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===== What's the main problem with classical growth models? ===== First of all we have to take a look to the **reality** of at least official numbers in economy: {{:gra1.jpg?500|}} The figure above just shows the sum of all assets (red line, in bn. €) in Germany and the GDP as well (green line) from years 1950 to 2010. Virtually the same effect we see in USA, with assets (red) and GDP (blue): {{:us-schulden_total.gif|}} Indeed there are __some problems with US-statistical data__ for years 1957 to 2007, as visa versa to Germany were the nominal numbers are published clearly, the US-GDP is hedonized since some ten years //(and thus to large in comparison with the real (nominal) numbers)//. And the sum of all assets can only be found by adding a lot of general datas about assets. But in general we see this kind of progressivly increasing differences between assets and GDP in every long term time series of any economy. The effect from this general trend is, that incresingly more money has to be involved to get a but more and more diminishing effect on GDP growth: {{:marginal.jpg?500|{{:marginal.jpg|}} The above figure is computed for the official data from Bundesbank for Germany and shows the same trend in the marginal productivity of debt (//ΔGDP/Δdebt//) as in the US too: As in the early years every new Euro or USD generated at least an Euro or Dollar growth in GDP, this value went down to 50 cents in the eighties, and now in crisis gets zero or just even below zero. Which finally means, the more assets the less GDP. **This very real effect is just the contrary to classical growth theory.** And it states in some simple words, what financial crisis really means: You have no real chance to solve problems of economies by creating just more debt in the long run. ===== What's should a growth model do? ===== Well, at [[theory_specials|the best it will describe and explain]] the underlying fuctions and connections and will produce realistic results. But the //very least// is of course to produce realistic results. Let's now have a look to some samples. {{:kzuy.jpg?500|}} The figure shows the socalled capital-coefficient, which is just the quotient of the whole of all assets //K// to the GDP //Y// of an economy. This depiction is much more convenient than to show the graphs of //Y// and //K// both, as we now have only one graph depicting the whole story and also we get rid of some scaling effects like inflation and thus the need of a logarithmic depiction (we could also depict the GDP-coefficient //Y/K//, but commonly //K/Y// is used). What we see is that the commonly used classical growth models underestimate the influence of captial growth very much. Those models but are indeed more or less customizeable to any desired result by changing there parameters arbitrarly. For example in the graph above we took into acount a realistic parameter for the IMF 2005 model to have at least a very approximate fit //(in its simplest form its uses gK-gY=0 and thus gets just the same simple line at the bottom)// but to fix the models with realistic parameters on one hand and a realistic time evolution on the other hand doesn't get any reliable results //(some mathematical reasoning about this topic we'll do at another chapter)//. Now we may have a look at the most simplified basis model of field theory. Which means, that a nationally closed economy (no international contributions) is assumed, with the only parameter included is an average interest rate. And of course its regarding all debt, including the both two fundamental financial parts as are the loans to the real economy and banks-own-business as well. The outcome for Germany is: {{:basismod.jpg?500|}} The deviation from the mid 90-ties to the first 2000-ties results from the influence of the socalled DotCom-bubble, which attracted a lot of international assets which are not included in this simplest possible model. =====Classical AK-model(s) and others===== The so called [[http://en.wikipedia.org/wiki/AK_model|AK-model]] is often used to introduce into the study of classical growth models. In its simplest form it just states **//Y=AK//**. Here //A// may be most simply a constant, which then states that GDP //Y// growths just like capital //K// does. This then indeed gives the straight bottom line in our figure above, as a simple rearrangement gives **//K/Y=1/A=const.//**, which is indeed in very much contrast to reality. More powerfull thus should be to define a //A=A(t)// as a function of time too. Of course this has been done in serveral manners by different assumptuions we will not refer to here in very much detail. As the problem is, that one can virtually take any desired finction //A(t)// to put in. The most "powerful" which would be just to take **//A(t):= 1 / MeasuredValuesof-K-over-Y(t)//** which then simply gets a pretty good approximation and interpolation to the near future as well. Indeed this is, from the mathematical and physical standpoint, the very best what you can do with such simple gadgets. As now what ever clever assumptions you invest to calculate the "real" //A(t)//, it is just putting in your own oppinions. And at the end of your mathematically, hopefully correct, calculations you will of course get exactly the answers you desired. More over one can show, that this is true for virtually every classical growth model, as regularly they all are rearrangeable to special AK-models. In most classical models there is the case of a [[http://en.wikipedia.org/wiki/Cobb-Douglas|Cobb-Douglas-production-function]] (CDPF) used to arrange things, with the formulation is: **Y=AK<sup>a</sup>L<sup>1-a</sup>**, Where //Y// represents the total production in an economy. //A// represents "total factor productivity", //K// is capital, //L// is labor, and the parameter //a// measures the "output elasticity" of capital. But, furthermore, because the work //L// is always a certain proportion //d(t)// of the GDP //Y//, we can rearrange the commonly used function Y=AK<sup>a</sup>L<sup>1-a</sup> easily to Y =c K<sup>a</sup> ( d•Y )<sup>1−a</sup> ⇔ Y /Y<sup>1−a</sup> =c d<sup>1−a</sup> K<sup>a</sup> ⇔ **Y/K = (c d<sup>1−a</sup>)<sup>1/a</sup>** which shows that the Cobb-Douglas model again is indeed a special AK-model with the choice **A(t):=(c d<sup>1−a</sup>)<sup>1/a</sup>** implicitly. All such AK-models have the option to represent by the choice of //A(t)=Values(t)// any desired result. =====Mathematical Fundamentals===== Besides the special problems mentioned above, there are some very simple findamentals of mathematics which state clearly that the above kind of modelling is virtually useless. First of all we should therefore look at the principle ansatz done in most classical growth models: **//Y = f(K)//** where in //f// is any functional of //K//. As for example the commonly known CDPF //Y=AK<sup>a</sup>L<sup>1-a</sup>// is exactly such a gadget. But what we are searching for are __two__ functions //Y// and //K//, which are indisputable connected in any way. From this follows directly, that we must have at minimum two linearly independent coupled equations for having a chance to solve the problem. And things gone more worse: it is also clear from mathematically known fundamentals, that one needs such a **system of at least two linearly independent coupled __differential__(!) equations** like [d<sup>(n)</sup>Y=f(d<sup>(k)</sup>Y, d<sup>(k)</sup>K) ∧ d<sup>(m)</sup>K=g(d<sup>(k)</sup>Y, d<sup>(k)</sup>K)] at minimum. Using an ansatz like //Y=f(K)// thus can never lead to satisfactory results, regardless how sophisticated and complex your mathematical work on it is done. The very simple reason why is, that __**the answers on functions are never functions!**__ {{:math-fundamentals.jpg?700|}} The problem thus is to ask mathematics the right questions to get correct answers. If you ask badly defined questions, you will get badly defined answers. If you ask for a function, you have to take step 3 and not step 2, which itself is an answer to step 3. Thus it is an inevitable constraint that we have to have at minimum //a system of at least two linearly independent coupled differential equations//, but this is still not a sufficient constraint on its own. To be sufficient we have to do some more work of course. **But we can say //__for sure__// that if you start with an equation like the CDPF or in general with an //Y=f(K)//, or introduce it at any crucial point, then you can //__never__// get reliable results just from simple mathematical fundamentals.** The best one can do in such a case is to make a more or less good guess, how //Y// relates on //K// or whatever else, and look how it works. But it is always nothing but a self-fulfilling-prophecy which depends crucial on the general acceptance of the assumption //Y=f(K)//. And this assumption must be crude, as to guess the correct answer instead of calculating it with the correct mathematical gadget, is indeed a very much unlikely lottery. ---- //work in progress...// --- //[[heribert.genreith@t-online.de|Heribert Genreith]] 2012/04/01 20:29//

 
classical_growth_models.txt · Zuletzt geändert: 2012/10/05 01:23 (Externe Bearbeitung)
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