tag:blogger.com,1999:blog-4799264811265759956.post7219559517416747388..comments2015-05-29T10:13:43.500-04:00Comments on Quantitative Finance: Foundations and Applications: Of course it's a model, duh! A final post on income, expenditure, and endogenous moneyMatheushttp://www.blogger.com/profile/05386153701958504638noreply@blogger.comBlogger30125tag:blogger.com,1999:blog-4799264811265759956.post-31943708594807668792013-06-21T06:30:34.612-04:002013-06-21T06:30:34.612-04:00Let me see if I understand the gist of it.
Please ...Let me see if I understand the gist of it.<br />Please forgive my ignorance, I'm just an uneducated naive whose a little rusty on his calculus.<br /><br />For simplicity, lets take a new economy, a blank slant.<br />A bank loans me $100 which I then spend on goods & services.<br />private Net financial assets = 0<br />money supply = 100<br />AD = 100<br /><br />Is this correct so far?<br /><br />Now if the rate loan creation is greater than the rate at which they are payed back then the volume of outstanding debt increases.<br />At this point in time hasn't this outstanding debt added to demand?<br />Now when the reverse occurs and the rate of loan creation become less than the rate at which loans are paid back, the volume of outstanding debt shrinks.<br />Does this not subtract from demand?<br />When viewed from a point in time after all the loans have been retired the effect of loan creation on demand would seem neutral.<br />But viewed at any point in time before this demand = income +/- change in debt.<br /><br />Is this every remotely correct?<br /><br />Continuing the example<br /><br />If the government then buys $100 in goods from me which I use to pay back the loan.<br />private NFA = 100<br />money supply = 100<br />AD = 100<br /><br />Although the net effect was zero, depending on the specific point in time, loan creation first has a positive effect on demand and then a negative effect when the loan is retired.<br />At all times debts = liabilities<br /><br />Vilhelmohttp://www.blogger.com/profile/04501940617030009190noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-69243233425752344972012-10-17T14:42:28.133-04:002012-10-17T14:42:28.133-04:00I know the integral is zero, that is the problem a...I know the integral is zero, that is the problem actually - this kind of function has no economic meaning. Debt incurred over time (even multiplied by a [1/t] constant) is an extensive quantity, it should integrate to the same number no matter how you discretize the time. Y has this property, debt (in your model) doesn't, yet uou mix the two. But let's drop this, as it doesn't matter for accounting.<br /><br />The problem remains: in instances where \Delta D <>0 your accounting doesn't work. In eqn. 1.20 Yi(t)<>Ye(t), at points when \Delta debt<>0, not good. Accounting doesn't know the concept of time or integration, it checks every instant. There is no Heisenberg principle that you can borrow energy (income) from another instant of time as long as dE*dt~h. ;)PeterPhttp://www.blogger.com/profile/02032621777697914182noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-4789478721803441682012-10-17T14:23:46.070-04:002012-10-17T14:23:46.070-04:00"PeterP, in the example \Delta D(t) is zero e..."PeterP, in the example \Delta D(t) is zero everywhere except at t, where it equals $10 (once more, \Delta D(t) is the actual change in debt, NOT it's rate of change). If you can't see that the integral of such a function is zero then there is nothing I (or math, or physics) can do to clarify it further"<br /><br />Matheus,<br /><br />Curious to know what sort of differential equations have functions which are zero everywhere except at a few points where they are non-zero and finite. <br /><br />Any link to such things anywhere on the internet?Ramananhttp://www.concertedaction.com/noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-50761371050143488082012-10-17T10:04:11.938-04:002012-10-17T10:04:11.938-04:00small world again ...
Witten received the Dirac m...small world again ...<br /><br />Witten received the Dirac medal in 1985<br /><br />http://en.wikipedia.org/wiki/Edward_WittenJKHhttp://www.blogger.com/profile/06322177539880818556noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-56631332319902570142012-10-17T09:42:36.597-04:002012-10-17T09:42:36.597-04:00neat!
yes - please bring him into this ASAP - phy...neat!<br /><br />yes - please bring him into this ASAP - physics may well be the gateway to a unified theory of accounting and economics<br /><br />Hawking too, when you get a minute<br /><br />:)<br />JKHhttp://www.blogger.com/profile/06322177539880818556noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-75350419390534632522012-10-17T09:05:11.344-04:002012-10-17T09:05:11.344-04:00PeterP, in the example \Delta D(t) is zero everywh...PeterP, in the example \Delta D(t) is zero everywhere except at t, where it equals $10 (once more, \Delta D(t) is the actual change in debt, NOT it's rate of change). If you can't see that the integral of such a function is zero then there is nothing I (or math, or physics) can do to clarify it further. <br /><br />JKH, your comment requires a fuller response, so I'll get back to it later - it's a very busy week here at the Institute, and I have to attend a few talks, including Ed Witten's, the world's greatest living physicist. I doubt he'll be interested in this discussion, but I can try...Matheushttp://www.blogger.com/profile/05386153701958504638noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-45598273517710166492012-10-17T08:59:45.940-04:002012-10-17T08:59:45.940-04:00That Harless post could be considered a "clas...That Harless post could be considered a "classic". It's been widely cited in related internet discussions since he posted it.<br /><br />An excellent demonstration of the required absorption of accounting discipline into economics - and by an economist - and for the most part, without inducing the equally classic paranoid reflex of the economics profession when confronted with the constraints of accounting logicJKHhttp://www.blogger.com/profile/06322177539880818556noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-13108536940805093502012-10-16T22:11:52.513-04:002012-10-16T22:11:52.513-04:00Fantastic comment by JKH.
Crucial point: if you h...Fantastic comment by JKH.<br /><br />Crucial point: if you have 1 single transaction in the period, the accounting has to check and it will. If A hands over $107 to B to buy a widget, this 1 transaction is recorded as: <br /><br />Income (collected by B) = $107<br />Expenditure (by A) = $107<br /><br />If you model transactions as income flow being delta functions, then you can always divide t2-t1 period into subperiods so that each subperiod has only 1 transaction. In each subperiod accounting will check as above.<br /><br />I only got it after reading this post, which has been a revelation for me:<br />http://blog.andyharless.com/2009/11/investment-makes-saving-possible.html<br />YMMV<br /><br />The guy shows that in a closed economy by the same logic S=I, no matter what interest rates are, equilibrium or not, no model involved, just recording each transaction *twice* ("double entry"): as income and as expenditure.PeterPhttp://www.blogger.com/profile/02032621777697914182noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-18756501001754642822012-10-16T12:23:20.952-04:002012-10-16T12:23:20.952-04:00Matheus:
I’m sure I’ve made the following points ...Matheus:<br /><br />I’m sure I’ve made the following points somewhere previously, but I’ll repeat them in the context of your useful example in a comment above:<br /><br />“Suppose we model the income flow before time t as 100$/min. Assume further that the stock of debt in the economy is say 5$. Now at time t someone takes up new debt in the amount of 10$ and spends it ...” <br /><br />a) I have no problem myself in general with your use of continuous time math the way you’ve described it, including the way it accommodates discrete events, etc.<br /><br />b) Relative to the quoted comment, the critical point for me is that expenditure generates an equivalent amount of income simultaneously. And this is the case whether you’re capturing these processes by measuring in discrete or continuous time. And the reason this is true is because of double entry bookkeeping. You can’t give me an example of a well defined original expenditure that won’t be converted simultaneously into income somewhere. And so the point there is that that expenditure/income simultaneity reflects coincident discrete (accounting) events that according to the design of your model should be able to be accommodated by your continuous time model. My preference would be to see this simultaneity reflected unambiguously in whatever equations or relationships you construct in your model.<br /><br />c) There is another critical point that is married to the one in b). In fact, I don’t know which one I would consider more important for this kind of work. But the additional point is that a loan/deposit creation event (i.e. an event of debt creation in banking) is distinct from any expenditure/income event that might be associated with it. Again, the fact that you may want to treat loan/deposit events as simultaneous with expenditure/income and even model either or both loan/deposit events and expenditure/income events as continuous events in themselves, doesn’t alter the fact that loan/deposit events are distinct from expenditure/income events. Loan/deposit events are instantaneous stock events; expenditure/income events are periodic flow events. And the fact that either type of event may be differentiated down to a continuous process doesn’t means that expenditure/income events somehow converge with and disappear into the content of stock events with which they might be associated but in fact they are still distinct from. It has always appeared to me that you want to use debt as a systematic proxy for an addition to expenditure/income. It is not in substance. It is a separate production process for a stock that becomes useful in a flow. And you can treat those separate stock and flow processes as occurring/starting at the same instant of time. But they are very separate events in reality and in accounting for reality.<br /><br />P.S.<br /><br />I trust the reading assignment is coming along.<br /><br />:)<br />JKHhttp://www.blogger.com/profile/06322177539880818556noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-68673050189860016262012-10-16T10:27:47.384-04:002012-10-16T10:27:47.384-04:00In your own example \int Delta(Debt)=$10, no matte...In your own example \int Delta(Debt)=$10, no matter how you distribute the debt increase in time, it is $10 no matter what.<br /><br />Plus, in the period you have the debt injection you have income<>spending, not correct. <br /><br />I *like* your model, but I think you mean to say:<br /><br />Spending(t+1)=Income(t)+D(Debt)(t), which would work. No need to "cheat" with accounting, you can accompany it with Spending(t)=Income(t), no need to integrate anything, checks any instant.<br /><br />You can also think about it this way:<br /><br />Say you have no income, only spend money<br /><br />Wealth change rate = Spending rate<br /><br />but hey, if you compress spending into a couple of infinitesimal time periods, the integral of RHS is zero! (for you, in reality it isn't) and you can spend indefinitely w/o adverse effects of wealth! Clearly this is wrong. <br /><br />Please please talk to some physicist you must have around at the Institute, I am unable to persuade you apparently.PeterPhttp://www.blogger.com/profile/02032621777697914182noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-58946844923555514802012-10-15T15:00:59.920-04:002012-10-15T15:00:59.920-04:00Oh dear... the ODE example was just to illustrate ...Oh dear... the ODE example was just to illustrate that a flow can peacefully coexist with a stock in the same equation, as long as there is a proportionality constant to convert one into another. But 1.20 is NOT a differential equation. <br /><br />Let me try to explain by continuing the example I started in my reply to Ramanan @8:38am. <br /><br />Suppose we model the income flow before time t as 100$/min. Assume further that the stock of debt in the economy is say 5$. Now at time t someone takes up new debt in the amount of 10$ and spends it, in ADDITION to any other income that was already modelled into the flow before t. Next suppose that the person who receives it also spends, and the next person, and so on and so forth. If nothing else happens, it's clear that the income flow is now 110$/min, since the extra 10$ are now circulating in the economy together with the 100$/min that were being spent just before the debt injection. <br /><br />What happens then is that an increase in the stock of debt equal to 10$ (debt went from 5$ to 15$ without any Dirac delta) induced a change in the flow of income equal to 10$/min. All you need for that is a constant in front of \Delta D in 1.20 equal to exactly 1/min. <br /><br />Of course other things can happen just after the debt injection, such as planned increases in real salaries, expansions, etc, but that is accounted for by the continuous part of the income flow after the debt injection. <br /><br />You can also say that debt itself SHOULD increase in a continuous way, but that is not the way we chose to model it in equation 1.20 - and that's just it, a modelling choice. <br /><br />As I mentioned, we do have a full model where debt changes continuously with a rate of change that depends on a whack of other things, including the current level of profit for business. But that's a different model, NOT the toy model presented in the appendix. <br /><br />Again, you might say it's simplified, that it stinks, or any other criticism of the sort, but the math (and accounting) are still correct (minus the proportionality constant). <br /><br /><br />Matheushttp://www.blogger.com/profile/05386153701958504638noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-85373390996047713862012-10-15T13:45:10.287-04:002012-10-15T13:45:10.287-04:00Thanks for offering to acknowledge me in the paper...Thanks for offering to acknowledge me in the paper (very appreciated!), I may accept, but when we agree what those constants need to be! :)<br /><br />I agree completely with your population example, yes, constants will fix the units. But where you say the constant is finite, i say it is infinite (if you want a non zero debt accrual).<br /><br />Say:<br /><br />dX/dt=k*f(Y)<br /><br />when you say: f(Y) is only nonzero in a couple of points, then you either:<br /><br />* have zero X population change (k is finite and the integral of RHS is zero)<br /><br />* k is infinite and X(T)-X(0)=\int k f(Y) >0 (you integrate over a Dirac delta)<br /><br />You argue both. Your accounting checks only if k is finite, but then you get zero debt accrual over the period! - the model goes out the window. <br /><br />It boils down to this: if you put a constant in front of dD term, you get a RATE of change in debt. What should this instantaneous rate be if you want to accrue finite debt on a measure zero set of times? It has to be infinite. <br /><br />Do you agree that \int k * Delta (debt) is the increase in debt over the period? <br /><br />If you do, does the economy accrue debt in your model?<br /><br />If it does, then this integral is >0 and your recorded income <> recorded expenditure.<br /><br />RegardsPeterPhttp://www.blogger.com/profile/02032621777697914182noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-90204539044068799832012-10-15T11:45:21.316-04:002012-10-15T11:45:21.316-04:00Ramanan, a continuous time formulation does not me...Ramanan, a continuous time formulation does not mean that all the variables are themselves continuous. Many can have discontinuities and it is up to the modeller to say where and how the discontinuities occur. <br /><br />Again, you might disagree with this way of modelling, but it doesn't mean that is CANNOT be done. <br /><br />As for equation (1.13), it is a typo, should also be \Delta D_s and \Delta D_k as implied by equation (1.12). Thanks for spotting that. I'd also be happy to include your name in the acknowledgements in the paper. Matheushttp://www.blogger.com/profile/05386153701958504638noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-51120557808460325932012-10-15T11:37:23.482-04:002012-10-15T11:37:23.482-04:00A continuous time formulation means
Y_E - Y_I = ...A continuous time formulation means <br /><br />Y_E - Y_I = dD/dt - dA/dt for a single sector. <br /><br />over an economy, the terms on the right hand side cancel out. <br /><br />You cannot have Y_E - Y_I = ΔD in a continuous time formulation. <br /><br />At any rate you do have equations in which dD/dt appear: see for example: 1.13Ramananhttp://www.concertedaction.com/noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-66751747620277767812012-10-15T11:29:12.245-04:002012-10-15T11:29:12.245-04:00PeterP, if I say that the rate of change of a give...PeterP, if I say that the rate of change of a given predator population X is proportional to the size of the prey population Y and write this down as dX/dt=Y would you say that this is stock/flow inconsistent and the ONLY way to make it correct is to replace Y with dY/dt? <br /><br />Of course not, you would say "Matheus you forgot a proportionality constant k in your model and should write dX/dt=k Y", which would be the right way to correct it. <br /><br />Same thing with our paper - I noticed a while ago that there are a few proportionality constants missing here and there, but decided not to worry about it until the final revisions. If you give me your full name I can add you to the acknowledgements section in the paper.<br /><br />Incidentally, in the fuller version of the model (see my paper "An analysis of the Keen model...") we have an ODE system exactly in the way I describe, with rates of changes galore (flows) depending in complicated ways on the actual values of the other variables (stocks), all according to nonlinear functions with a bunch of constants to guarantee that the units match.Matheushttp://www.blogger.com/profile/05386153701958504638noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-62764018166228505112012-10-15T11:16:30.972-04:002012-10-15T11:16:30.972-04:00Could you tell me where in the paper we have Y_E-Y...Could you tell me where in the paper we have Y_E-Y_I=dD/dt? I Just re-read it and couldn't find. <br /><br />Matheushttp://www.blogger.com/profile/05386153701958504638noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-53535421173852033432012-10-15T11:08:16.769-04:002012-10-15T11:08:16.769-04:00Well, you have Y_E - Y_I = dD/dt in one place and ...Well, you have Y_E - Y_I = dD/dt in one place and ΔD in another place. <br /><br />You cannot have the two things unless you have different meanings for the Ys. <br /><br />The one with ΔD is a discrete time formulation and hence (for a single sector at any rate) will work only if Income flows are over a period and not in one point in time. Income flow in a period is a sum of flows and in a mathematical sense equivalent to integration. <br /><br />But more generally the basic point remains. I don't know why you want to claim the difference between income expenditure is both zero and equal to the change in debt. <br /><br />I like this point by PeterP:<br /><br />"...Otherwise \int dD=0 and you add... zero debt over the period and your model is vacuous."<br /><br /><br />Ramananhttp://www.concertedaction.com/noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-82635019057801584332012-10-15T11:02:47.063-04:002012-10-15T11:02:47.063-04:00You can't add a stock (\Delta D(t)) to a flow ...You can't add a stock (\Delta D(t)) to a flow (Y(t)), units don't check out. The only way to do it is to look at the flow of debt accrual, which is dD/dt and which in this case would be a Dirac delta times delta D.PeterPhttp://www.blogger.com/profile/02032621777697914182noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-9415665701227211622012-10-15T10:59:51.488-04:002012-10-15T10:59:51.488-04:00"As I said in my reply to both you and Ramana..."As I said in my reply to both you and Ramanan, equation (1.20) has the actual change in debt occurring discretely at time t, NOT it's rate of change. "<br /><br />If it is not the rate of change then you are stock-flow inconsistent. Y is a flow, debt is a stock, can't add them. Units of Y are [$/time], units of debt are [$], you cannot add speed to distance which is what you are in such a case doing in eqn. 1.20.<br /><br />Y(t) is a *rate* of wealth accrual, if you want to add debt-anything to it, has to be a rate as well. PeterPhttp://www.blogger.com/profile/02032621777697914182noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-41266697454083802962012-10-15T10:44:26.893-04:002012-10-15T10:44:26.893-04:00Hmm, let's see. Just after equation (1.20) we ...Hmm, let's see. Just after equation (1.20) we say:<br /><br />"That is, the flow of expenditure always equals the ex-post flow of income, which in turn equals the flow of income plus change in the stock of debt."<br /><br />So it's blatantly obvious that we are talking about the flow of income, which hasn't yet been integrated. Then we move to equation (1.21) and take the integral of both sides of (1.20), which agree because \Delta D(t) is the size of a finite discontinuity. <br /><br />Again, how much clearer can this be?<br />Matheushttp://www.blogger.com/profile/05386153701958504638noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-29239153091876344012012-10-15T10:38:23.808-04:002012-10-15T10:38:23.808-04:00Matheus @October 15, 2012 8:36 AM,
Thanks for a d...Matheus @October 15, 2012 8:36 AM,<br /><br />Thanks for a detailed response. <br /><br />Believe me, I did see the ΔD term instead of there being a dD/dt term. <br /><br />However, the way the equation has been written means income flows have already been integrated once (although you don't say it) and don't see the point of integrating them again. <br /><br /><br />Ramananhttp://www.concertedaction.com/noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-35534067805028907682012-10-15T09:54:13.231-04:002012-10-15T09:54:13.231-04:00Peter, that's becoming a bit repetitive now.
...Peter, that's becoming a bit repetitive now. <br /><br />As I said in my reply to both you and Ramanan, equation (1.20) has the actual change in debt occurring discretely at time t, NOT it's rate of change. So there is no Dirac delta and the contribution to the integral is zero. I don't know how much clearer I can be on this.<br /><br />As for the integrals, that is part of the model. You can go around and record every single transaction in the universe (call it a Pacioli demon instead of a Maxwell demon) and add them up, so evidently there is no need for integrals, because you are not modelling anything, but rather measuring an awful lot of transactions. Alternatively, you can pretend (another word for modelling) that there is a hypothetical flow of income and then calculate the recorded income over a period as the integral of the flow. To check if the model is a good representation of reality all you have to do is compare the value of the integral with the value obtained by your Pacioli demon at the end of a period. Matheushttp://www.blogger.com/profile/05386153701958504638noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-56144635185457022902012-10-15T09:37:29.911-04:002012-10-15T09:37:29.911-04:00Matheus,
Than you for the response, but I still d...Matheus,<br /><br />Than you for the response, but I still disagree.<br /><br />I think you make an error twice to come up with integrals of Ye and Yi over the time period being equal.<br /><br />If the change of debt Delta D is nonzero during this period (meaning \int dD>0) then compressing the change of debt into an instant WILL give you a Dirac Delta, no way around it. Otherwise \int dD=0 and you add... zero debt over the period and your model is vacuous. Then I agree: if injection of debt is zero, then integrals of Ye and Yi are the same, but then what is the point? You are trying to argue that your accounting (Ye=Yi) checks with debt injections, it doesn't if you add nonzero debt.<br /><br />Btw, you should NOT need to do integrals to check accounting, accounting has to check at any instant, as it checks in EVERY TRANSACTION. Can you invent a transaction in which income<> spending? No, so doing integrations (which is summing these transactions) should not be something you need to resort to.PeterPhttp://www.blogger.com/profile/02032621777697914182noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-27252896970968216022012-10-15T08:36:13.484-04:002012-10-15T08:36:13.484-04:00Ramanan, if you actually read equation (1.20) you ...Ramanan, if you actually read equation (1.20) you will see that it contains the term \Delta D(t), which does NOT represent the derivative of debt, but the actual change in debt that occurs at time t (in your own terms, the actual jump that occurs when you go from one step to the next in the step function, or the height of the step if you will). This is of course not the same thing as the derivative of debt, which would be the instantaneous rate of change necessary to produce this jump, and is not well defined (although the intuition is capture by an "infinite" Dirac delta at the point). So you were the one who jumped (no pun intended) to the conclusion that we were talking about the rate of change in debt, when what we actually wrote was the change in debt itself. <br /><br />Now about the modelling. First of all, we are free to model things anyway we like. If we think that some variables are well approximated by continuous flows and others by discrete jumps then that's the way we are going to model them. You started this whole critique saying that we violated the laws of accounting (we didn't), then switched to the math being wrong because of Dirac deltas and what not (it isn't) and now resort to some ill-defined principle of defining smoothness of variable "self-consistently". Do you see that you are just moving in the direction of saying that the model stinks because you happen to disagree with it? That's of course something that you are entitled to say, but it's a whole lot weaker than saying that it's either inconsistent with accounting or mathematically wrong.<br /><br />Having said that, the reason we model things this way is purely didactic. We wanted to conceptualize what happens with the economy when there is a single debt injection, just as a thought experiment. Suppose then that the income flow is continuos (some sort of average for salaries, contract, etc, paid at a certain rate, say $100/minute). Then suppose someone takes out a single loan and spends it, and also that the person who received the money proceeds to spend it as well (i.e no savings or debt repayment). The effect is that the flow of income itself becomes higher (say $110/minute) from that point onwards. <br /><br />Of course the next step in the modelling is to add many more such debt injections, until they become themselves a continuous variable in the model, but that leads to the ODE system that I call the "Keen model", where the rate of change of output (dY/dt) is related in a somewhat complicated and nonlinear way with the rate of change of debt (dD/dt). <br /><br />Again, you might not like this reasoning, but it doesn't make it either inconsistent or mathematically wrong. <br /><br />Matheushttp://www.blogger.com/profile/05386153701958504638noreply@blogger.comtag:blogger.com,1999:blog-4799264811265759956.post-64551825135634925102012-10-15T00:49:10.444-04:002012-10-15T00:49:10.444-04:00I agree with PeterP somewhat - at least in spirit....I agree with PeterP somewhat - at least in spirit. <br /><br />The derivative of a step function is the delta function. (Ok not a function but I don't see that digression affecting anything). <br /><br />So you should treat the smoothness and non-smoothness of variables self-consistently. In real life nothing is smooth. <br /><br />You treat debt injection as a jump - something not smooth but then you should also treat income flows as non-smooth. <br /><br />And I guess the reason PeterP mentioned Delta functions is that it is the derivative of the step function. <br /><br />And as PeterP points, the jump dD/dt is infinite and the inequality still exists after integration if you were to start with a dD/dt term in the equation. <br />Ramananhttp://www.concertedaction.com/noreply@blogger.com