Ponzi Maths–Part 3

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This is getting a bit like Star Wars, but I promise–this will be the last post in this series. In the previous two, I constructed a model of a pure credit economy in which the money supply and economic activity can expand smoothly of time.

Of course, that’s not the real world. As we know from the bitter experience of the financial crisis that is this blog’s raison d’etre, finance characteristically destabilises an otherwise healthy economy. Part of the reason for that is the existence of Ponzi Financing–something that the recent Bernie Madoff scandal has thrown into $50 billion high relief.

In reality, given our current financial system and how assets are valued and defined, there is a Ponzi potential that the system almost inevitably succumbs to. But it is a “Type II” Ponzi Scheme: given the potential to make unearned profits by buying and selling assets on a rising market, the game of leveraged asset price speculation seems to inevitably take hold–and in our epoch, this has reached heights that have never before been seen.

But this series of posts is about a pure “Type I” Ponzi Scheme: one in which a promoter accepts “investments” and provides fantastic returns from the simple expedient of returning the principal (with a time lag) and calling it interest. So long as the inflow of funds continues to grow, such a Scheme seem like an easy road to riches, which normally inspires “investors” to reinvest their “profits”–and possibly even more–as the Scheme continues. But ultimately the Scheme must come a-cropper, as either the inflow becomes insufficient to finance the level of promised outflows, or redemptions rapidly empty the Scheme of its trivial actual reserves.

It’s the latter mechanism that I model here.

Firstly we need our legitimate financial system, as described in the last table of the previous post:

Type 1 0 -1 -1 -1
Account Firm Loan (FL) Bank Reserves (BR) Firm Deposit (FD) Bank Deposit (BD) Worker Deposit (WD)
Interest on Loan +A        
Interest on Deposit     +B -B  
Pay Interest on Loan -C   -C +C  
Pay Wages     -D   +D
Interest on Deposit       -E +E
Consume     +F+G -F -G
Repay Loan -H +H -H    
Relend Reserves +I -I +I    
Extend Credit +J   +J    

To this we need to add one more column to record the actual deposits of a Ponzi Scheme:

Type 1 0 -1 -1 -1 -1
Account Firm Loan (FL) Bank Reserves (BR) Firm Deposit (FD) Bank Deposit (BD) Worker Deposit (WD) Ponzi Deposit (PD)
Interest on Loan +A          
Interest on Deposit     +B -B    
Pay Interest on Loan -C   -C +C    
Pay Wages     -D   +D  
Interest on Deposit       -E +E  
Consume     +F+G -F -G  
Repay Loan -H +H -H      
Relend Reserves +I -I +I      
Extend Credit +J   +J      

The first action in this extended table is a transfer from firms (for now I’ll pretend that firms do the Ponzi investing–it’s actually capitalists themselves who tend to do it, as the Madoff Scandal has made clear, but that would add yet another column to an already complicated table) of funds to invest in the Ponzi Scheme. This adds an additional row in which the sum K is transferred from the firm’s deposit account to the Ponzi’s.

We now need a second table to record the actions of the Ponzi Scheme itself. Here there are only two columns: one mirroring the Ponzi Deposit above, which records the actual money the Ponzi Scheme has; the other a column showing the returns the Ponzi Scheme alleges it is making from its fantastic scheme (the original Charles Ponzi promised to make the money by exploiting arbitrage on postage coupons), but which are in fact entirely fictional.

The essence of a Ponzi Scheme is to return principal to the investors, and call it interest. In fact, the only interest the Ponzi is making is coming from the banks’s payments of interest on the Ponzi’s deposit account at the bank–so the two entries in this second row (L for the actual return and M for the fictional) are very different.

It’s important for a Ponzi promoter to appear fabulously wealthy, so there is a substantial consumption from that is shown to come from both the actual and the fictional. This is the deduction N in the third row.

An essential step in a successful Ponzi Scheme is to give investors what they expect; so the next row shows the Ponzi Scheme making a payment from its account to the firm’s account, where the rate of return is much higher than that on standard financial products; this is the deduction O in the fourth row.

The final step is the reinvestment of these profits by the firms with the Ponzi Scheme–and even more funds, given how “successful” the Ponzi Scheme is in comparison to other investments. A transfer of P occurs

All these steps in a Ponzi process are mirrored in the accounts kept by the banks–except for the fictional balance of course, as again the Madoff scandal has indicated. So our table for legitimate finance now is:

Type 1 0 -1 -1 -1 -1
Account Firm Loan (FL) Bank Reserves (BR) Firm Deposit (FD) Bank Deposit (BD) Worker Deposit (WD) Ponzi Deposit (PD)
Interest on Loan +A          
Interest on Deposit     +B -B    
Pay Interest on Loan -C   -C +C    
Pay Wages     -D   +D  
Interest on Deposit       -E +E  
Consume     +F+G -F -G  
Repay Loan -H +H -H      
Relend Reserves +I -I +I      
Extend Credit +J   +J      
Ponzi Invest     -K     +K
Ponzi Scheme       -L   +L
Ponzi Consume     +N     -N
Ponzi Return     +O     -O
Ponzi Reinvest       -P   +P

and the table for the Ponzi Scheme is:

Type 1 -1
Account Ponzi Deposit (PD) Ponzi Scheme (PS)
Ponzi Invest +K +K
Ponzi Scheme +L +M (where M>>L)
Ponzi Consume -N -N
Ponzi Return -O -O
Ponzi Reinvest +P +P (where P>O)

One important aspect of the Scheme is that, from the bank’s point of view, everything seems legitimate. All flows from the Scheme on the bank’s books are balanced–and the Ponzi investor is a major source of demand for the output of the firm sector too.

As Madoff showed, a Scheme like this can go on for a very long time, so long as the returns being promised are not too outrageously good. His delivery of effectively 12% a year through good times and bad was much more sustainable than Charles Ponzi’s promise of a 50% return in 45 days.

But trouble comes when either the inflow needed to sustain the (lagged) outflow from the Scheme exceeds the capacity of the legitimate economy, where actual monetary profits are being made, or when trouble in the economy mean that investor stop putting their funds into the Scheme–or worse still, as happened to Madoff, start to make withdrawals from the Scheme to meet debt commitments elsewhere in the economy.

I model this with a simple change in parameter values: when a crisis hits, the inflow and reinvestment in the Ponzi Scheme drop to zero (in practice, as we also know from Madoff, “investors” actually withdraw their funds–something a Ponzi Scheme has to allow to avoid suspicion, in the hope that it will survive the downturn.

No such luck. The question everyone was asking after Madoff failed was “where did the money go?” The answer is (a) that the amount of money that was actually “there” was a lot less than Madoff claimed–he didn’t make the returns he claimed to be making, so the size of the fund itself (consisting largely of “reinvested” “profits”) was much smaller than he claimed; and (b) it was repaid to investors who hung on to it–and spent it, or used it to pay other debts–rather than reinvesting it. So it evaporates in a comparative flash:

It is also the case that, for its entire history, the Ponzi is effectively insolvent. The accumulated funds it claims to have are its debts to its “investors”; the funds it has on hand at the bank are its actual assets. The former far exceeds the latter, either side of the crisis that brings the Scheme to an end:

So that’s the basic mathematics of a Ponzi Scheme. In my forthcoming book (still a long way off!), this will be generalised to include a process by which the shift in sentiment occurs that brings a Ponzi Scheme to a crashing halt. This however is the bare bones of how such Schemes work, and how they fail.

Appendix: the actual mathematics

I hope the above wasn’t too intimidating to non-mathematical readers! In my February Debtwatch, I’ll go much more slowly and explain why a model like this–which allows for the endogenous expansion of credit independent of any manipulation of the money multiplier relation by the central bank–is needed to explain the empirical data on money.

Below are the full the full equations for the model (in tabular layout). The models are simulated in Mathcad (though they could easily be done in any mathematics program that simulates systems of Ordinary Differential Equations, such as Mathematica or Matlab). I’ll gladly share them with any researcher interested in exploring the technique and the models. Formatting equations in WordPress is a bit tricky, and I haven’t mastered the tables yet, so to make them look not too dreadful I’ve put line breaks inside some of the terms. I’ll explain all this in more detail in February–for now I’m buggered and it’s time I got back to some academic papers!

Legitimate Finance

 

Type 1 0 -1 -1 -1 -1
Account Firm Loan (FL) Bank Reserves (BR) Firm Deposit (FD) Bank Deposit (BD) Worker Deposit (WD) Ponzi Deposit (PD)
Interest on Loan +rL×FL          
Interest on Deposit     +rD×FD -rD×FD    
Pay Interest on Loan -rL×FL   -rL×FL +rL×FL    
Pay Wages     -(1-s)/tF×FD   +(1-s)/tF×FD  
Interest on Deposit       -rD×WD +rD×WD  
Consume     +BD/tB+WD/tW -BD/tB -WD/tW  
Repay Loan -FL/tLR +FL/tLR -FL/tLR      
Relend Reserves +mR×BR -mR×BR +mR×BR      
Extend Credit +nM×FD   +nM×FD      
Ponzi Invest     -pI×FD     +pI×FD
Ponzi Scheme       -rD×PD   +rD×PD
Ponzi Consume     +PS/tP     -PS/tP
Ponzi Return     +rP× PS     -rP×PS
Ponzi Reinvest       -pR×rP  

×PS

  +pR×rP  

×PS

Ponzi Scheme

 

Type 1 -1
Account Ponzi Deposit (PD) Ponzi Scheme (PS)
Ponzi Invest +pI × FD +pI × FD
Ponzi Scheme +rD × PD +rP × PS
Ponzi Consume -PS/tP -PS/tP
Ponzi Return -rP × PS -rP × PS
Ponzi Reinvest +pR × rP × PS +pR × rP × PS

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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One Response to Ponzi Maths–Part 3

  1. Is this a 6 variable system or a 7 variable one?

    Do you need dPD/dt as well as dPS/dt?

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