. A table was published in the Australian around the same time a year ago.

    Craig James predicted 7550 for the ASX and 15400 for the DOW. Not bad Craig! Only out by a bit. At the time of Craig’s predition the ASX was at 6423. Craig predicted a rise of 1127 (17.5%. This time he is predicting a rise of 978 (26.3%). Craig James is the eternal bull. I can’t wait to see his prediction next year.

    I think the ASX will be in the mid to low 1000s by the end of next year. When the point of recognition takes over in Australia, I’m keen to see if Craig James changes to being a bear.

    When Craig and his crew become bearish the market will be approaching a major bottom. That’s my prediction.

  5. Steve Keen says:

    It’s not a steady-state model Bullturnedbear; while it describes one (and shows that it is feasible at least at the level of finance), it’s just a prelude to a model including growth, financial instability, and all the other significant paraphernalia of the real economy. I’m still working on Parts 2 and 3–they should turn up in the next couple of days.

  6. Steve Keen says:

    Excellent. It would be great if someone (not me–I don’t have the time I’m afraid!) put together a table showing the key predictions of these characters versus the outcomes over time.

  7. Hoss says:

    Jim Cramer’s predictions for 2008:

    What follows are my top-ten financial predictions for 2008—some mortal locks, others long shots, in that order.

    1. Goldman Sachs makes more money than every other brokerage firm in New York combined and finishes the year at $300 a share. Not a prediction—an inevitability. In fact, it’s only January, and I think it’s already come true.

    2. Oil goes much higher, maybe as much as $125 a barrel. That sends gasoline to $5 a gallon, even at those terrific service stations outside the Holland Tunnel. Pundits keep blaming the endless rise on geopolitics, but in the latter half of 2007, we saw reduced tension in Iraq, Iran, and Venezuela, plus flat-out production by the Saudis and the Russians, and all that happened is the price went from the $70s to the $90s. We are running out of oil more quickly than people can imagine, and that means great returns for oil companies. Just buy the stock of the company you filled up at today or buy a driller (Transocean’s my favorite), then sit back and make money. The odds oil will rise? Two to 1. The $125-per-barrel target might be pushing it, but higher oil is pretty much a sure thing.

    3. The Fed arranges an Arabic Heimlich maneuver on Citigroup, so the banking giant doesn’t choke on the worst mortgage portfolio in the country. Rather than face the demise of the biggest U.S. bank, and the panic its fall could trigger, Congress looks the other way as Arab investors buy 51 percent of the somnambulant bank. Unfortunately for Citigroup, I’d lay 3 to 1 on this happening. I say “unfortunately,” but I shouldn’t. It’s unfortunate that a proud institution basically has to give up its autonomy, but its stock would go up considerably once it got that capital.

    4. Verizon becomes your cable provider. In one of the most remarkable frog-to-prince transformations I’ve seen, Verizon CEO Ivan Seidenberg offers an alternative, Fios, that is better and cheaper than anything Time Warner, Cablevision, or Comcast can produce. Throw in Verizon’s growing cell-phone business and growth accelerates dramatically, making VZ the best-performing stock in the Dow Jones averages. Time Warner and Comcast hit new lows, and the retreat of cable begins. Sorry, cable guys: We’re looking at 4 to 1 here.

    5. In the first real debacle of the private-equity era, Cerberus Capital Management, the quiet hedge-fund king, fails in its bid to resuscitate Chrysler—not a surprising turn, given that it picked Bob “I ruined Home Depot and all I got was $200 million” Nardelli to run the country’s worst car company. The combination of Chrysler and the 51 percent of GM’s lousy mortgage business that it paid top dollar for forces former Treasury secretary John Snow to seek a bailout for Cerberus. Amazingly, given the love of hedge-fund contributions by both parties, Congress agrees and writes checks for billions to save Cerberus’ wealthy investors. Call the Chrysler failure a lock. The bailout? I’d say 5 to 1.

    6. Google continues its dominance and becomes one of the top three companies in the U.S. in market capitalization. It doubles its advertising share, at the expense of television and print. It also successfully challenges Microsoft for operating-system dominance. Microsoft calls for a government investigation of Google’s power, but no one cares because Microsoft is just too hated for anyone in Washington to champion. The stock roars to $1,000. I like Google enough to put this one at 7 to 1. If you use an $800 target, make it 5 to 2.

    7. European companies, eyeing the weak dollar, snap up New York real estate, and offer to buy Merrill Lynch and JPMorgan. John Thain and Jamie Dimon, the companies’ respective CEOs, agree to the bids (Thain sold a chunk of stock to a foreign entity just last week). Colgate, Clorox, Whirlpool, and Black & Decker get snapped up, too. All six companies’ stock prices head north. Lots of moving parts, but let’s put the odds of at least one of these deals happening at 3 to 1. A perfect Pick Six pays 50 to 1.

    8. Apple completes its dominance of the music business, as the music producers decide no longer to produce new CDs. It’s just too expensive for them. Warner Music Group files for bankruptcy. Apple goes to $300. Okay, these may not be 2008 events, but they will happen, sooner rather than later. This year: 25 to 1. Next year: 5 to 1.

    9. The New York Times, after spending several hundred million dollars buying back its stock while it was in the $30s and $40s, slashes its dividend in half because of a cash shortage. The stock drops to $10. To save the world’s greatest newspaper, the company accepts a buyout offer from Mayor Michael Bloomberg at $20 a share. Don’t be so quick to scoff: The cash is spare change for Bloomberg, who, don’t forget, already owns a small media company. I’d say the $10 share price is even money. That’s how bad it is at the Times. The Bloomberg buyout is probably a 100-to-1 shot, but may be less if he decides not to run for president and needs something else to do this year.

    10. An Army of the Foreclosed marches on the White House, then launches a siege at the Federal Reserve, before camping out in front of the Washington Monument. The army demands relief from eviction. Bernanke, recognizing that he did nothing to regulate the mortgage mess in 2006 and then did not cut rates fast enough in ’07, resigns. The siege ends, the new guy slashes rates, and the market takes off. Here, the odds are 1,000 to 1 (as Marx taught us, people have a hard time losing their chains). But if Bernanke or a future Fed chair does cut rates meaningfully, here’s a sure bet: That’s the time to start buying.

    Can’t wait for the 2009 predictions.

  8. Zulu says:

    Good starting point Steve, I look forward to the rest of your blog. I’ve been reading your blogs for over a year now, but this is my first post.

    They tried to teach me Economics at Uni, I understood what they were saying but I could never see how it would work. Now that I’ve found your website (& I’m older and don’t trust everything I’m told even by professors etc.) Now I understand why! They were wrong! You are one of the few who actually makes sense to me.

    As someone who’s worked primarily as an engineer I’m amazed to read that most economists can’t understand things like differential equations & system dynamics!! You can’t possibly understand the financial system, or similar complex systems if you do not know how to make a realistic model.

  9. Hoss says:


    You might be interested in this about your colleagues in America.


  10. clive says:

    This is Michael Pascoe’s predictions DEC 19 2007
    “The US recession is a bit like religion and global warming – people don’t really believe”


    And this one on DEC 12 2007

    I must keep a look out for his 2009 predictions. I wonder what brand of crystal ball he’s using?

  11. clive says:

    Shane Oliver on 2009

    ‘The All Ordinaries index plunged as much as 53 per cent last year from its high in November 2007.

    AMP Capital Investors chief economist Shane Oliver says the sharemarket could remain volatile in the early part of 2009.

    But he believes the market could rise more than 25 per cent this year.

    “It’s also quite conceivable that we’ll might see more lows in the market earlier in the year but I think as the year progresses the sharemarket will start to get on to a sustainable upswing,” he said.

    “My view is that global growth and Australian growth should start to look a bit healthier through the second half of the year and going in to 2010.”‘

    With over a billion of peoples money locked up for a year, you would be hoping 2009 was better… wouldn’t you?


  12. marcf999 says:

    Hello Steve,

    I notice that D (wages) has evolved since your older papers. In the older ones it was a fraction of firms deposits here you use the (1-s)/tF formulation.

    For consumption you use fractions of deposits multiplied by 1/tB and 1/tW respectively for bank and workers consumption.

    I am not familiar with this formulation for consumption and wages can you expand a little bit how one comes to these? is this standard in the literature? Is there a textbook entry on these and what they represent?


  13. Steve Keen says:

    Hi marcf999,

    Thanks for paying that post such close attention!

    Nothing I do is standard in the economics literature, unfortunately–I’m trying to influence them to get them to take up a strictly dynamic approach to modelling, but that will take a very long time. But the formulation is standard in systems engineering literature, where a time lag is represented as one over the fraction of the time period over which it occurs.

    Thus if the rate of flow of water out of a lake is such that, if there were no flow into it and this rate kept up linearly, it would empty in 3 months, the time lag is given as 1/4 of a year and the expression would be d/dt LevelOfLake = -1/(1/4) * LevelOfLake.

    This generates an exponential decay in the level of the lake–not a linear one–which describes most processes of that nature.

    The actual terms are “standard” in a way, but the practitioners on those areas have forgotten them. “s” is the term Marx used for the rate of surplus value’ I’ve cheated here a bit by using the same term to describe the share of the surplus generated in production that goes to capitalists (so that the share [1-s] goes to workers). Marx also spent considerable time talking about the production period of capitalism–the time between forking out the M and getting the M+, in his brilliant explanation of why Say’s Law doesn’t apply in a capitalist economy.

    This can be expressed as a time lag–which is how I’ve used the tau_F in that paper–between the start of a process (hiring workers effectively) and the end of it (receiving revenue from the sale of the commodities the workers produced).

    In equilibrium–and only in equilibrium–this flow to capitalists (s/tau_F) equals P*Q – Wages. At other times it will be greater (if output is expanding) or less (v.v.). See the paper “Not Keen on Bailouts” on the Research page for a table that illustrates these dynamics in the context of a credit crunch (but without the Ponzi component of that paper).

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