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
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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.