I believe it was more or less a year ago that Michel Bauwens pointed me to the International Currency System Engineering Group, ICSEG (http://www.icseg.org/) which was running a discussion list about the architecture of monetary systems.
I joined the list and the discussions were of great interest. There has now been a conclusion. Just recently, a dedicated web page was put up with the currently available documents, including a proposed standard that would ensure stability of any currency.
With several p2p currencies on the drawing board, it would seem important to take a look at what may be the fatal flaw of the bank money we are using universally today, with a view of how to avoid those same pitfalls when discussing currency alternatives. The proposed standard could be used in any kind of currency, whether issued by governments, banks or of private/collective origin.
BIBO – Bounded-Input-Bounded-Output is an engineering term. In Control Systems Theory it signifies that any system, to be stable, must respond to a bounded input with a bounded output.
Bank money is not BIBO compliant, it is argued on http://bibocurrency.org/
From a rigorous control system theory stability analysis of the current world de facto standard currency system we identify a root instability in the form of the growth component of Debt associated with the money creation process. We thus establish the inherent instability of Common Lending Practices (application of interest). Then we further chart the logical consequences of said root instability as it affects the economy as a whole and we identify how it provokes a systematic divergence between debt and value attributed to wealth in past cycles with the minimum value required in current and future cycles as those incorporate past unpaid debt. i.e. systematic compounding of debt. We also identify how the only means available within the current system design for staving off inflation is through the continued contribution of collateral wealth as guaranty for the creation of new principal debt money commensurate with past debt growth. Finally, we illustrate how compounding debt inevitably leads to a point where the inability to provide new wealth to guaranty new money to keep up with past debt growth becomes chronic at which point either runaway inflation or a definitive collapse of the system inevitably ensues.
Financial System walkthrough
1. Wealth is generated by ingenuity, human effort and resources made available through past investment of units of currency.
2. Through the process of asset evaluation, a fixed amount of existing wealth is attributed a fixed collateral value in the form of a sum of units of currency.
3. The fixed collateral sum is used as the basis for the creation of new currency in the form of a second fixed value i.e. the principal sum of loans issued into circulation through current account entries. Since both the collateral and principal loan sums are fixed, they maintain a constant ratio to the wealth pledged.
4. Current account units are distributed back to wealth producers through purchasing transactions or may be saved or stored (at a compounding interest rate) or used to cancel debt thus reducing the total amount of money in circulation.
5. Total debt due is the principal sum entered as a negative number in a loan account to which interest is added such that the debt grows as a function of time.
6. Because the total debt created always exceeds the amount of money available to satisfy it, the system produces a minimum residual debt that must be refinanced in subsequent cycles thus compounding it.
There are three documents available at the bibocurrency site:
Formal Stability Analysis of common lending practices
Passive BIBO Currency Rationale
Draft Passive BIBO Currency Specification
Documents may be updated with time, so to get the latest version, check at the source:
Roberto Verzola writes, on February 10th, 2011 at 2:04 pm:
What Ardeshir has been trying to argue, I think, is that your simple model of interest is open-loop, and admittedly unstable. But just as pure inductors are never connected alone across a DC voltage source because current flow will increase indefinitely, unpaid interest is never allowed to grow indefinitely in any financial model either.
Quite right – or, more accurately, interest is never allowed to grow indefinitely in THE REAL WORLD either.
Cheers.
# Ardeshir Mehta Says:
February 10th, 2011 at 10:33 pm
Roberto Verzola writes, on February 10th, 2011 at 2:04 pm
Ardeshir:
P may stand for population of some organism
t is time
e is a number, 2.71… that occurs frequently in the natural sciences (it also crops up when interest is compounded continuously
The logistic equation usually closely describes organic growth that initially appears exponential but then approaches an upper bound a time approaches infinity. Its graph appears like an “S” that is stretched forward.
Yes, indeed. I got that when Marc said (later) that his was the equation for a logistic curve.
MG: False I indicated it was the logistic curve at the onset you just didn’t bother to look it up.
Marc
# Ardeshir Mehta Says:
February 10th, 2011 at 10:07 pm
Marc writes, on February 9th, 2011 at 1:22 am:
MG: You have to do much better than this to be so brash in your criticism clearly without the requisite technical acumen.
AM: (Is there really cause for ad-hominem remarks here?)
MG: No ad-hominem, just pointing out that you tried to equate unbounded interest debt growth that Roberto concedes “is admittedly unstable” with natural growth that is modeled by the logistic curve and that exemplifies bounded growth! This proves that you don’t know what you are talking about, yet you insist on critiquing the work of others who do know what they are talking about.
AM: The relevant point I was trying to make is that in Marc’s formula
Yk = P(1+kr1)
… the term k is bounded in THE REAL WORLD.
MG: No, time is not bounded it precisely goes on forever. This is the point, there are functions whose output is finite as t (time) goes to infinity and then there are those whose output is not finite as t goes to infinity. You confuse the fact that the function for whatever reason does not go on forever with time not going on forever. The whole point of using time as a variable to study relations is to see what happens to the output AS TIME GOES TO INFINITY. And precisely those functions that continue to grow are considered unstable and those that reach equilibrium AS TIME GOES TO INFINITY are considered stable.
The functions that will increase to infinity as t goes to infinity are precisely what is considered as unbounded functions and therefore are unstable in nature.
Consider a grenade explosion its expansion is defined by an unbounded function but that doesn’t mean it will explode forever. But you can’t argue that because the grenade does not explode forever it is therefore stable, which is what you have been doing.
Marc
Marc
But that is not what defines instability. As I pointed out many times you are confusing the unbounded nature of the function
If anyone – including Marc – wishes to deny this, let them show us a loan IN THE REAL WORLD of which the period k has extended beyond 5,000 months!
If no such loans exist in the REAL WORLD, then it stands to reason that k must at all times be less than 5,000 months; and since R1 is also less than 50%, Y – the total debt – can never be unbounded in the REAL WORLD.
Best wishes.
Resubmitting the last post deleting the unintended text left over after my signature
Marc Says:
February 11th, 2011 at 9:55 am
# Ardeshir Mehta Says:
February 10th, 2011 at 10:07 pm
Marc writes, on February 9th, 2011 at 1:22 am:
MG: You have to do much better than this to be so brash in your criticism clearly without the requisite technical acumen.
AM: (Is there really cause for ad-hominem remarks here?)
MG: No ad-hominem, just pointing out that you tried to equate unbounded interest debt growth that Roberto concedes “is admittedly unstable” with natural growth that is modeled by the logistic curve and that exemplifies bounded growth! This proves that you don’t know what you are talking about, yet you insist on critiquing the work of others who do know what they are talking about.
AM: The relevant point I was trying to make is that in Marc’s formula
Yk = P(1+kr1)
… the term k is bounded in THE REAL WORLD.
MG: No, time is not bounded it precisely goes on forever. This is the point, there are functions whose output is finite as t (time) goes to infinity and then there are those whose output is not finite as t goes to infinity. You confuse the fact that the function for whatever reason does not go on forever with time not going on forever. The whole point of using time as a variable to study relations is to see what happens to the output AS TIME GOES TO INFINITY. And precisely those functions that continue to grow are considered unstable and those that reach equilibrium AS TIME GOES TO INFINITY are considered stable.
The functions that will increase to infinity as t goes to infinity are precisely what is considered as unbounded functions and therefore are unstable in nature.
Consider a grenade explosion its expansion is defined by an unbounded function but that doesn’t mean it will explode forever. But you can’t argue that because the grenade does not explode forever it is therefore stable, which is what you have been doing.
Marc
Marc writes, on February 11th, 2011 at 1:08 pm, inter alia:
AM: The relevant point I was trying to make is that in Marc’s formula
Yk = P(1+kr1)
… the term k is bounded in THE REAL WORLD.
MG: No, time is not bounded it precisely goes on forever. […etc….]
Please note that in your paper, the term “k” does NOT denote MERELY time, but “the k-th PERIOD OF THE LOAN whatever the period a week a month etc.” (My emphasis, but YOUR words, written on page 4 of your paper.)
But there is NO such thing, and never HAS BEEN any such thing, nor is it likely that there WILL EVER BE any such thing, as the 5,000th period of a loan – when k is measured in months – of ANY loan in the REAL world.
Thus it is ABUNDANTLY clear that your above objection does NOT hold, and neither does your formula, in the REAL world.
Cheers.
Marc writes, on February 11th, 2011 at 9:30 am:
# Ardeshir Mehta Says:
February 10th, 2011 at 10:33 pm
Roberto Verzola writes, on February 10th, 2011 at 2:04 pm
Ardeshir:
P may stand for population of some organism
t is time
e is a number, 2.71… that occurs frequently in the natural sciences (it also crops up when interest is compounded continuously
The logistic equation usually closely describes organic growth that initially appears exponential but then approaches an upper bound a time approaches infinity. Its graph appears like an “S” that is stretched forward.
Yes, indeed. I got that when Marc said (later) that his was the equation for a logistic curve.
MG: False I indicated it was the logistic curve at the onset you just didn’t bother to look it up.
Marc
I am not going to get side-tracked onto irrelevant issues. The RELEVANT issue is whether your equation:
Yk = P(1+kr1)
… EVER stands for a situation in the REAL world when k is either greater than or equal to 5,000 months. I presume, from your silence hitherto on this, that you cannot demonstrate that it does. If it does not, CLEARLY the term Yk – debt at period k when k is greater than or equal to 5,000 months – does not exist in the REAL world either. It must, therefore, be an entirely theoretical concept, having no application to REALITY!
Cheers.
Cheers.
Ardeshir wrote:
The RELEVANT issue is whether your equation:
Y – subscript- k = P(1+kr1)
… EVER stands for a situation in the REAL world when k is either greater than or equal to 5,000 months.
MG: The above is equivalent to saying that since there has never been a nuclear explosion that goes on for more than 5,000 months, therefore the explosion is not unbounded and therefore it is not unstable.
This statement of clearly absurd and further illustrates your complete lack of understanding of math, science and stability. Here is why:
Nothing in the real world goes on forever, we all know that, it is a given. But that does not prevent us from distinguishing that there are types of growth whose nature is that their output approaches infinity AS time approaches infinity which are classified as unstable and there are other functions that approach a fixed sum as time approaches infinity which are classified as stable.
Unstable growth is unstable for any period of time that the function is active, the term of the growth is irrelevant to whether the function is stable or unstable. You need to understand better what “..approaches infinity as time approaches infinity” means in math, this statement is true for any output and for any term of the growth function. That is instability is defined for the debt function for a any term whether it is a day or 5000 months or infinity for that matter.
In standard Control System’s engineering the term unbounded output refers to a growth that continues to grow AS time continues to approach infinity (which it always does), bounded refers to a growth that approaches a finite sum AS time approaches infinity (which it always does). The unbounded nature is true for any term no matter how small or large that term is. This is the root of your confusion.
Unstable growth can go on for a day a week or infinity and for each of those periods the growth remains unstable by its unbounded nature.
Therefore the term of the growth is entirely irrelevant to whether the growth is stable or not. Again you have not taken the trouble to study the prerequisites for this discussion and are confusing the discussion because you don’t understand that “approaches infinity as the independent variable time approaches infinity” is true for any term that the function.
BTW the definitions I use are not mine they are standard definitions from Control System’s Engineering. The difference is I have taken the time to study them while you obviously haven’t as you believe that they are my definitions.
I am answering Ardeshir again as it seems that my previous response from this morning did not get posted:
Ardeshir wrote:
The RELEVANT issue is whether your equation:
Yk = P(1+kr1)
… EVER stands for a situation in the REAL world when k is either greater than or equal to 5,000 months.
No, the term of the growth function is entirely irrelevant to the question of whether or not it is stable or not as the The term of a function is entirely irrelevant to whether the FUNCTION itself is bounded or unbounded.
To illustrate how absurd Ardeshir’s reasoning is:
A nuclear explosion is unstable yet it has a finite term. Following Ardeshir’s argument, since the term of the explosion is finite, the explosion is therefore bounded and therefore the function is bounded and therefore it is stable!
This is wrong because what defines the instability of the explosion is not its duration in time (i.e. its term) but that the growth is directly proportional to the progression of time i.e. the function approaches infinity as time approaches infinity (as it always does). Whereas stable growth diverges away from times immutable march towards infinity and therefore towards a finite sum.
We all know that nothing grows forever that is a given. But that is not what defines the unbounded or bounded nature of a function. It is the behaviour of the function with respect to time that determines its bounded or unbounded nature. This is what Ardeshir does not understand.
Therefore the arbitrary term of 5,000 months is entirely irrelevant to the question of whether or not the debt function is stable or not.
These are not my definitions they are entirely those of standard Control Systems Engineering.
What Ardeshir clearly doesn’t grasp is that in the language of mathematics, the statement “function A approaches infinity as time approaches infinity” means that the function’s growth is directly proportional to time’s immutable progression towards to infinity irrespective of the term of the function.
So in conclusion, Control Systems Engineering defines any unbounded growth as unstable no matter what the term of that growth. Therefore, the term of a growth function (one day a month, 500 months or n months) is entirely irrelevant to whether or not the function is stable or not.
Marc
In another list I was explaining Ardeshir’s error in reasoning and I thought it might be helpful to compliment what has been said up until now:
In short Ardeshir tries to prove that Debt = f(k) = P(1+ik) is bounded because in real life k, a periodic measure of time, is never greater than say 5000. He therefore concludes that since k is never allowed to progress past the finite arbitrary value of 5000, then f(k) in the real world is limited and therefore bounded. As a consequence f(k) is not unstable because it is not unbounded. Sounds good but it is dead wrong, let us give Control System’s Engineers a little more credit than Ardeshir does.
Lets look at Ardeshir’s argument with respect to a nuclear explosion. The exponential growth of the explosion doesn’t go on forever either and probably does not go past 5000 minutes ever. But does that make the unbounded function that accurately describes the explosion bounded and therefore mathematically stable? Does it make the explosion itself stable? No it doesn’t and no it doesn’t. So where is Ardeshir’s error?
His error is that he doesn’t understand the following statement with the requisite rigour necessary to truly understand the mathematics being used in the standard Control Systems Engineering definition of stability:
“Function f(t) (the dependent variable) approaches infinity as time (t) (the independent variable) approaches infinity.”
This is what is precisely referred to as “unbounded” growth. Note that this statement is true for any value of f(t) in the range of values f(t) can take. Why, because time (t) always approaches infinity (it might help to think of time’s target ALWAYS being infinity) and f(x) grows in direct proportion to time’s immutable progression towards infinity at all points and all arbitrary segments of time within the infinite range of values that f(t) can take. The fact that the function also has no limit as t approaches infinity, is precisely what proves the exact correlation of the growth with time and hence its inherent instability for all values of f(t) in the range of f(t).
Now, let us take another function such as the logistic curve P(t) = 1/ (1+(1/e^t). What is the difference between this function and the one above?
Well as we already have shown in a previous post, this function’s range is between 0 and 1, therefore the function does not approach infinity as t approaches infinity but rather it diverges from “times immutable march towards infinity” towards fixed values (0 and 1). This is precisely the opposite from the above and therefore in Control Systems Engineering, such a function is considered to be bounded, in this case between f(t)=0 and f(t)=1.
In short, the term or duration of a function in reality says nothing about whether that function is bounded or unbounded and hence whether it is stable or not. This is not according to my personal definitions but it is according to Control Systems Engineering’s definitions.
If Ardeshir’s argument were valid then a nuclear explosion would have to be considered stable which we all know is absurd.
Marc writes, on February 12th, 2011 at 9:29 pm (I have abbreviated the message to prevent repetition):
In another list I was explaining Ardeshir’s error in reasoning and I thought it might be helpful to compliment what has been said up until now:
In short Ardeshir tries to prove that Debt = f(k) = P(1+ik) is bounded because in real life k, a periodic measure of time, is never greater than say 5000. He therefore concludes that since k is never allowed to progress past the finite arbitrary value of 5000, then f(k) in the real world is limited and therefore bounded. As a consequence f(k) is not unstable because it is not unbounded. Sounds good but it is dead wrong, let us give Control System’s Engineers a little more credit than Ardeshir does.
[…]
His error is that he doesn’t understand the following statement with the requisite rigour necessary to truly understand the mathematics being used in the standard Control Systems Engineering definition of stability:
“Function f(t) (the dependent variable) approaches infinity as time (t) (the independent variable) approaches infinity.”
[…]
In short, the term or duration of a function in reality says nothing about whether that function is bounded or unbounded and hence whether it is stable or not. This is not according to my personal definitions but it is according to Control Systems Engineering’s definitions.
If Ardeshir’s argument were valid then a nuclear explosion would have to be considered stable which we all know is absurd.
Marc seems to be contradicting himself. Earlier he defined the terms “bounded” and “stable” thus:
[QUOTE]
1) Anything that NEVER has unbounded inputs or outputs is inherently STABLE.
2) As a corollary to 1) Anything whose design CAN produce unbounded inputs or outputs for any reason is inherently UNSTABLE.
Passive Stability: A stable system whose Outputs never exceed Inputs.
Unbounded: Unbounded means any growth over time that approaches infinity..
[END QUOTE]
Using these definitions, it is clear that in his formula
Yk = P(1+kr1)
… the term k is bounded in THE REAL WORLD, since it never exceeds 5,000 months. And if it is bounded, so must Yk be – and Yk being the output of Marc’s system, his system must be stable. There is no escaping this conclusion, GIVEN Marc’s own above-quoted definitions of “bounded” and “stable” (and, of course, of “unbounded” and “unstable” also).
But now he seems to be implicitly changing the definitions of the terms “bounded” and “stable” (and, of course, of “unbounded” and “unstable” also). Perhaps he would like to explicitly give the definitions of those terms de novo, and after that, stick to just ONE SINGLE set of definitions … ? Otherwise, how can he be said to be consistent?
And then, suppose we do change the definitions of “bounded” and “stable” so that they may fit the meaning of Marc’s latest posts – then of what value is his function Yk = P(1+kr1) in illustrating anything in economics as it functions in the REAL world? Or in other words, what exactly is expected to HAPPEN in the REAL world of economics, if the function Yk = P(1+kr1) applies to it, and if as a result there is indeed an instability of the kind which Marc claims there is?
Marc can surely not claim that such an instability would lead to inevitable defaults or foreclosures, for I have conclusively demonstrated at
http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html
… that defaults and foreclosures are definitely not inevitable.
Cheers.
All,
The new line of Ardeshir’s obfuscation is the following:
AM:
Marc seems to be contradicting himself. Earlier he defined the terms “bounded” and “stable” thus:
[QUOTE]
1) Anything that NEVER has unbounded inputs or outputs is inherently STABLE.
2) As a corollary to 1) Anything whose design CAN produce unbounded inputs or outputs for any reason is inherently UNSTABLE.
Passive Stability: A stable system whose Outputs never exceed Inputs.
Unbounded: Unbounded means any growth over time that approaches infinity.
MG: He claims that these definitions contradict the definition I gave in my last post:
“Function f(t) (the dependent variable) approaches infinity as time (t) (the independent variable) approaches infinity.”
MG: I call on Ardeshir to explain exactly where and how these definitions contradict each other or are in anyway inconsistent?
……
He then asks the following:
AM: Or in other words, what exactly is expected to HAPPEN in the REAL world of economics, if the function Yk = P(1+kr1) applies to it, and if as a result there is indeed an instability of the kind which Marc claims there is?
MG: To answer the question of what occurs in the economy due to the instability, I will simply quote Ardeshir’s own words following mine from a previous post:
Ardeshir Mehta Says:
February 6th, 2011 at 5:08 pm
Marc writes, on February 5th, 2011 at 10:49 am:
[QUOTE]
This last point can be confirmed by the simple fact that interest is the only source of a cost increase that is not associated with any corresponding contribution of wealth. It is a source of systematic inflation that if refinanced would make BIBO practices non BIBO at the aggregate level.
[END QUOTE]
AM: This is not an aspect of your paper with which I am in disagreement. I do not deny that interest causes inflation
MG: So Ardeshir agrees that the interest causes inflation as our paper shows and that causes instability as the inflated wealth gets refinanced.
Again Ardeshir doesn’t understand the language of mathematics, he doesn’t understand that unbounded refers to all values that an unbounded function can take in its range.
He simply cannot grasp that an unbounded function is one that progresses along with the independent variable time towards infinity at all points and segments of time.
He simply does not understand the use of the term “approaches”. As I said, time always “approaches” infinity but some functions do so AS time does and others do not approach infinity and therefore must approach a finite sum. Those that approach infinity AS time does are unstable those that do not i.e. approach a finite sum are stable for All values of the function.
The term or duration of the function is therefore entirely irrelevant to whether the function is stable or not.
This I showed graphically using the example of an nuclear explosion. Its term is finite yet nonetheless it is still unstable. According to Ardeshir’s argument because the duration of the growth function is finite then it is not unbounded and therefore it is stable.
All,
Some might be wondering why I continue to respond to Ardeshir, so I thought the following explanation might be useful.
I take the time to respond to Ardeshir’s nonsense, only because the understanding of stability and instability is so crucial and his types of objections are useful to help others learn the importance of applying the rigour of Control Systems Engineering definitions. If we don’t have a rigorous and unequivocal understanding of stability we will forever be led down the garden path, with false theories such as those that are the foundation of conventional economics.
Notice that Ardeshir began his onslaught by declaring in unequivocal and authoritative terms: “There are some serious errors in the document entitled “Formal Stability Analysis of common lending practices””.
He didn’t ask for clarifications as did Roberto for example, providing useful references in the discipline so that we can all better understand. No, Ardeshir like a bull in the china shop, brashly declared that the stability analysis was wrong. Yet we all have been able to see clearly that Ardeshir knows nothing about Control Theory, formal stability and lacks basic understanding of mathematics.
In short his attack on the basic Control Systems definition of stability boils down to the absurd conclusion that:
– If the term of the growth is finite the growth is finite and therefore bounded and therefore it is stable-
Yet we all know that can’t be because we all know that a fire cracker is inherently unstable and its duration is less than seconds.
Similarly, conventional economic theory is completely unscientific, logically incongruous, it is a false creed basically a cult. It depends on a class of implementers that precisely lack the ability to think independently logically and scientifically in a mathematically consistent manner. It benefits from the intellectually slack, those who would modify their conclusions by mutilating pristine logic for short term personal gain and status. In fact, this false creed promotes as many quack theories as it can, as all serve to lead us away from any truth about the true nature of the economic design.
This is why it is so important to apply engineering stability analysis as we have done and as others also have in the strictest formal mathematics. By doing so, we prove that debt growth by interest as applied in today’s common lending practices, is a root source of instability and we also show that an economy is not only possible without interest, but it is infinitely more stable and therefore viable.
The only apparent utility that interest has, is to manipulate the free market and centralise control of it. But unfortunately that very mechanism destabilises the whole economy and sends it into ever increasing oscillations that inevitably lead the whole world into a state of mad desperation, which ultimately becomes the underlying justification of the greatest horrors humans can be capable of. All wars have economic foundations, before a first bullet is fired economic killing has taken place. All torture and loss of human rights are accepted on the basis of economic fear and insecurity. In short, most if not all extreme actions that contradict the notion of respect for the individual and their inalienable rights are founded in dire economic circumstances.
We therefore cannot allow that scientifically unsound arguments of the like Ardeshir puts forth, prevail unchallenged as it is precisely this type of false reasoning that our common cult of economics depends on to continue to misinform and distract us all from the simple solution of zero interest i.e. a Passive BIBO Currency System.
It preys on people’s good will to accept all propositions as “good willed” subtly confusing concepts such as interchanging “everyone’s equal right to an opinion” with “everyone’s opinion is equally valid”. No, everyone’s opinion is not equally valid and to claim so by rote is to abuse in the most subtle way constructive dialogue as it only serves to debase the quest for common understanding precisely between good willed contributors. It acts to send the message to those that would offer their best that doing so is futile because in the end any half backed theory will prevail equally and alas the majority of people’s opinion, whether intentional or not, are incomplete or misinformed half backed conjecture. This is due mostly because of how society is becoming more and more dumbed down.
Therefore, I call on all those with the requisite intellectual and mathematical tools, to apply them now to the issue of money systems. As I said, sorting the economic issue out is of paramount importance because it literally will save millions if not billions of lives that do not need to be extinguished either in abject poverty and suffering or for economically motivated armed conflict undoubtedly founded in the root inequity of the present de facto economic standard.
Finally, it is not just a question of putting a stop to the symptoms of this root instability by removing peoples freedoms, rights and autonomy, it is a question of delivering a solution that precisely requires and promotes such rights and freedoms.
It is not only possible to do so, but it must be in the interest of all included the deluded keepers of this false system and creed based on this inherently unstable financial design. Because once the keepers realise that it is not a question of personal opinion and that their creed is not a rational proposal based on a sound assessment of human nature as they would have us believe. But rather, it is the faulty and logically incongruous design of their belief system that is destroying society and nature, the must at least for their children’s sake accept the simple technical solution that is so trivially proposed by a Passive BIBO Currency solution. All it takes to fix the world economic crisis is for all of us to simply insist the root instability be removed, and we are done.
No one needs be punished or persecuted or pursued we all are responsible for the state of the world whether for our unwillingness to defend the or for propagating what ultimately is false. But we have no alternative but to fix and it it going to take as many of us capable of standing fast in support of sound scientific reasoning against the smoke screens and illusions that attempt to confuse us.
Best regards and warmest wished to all,
Marc
For some reason not all the posts I make are visible to me after I post them, then they become so after I make a subsequent post, often thinking that the previous post hasn’t gotten through.
I am sending this one to see what happens.
Regards,
Marc
Yes the missing post appears before the the latest. It must have to do with the approval process. Sorry for any inconvenience or redundant posts.
Hi Marc, I try to moderate twice a day, mornings and evenings thai time. We tried non-moderation and post-publishing-moderation, but both let through enormous amount of spam. Together with the automated spam filters, and the daily average of a dozen or so I have to remove manually, we’re reaching 1.5m spam items … just to offer a perspective from this side of the fence ..
Michel
Marc,
I am still awaiting your answer to my very important question: if a component that shows inherently unstable behavior under open-loop conditions is placed in a larger system that closes the loop with negative feedback, to make the larger system stable, would you still say that the inherent instability of one particular component makes the whole system unstable?
(By the way, may I just make a small correction: unbounded input does not prove instability, only unbounded output and — I would add — unbounded state variables. Unbounded input is an off-spec input and cannot be blamed on the system being tested.)
I gave the simple example of a resistor and inductor in series, driven by direct current. You said however that this does not represent your interest model and that “the model of what takes place in the economy is more akin to turning the inductor on for a period of time without any resistance and then turning it off”, I only cited the resistor-inductor circuit to give you the simplest example I can find of a component (inductor) with unstable behavior under direct current, but which can be made stable with the simple addition of a second component, my point being that showing that a component like an inductor (or an interest-bearing loan) is unstable by itself is not enough to prove that the larger system of which it is a part of is also unstable.
Let me go to the more complex example which you yourself gave, a zero-resistance inductor that is periodically connected to a direct current source. You said this model is closer to what takes place in the economy. I agree. In fact, this is closer though not yet the exact equivalent of an compound interest-bearing loan. Very common in oscillators, this circuit generates triangular pulses — an exponentially-rising pulse which quickly drops to zero, and then repeats itself. Using a capacitor-resistor combination in place of inductors, this is also very common in flip-flops and multivibrators, which are at the heart of digital technology.
May I know if you consider such output (a train of fixed-amplitude triangular pulses) BIBO-stable?
I think you can see where I’m leading at. There are many engineering situations where individual components may show unstable behavior, but such unstable behavior may be controlled through negative feedback. Thus the larger closed-loop system, if properly designed, can pass stability analysis and be considered stable.
Let us go to your interest-bearing loan model. You modelled it as an amount that increases linearly with time (very much like an inductor connected across a DC source). While it is trivial to show that it has unbounded output, just as an inductor across a DC source will have unbounded output, this will not prove that the system of which it is a part is also unstable. I have already given you two examples of circuits that can be stabilized through negative feedback.
In all financial systems I am aware of, when interest-due accumulates, this will trigger negative feedback which can result in at least 3 possibilities: 1) borrower is forced to pay interest-due regularly, making the overall debt bounded once more; 2) foreclosure, with the lender taking less profit or a partial loss, liquidating the debt, or 3) writing off as bad debt, with the lender absorbing the loss through bad debt accounts, also liquidating the debt. Standard accounting practices will not allow bad debts to grow without limit. So the larger system itself prohibits it and has mechanisms for keeping bad debts bounded.
As you said yourself, this is very much like an inductor whose current increases linearly, but at a certain point is disconnected (turned off). If you model the economic system this way (as your statement I quoted seems to indicate), then its output will be a train of triangular pulses (i.e., aborted exponential curves). This is BIBO to me.
But I am not ready to give up on your project. I do think, as you do, that something is causing instability in the financial system. I am also looking for it.
But I am not convinced by your argument that because an open-loop unpaid interest-bearing loan is unbounded, this is sufficient proof that the financial system itself is unbounded.
Greetings,
Roberto
Roberto,
Thanks for your thoughtful and knowledgeable posts.
Yes of course, if a system that has an unstable component attenuates the instability for all cases and circumstances such that the output of the greater system is unbounded in all circumstances, then yes the greater system would be stable.
But such is not the case with the interest debt function at hand. The three measures you mention and as we agree simply represent the setting of finite terms to the manifest unbounded growth but do not compensate the instability. What you have not considered are the other eventualities that we discuss in detail in our paper:
1) Refinancing of debt making the output guarantying a minimum exponential debt seed.
2) Incorporating the interest into the cost of production that in turn is refinanced.
For example, producer A finances production and distributor B finances the purchase and distribution of A’s production and finally consumer C finances the purchase of the production, distribution and consumption of the final product.
I think it is clear that the root instability in spite of having a fixed term still produces unbounded repercussions in the form of positive feedback.
The only practical solution is to implement a certifiable Passive BIBO Currency as we suggest in our paper, that contract and expands as a function of the real wealth brought to the market expanding and contracting.
Best regards,
Marc
Roberto,
I will rephrase the conclusion of my last post:
I think it is clear that the root instability IN THE CASE OF A LOAN AGAINST A FIXED COLLATERAL VALUE and in spite of having a FINITE terms FOR INDIVIDUAL LOAN CONTRACTS, still produces unbounded repercussions ON THE AGGREGATE in the form of INCREASES IN PRICES that act as positive feedback to subsequent future lending AGAINST THAT SAME COLLATERAL.
The only practical solution is to implement a certifiable Passive BIBO Currency as we suggest in our paper, that contracts and expands as a function of the real wealth brought to the market expanding and contracting AND NOT THE OTHER WAY ROUND.
Regards,
Marc
Marc wrote, on February 14, 2011 at 11:22 am:
All,
The new line of Ardeshir’s obfuscation is the following:
AM:
Marc seems to be contradicting himself. Earlier he defined the terms “bounded” and “stable” thus:
[QUOTE]
1) Anything that NEVER has unbounded inputs or outputs is inherently STABLE.
2) As a corollary to 1) Anything whose design CAN produce unbounded inputs or outputs for any reason is inherently UNSTABLE.
Passive Stability: A stable system whose Outputs never exceed Inputs.
Unbounded: Unbounded means any growth over time that approaches infinity.
MG: He claims that these definitions contradict the definition I gave in my last post:
“Function f(t) (the dependent variable) approaches infinity as time (t) (the independent variable) approaches infinity.”
MG: I call on Ardeshir to explain exactly where and how these definitions contradict each other or are in anyway inconsistent?
I explained that already. I had written:
[QUOTE]
Using these definitions [i.e., the ones he had given earlier], it is clear that in his formula
Yk = P(1+kr1)
… the term k is bounded in THE REAL WORLD, since it never exceeds 5,000 months. And if it is bounded, so must Yk be – and Yk being the output of Marc’s system, his system must be stable. There is no escaping this conclusion, GIVEN Marc’s own above-quoted definitions of “bounded” and “stable” (and, of course, of “unbounded” and “unstable” also).
[END QUOTE]
To elaborate on what I had written, in case Marc didn’t understand it: If “Unbounded means any growth over time that approaches infinity”, as Marc says, then in the real world, the growth of a debt at t = 5,000 months after the issuing of the loan is ZERO, and thus CANNOT approach infinity thereafter. Therefore by Marc’s own definition, it MUST be bounded.
But now he is claiming that debt is NOT bounded. This is a blatant contradiction.
Marc continued:
He then asks the following:
AM: Or in other words, what exactly is expected to HAPPEN in the REAL world of economics, if the function Yk = P(1+kr1) applies to it, and if as a result there is indeed an instability of the kind which Marc claims there is?
MG: To answer the question of what occurs in the economy due to the instability, I will simply quote Ardeshir’s own words following mine from a previous post:
Ardeshir Mehta Says:
February 6th, 2011 at 5:08 pm
Marc writes, on February 5th, 2011 at 10:49 am:
[QUOTE]
This last point can be confirmed by the simple fact that interest is the only source of a cost increase that is not associated with any corresponding contribution of wealth. It is a source of systematic inflation that if refinanced would make BIBO practices non BIBO at the aggregate level.
[END QUOTE]
AM: This is not an aspect of your paper with which I am in disagreement. I do not deny that interest causes inflation.
MG: So Ardeshir agrees that the interest causes inflation as our paper shows and that causes instability as the inflated wealth gets refinanced.
So what? What is wrong with some kinds of inflation? In fact, inflation helps the borrower, does it not, if it is equal to or greater than the (nominal) interest rate? If that happens – say if the interest rate charged on a loan is 5% per annum while inflation is also 5% per annum, the result in REAL terms is as if there were no interest on the loan at all! So why is it NECESSARILY a bad thing – or in other words, what sort of “instability” would it cause?
Marc continued:
Again Ardeshir doesn’t understand the language of mathematics, he doesn’t understand that unbounded refers to all values that an unbounded function can take in its range.
He simply cannot grasp that an unbounded function is one that progresses along with the independent variable time towards infinity at all points and segments of time.
He simply does not understand the use of the term “approaches”. As I said, time always “approaches” infinity but some functions do so AS time does and others do not approach infinity and therefore must approach a finite sum. Those that approach infinity AS time does are unstable those that do not i.e. approach a finite sum are stable for All values of the function.
The term or duration of the function is therefore entirely irrelevant to whether the function is stable or not.
I said nothing about the FUNCTION as such. All I am saying is that a DEBT (or LOAN) cannot approach infinity. If Marc’s mathematics shows that it can, then the mathematics must be clearly faulty, because simple, ordinary observation – which any non-mathematician can carry out – reveals that it never does.
Marc added:
This I showed graphically using the example of an nuclear explosion. Its term is finite yet nonetheless it is still unstable.
According to Ardeshir’s argument because the duration of the growth function is finite then it is not unbounded and therefore it is stable.
I do not want to get sidetracked by irrelevant arguments, but since Marc keeps bringing it up, may I ask in what way, ACCORDING TO MARC’S OWN DEFINITIONS, is a nuclear explosion unstable? Its input is bounded and its output is also bounded, and therefore, by Marc’s OWN definitions given above, it must be STABLE!
However, nuclear explosions are irrelevant to my argument, and I won’t go into them again. I would like Marc to stick to the issue, and to show us a single DEBT in all of history that had no limits, no matter how much time passed. (I might add that according to most dictionaries, the term “unbounded” means having no limit – Marc can look it up himself if he wishes.) If he cannot show us such a debt, then WHAT can he be talking about when he claims that Yk, the output of the system, is “unbounded”?
Cheers.
Roberto Verzola writes, on January 25th, 2011 at 11:27 am:
QUOTE
In PRACTICE the output of a financial system is NOT unbounded: loans do not usually continue for much beyond 30 years. Hence it is not true in PRACTICE that – as the document says on Page 6, “[…] if no reduction of principal is produced, the total debt grows to infinity in a linear fashion.”
ENDQUOTE
There is one type of debt that in practice is meant to be perpetual: funds where only a portion of the interest income are withdrawn and the principal, programmed to grow over time, are meant to be kept in perpetuity.
Sorry Roberto, I had missed this post of yours.
However:
1.
Can you give us a historical example of such a debt that has been in force for, say, 5,000 months or more, and is still continuing in force?
2.
Even if you can, would you agree that the vast majority of debts are not like that, and that these kinds of debt play a very minor role, if any, in the economy of the real world?
Cheers.
Here is some actual, real world statistical data that shows the current economic system isn’t sustainable and that debt and interest ARE spiraling out of control, at least in the U.S. … through what seems to be an unbounded output of the system:
10 Charts That Embody Everything That’s Wrong With The U.S. Economy
http://www.businessinsider.com/charts-debt-unemployment–2011-2?slop=1#ixzz1ECYXRViM
Ardeshir wrote:
“I said nothing about the FUNCTION as such. All I am saying is that a DEBT (or LOAN) cannot approach infinity.”
Not only can the interest-debt function approach infinity but it does for all values of the function within the range of values the function can take, irrespective of any arbitrary term that the function may have in reality.
Your refusal to recognise your error in interpreting the use of the word “approach” is at the heart of the fallacy you are trying to perpetrate on everyone. Approach indicates a particular trajectory with respect to time that can be determined for any finite point along that trajectory.
The way to determine this is by taking the derivative with respect to time at every point of the finite term of the growth in question. If for every point in time of any finite growth the derivative or rate of change is not zero, then it is “approaching infinity” at those points in time.
The explosion and interest-debt growth examples both share the property of approaching infinity but as is the case with almost any growth in the real world, the duration or terms of growth in each case are finite. But that does not imply that the growth is not of an unbounded nature as Ardeshir tries to superficially conclude.
So Ardeshir’s insistence on the term of growth being relevant to whether the growth that takes place within that term is bounded or not is simply wrong. The term of the growth says nothing about its bounded or unbounded nature but rather it is the derivative of the growth that determines whether the growth is bounded or not.
The interest-debt function is unstable period.
Marc
Marc
All some very important and revealing observations:
Ardeshir wrote:
AM: However, nuclear explosions are irrelevant to my argument, and I won’t go into them again.
MG: But Ardeshir has never addressed the example of the instability of an explosion.
I therefore conclude that Ardeshir lacks the requisite integrity to maintain a constructive discussion on the issues.
We are talking about the rigorous definition of stability and instability and therefore any example of stability or instability is relevant. To say otherwise is to evade the issue. But to make patently false assertions such as Ardeshir has, shows that he is either not being honest or is not playing with a full deck.
He attempts to wing the insinuation that he has “dealt” with an issue when he blatantly never has. This is clearly designed to give the false impression to new comers that his position is thorough and complete, when it is not.
This is frankly unacceptable and if Ardeshir is not capable of producing the text where he “deals” with the example of the instability of an explosion, then he should be required to immediately apologise for trying to mislead all of us.
I must demand the utmost standard in integrity in discussing this issues because of what I have said in a previous post and repeat here:
“All wars have economic foundations, before a first bullet is fired economic killing has taken place. All torture and loss of human rights are accepted on the basis of economic fear and insecurity. In short, most if not all extreme actions that contradict the notion of respect for the individual and their inalienable rights are founded in dire economic circumstances.”
The issue we are discussing is whether or not interest is at the root of “dire economic circumstances” and within such a debate where so much is at stake for so many, Ardeshir attempts to support his technically and intellectually bankrupt position by lying!
He knows very well that he hasn’t dealt with the very relevant example that blows his whole argument out of the water i.e. an explosion is a clear example of instability of a growth with a finite term, therefore the term is irrelevant to the question of stability. This is why he most make the false claim that the example is irrelevant. But if it is irrelevant then let him show us how it is so, as he falsely claims he already has.
Finally, it is worth noting that when called on to admit that his own words recognise that the root instability of interest results in inflation (another unbounded consequence of the root instability of interest-debt growth). He answers with and entirely evasive response in the form of an apology for inflation!
“So what? What is wrong with some kinds of inflation? In fact, inflation helps the borrower, does it not, if it is equal to or greater than the (nominal) interest rate?”
What a cold and callous remark to make, when in Egypt the average family spends 80% of their income on food of which wheat products are a staple, and the recent increase of prices of wheat as a function of the inherent instability of the financial system is pushing millions into absolute and dire poverty and famine. But that doesn’t prevent Ardeshir from seeing the positive side of inflation.
IN ANY EVENT, WE ARE NOT TALKING ABOUT THE GOODNESS OR BADNESS OF INFLATION, WE ARE DISCUSSING THE FACT THAT INTEREST-DEBT GROWTH HAS REPERCUSSIONS THAT ARE ALSO UNBOUNDED I.E CAUSE INFLATION. AND ARDESHIR’S SUPERFICIAL APOLOGY FOR INFLATION IS ENTIRELY IRRELEVANT TO THE ISSUE OF UNBOUNDED REPERCUSSIONS OF INTEREST-DEBT GROWTH.
Is this another error of his? Or is it that Ardeshir really is an apologist for the current system and has an agenda to at all costs debunk the real science behind the system that is creating such dire circumstances for so many.
What a brazen tactic of camouflaging himself as anti interest and then proceed to ignore control systems engineering, math and plain logic to argue against all the major and sound technical proofs that show the true nature of interest-debt and its violent repercussions on society at all levels.
In summary, Ardeshir professes to be against interest but thinks that it is not unstable and that inflation an unbounded result of interest-debt, according to him, is just dandy because it helps the borrower!
On 17-Feb-11, at 8:19 AM, P2P Sepp wrote:
Here is some actual, real world statistical data that shows the current economic system isn’t sustainable and that debt and interest ARE spiraling out of control, at least in the U.S. … through what seems to be an unbounded output of the system:
10 Charts That Embody Everything That’s Wrong With The U.S. Economy
http://www.businessinsider.com/charts-debt-unemployment–2011-2?slop=1#ixzz1ECYXRViM
My response:
1.
The charts at that site are misrepresentations of reality, as the charts themselves demonstrate if analysed. The chart showing the change in the Consumer Price Index shows that between 1970 and today the dollar has decreased in value about 6 times, while the chart showing the total household debt (the one entitled “Household Sector: Liabilities”) shows that the debt has gone up around 10 times. Thus the amount of increase in REAL debt (i.e., debt measured in dollars of constant value) is only about 66 per cent – which amounts to only about 2% per annum. This is hardly serious, and can be accounted for by the fact that today, due to credit cards (which hardly existed in the 1970s) Americans are much more ready to incur debt than they were then.
As for the chart showing “Future Interest Costs”, it is just a projection after the year 2011, not data at all. The actual data on this chart shows the interest on past debt, especially since the year 2000, as being quite low; and it shows absolutely NO interest on new debt at all. So how the author of the chart got his projections of huge amounts of interest on future debt is, to put it mildly, questionable.
2.
In any case, however, I am against interest, as indeed I am against debt, as my papers at
http://homepage.mac.com/ardeshir/AbolishInterest.html
and
http://homepage.mac.com/ardeshir/Debt-FreeMoneyIsBetterThanDebt-BasedMoney.html
… clearly show.
This is not to say that all arguments against interest are valid, as I have had occasion to point out in my paper at
http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html
In fact, those who make invalid arguments against interest do some disservice to those who are genuinely opponents of interest, because it allows proponents of interest to point out the obvious flaws in those arguments, and then argue – albeit spuriously – that interest can’t be all that bad after all.
Cheers.
Dear Marc,
You wrote this in reply:
Quote
The three measures you mention and as we agree simply represent the setting of finite terms to the manifest unbounded growth but do not compensate the instability. What you have not considered are the other eventualities that we discuss in detail in our paper: 1) Refinancing of debt making the output guarantying a minimum exponential debt seed. 2) Incorporating the interest into the cost of production that in turn is refinanced.
End quote
I’ve gone through your paper and you only use the z-transform / pole-zero analysis on the simple open-loop loan models (simple interest, pp.2-8; compound interest, pp.9-13), which grow without bound because of your own choice of simplistic assumptions.
Your paper did not model, and of course did not do z-transform analysis, on a more realistic loan model where standard practices like foreclosing on the collateral or writing off bad debts are involved. If you had modeled such standard practices and did a stability analysis, I suspect you would have come up with triangular pulses as output, which will of course be bounded. You write above that these practices “do not compensate the instability”, but where is your z-transform analysis of these more realistic closed-loop models I suggested, that would justify your reply? There is none in your paper.
You did show a larger systemic model of the financial system in p. 14 of your paper (Fig. 6), right after the open-loop compound interest model, but it was just a drawing, with no mathematical modelling and no z-transform / pole-zero analysis. Those “other eventualities” you refer to above were never rigorously analyzed for stability in your paper either.
So, my question is where did you get your conclusion that financial system itself is unstable? Based on what I’ve seen so far, this seems to be an a priori conclusion rather than a result of the use of stability analysis using z-transforms and similar methods of control systems engineering.
Roberto
Ardeshir Mehta Says:
February 17th, 2011 at 9:18 pm
10 Charts That Embody Everything That’s Wrong With The U.S. Economy
http://www.businessinsider.com/charts-debt-unemployment–2011-2?slop=1#ixzz1ECYXRViM
My (Ardeshir) response:
AM: The chart showing the change in the Consumer Price Index shows that between 1970 and today the dollar has decreased in value about 6 times, while the chart showing the total household debt (the one entitled “Household Sector: Liabilities”) shows that the debt has gone up around 10 times. Thus the amount of increase in REAL debt (i.e., debt measured in dollars of constant value) is only about 66 per cent – which amounts to only about 2% per annum.
MG: First Ardeshirs calculations are wrong, x times = x 100%. The graph shows over 500% change in CPI since 1976 100 ->. The household debt is more than 14 times between 1970 and now. 1400%- 500% is a total of 900%.
MG: So we can quickly throw out Ardeshir’s pseudo authoritative estimations as the difference is nothing close to 66%.
MG: Second, the issue was instability an in almost all of the charts the output shows an exponential output. That is sufficient to prove that the economic output is unbounded, which is the unquestionable underlying issue.
MG: Only pseudo scientists would be so clued out as to rationalise an abuse using questionable statistical indexes such as CPI to conclude that the theoretical result, although not zero is “ummm quite reasonable”. But as Ardeshir illustrates the theoretical “calculations” leaves a lot to be desired.
MG: The whole point of this stability discussion is to eliminate the thrashing in the first place! And no matter how you massage the raw data a posteriori, it is rising exponentially!
Marc
Ardeshir wrote:
“In fact, those who make invalid arguments against interest do some disservice to those who are genuinely opponents of interest,because it allows proponents of interest to point out the obvious flaws in those arguments, and then argue – albeit spuriously – that interest can’t be all that bad after all.”
MG: Right! Like interest is not unstable when its derivative is a always positive and inflation is no problem because it helps the borrower even when it starves millions. And that debt growth even if exponential is no problem that the good concocted CPI cannot compensates. Yup we all no who and what you mean.
Thank God such spurious arguments can be easily seen for what they are, just spurious not to mention vexatious arguments. But what a pain they are to have to constantly rebut, don’t you think Ardeshir?
# Roberto Verzola Says:
February 18th, 2011 at 1:41 pm
Roberto,
Before I answer your post I need to clarify something with you first.
Do you believe that because the term of an unstable growth is finite the result is therefore bounded and hence the function is stable?
Marc
Marc
I find it utterly unnecessary to respond to Marc’s latest round of responses, since they have nothing to do with his published equation, namely:
Yk = P(1+kr1)
… so as to explain how it must apply to the REAL world, where k, the period of a loan, is always limited in duration, and which therefore mathematically entails that Yk, the debt resulting from the loan, cannot possibly be unbounded.
In fact, I would be very surprised if Marc could show us as few as three professors of control systems engineering who would unanimously agree with him that in the formula given above, when the input, k, is limited in duration (as it always is in the REAL world), the output, Yk MUST be unbounded. Just THREE such professors unanimously agreed that Marc is right in claiming the above! He surely knows at least that many.
If he can’t produce even three such professors who all agree with him in his above-noted claim, then I submit that Marc has simply not proven his case.
Cheers.
All,
For those that haven’t read our paper, Roberto’s remark:
“You did show a larger systemic model of the financial system in p. 14 of your paper (Fig. 6), right after the open-loop compound interest model, but it was just a drawing, with no mathematical modelling..”
Is not quite fair, we did provide a very extensive analysis of the diagram that included a series of proportions to establish relationships between necessarily fixed (stable) sums ((w) wealth, (C)Collateral, (P) Principal debt and total debt due (P+I)). These were followed by further text that essentially concluded the following:
Either the system fails which is equivalent to Roberto’s triangular pulse output i.e. the system abruptly stops or, refinancing must take place leading necessarily to an exponential output. In the case of this second eventuality, there is no need for further z-transform analysis as it would simply be a repeat of the analysis already given.
It may seem simplistic, but the system is dead simple so there is no need for more convoluted analysis. The economy either completely stops (breaks down) i.e. Roberto’s triangular pulse output, or it maintains an unbounded output that as it turns out necessarily becomes exponential when debt is refinanced. The empirical data that Sepp provided clearly shows that in reality, all system debt outputs are exponential, confirming our conclusions.
Marc
Re. the following exchange, which has nothing to do with Marc’s paper, but was initiated by Sepp on February 17th, 2011 at 10:12 am:
[SEPP]: Here is some actual, real world statistical data that shows the current economic system isn’t sustainable and that debt and interest ARE spiraling out of control, at least in the U.S. … through what seems to be an unbounded output of the system:
10 Charts That Embody Everything That’s Wrong With The U.S. Economy
http://www.businessinsider.com/charts-debt-unemployment–2011-2?slop=1#ixzz1ECYXRViM
[MYSELF, RESPONDING TO SEPP]: The charts at that site are misrepresentations of reality, as the charts themselves demonstrate if analysed. The chart showing the change in the Consumer Price Index shows that between 1970 and today the dollar has decreased in value about 6 times, while the chart showing the total household debt (the one entitled “Household Sector: Liabilities”) shows that the debt has gone up around 10 times. Thus the amount of increase in REAL debt (i.e., debt measured in dollars of constant value) is only about 66 per cent – which amounts to only about 2% per annum. This is hardly serious, and can be accounted for by the fact that today, due to credit cards (which hardly existed in the 1970s) Americans are much more ready to incur debt than they were then.
[MARC GAUVIN, COMMENTING ON MY RESPONSE TO SEPP]: First Ardeshir’s calculations are wrong, x times = x 100%. The graph shows over 500% change in CPI since 1976 100 ->. The household debt is more than 14 times between 1970 and now. 1400%- 500% is a total of 900%.
So we can quickly throw out Ardeshir’s pseudo authoritative estimations as the difference is nothing close to 66%.
[MY RESPONSE TO MARC’S ABOVE COMMENT]: Marc commits multiple errors here: some of them quite elementary. It seems almost unnecessary to point them out: anyone capable of doing even high school mathematics can easily spot them.
To begin with, he compares the Consumer Price Index (CPI) change from 1976 to the present, while he compares household debt from 1970 to the present. Those comparisons do not both cover the same period of time, and are therefore invalid.
Secondly, the chart shows that CPI increase from 1970 to the present is about 6 times, which means that today’s dollar is worth about 16.7 cents in 1970s dollars. The household debt in 1970 is shown by the other chart to be somewhere around $1.4 trillion, while today it is around $14 trillion. That represents an approximately 10-fold increase, not a 14-fold increase, as Marc claims.
Thirdly, and most importantly, a household debt of 14 trillion of today’s dollars would amount to (14*0.167=2.338) trillion in 1970’s dollars. Since the household debt in 1970 was, as the chart shows, about 1.4 trillion in 1970’s dollars, that is equivalent to an increase of 67% in dollars of constant value, as I had originally said … and not 900%, as Marc most erroneously claims.
Cheers.
Ardeshir says that no loan is unstable, because the time of each loan is limited.
The problem with this view is that it overlooks what is happening in the REAL world.
Some 90% or better of our modern day money supply is based on credit created by the commercial banks. Without that credit money (or debt-money as some call it) we would not have any economic activity, our economy would come to a standstill. So whether we like it or not, we must consider that “current lending practices”, for all intents and purposes, are the life blood of all economic activity and we cannot do without them. Consequently, we must also consider that when one loan gets paid back, another loan MUST be made just to keep the economy going in a steady state.
Considering that any loan gives rise to interest, which must ALSO be paid, it is not enough for one loan to be followed by another in linear succession. Each loan that gets paid back in a time period of – let’s say – 30 years as modeled in Ardeshir’s article, necessitates payback of not only the originally created sum, but of MORE THAN DOUBLE that sum.
So each loan that is paid back must be replaced by a new loan, making “current lending practices” a permanent arrangement necessary to keep our economy running – on loaned money. It is therefore not logical to argue that each loan is only active for a limited time span. The succession of loans acts like a continuous refinancing, always subject to interest.
As Ardeshir documents in his article, an average 30 year lifespan of a loan (let’s say a mortgage) gives rise to interest payments greater than the original amount loaned. This interest tends to push money into speculative ventures, resulting in economic bubbles and eventually in failures of economic institutions.
Not stable in my book.
Ardeshir,
You are correct my calculation is incorrect. I should have divided by 6, why I subtracted 600% from the 1400% increase in debt(which is correct see below) I don’t know.
But your reading of nominal debt growth is not correct. In the graph of household debt at:
http://www.businessinsider.com/charts-debt-unemployment–2011-2?slop=1#ixzz1ECYXRViM
the vertical axis has increments of 2 Trillion the value for 1970 indicates a value of around half way between 0 and 2 trillion i.e. about 1 trillion so the increase is 14 times not 10 times.
Therefore, the nominal increase of debt is around 1400% which is sufficient to show that the output of the system is exponential and therefore undoubtedly unstable.
The CPI is only an estimate and no error margins are given so at best it is an artificial calculation to try to invoke here against the hard data. In any event, even using this artificial index it still gives you a 67% increase in interest debt output.
Either way take your choice of values between the nominal 1400% and the adjusted value of 67% and anywhere in between, it still proves the point that over a 40 year period the system still has an unbounded output.
But under no circumstances can you argue that the adjusted value supplants the unit. All it says is that inflation has taken place which we all agree is an expected outcome of the system due to the instability of interest debt growth.
Marc
Sepp wrote, on February 20, 2011 9:15:11 PM EST (CA):
Ardeshir says that no loan is unstable, because the time of each loan is limited.
More accurately, I say no loan is necessarily unstable; and it’s not only because the time of each loan is limited, but because the repayment schedules of all loans made using common lending practices are such that every loan can be repaid, with interest.
I have, in fact, proved it in my paper at http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html
The problem with this view is that it overlooks what is happening in the REAL world.
Some 90% or better of our modern day money supply is based on credit created by the commercial banks. Without that credit money (or debt-money as some call it) we would not have any economic activity, our economy would come to a standstill. So whether we like it or not, we must consider that “current lending practices”, for all intents and purposes, are the life blood of all economic activity and we cannot do without them. Consequently, we must also consider that when one loan gets paid back, another loan MUST be made just to keep the economy going in a steady state.
Yes, of course. This is very true. Indeed, I myself have pointed this out in my article at http://homepage.mac.com/ardeshir/Debt-FreeMoneyIsBetterThanDebt-BasedMoney.html
Considering that any loan gives rise to interest, which must ALSO be paid, it is not enough for one loan to be followed by another in linear succession. Each loan that gets paid back in a time period of – let’s say – 30 years as modeled in Ardeshir’s article, necessitates payback of not only the originally created sum, but of MORE THAN DOUBLE that sum.
Yes, but as my article at http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html shows, it can be done without additional money being created.
So each loan that is paid back must be replaced by a new loan, making “current lending practices” a permanent arrangement necessary to keep our economy running – on loaned money.
Yes, of course. As I pointed out in my article at http://homepage.mac.com/ardeshir/Debt-FreeMoneyIsBetterThanDebt-BasedMoney.html , “In a world in which all money is created via debt, it stands to reason that if all debt were gone, all money would be gone too.”
But how much is actually going to be issued in new loans will depend upon the amounts that people are ready to borrow. If people become more reluctant to borrow than they were before (say because interest rates are too high, or because banks are not lending as is happening now, or indeed for any other reason whatsoever), the amount in new loans will be less than a trillion dollars, and thereby there will be less money replenishing the money supply, resulting in deflation. If people are more willing to borrow, for any reason, than they were before, more than a trillion dollars worth of loans are issued, there will be an increase in the money supply, which could lead to inflation.
It is therefore not logical to argue that each loan is only active for a limited time span. The succession of loans acts like a continuous refinancing, always subject to interest.
Not exactly. In a refinancing, there is an unpaid balance which is rolled over. In a succession of loans there is no such thing.
The succession of loans is dependent, not on the lenders (banks), but on the borrowers (the people). If people were to stop borrowing, there would be no succession of loans.
Around a hundred years ago few people used to borrow. That was probably because it was a social stigma to borrow. Today no such stigma exists, so people are willing to go into debt much more readily.
To replenish the money via loans, the succession of loans need not be an ever-increasing succession. As the loans issued 30 years ago get paid back, all that’s needed to keep the money supply constant is to issue additional loans in the same amount as those which are being paid back.
Remember that according to the generally-accepted theory, when a loan is paid back, all the money that was created when the loan was issued is “extinguished”, or destroyed. So all that’s needed is to replenish the money that was destroyed, which means issuing another loan of the same magnitude.
So for instance if in a given time period, a trillion dollars worth of loans issued earlier are paid back, only a trillion dollars worth of new loans would have to be issued to keep the money supply constant.
As Ardeshir documents in his article, an average 30 year lifespan of a loan (let’s say a mortgage) gives rise to interest payments greater than the original amount loaned. This interest tends to push money into speculative ventures, resulting in economic bubbles and eventually in failures of economic institutions.
Not stable in my book.
Of course. But this is not to say that loan(s) is/are necessarily unstable. I said earlier that no loan is necessarily unstable: and I proved it in my paper at http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html .
But the fact that interest “tends to push money into speculative ventures, resulting in economic bubbles and eventually in failures of economic institutions”, as Sepp says, is quite correct.
Indeed, as I said, I am against interest myself, and most arguments against interest are totally valid, including of course the one made by Sepp above.
But when Sepp wrote, at the beginning of his post,
Ardeshir says that no loan is unstable, because the time of each loan is limited. The problem with this view is that it overlooks what is happening in the REAL world.
… there no neccessary connection between these two sentences. In the real world there is an instability due to interest, but it is, as Sepp himself notes, due to interest resulting in money being pushed into speculative ventures, and not due to loans themselves being unstable.
In general terms, not all criticisms of interest are valid. Some are, and some aren’t. It is not correct to lump all of them together.
Cheers.
Sepp wrote, on February 20, 2011 9:15:11 PM EST (CA):
Ardeshir says that no loan is unstable, because the time of each loan is limited.
More accurately, I say no loan is necessarily unstable; and it’s not only because the time of each loan is limited, but because the repayment schedules of all loans made using common lending practices are such that every loan can be repaid, with interest.
I have, in fact, proved it in my paper at http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html .
Sepp added,
The problem with this view is that it overlooks what is happening in the REAL world.
Some 90% or better of our modern day money supply is based on credit created by the commercial banks. Without that credit money (or debt-money as some call it) we would not have any economic activity, our economy would come to a standstill. So whether we like it or not, we must consider that “current lending practices”, for all intents and purposes, are the life blood of all economic activity and we cannot do without them. Consequently, we must also consider that when one loan gets paid back, another loan MUST be made just to keep the economy going in a steady state.
Yes, of course. This is very true. Indeed, I myself have pointed this out in my article at http://homepage.mac.com/ardeshir/Debt-FreeMoneyIsBetterThanDebt-BasedMoney.html .
Sepp also added,
Considering that any loan gives rise to interest, which must ALSO be paid, it is not enough for one loan to be followed by another in linear succession. Each loan that gets paid back in a time period of – let’s say – 30 years as modeled in Ardeshir’s article, necessitates payback of not only the originally created sum, but of MORE THAN DOUBLE that sum.
Yes, but as my article at http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html shows, it can be done without additional money being created.
Sepp added once more,
So each loan that is paid back must be replaced by a new loan, making “current lending practices” a permanent arrangement necessary to keep our economy running – on loaned money.
Yes, of course. As I pointed out in my article at http://homepage.mac.com/ardeshir/Debt-FreeMoneyIsBetterThanDebt-BasedMoney.html , “In a world in which all money is created via debt, it stands to reason that if all debt were gone, all money would be gone too.”
But HOW MUCH is actually going to be issued in new loans will depend upon the amounts that people are ready to borrow. If people become more reluctant to borrow than they were before (say because interest rates are too high, or because banks are not lending as is happening now, or indeed for any other reason whatsoever), the amount in new loans will be less than a trillion dollars, and thereby there will be less money replenishing the money supply, resulting in deflation. If people are more willing to borrow, for any reason, than they were before, more than a trillion dollars worth of loans are issued, there will be an increase in the money supply, which could lead to inflation.
Sepp added, again:
It is therefore not logical to argue that each loan is only active for a limited time span. The succession of loans acts like a continuous refinancing, always subject to interest.
Not exactly. In a refinancing, there is an unpaid balance which is rolled over. In a succession of loans there is no such thing.
The succession of loans is dependent, not on the lenders (banks), but on the borrowers (the people). If people were to stop borrowing, there would be no succession of loans.
Around a hundred years ago few people used to borrow. That was mostly because it was a social stigma to borrow. Today no such stigma exists, so it would seem that people are willing to go into debt much more readily nowadays.
Moreover, to replenish the money via loans, the succession of loans need not be an ever-increasing succession. As the loans issued 30 years ago get paid back, all that’s needed to keep the money supply constant is to issue additional loans in the same amount as those which are being paid back.
It is to be remembered that according to the generally-accepted theory, when a loan is paid back, all the money that was created when the loan was issued is “extinguished”, or destroyed. So all that’s needed is to replenish the money that was destroyed, which means issuing another loan of the same magnitude.
So, for instance, if in a given time period, a trillion dollars worth of loans issued earlier are paid back, only a trillion dollars worth of new loans would have to be issued to keep the money supply constant.
Sepp also said, in closing:
As Ardeshir documents in his article, an average 30 year lifespan of a loan (let’s say a mortgage) gives rise to interest payments greater than the original amount loaned. This interest tends to push money into speculative ventures, resulting in economic bubbles and eventually in failures of economic institutions.
Not stable in my book.
Of course. But this is not to say that any given loan(s) is/are necessarily unstable. I said earlier that no loan is necessarily unstable: and I proved it in my paper at http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html . But the fact that interest “tends to push money into speculative ventures, resulting in economic bubbles and eventually in failures of economic institutions”, as Sepp says, is quite correct.
Indeed, as I said, I am against interest myself, and most arguments against interest are totally valid, including of course the one made by Sepp above.
But when Sepp wrote, at the beginning of his post,
Ardeshir says that no loan is unstable, because the time of each loan is limited. The problem with this view is that it overlooks what is happening in the REAL world.
… there no neccessary connection between these two sentences. In the real world there is an instability due to interest, but it is, as Sepp himself notes, due to interest resulting in money being pushed into speculative ventures, and not due to loans themselves being unstable.
In general terms, not all criticisms of interest are valid. Some are, and some aren’t. It is not correct to lump all of them together.
Cheers.
Ardeshir,
I forgot to add that another error in you calculations is that given that the debt in 1970 is not 1.4 trillion but far below 1 Trillion, this can be corroborated by:
http://www.paulvaneeden.com/Household.debt
Where it states that in 1970 US the household debt was approx 38% of GDP (a reasonable number).
And at this site:
http://www.nationmaster.com/graph/eco_gdp_in_197-economy-gdp-in-1970
We find that GDP was 1.025 Trillion source (OECD), so house hold debt in 1970 was close to 0.3895 Trillion making the difference much greater in the nominal increase but also in your CPI adjusted “constant dollars”, today’s debt 14 T * 0.167 = 2.338, 2338 – 0.3895 = 1.9485 trillion increase. So the percentage increase in constant dollars is 1.9485/0.3895 * 100 = 500.25% far from your loose 67% increase.
I think that the OECD stats are hard to argue with and the assumption that 40% of the GDP being household debt seems more than reasonable. Therefore even in constant dollars the debt increase is tremendous.
Marc
Quite an unbounded sum
Also the difference between 14 trillion and 0.3895 Trillion is massive nominal terms
Ardeshir wrote:
Of course. But this is not to say that loan(s) is/are necessarily unstable. I said earlier that no loan is necessarily unstable: and I proved it in my paper at http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html .
Marc: The paper in question DOES NOT analyse the stability of loans. It does not even analyse the debt growth equation. What Ardeshir confuses with “stability” is a hypothetical case of being able to pay P+I. He of course has no mathematical proof to support his conjecture but instead uses repayment schedules designed to theoretically allow for all P+I to be paid. This is not a proof.
Soon following this post will come a proof for the instability of any interest bearing loan by Dr. Sergio Dominguez Professor of control Systems Engineering and Author of texts on State Space Control. He will be referencing the most recognised authorities in Control Systems Engineering and corresponding mathematical theorems in his proof of the absolute instability of both simple and compound interest growht. He will also show that stability categorically cannot depend on the finite term of a growth function but rather by the behaviour of the function near and about the equilibrium point that manifests as unbounded output. The criteria for determining stability or instability will be shown to the derivative (rate of change) of a function.
Marc
Following is the text sent to me by Dr. Sergio Dominguez:
Author: Ardeshir Mehta
Comment:
I find it utterly unnecessary to respond to Marc’s latest round of responses, since they have nothing to do with his published equation, namely:
Yk = P(1+kr1)
… so as to explain how it must apply to the REAL world, where k, the period of a loan, is always limited in duration, and which therefore mathematically entails that Yk, the debt resulting from the loan, cannot possibly be unbounded.
Dear Marc,
With regards to the statements above posted by Ardeshir Mehta:
First of all, infinite is a concept, not a number, so there’s no point in trying to reach it by merely counting one by one. But that cannot be used to deduce that there is NO theoretic possibility for the existence of unstable systems as this concept (NOT number) represents a trend or behavior. Normally instability will always translate into a collapse of the system, since no natural system can bear infinite input/outputs, and is that not precisely where we are right now?
With regards to the affirmation that the “duration of a loan always is limited”, there are several things to say: ALL intervals that can be observed are limited; the conclusion that, since all existing time intervals are finite (even the Universe life time), monotonically growing systems will never reach infinite, and therefore the concept of infinite itself is useless or absurd, brings us back to the absurd conclusion of negating the existence of unstable systems (and of course, negating countless useful results on philosophy, math, physics, …)
The above statement makes recourse to the negation of all and any instability on the grounds that nothing is infinite; negating the instability of a system by affirming that infinity does not exist, is an inadmissible simplification based on our limited observation ability (limited time intervals only) and not on the intrinsic nature of things. But going even further, an inverted pendulum is an unstable system (I doubt this can be negated), and its output does not reach infinity (a review of Lyapunov’s theory of instability might be in order). Hence, there are examples of instability where the variables involved are not required to reach infinity. Although it is no doubt a simplistic generalization (I know that), we can therefore associate instability to the fact that a small perturbation in the point of equilibrium (zero debt), will result in a large modification to the output of the system (exponential growth in our case).
Moreover, it is true that a system in a finite domain always will give a limited output (thus returning to the absurd conclusion that all existing systems are stable). But the fact that a finite domain limits the output really has no bearing on the instability of a system (review Lyapunov theory). It is also true that there exist systems that exhibit a tendency to infinity for a finite term (finite escape time systems), but logically, we cannot limit the definition of unstable systems to these cases, because we would return to the absurdness of “infinity is never reached”. Therefore we cannot confuse the analysis of stability of a given equation set with the finite duration those functions are ruling physical phenomena in our real life experiments or experience.
Finally, and to make recourse to the sources that so far have not been refuted, Lyapunov’s (local) theorem of stability is based on the establishment of a definite positive energy-like function (not on the bounded and unbounded criteria) and in the behavior of its derivative (again and not on the bounded unbounded criteria) and THIS is the more general proof of instability known so far. And I cite (“Applied nonlinear control”, Slotine and Li, p. 62):
Lyapunov theorem for local stability
“If, in a ball B_R0, there exists a scalar function V(x) with continuous first partial derivatives, such that:
– V(x) is positive definite (locally in B_R0)
– V'(x) is negative semi-definite (locally in B_R0)
then the equilibrium point 0 is stable. If, actually, the derivative V'(x) is locally negative definite in B_R0, then the stability is asymptotic”
So, and as can be seen in this theorem bounded/unbounded is not mentioned (since that is just the manifestation of the intrinsic behavior described by this theorem for linear systems). Hence, according to the maximum authority in the study of stability to date, instability is not a question of limiting variables, but rather it is concerned with the behavior of a representative function around the point of equilibrium, reflected by the conditions that the theorem poses.
Following this theorem (to date not refuted): as it is IMPOSSIBLE to generate a V(x) that exhibits this behaviour for the system at hand (application of interest to a debt, Yk=P(1+k r1)), the system is irrefutably unstable. Furthermore, the INSTABILITY can be proved immediately by applying directly Lyapunov’s instability theorem.
For simple interest:
y=P(1+k r) transformed to the continuous case would give
y=P(1+ t r) => y’=Pr
V(y)=y^2 => V'(y)=2 y y’=2 P r y>0,
given that P>0, r>0 y>0.–> The system is UNSTABLE
For compound interest:
y’=P r y
V(y)= y^2 => V'(y)= 2 y y’=2 P r y^2>0,
given that P>0, r>0, y^2>0 –> The system is UNSTABLE
Some final remarks: this proof shows that the systems represented by these equations are globally unstable, which means that show this behavior for any value of debt (from one cent to trillions). If you happen to find a refutation of this, or Mr. Mehta find some flaw on the proof, please let me know.
All,
Our work is now well on the way to becoming mainstream science. It is now very easy to state that the interest function is intrinsically unstable as both our “Formal Stability Analysis” and John Turmel’s exponential estimation model both prove IRREFUTABLY!!!
Also, and this is where the discussion on the consequences in our “Stability Analysis” comes in, the root instability of the interest growth function is not constrained in today’s “real world” because the only way it is dealt with is for the system to either collapse or for it to continue with an output whose derivate (rate of change) is always positive i.e. continues with unbounded debt growth. Furthermore the re-financing in series down the value chain of initial interest amounts creates a massive exponential debt explosion that can easily be attested to by the following hard data graphs.
http://www.businessinsider.com/charts-debt-unemployment–2011-2?slop=1#ixzz1ECYXRViM
The significance of this is tremendous. What this says, is that stability in our economy will never be reached if it supports any debt instability or ANY DEBT GROWTH.
The more profound issue is the underlying mind game we have all been subjected to (including the keepers of interest) that has allowed us to rationalise its use in what can only be described as a completely irrational discipline of social and economic PSEUDO “control” or rather “systematic abuse” that is intrinsically foreign to our nature. This mind game must cease immediately and the way to do that is to understand that the physical consequences of the current crisis are purely a function of the design of the system. Hence, the following axiom:
The design (rules) of the system determine the behaviour of users but the behaviour of the users does not affect system design unless it acts to alter or replace that design.
This axiom states that it is the design alone that dictates the outcome and the only thing we can do is to act to change or replace the design. Given the nature of the design (intrinsic instability) and the behaviour it provokes in us within that design (systematic abuse), it is categorically essential that WE ALL ACT TO ALTER OR REPLACE THE SYSTEM. Of course it goes without saying “replace” it with a stable model. The Passive BIBO Currency Specification is such a specification.
For now, it is important that everyone simply pass the word on that we want Passive BIBO Currency now. Rest assured all of you, that Passive BIBO Currency is now fully certifiable by any Control Systems Engineer as being Stable.
So and to conclude, we no longer have the excuse of ignorance of the system’s inherent instability and the problem is not Debt, nor does it have to do with any political dogma. It simply has to do with Debt Growth and how we treat one another on the basis of being profoundly affected by the constant perturbation or cattle prod if you will of the artificial scarcity brought about by the root instability of interest (simple or compound) on debt.
Twitter and Face book: “We want “Passive BIBO Currency now”. Now!!”
Author: Marc
Comment:
Following is the text sent to me by Dr. Sergio Dominguez:
[Comment by Ardeshir]:
Please note that Dr Dominguez is co-author with Marc of the paper I am critiquing, so his testimony is hardly unbiased.
[Marc wrote]:
Author: Ardeshir Mehta
Comment:
I find it utterly unnecessary to respond to Marc’s latest round of responses, since they have nothing to do with his published equation, namely:
Yk = P(1+kr1)
… so as to explain how it must apply to the REAL world, where k, the period of a loan, is always limited in duration, and which therefore mathematically entails that Yk, the debt resulting from the loan, cannot possibly be unbounded.
[Dr Sergio Dominguez wrote]:
Dear Marc,
With regards to the statements above posted by Ardeshir Mehta:
First of all, infinite is a concept, not a number, so there’s no point in trying to reach it by merely counting one by one. But that cannot be used to deduce that there is NO theoretic possibility for the existence of unstable systems as this concept (NOT number) represents a trend or behavior.
[Comment by Ardeshir]:
As we see here, Dr Dominguez says that it is a THEORETIC possibility – a statement with which I agree. I have, however, been talking about the REAL world, as my words above clearly indicate: not THEORETIC possibilities.
[Dr Sergio Dominguez wrote]:
Normally instability will always translate into a collapse of the system, since no natural system can bear infinite input/outputs,
[Comment by Ardeshir]:
I do not deny this.
[Dr Sergio Dominguez wrote]:
[…] and is that not precisely where we are right now?
[Comment by Ardeshir]:
Although I do not deny that the economic system is unstable, I say the REASONS given in Marc’s and Dr Dominguez’s paper are not the reasons for its instability. Dr Dominguez has not proven here that those reasons are valid, except as a THEORETIC possibility.
[Dr Sergio Dominguez wrote]:
With regards to the affirmation that the “duration of a loan always is limited”, there are several things to say: ALL intervals that can be observed are limited; the conclusion that, since all existing time intervals are finite (even the Universe life time), monotonically growing systems will never reach infinite, and therefore the concept of infinite itself is useless or absurd, brings us back to the absurd conclusion of negating the existence of unstable systems (and of course, negating countless useful results on philosophy, math, physics, …)
[Comment by Ardeshir]:
Dr Dominguez is conflating two meanings of the word “limited” here. In the sense in which I use the word, the duration of a debt (and therefore of the interest on it) comes to an END, after which there is NO further growth of the debt. In the sense in which Dr Dominguez is using the word, all intervals are limited in duration by the PRESENT, regardless of whether the interval continues into the future or not.
The difference is, for example, the difference between the age of the dinosaurs, which has come to an END, on the one hand, and on the other, the orbiting of an electron around the nucleus of its atom, which is still continuing, and will always continue (at least as long as the atom remains an atom). Both are limited in duration, but the latter is continuing and will always continue, while the former has come to an end.
Likewise, in the REAL world the duration of a debt – and thus the interest on it – will ALWAYS come to an end, and therefore the debt will ALWAUS be BOUNDED. I am not sure how Dr Dominguez can respond to this, EXCEPT by saying that there is a THEORETIC possibility for a debt to be unbounded (which is not something I deny).
[Dr Sergio Dominguez wrote]:
The above statement makes recourse to the negation of all and any instability on the grounds that nothing is infinite; negating the instability of a system by affirming that infinity does not exist, is an inadmissible simplification based on our limited observation ability (limited time intervals only) and not on the intrinsic nature of things. But going even further, an inverted pendulum is an unstable system (I doubt this can be negated), and its output does not reach infinity (a review of Lyapunov’s theory of instability might be in order). Hence, there are examples of instability where the variables involved are not required to reach infinity. Although it is no doubt a simplistic generalization (I know that), we can therefore associate instability to the fact that a small perturbation in the point of equilibrium (zero debt), will result in a large modification to the output of the system (exponential growth in our case).
[Comment by Ardeshir]:
No doubt an inverted pendulum is unstable, but its instability is not explained by the equation Yk = P(1+kr1), is it?
[Dr Sergio Dominguez wrote]:
Moreover, it is true that a system in a finite domain always will give a limited output (thus returning to the absurd conclusion that all existing systems are stable). But the fact that a finite domain limits the output really has no bearing on the instability of a system (review Lyapunov theory).
[Comment by Ardeshir]:
I discussed this above. It is a conflating of two different meanings of the term “Limited Duration”; and as such, irrelevant here.
[Dr Sergio Dominguez wrote]:
It is also true that there exist systems that exhibit a tendency to infinity for a finite term (finite escape time systems), but logically, we cannot limit the definition of unstable systems to these cases, because we would return to the absurdness of “infinity is never reached”.
[Comment by Ardeshir]:
This does not refute my argument.
[Dr Sergio Dominguez wrote]:
Therefore we cannot confuse the analysis of stability of a given equation set with the finite duration those functions are ruling physical phenomena in our real life experiments or experience.
[Comment by Ardeshir]:
The meaning of this sentence is unclear.
[Dr Sergio Dominguez wrote]:
Finally, and to make recourse to the sources that so far have not been refuted, Lyapunov’s (local) theorem of stability is based on the establishment of a definite positive energy-like function (not on the bounded and unbounded criteria) and in the behavior of its derivative (again and not on the bounded unbounded criteria) and THIS is the more general proof of instability known so far. And I cite (“Applied nonlinear control”, Slotine and Li, p. 62):
Lyapunov theorem for local stability
“If, in a ball B_R0, there exists a scalar function V(x) with continuous first partial derivatives, such that:
– V(x) is positive definite (locally in B_R0)
– V'(x) is negative semi-definite (locally in B_R0)
then the equilibrium point 0 is stable. If, actually, the derivative V'(x) is locally negative definite in B_R0, then the stability is asymptotic”
So, and as can be seen in this theorem bounded/unbounded is not mentioned (since that is just the manifestation of the intrinsic behavior described by this theorem for linear systems). Hence, according to the maximum authority in the study of stability to date, instability is not a question of limiting variables, but rather it is concerned with the behavior of a representative function around the point of equilibrium, reflected by the conditions that the theorem poses.
Following this theorem (to date not refuted): as it is IMPOSSIBLE to generate a V(x) that exhibits this behaviour for the system at hand (application of interest to a debt, Yk=P(1+k r1)), the system is irrefutably unstable. Furthermore, the INSTABILITY can be proved immediately by applying directly Lyapunov’s instability theorem.
For simple interest:
y=P(1+k r) transformed to the continuous case would give
y=P(1+ t r) => y’=Pr
V(y)=y^2 => V'(y)=2 y y’=2 P r y>0,
given that P>0, r>0 y>0.–> The system is UNSTABLE
For compound interest:
y’=P r y
V(y)= y^2 => V'(y)= 2 y y’=2 P r y^2>0,
given that P>0, r>0, y^2>0 –> The system is UNSTABLE
[Comment by Ardeshir]:
(Some of the mathematical terms do not seem to have been reproduced properly here.)
Again, all this is THEORETICAL only. I don’t see how Dr Dominguez has shown above how, if k, the period of a loan, is of finite duration (as it always is in the real world), Yk, the debt that comes into existence as a result of the loan, can be unbounded.
[Dr Sergio Dominguez wrote]:
Some final remarks: this proof shows that the systems represented by these equations are globally unstable, which means that show this behavior for any value of debt (from one cent to trillions). If you happen to find a refutation of this, or Mr. Mehta find some flaw on the proof, please let me know.
[Comment by Ardeshir]:
I shall be pointing out further errors in the paper shortly, quoting the words of the paper itself, and irrefutably proving that the paper itself is unsound as a description of the REAL world.
Best wishes.
As I said earlier (January 20th, 2011 at 11:03 pm), the paper “Formal Stability Analysis of Common Lending Practices and Consequences of Chronic Currency Devaluation” by Sergio Dominguez and Marc Gauvin contains not just one but several errors.
Now I shall enumerate one more.
Reading the paper reveals that EVERY SINGLE SIMULATION OF THE MODEL is based on hypotheses (i.e., assumptions) that simply do not hold in the REAL world of common lending practices.
Consider the simulation of the Simple Interest Model (page 5). It reads:
[QUOTE]
Regular interest loan
The hypothesis for this simulation is a loan where only the regular interest is ypothesis for thist the end of each an is provided.(8)
?[NOTE 8 READS]: Mathematically, it is completely equivalent to the evolution of the debt where no principal interest are paid, but no penalty interest nor interest composition (anatocism) are applied.
[END QUOTE]
This is illustrated in Fig. 1 on page 6.
The caption under the figure reads:
[QUOTE]
From the diagram above it can be clearly stated that, if no reduction of principal roduced, the total debt grows to infinity in a linear fashion.
[END QUOTE]
The figure shows the debt resulting from the loan increasing continually.
Now consider the Simulation of a Loan with regular and penalty interest, as given on page 7. It reads:
[QUOTE]
The hypothesis for this simulation is a loan where no funds are paid at any time, i.e. no principal, regular or penalty interests are paid.
[END QUOTE]
This is illustrated in Fig. 2. The sentence below the figure says:
[QUOTE]
?From Fig. 2 above it can be cltated that, if no reduction of principal is produced and as the regular interest remains unpaid, the total debt grows to infinity in a quadratic fashion.?
[END QUOTE]
The figure again shows the debt resulting from the loan increasing continually.
And then there is the Simulation of a Compound Interest Loan Model. The simulation of the model is based on the following hypothesis (page 10):
[QUOTE]
The hypothesis for this simulation is a loan where no interest is paid and no reduction on the principal of the loan is provided.
[END QUOTE]
It is illustrated in Fig. 4. This figure also shows the debt resulting from the loan increasing continually.
However, as anyone with REAL WORLD loan experience knows, loans are NEVER paid back in the way the hypotheses quoted above all declare explicitly. Quite the contrary: ALL loans, according to common lending practices in the REAL world, are issued with repayment schedules, ALL of which include repayments of both principal AND interest, in such a way that with regular such payments, the entire loan is paid off at the end of the term of the loan. To illustrate this: the web page at
http://www.drcalculator.com/mortgage/
… shows the graphs generated by a typical $250,000 loan at 5% interest issued for a term of 30 years. The “Balance” graph shows the balance going down steadily from an initial $250K to zero over 30 years.
Since the hypotheses used for simulating the models in the paper “Formal Stability Analysis of Common Lending Practices and Consequences of Chronic Currency Devaluation” by Sergio Dominguez and Marc Gauvin do NOT fit the ACTUAL common lending practices, it is clear that the conclusions drawn by these simulations are not valid in the REAL world.
Best wishes.
Marc wrote, on February 22nd, 2011 at 10:08 am:
Ardeshir wrote [earlier]:
Of course. But this is not to say that loan(s) is/are necessarily unstable. I said earlier that no loan is necessarily unstable: and I proved it in my paper at http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html .
Marc writes:
The paper in question DOES NOT analyse the stability of loans.
Response by Ardeshir:
This is clearly rebutted by anyone reading the paper itself. I hardly need to point out sentences on Pages 5, 6 and 8, which read as follows:
“Regular interest loan”
“Loan with regular and penalty interest”
“it is assumed that the interest rate is kept constant for the life of the loan …(etc.)”
“The hypothesis for this simulation is a loan where …” (etc).
Marc writes:
What Ardeshir confuses with “stability” is a hypothetical case of being able to pay P+I. He of course has no mathematical proof to support his conjecture but instead uses repayment schedules designed to theoretically allow for all P+I to be paid. This is not a proof.
Response by Ardeshir:
What I have illustrated in my article at http://homepage.mac.com/ardeshir/DebunkingTheDebt-VirusHypothesis.html are COMMON LENDING PRACTICES. Marc’s paper purports to reflect “common lending practices”, but it clearly does NOT.
Cheers.
Marc wrote, on February 21, 2011:
Ardeshir,
You are correct my calculation is incorrect. I should have divided by 6, why I subtracted 600% from the 1400% increase in debt(which is correct see below) I don’t know.
But your reading of nominal debt growth is not correct. In the graph of household debt at:
http://www.businessinsider.com/charts-debt-unemployment–2011-2?slop=1#ixzz1ECYXRViM
the vertical axis has increments of 2 Trillion the value for 1970 indicates a value of around half way between 0 and 2 trillion i.e. about 1 trillion so the increase is 14 times not 10 times.
Therefore, the nominal increase of debt is around 1400% which is sufficient to show that the output of the system is exponential and therefore undoubtedly unstable.
Comment by Ardeshir:
But there is nothing in the data to connect this with interest. There is no evidence to show that if interest were abolished, debt would not grow just as fast if not even faster.
Cheers.
Ardeshir wrote:
“Although I do not deny that the economic system is unstable, I say the REASONS given in Marc’s and Dr Dominguez’s paper are not the reasons for its instability.”
Sure, sure where is your mathematical model of the economic instability? I have asked for it before and you have come up with nothing. If you don’t have any how can you make such a statement?
Why don’t you just admit that:
1) You haven’t a clue about Control Systems Engineering as applied to stability and instability.
2) Have never studied the subject.
3) Know nothing about when it is relevant to reality or just theoretical
4) Lack the math to follow other people’s work on the subject
5) Lack the integrity to admit your ignorance in the subject.
All the above is amply exhibited by your own comment:
“(Some of the mathematical terms do not seem to have been reproduced properly here.)”
This statement proves all I say above because if you had the acumen to follow Dr. Dominguez’s discourse you would not have made that comment without first checking the ample on-line resources to bring you up to speed on the seminal theoretical framework on stability and instability by Lyapunov and other experts with respect to the engineering of real life systems.
But since you don’t have sufficient acumen AT ALL you resort to the very cheap and ungentlemanly tactic of sewing doubt over the math you CLEARLY don’t understand because if you did understand, you could never have made that comment. YOU would have either corrected it or made no comment at all. Hence, such a clumsy comment on your part only reveals your complete incompetence in math and hence in the subject matter!
All, of Ardeshir’s comments on all his posts are riddled with this level of reasoning.
Of course I do not concede a word but I have better things to do than to respond to such rubbish.
Marc
All,
I relayed Ardeshir’s comments to Sergio and here is his response. I think it is in order to take special notice of Sergio’s response to Ardeshir’s carelessly and profoundly offensive remark:
[Comment by Ardeshir]:
“Please note that Dr Dominguez is co-author with Marc of the paper I am critiquing, so his testimony is hardly unbiased.”
Sergio: Your answer could have hardly started worse. This assertion is deeply offensive. No bias is reasonable, just for two reasons:
First, this is not a competition, I’m not trying to win anything or to simply ‘be right’. we’re trying to disclose whether the financial system as it is designed so far is jeopardizing our future and our children’s by leading us to another financial breakdown, throwing them to unemployment, misery and starvation. I really don’t want to be right, I’m really longing to receive a clear proof that everything is OK as it is right now, and that our present crisis is just an accident. But in achieving this goal you’re definitely not on the right track.
Second, these are mathematical arguments, not subject to opinion or bias. If you happen to find a flaw, express it in the same (mathematical) language, not in a pseudo-logical blah, blah, blah, based exclusively in your (limited) view of reality.
If you want to be the winner of something, the alpha male of some herd or whatever, just do yourself up, go to your local disco bar and best of luck to you!
………..
[Comment by Ardeshir]:
“Although I do not deny that the economic system is unstable, I say the REASONS given in Marc’s and Dr Dominguez’s paper are not the reasons for its instability. Dr Dominguez has not proven here that those reasons are valid, except as a THEORETIC possibility.”
Sergio: Have you any such theoretic refutation? You don’t. So don’t bother us with OPINIONS and please contribute something SOUND.
…………
[Comment by Ardeshir]:
“Again, all this is THEORETICAL only. I don’t see how Dr Dominguez has shown above how, if k, the period of a loan, is of finite duration (as it always is in the real world), Yk, the debt that comes into existence as a result of the loan, can be unbounded.”
[Dr Sergio Dominguez wrote]:
Some final remarks: this proof shows that the systems represented by these equations are globally unstable, which means that show this behavior for any value of debt (from one cent to trillions). If you happen to find a refutation of this, or Mr. Mehta find some flaw on the proof, please let me know.
[Comment by Ardeshir]:
I shall be pointing out further errors in the paper shortly, quoting the words of the paper itself, and irrefutably proving that the paper itself is unsound as a description of the REAL world.
Sergio: I’ll override this whole bunch of absurdities, lack of knowledge and/or understanding, contradictions and naive opinions that you have used to annoy us with, and will focus only on your last and unbelievable attempt to put Lyapunov’s theorem down. The only thing you have proven with this last (specially this last, but also all the rest of your arguments) is your IGNORANCE. Sadly, you have tried to cover it up with your BOLDNESS; these two qualities (ignorance and boldness) become very dangerous when put together, because they always lead to the RIDICULOUS, exactly where you are now.
Please, learn maths, I have no time to teach you, nor have I committed to doing so.
Have a nice, long life full of happiness .
Bye forever.
Sergio
Ardeshir wrote:
Since the hypotheses used for simulating the models in the paper “Formal Stability Analysis of Common Lending Practices and Consequences of Chronic Currency Devaluation” by Sergio Dominguez and Marc Gauvin do NOT fit the ACTUAL common lending practices, it is clear that the conclusions drawn by these simulations are not valid in the REAL world.
MG: Of course we need to simulate the full extent of growth to fully and rigorously analyse the phenomenon. But that by no means should imply that we assume for a moment that in the real world partial loan payments are not made. The point is to show the inherent instability of the phenomenon.
Again Ardeshir confuses the definition and hence the identification of instability with the result or practices to somehow contain that instability. This is an elementary error that further evidences his complete lack of knowledge and experience in control systems engineering.
The key points of our paper can be summarised as follows:
1) The interest growth is inherently unstable
2) If for whatever reason either part of the interest or principal is not paid the debt will continue to grow
3) Because of 1) and 2) inevitably a minimum residual debt will be produced at some point in time.
4) When refinancing takes place the linear growth becomes exponential planting a minimum debt seed in the economy.
5) With respect to a fixed collateral the refinancing of debt will lead to the system breaking down or to inflation.
6) Increasing collateral will stave off the breakdown and/or inflation but that has physical limits that will always be reached in the real world.
A far cry from Ardeshir’s superficial reading of our paper. But what can we expect from someone who ventures to attempt to debunk the work of others in a discipline where he has no knowledge or experience and where he doesn’t understand the language.
As pointed out by Sergio, Ardeshir has revealed that he takes this as some sort of competition. I have before and now Sergio has also, made it clear that our motive is to analyse the system as rigorously as possible because of the dire consequences it has on the future for everyone particularly children.
I don’t have a problem with his lack of knowledge but I do have a problem with his complete lack of integrity made abundantly obvious by such brazenly clumsy comments.
So with no restraint and to make it absolutely clear to everyone, how absurd his proposition is, I will spell it out in the simplest terms:
Ardeshir’s statement that:
“because the term of a function is finite the output is bounded and therefore not unstable”
Is absolutely inane, because it therefore implies that nothing can be unstable ever BECAUSE EVERYTHING HAS A FINITE TERM IN REAL LIFE.
To assume that the standard Stability Analysis does not fully address this elementary observation and to have not looked it up, further attests to the stupidity of the assertion.
But what is even more extraordinary is that he expects to apply such a statement to the isolated case of Y = P(1+r1k) without realising that in doing so he is implying that instability can never ever exist for any case ever. But then turns around to say that he believes the system is unstable just not for the reasons we give. Of course he is completely unable to provide the formal stability analysis to support his claim, but to the like of Ardershir that isn’t necessary, because all that matters is that he berates other people’s work while seemingly not losing his composure, before what he hopes are people unable to assess his deeply flawed logic.
All this without realising that no one has denied that a growth function with a finite term will produce a finite output. Just that unbounded is not the same as infinite.
We then both in different posts proceeded to explain that a finite term is irrelevant to the question of stability, that unbounded does not mean infinite but does mean persistent growth towards infinity which is another way of saying will grow as long as you don’t stop it. Which is exactly what the interest function does it keeps growing until you stop it.
We then explained that alternatively stability can be determined just by observing the behaviour around the point of equilibrium that wrt to interest growth necessarily is zero debt, by pointing out that the derivative or rate of change of the function is what determines stability, if, as long as the function exist (when it stops there is no function nor ouput) the derivative is positive then the function is unstable.
So Ardeshir has no legs to stand on. We all know and always knew that things are finite or never infinite, and all of us except Ardeshir also know by now that that observation proves nothing wrt to the stability or instability of a system.
When you are in a ditch and realise you can’t get out very easily it is wise to stop digging 😉
Marc
I am at a loss to understand how Marc’s latest comments refute anything I said.
I have asked Marc to produce just THREE professors of Control Systems Engineering who would unanimously agree with him that in the formula Yk = P(1+kr1), when the input, k, is limited in duration (as it always is in the REAL world), the output, Yk MUST be unbounded. (By “limited” I mean, of course, “always comes to an end”.)
I asked for professors, so that their academic reputations would be on the line if they were to say something utterly absurd. Marc has not been able to produce them, which seems to indicate that professors of Control Systems Engineering would NOT agree with him.
Cheers.
Dear Marc,
Yes, I should have said “a flow diagram with a one-page description of the diagram”, to be exact. But this does not change a bit the problem with your paper: it had no mathematical model or systems of equations to formalize the flow diagram (Fig. 6). No formal stability analysis of this flow diagram was done. I will leave it to others to conclude whether your one-page description is a “very extensive analysis”.
You used heavy mathematical artillery to analyze a very simple open-loop model (whose solution could have been easily derived with using standard differential/difference equation methods). Laplace and z-transforms are precisely used best for models of complex systems with many interconnected components and subsystems, whose simultaneous differential/difference equations would be very difficult to solve directly (though computers can now do so numerically).
But instead, you skipped the use of these transforms and associated pole-zero analysis where they would have been most useful — in analyzing the *full model* (Fig. 6), not just one simple component of the model.
Your defense is that “the [full] system is dead simple so there is no need for more convoluted analysis”. I don’t consider the financial system (or even your simplified model of it in Fig. 6) “dead simple” at all. The open-loop loan model you analyzed with the transform method is so much simpler, yet you subjected the latter to transform analysis. Why not do formal analysis of the full system, if it is also simple? Why denigrate as “convoluted” the use of method where it will be most useful?
Your defense seems to be that, having shown one component (your variations of linear, quadratic and exponential loan models with positive slopes and non-negative second derivatives) of the system to be unbounded, then the whole system is therefore unbounded.
This is plain wrong. The way components and subsystems are interconnected is critical in determining if the overall system is bounded or not, is stable or not. (By the way, there’s a difference between bounded and stable: oscillatory behavior is unstable but may be bounded.) Foreclosure of collaterals and/or writing off a bad debt are part of what you call “common lending practices”. They create a negative feedback loop in the loan model. If you don’t include such feedback loop in your formal stability analysis, your conclusions cannot apply to systems (theoretical or real-world) which contain such feedback loops.
I earlier surmised that the output of a closed-loop loan model might be a triangular pulse. You have misinterpreted this as total breakdown of the economy, which is not correct. It means that a particular instance of debt is extinguished, the money supply shrinks slightly. You have not shown at all thru formal stability analysis the effect of such a triangular pulse (or a succession of them) on the larger financial system.
The financial system might indeed be inherently unstable (as many of us believe), but because of your paper’s defects, it unfortunately fails to show that this is so.
You can improve your paper significantly if you: 1) replace your open-loop loan model with a more realistic one that takes into account foreclosure of collaterals and/or the writing off of bad debts then do formal stability analysis of this more realistic closed-loop model, 2) use this closed-loop loan model, not the open-loop one, as your component for your larger financial system model, and 3) do a proper mathematical model of your flow diagram of the simple financial system in Fig. 6 and subject this model to formal stability analysis.
Your question whether “the term of an unstable growth is finite” is too ambiguous. Give the equation of growth, and point out which term in the equation you are referring to.
Let me finally point out that you have, as some mainstream economists often do, 1) misused maths to bludgeon those like Ardeshir who are less familiar with its more advanced methods but are critical of an approach because they intuitively sense something wrong with it, and 2) made simplifying but unrealistic assumptions based on open-loop models leading to conclusions which cannot be applied to more complex closed-loop models (theoretical or real-world).
To Ardeshir: I disagree with you about the three professors. Invoking titles (or math skills) might win an argument, but does not help establish the truth.
Greetings to all,
Roberto
I had pointed out earlier that the paper “Formal Stability Analysis of Common Lending Practices and Consequences of Chronic Currency Devaluation” by Sergio Dominguez and Marc Gauvin contains not just one but several errors which make it inapplicable to the real world. Below I shall point out one further such error in the paper.?
On page 3 of the paper we see the following, expressed in table form:
[QUOTE]
?Model Variables:
P Principal of the loan??Y Total debt in each period
I Cumulative regular interests produced by the loan
X Total funds paid to cancel I
R1 Total regular interest not covered by X
D Cumulative penalty interests produced by R1
W Total funds paid to cancel D
R2 Total penalty interest not covered by W
[END QUOTE]
The paper also says that k denotes “the period for which the variables take their present value” (see page 3), and clarifies it further as follows:
“The parameter k is an integer, indicating the k-th period of the loan, whatever the period a week, month etc. This means that wherever the k subscript appears it indicates that the variable takes on its value for the k?th period portion of the loan’s life” (see page 4).
The paper then goes on to say (at page 4):
[QUOTE]
Evolution of total debt
Yk = Pk + R1k + R2k = Pk + (Ik – Xk) + (Dk – Wk)
[END QUOTE]
Up to this point I have only quoted the paper. Now comes my critique thereof.
To anyone familiar with common lending practices, what stands out immediately from the paper is the LACK of any variable representing the total funds paid to cancel the Principal P.
Let us call such a variable F, and define it as:
F Total funds paid to cancel P
Given that k denotes “the period for which the variables take their present value” (as it is defined on page 3 of the paper), the evolution of the total debt generated by a loan in the REAL world would ACTUALLY be as follows:
Yk = Pk + R1k + R2k = (P – Fk) + (Ik – Xk) + (Dk – Wk)
… where P (without a k) denotes the INITIAL Principal of the loan. (Thus Pk = P – Fk).
In other words, the total debt at the k-th period would be equal to the amount of the initial principal of the loan at the k-th period, plus the cumulative amount of interest – of every kind – at the k-th period, from which sum must be subtracted the total funds paid up to the k-th period to cancel the initial Principal as well as the total funds paid to cancel all the interest due up to the k-th period.
This is the common lending practice, as may be confirmed by examining any detailed loan schedule.
If, therefore, the loan is for a term of j periods, and all the required payments to repay the loan have been made, then at the j-th period, that is, when k = j,
Yk = (P – Fk) + (Ik – Xk) + (Dk – Wk) = 0
… which is to say, the total debt Yk when k = j will be zero.
This also is the common lending practice, as may also be confirmed by examining any detailed loan schedule.
The diagrams illustrated in Figures 1, 2, 3 and 4 of the paper are clearly, therefore, rendered invalid by this calculation and conclusion.
Best wishes.