Not all science is so familiar with the method of mathematical modeling as theoretical physics. For many readers of this blog, especially among classical economists, there are therefore some basic Questions about DMWM presented here.
These are:
Why is it at all a mathematical modeling, which has the fundamental advantage over qualitative and empirical (statistical) considerations?
How to do such a thing as a meaningful way?
Why can because then any model A, a complaint against another model B, a greater claim to accuracy for themselves? Why are good especially the 2005 model IMF or the Harrod-Domar model so little?
Well, why do you do mathematical modeling? It's like that in any modern Science experiment and theory can complement each other and need. This interaction process works by empirical observations that can be targeted experiments, or just statistics (such as the Federal Statistical Office and the Bundesbank), or just polls. The success of such experiments is a knowledge of the effects, yielding a given system. So in our case about the statistically proven fact that the capital stock more generally is growing exponentially, and by far as the underlying GDP, which in contrast leads to a finite period of stagnation. Or with the growth of the capital stock on the one hand, of course, increases the (nominal) wealth, but also in practical lockstep poverty. Coincidence or rule?
is at this point is a theory. Because if you get based on the data to an idea about the underlying causes, can be something relatively easy to formulate a differential equation that must be integrated only. Ideally, the successful and analytically, in very many cases, but numerically we integrate what is practically forever. Except for such complex models, such as climate models that still the computing power of even the most advanced computer on demand.
The advantage of the theory is this return of a system to its basic causes. For only if the sufficient understood, one can begin effective action on the roots of the problem, instead of doctoring are only the symptoms have around. Add to that the experience comes from many scientific problems that extremely complex systems usually follow from the combination of a very few and most simple causes (so-called emergence). This is the theory makes it so much. In particular, develop from the then mathematically rigorous analytical approach possible new relationships, which in turn can be verified by experiments, or falsified, and must. And then the carousel starts again, until, as everything is consistent. Now, how it's done in strict Way? Unfortunately, this is at the universities, so my experience is not as consistent theme. Also just for theoretical physicists, more than any other natural scientists are thoroughly trimmed to mathematical modeling, which is chewed through in every variation, but ultimately left to the common intuition that practice and practical experience. This is what one models quickly "from the gut" on, as you just in urine, as it works best. So I did it when DMWModell also made first.
to lower grounds of the DMWM, and in particular also as an idea quarry for young economists, I write this contribution to the understanding of the methodology and as Call for development by interested economists. For there is, as we shall see, after much air above it. The following makes but well ahead of mathematical understanding: Who will be a natural scientist or engineer with such a modeling deal regularly knows this. Economics is in principle in fact, also an engineering science, it is however often located more in the philosophical corner and it makes even a distinctive cross-faculty ( WiSo Faculty ) what the matter is, however, are less well served, and many currently evident become defects explained.
Well, what fundamental rights agency I, a mathematical model (theory)?
First course, it is in the experiment (Statistics) to reflect actual observed behavior with sufficient accuracy. Globally and, when possible, that is, over the entire course of the system, not only locally (ie only for a smaller area).
Second It must keep consistency considerations stood. For example, should be preserved and conserved quantities. The express themselves in so-called continuity equations.
Third In the first approach, we are of course with only a global agreement between theory and experiment been satisfied, without having to explain every detail do the same (scientific economic action: first the coarse, the fine-tuning will come later). Ideally, a Kascade the mutual influence between theory and experiment, but vote at the end of the theoretical and measured values match practice, because every good model can be explained by other factors (causes) and adjustments to improve add (impact statement). So for the beginning is always as simple as possible. But not simpler! (For the latter leads directly to singular and meaningless equations ).
Fourth not mathematically principally important that the alleged causes should be justifiable and comprehensible ("common sense"). And of course it mathematically (physically) be clean, but that is actually nothing more than a matter of course (a necessary but not sufficient condition).
So let us to our specific problem: Find a solution to the relationship between capital stock is (assets and liabilities, K (t)) and the gross domestic product (GDP, Y (t)) over time an economy.
That is a relationship between capital stock and GDP is quite obvious and is generally accepted. But how does he look like?
The "gut feeling" modeling is this: I have a reasonable idea about the causes (rates of change df / dt), and then I write just as a differential equation (ODE), and since we use the dependence of two sizes seeking as a differential equation system (DGLS) out. I also try first a linear approach because of the experience, is easier and often brings the solution. Then I test the solution for consistency, if it worked, it is ok, if not, what do I need to stop at my approach, in which I make about me again thinking about the deeper causes.
But you can formulate a mathematically rigorous: First, we have two functions that are being sought. The rates of change d / dt of the functions of the left side are the effects of the functions on the right sides of this (allegedly) responsible causes.
dY / dt = F (t, Y (t), K (t), pi (t )...) and dp / dt = G (t, Y (t), K (t), qi (t )...)
The functions F and G are, if you have no great idea, of course, be completely unknown. The only certainty is that by the time t, and each of the other influencing factor Y (t) and / or K (t) must be dependent. In addition there are other parameters, namely the, possibly time-dependent functions pijkl (t) and qijkl (t), the percentages or what can also be accessible for parameters. Aijkl and Bijkl are simple constants.
Now you can make a mathematical trick that every proper analytic function fluppt well: namely, a representation as Taylor series :
dY / dt = Sum (Aijkl pijkl (t) Y ^ (i) j K ^ ( k) l, ...)
dK / dt = Sum (Bijkl qijkl (t) Y ^ (i) j K ^ (k) l, ...)
The genius of the Taylor development is that through each infinitely differentiable function a replace series of converging approximations can. So far so good, but Now you can first use the leading linear terms. The more severe terms of the order of play at the point of development is little or no role (or Aijkl Bijkl = 0). Out of this well-established fact, I may actually try first a simple linear approach. The fact is at least locally, and if I'm lucky, also even globally valid. So now only the leading terms of Taylor expansion:
dY / dt = A000 * P0000 * P0100 + A0100 + A0001 * Y * P0001 * K + ... ..
dK / dt = B000 + B0100 * * q0000 q0100 q0001 * B0001 * Y + * K + ... ..
So this is the first stage of simplification, because terms such as P0101 * Y * K or p0204Y ^ 2 * K ^ 4 or p2201 d2Y/dt2 * * K etc. pp. falling away first, because they at least locally, at least on a very narrow range delta-t, play no role.
The next step useful simplification, we drop the Aijkl and Bijkl just in the parameter functions pijkl and qijkl pure, or write:
dY / dt = y0 + p1 * y + p2 * K
dK / dt = k 0 + p3 * Y * K + p4
This is now the general linear model (*). here must now, economically justified, use discussion about the meaning and form of the parameter functions y0, k0, p1, p2, p3, p4 (t). These functions are generally functions the time, but if there are pseudo-constants (almost constant), or even constants, in particular, they can also be identically zero.
flow into the next considerations now an economic and mathematical consistency considerations. So let us consider what the parameters are y0, k0, p1, p2, p3, p4 for a meaning. The parameters y0 k0 and make no great sense, as their importance would be both of Y and K of totally independent growth. Here one could possibly but with the effect of shadow economies, as defined barter or counterfeit currency printing sense into it. In a somewhat righteous society we can rightly set to zero because they are thought to play only a minor role:
dY / dt = p1 * y + p2 * K
dK / dt = p3 * Y + p4 * K
There remain the parameter functions p1 to p4 . Let's start with the first equation. When the economy grows, without capital, from itself? Well, if you love each other quite has. Namely, through population growth. At least for the FRG can not neglect against the GDP growth through capital investment, as the population has been in the post-war years until today amazingly stable. Unlike in traditional immigration countries like the U.S., where immigration outweigh the natural growth. We can therefore p1 = 0. Should we not, one can use an appropriate formulation of p1 (t) is also to study the effect of population growth of the working population on GDP. But here now: dY / dt = p2 * K
dK / dt = p3 + p4 * Y * K
For consistency, should now be p2 =- p4, as capital grows through interest rate of the GDP. Besides the interest from investments, it grows naturally or by the savings from the income of the GDP, ergo is p3 = ps, the savings rate in the economy, often with S (= ps * y). The savings rate is again a pseudo-constant, because the German savings ratio is hardly different from many years of around 10%, thus ps (t) = ps = const. This one is almost finished:
dY / dt =-pn (t) * K
dK / dt = ps * Y + pn (t) * K
In the specific formulation of pn (t) (in the IMF model letter g) in turn requires an assumption about the relationship between investment in the real economy to the 'transactions of the financial system. With
pn (t) = pv - pr (t): = pv - pv * a * exp (- (tT) / T) / e
I now have just used a simple form, which by the empirically proven statement premise that has exponentially increased with time and capital stocks, GDP does not have enough profitable sources of return, so that the proprietary trading, ie, the investment banking continues to rise. Instead, one can of course also tickle from statistical data, the actual distribution between real economic and investment units out and invest in the model instead of the above formula.
At this point it must be admitted, however, that there are definitely ways to make the functions of p2 to p4. want as classical economists do not like to see the equality of p2 =- p4. For the classic view is that in p2 only the amount of directly invested in the real economy, capital flows, the business of the inter-bank trading in derivatives bets and etc. pp. are not. Ergo, should the amount of be significantly larger than that of p2 can, which would significantly mitigate the negative effect of too much capital. However, contrary to the holding, which is also the bank proprietary trading profit always has to come from the GDP. Because someone has to pay at the end of the chain of sellers and buyers the papers. And the last cut in the chain while the GDP of capital, at least by the amount he has to the person place on it, which he sold the securities at a profit. The interest also. And the money he is then no longer sufficient for a new car, but the Lehman certificates placed in the vault. Or a company that invests its profits in those papers, instead of buying new machines or people stop because they higher profits with amino debt promise.
other hand, Try out and put other's functions. But such models are indeed right there. And then see if the real development of the system is approximated better or worse.
And see if the system is then still consistent! For just a continuity equation, the so-called quantity theory of economics M * v = x * P dar.
And that is the Dead meets Man Walking model in fact, the ratio Mv / xP = 1, namely conservation of economy and is actually all the time constant (blue curve) and runs only out of control when the system finally collapses. The very slight differences result from the fact that the DMWM was integrated numerically with a Excelltabelle, which always leads to slight rounding errors.
Compare this information and the resulting from the DMWM course of the orbital velocity (yellow line) with the statements of Wikipedia to the empirical money in circulation : "... So that an economy is functioning normally, the velocity of money are constant. In fact, the velocity of circulation in Germany decreased steadily since 1981 ... ", also in DMWM is the velocity of circulation since the 1980s. And again, "... For V2 has been demonstrated for a number of OECD countries, a long U-shaped course ..." . Well, that's exactly the qualitative course as in DMWM.
A deeper reason is replaced by the old economic wisdom Mv / xP = 1 by a comparison with physical problems. For, let us see again:
dr / dt + div j = 0
This is a normal equation of continuity, we know from physics. That is, in this case that the source for a current j r are the charges. And, it means that the charge is conserved. This is true in all systems with conserved quantities. One can write the equation as a four-vector:
Div (j, r) = 0
The four-divergence Div (j, r): = (d / dx, d / dy, d / dz, d / dt) * (jx, jy, jz, r) = (ve * j + dr / dt) = 0, ve: = unit rate.
applies in economics now
vM-xP = 0
The units for M and P are respectively € for v and p are there events 1/year. Rewritten in as a four-vector stands now:
Div (M-P) = 0
It is therefore in the quantity theory well the continuity equation, as we know from physics to electrical currents Div (j, r) = 0, where it is here, but money flows. With prices to the function of a (source of charge) have. In other words, the prices (of consumption) the source of financial flows (assets) are. Who would have guessed it ... more precisely: It says minus P, where P is the price level. Ergo increases the price level (inflation) so that damages the property, it goes down (deflation) it strengthens the power and damage of course the GDP.
( supplement 05:10. As you can see from the above plot (blue curve), lost the source of freedom of the system is finite in the crisis phase of this content. of course, because at the end of the system very much capital is fed into the system. To print this by now all Western central banks pour money. This is really interesting but now: The loss of the source freedom is inherent in the system. you go in the end lost because of the exponential compounding effect in any case. Either more people buy more wobbly papers, and if it do not, then the state must take this stuff so that the system does not collapse. And again, you can see: Due Div (M-P) = 0 you get the system only re-source-free if the price level P is increased, it is strong, that is inflated. What policy and finance to take time, therefore, is only the inevitability of the system due. Where they hope to escape the ultimate consequences of inflation and hyperinflation. Only that one can hope abschminken to eventually turns the system back swelling free. What formulas are unfortunately unable to say whether this is fairly painless border or in the chaos.)
come
course, such analogies between physics and economics is not reason, since the mathematics and theoretical physics is concerned with structures. Whether charge or money flows, it makes no difference to be smoke and mirrors. The structures are crucial. And the macro economy is a typical mechanical structure of streams and springs.
is to be noted here that the methodology described here and the DMWM of course be expanded by the addition of other factors, especially foreign trade imbalances. This is quite simply another coupling parameter to the neighboring economy. Other variants include the testing of economies that fail to investment banking or other financial systems have, etc. pp. Of course this will be more complex and is killed only with a lot of numerical analysis.
Thus we come to the final question of why a model A is better or worse than model B? The answer arises from mentioned above. The acid test is always the question of that is the test of whether the reality will be given globally and with reasonable accuracy once again, and whether the system as a whole, both mathematically as arguments regarding the causes, consistent.
The IMF model we had already in focus, it is simply unique and meaningful free. A comparison with the most general linear model (*) shows the abstrusity clear: First mentioned equations (1 ') dY / dt = g * Y (2') dI / dt = g * I (3 ') dK / dt = g * K no mutual dependencies, which already is an exclusion criterion, and secondly it is in fact the rate of the working population equal to the rate of exponential growth g of the capital stock. In other words, ensure that the conclusions of the IMF's rise approximately, we would have today a factor of 3,25 / 0,4 * Population 1950 population, or about 560 million, instead of 81st And that would also have an employment rate as in the economic boom times.
Let's come to the Harrod-Domar model. Well, the model to bring conflicting results that it has found no major legal recognition. Ditto a number of similar systems. In terms of our general scheme one can see the error again with the basic equations: Thus, the derivation begins with dY / dt = 1 / v * dp / dt. Mathematically, he has thus first set all linear terms to zero. 1 / v is a constant, corresponding to p0011. So what could we do if they had previously discussed the leading terms actually fit away. Did he not, but made only a vague and ignores the presumption is not empirically verifiable effects. Although it is Y = Y (K) depends ever formulated, ok, but what is it for: GDP grows uniformly with the capital stock. In short: Here, the requirement of the claim is made then the following derivations.
Well, as Einstein said Master olden times: the core of the scientist is that he can still be surprised.
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