The genesis and the path towards what has come to be called the DSGE model is traced, from its origins in the Arrow-Debreu General Equilibrium model (ADGE), via Scarf’s Computable General Equilibrium model (CGE) and its applied version as Applied Computable General Equilibrium model (ACGE), to its ostensible dynamization as a Recursive Competitive Equilibrium (RCE). An outline of a similar nature, albeit very briefly, of the development and structure of Agent-Based Economics (ABE) is also included. It is shown that these transformations of the ADGE model are computably and constructively untenable. Suggestions for going ‘beyond DSGE and ABE’ are, then, outlined on the basis of a framework that is underpinned -from the outset- by computability and constructivity consideration

Velupillai’s point is that while there are theorems that show that an equilibrium exists for DSGE models, it is impossible to find it, in particular computationally as computers work on a finite number of points. In his words. it follows that formulating optimal decision problems as dynamic programming problems is impossible. This comes about because DSGE rely in their proofs on various theorems, some of which make assumptions leading to non-computability or non-constructivity. The only way out it to assume that the computer is solving a model economy where agents themselves are computers.

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8 Responses to DSGE And Beyond – Computable And Constructive Challenges

If it is impossible to find these DSGE equilibria, then what have all the papers featured here found? Or is the point that it is impossible to find the exact solution to those models?

Dear M.H,
You ask an important question and also provide an interesting conjecture as a possible answer. A full and satisfactory answer to your intriguing query will require too much space and unnecessary technicalities for this medium of communication.
Let me suggest an answer, albeit indirectly.

I was motivated to ask the questions posed in ‘DSGE and Beyond’ more than a quarter of a century ago, when reading Scarf’s classic of 1973, THE COMPUTATION OF ECONOMIC EQUILIBRIA.I think a concise answer to your query can be found in the paragraph immediately below Theorem 2.6.1, on p.52 of this classic book.

Juxtapose the observation in this paragraph with the content of the first full paragraph, on p.17 of the book edited by Arrow, Scarf and Karlin: STUDIES IN THE MATHEMATICAL THEORY OF INVENTORY AND PRODUCTION (chapter 2).

One minor caveat, though. Scarf’s statement that ‘The passage to the limit is the nonconstructive aspect of Brouwer’s theorem …’, is an uncharacteristic slip in the otherwise impeccable work by Scarf. I have pinpointed the various ‘nonconstructive’ aspects of Brouwer’s theorem’ in a series of papers in the past decade – and more.

Finally, from another – more economically intuitive – point of view, your question is a reflection of the famous ‘dispute’ between Jacob Viner and his uncompromising, distinguished, ‘draftsman’, Dr Y.K. Wong! See p.34 of Sauelson’s ‘FOUNDATIONS’ (or the original Viner article in the ZEITSCHRIFT …, of 1932).

In a sense all of this goes back to the ancient problems of trying to construct, with specified means, solutions for well specified problems (Turing comes down in a list that does not even begin with Gauss, Galois and Abel…)!!
Thank you and Best wishes,
K.V.V

Dear M.H,
If someone suggested to you that it is ‘impossible to double the cube’, it would be natural for you to wonder (perhaps aloud): ‘Impossible to do so with what?’ The answer, most likely, would be: ‘With straight edge and compass’.
Similarly, if I told you the answer to your query (in its telegraphic form) is (b), then I suppose I expect you to wonder: ‘Find with what or how’?
My answer should (or would) have to be: ‘Using constructive mathematics or computability theory’ – but only for starters.
This is why: ….. Delian’s Problem, ….. Gauss, Abel,…., Turing, …. CGE, …., DSGE,…!
It would, in any case, be useful if you had ‘looked at’ – at least – the illuminating Scarf reference in the second paragraph of my previous ‘reply’.
Best wishes,
K.V.V

For what it’s worth, the two paragraphs immediately below Theorem 2.6.1, on p.52, of Scarf’s book are:

“2.6.1, [Brouwer’s theorem] ‘A continuous mapping of the simplex S
into itself has at least one fixed point.’

In applying the algorithm it is, in general, impossible to select an ever finer sequence of grids and a convergent sequence of subsimplices. An algorithm for a digital computer must be basically finite and cannot involve an infinite sequence of successive refinements. Moreover, and this is an important point, there is no theoretical argument that prevents these subsimplices from being located at quite different regions of S as the grid is refined. The passage to the limit is the nonconstructive aspect of Brouwer‘s theorem, and we have no assurance that the subsimplex determined by a fine grid of vectors on S contains or is even close to a true fixed point of the mapping.

However, in all of the applications of Brouwer‘s theorem with which
we are familiar, there is never any necessity to determine a vector that is close to a true fixed point. We are invariably concerned instead with the determination of a vector that is almost a fixed point, in the sense of being quite close to its own image. As the following argument indicates, if the grid is sufficiently fine, any vector in the subsimplex with distinct labels will have this property.”

Thank you very much for making these remarks by Scarf available to a broad and serious audience.
However, even the Gods, sometimes, ‘nod’!
It is not ‘the passage to the limit that is the nonconstructive aspect of Brouwer’s theorem’ – but a particular kind of ‘passage to the limit’ associated with the use of the Bolzano-Weierstrass Theorem.
Moreover, it is not ONLY the ‘passage to the limit’, but also the use of the tertium non datur (the law of the excluded middle) at various steps of the proof of the theorem that makes the theorem nonconstructive.
Forty – or so – years after he first proved the fix-point theorem nonconstructively, Brouwer gave an impeccable intuitionistic (constructive) proof of his celebrated theorem.
This paper, and other papers, on some of the other mathematical infelicities of CGE claims on the constructivity of Scarf’s algorithm (and other so-called constructive and computable algorithms) are available in the forthcoming Elgar Companion to Computable Economics (edited by Velupillai, Zambelli & Kinsella).
In passing I may wonder, I think, what meaning (mathematical, computable or economic) could be attached to a ‘true fixed point’ which is provably nonconstructive and uncomputable?
KVV

Dear ‘Agentmodelling’,
I did try to leave a response to your kind and informative posting of the relevant paragraphs from Scarf’s book. Somehow, it does not seem to have succeeded in getting ‘posted’.
So, here’s another try!
First of all, thank you for making available these important paragraphs from Scarf’s book.
Secondly, it may be apposite to point out that even the Gods, sometimes, ‘nod’!
It is not ‘the passage to the limit’ that ‘is the nonconstructive aspect of Brouwer’s theorem’ – but a particular kind of ‘passage to the limit’: that which is associated with the invoking of the Bolzano-Weierstrass Theorem.
Moreover, it is not ONLY ‘the passage to the limit’ that ‘is the nonconstructive aspect of Brouwer’s theorem’. The appeal to various nonconstructive logical laws – such as the ‘tertium non datur (law of the excluded middle) – are also responsible for the ‘nonconstructive aspects of Brwouer’s theorem’.
Finally, I suppose it is reasonable to wonder what meaning – mathematical, computable or economic – could be attached to the notion of a ‘true fixed point’, which is provable nonconstructive and uncomputable?
Many thanks.
Best wishes,
K. Vela Velupillai

If it is impossible to find these DSGE equilibria, then what have all the papers featured here found? Or is the point that it is impossible to find the exact solution to those models?

Dear M.H,

You ask an important question and also provide an interesting conjecture as a possible answer. A full and satisfactory answer to your intriguing query will require too much space and unnecessary technicalities for this medium of communication.

Let me suggest an answer, albeit indirectly.

I was motivated to ask the questions posed in ‘DSGE and Beyond’ more than a quarter of a century ago, when reading Scarf’s classic of 1973, THE COMPUTATION OF ECONOMIC EQUILIBRIA.I think a concise answer to your query can be found in the paragraph immediately below Theorem 2.6.1, on p.52 of this classic book.

Juxtapose the observation in this paragraph with the content of the first full paragraph, on p.17 of the book edited by Arrow, Scarf and Karlin: STUDIES IN THE MATHEMATICAL THEORY OF INVENTORY AND PRODUCTION (chapter 2).

One minor caveat, though. Scarf’s statement that ‘The passage to the limit is the nonconstructive aspect of Brouwer’s theorem …’, is an uncharacteristic slip in the otherwise impeccable work by Scarf. I have pinpointed the various ‘nonconstructive’ aspects of Brouwer’s theorem’ in a series of papers in the past decade – and more.

Finally, from another – more economically intuitive – point of view, your question is a reflection of the famous ‘dispute’ between Jacob Viner and his uncompromising, distinguished, ‘draftsman’, Dr Y.K. Wong! See p.34 of Sauelson’s ‘FOUNDATIONS’ (or the original Viner article in the ZEITSCHRIFT …, of 1932).

In a sense all of this goes back to the ancient problems of trying to construct, with specified means, solutions for well specified problems (Turing comes down in a list that does not even begin with Gauss, Galois and Abel…)!!

Thank you and Best wishes,

K.V.V

Thanks for your response. I was hoping for a more straightforward answer. Something like:

a) the equilibrium cannot be found, be we can get arbitrarily close to it (say, by iterating long enough);

b) the equilibrium cannot be found, and we have absolutely no idea in which neighborhood it may be;

c) something else.

So is it a), b) or c)?

Dear M.H,

If someone suggested to you that it is ‘impossible to double the cube’, it would be natural for you to wonder (perhaps aloud): ‘Impossible to do so with what?’ The answer, most likely, would be: ‘With straight edge and compass’.

Similarly, if I told you the answer to your query (in its telegraphic form) is (b), then I suppose I expect you to wonder: ‘Find with what or how’?

My answer should (or would) have to be: ‘Using constructive mathematics or computability theory’ – but only for starters.

This is why: ….. Delian’s Problem, ….. Gauss, Abel,…., Turing, …. CGE, …., DSGE,…!

It would, in any case, be useful if you had ‘looked at’ – at least – the illuminating Scarf reference in the second paragraph of my previous ‘reply’.

Best wishes,

K.V.V

For what it’s worth, the two paragraphs immediately below Theorem 2.6.1, on p.52, of Scarf’s book are:

“2.6.1, [Brouwer’s theorem] ‘A continuous mapping of the simplex S

into itself has at least one fixed point.’

In applying the algorithm it is, in general, impossible to select an ever finer sequence of grids and a convergent sequence of subsimplices. An algorithm for a digital computer must be basically finite and cannot involve an infinite sequence of successive refinements. Moreover, and this is an important point, there is no theoretical argument that prevents these subsimplices from being located at quite different regions of S as the grid is refined. The passage to the limit is the nonconstructive aspect of Brouwer‘s theorem, and we have no assurance that the subsimplex determined by a fine grid of vectors on S contains or is even close to a true fixed point of the mapping.

However, in all of the applications of Brouwer‘s theorem with which

we are familiar, there is never any necessity to determine a vector that is close to a true fixed point. We are invariably concerned instead with the determination of a vector that is almost a fixed point, in the sense of being quite close to its own image. As the following argument indicates, if the grid is sufficiently fine, any vector in the subsimplex with distinct labels will have this property.”

Thank you very much for making these remarks by Scarf available to a broad and serious audience.

However, even the Gods, sometimes, ‘nod’!

It is not ‘the passage to the limit that is the nonconstructive aspect of Brouwer’s theorem’ – but a particular kind of ‘passage to the limit’ associated with the use of the Bolzano-Weierstrass Theorem.

Moreover, it is not ONLY the ‘passage to the limit’, but also the use of the tertium non datur (the law of the excluded middle) at various steps of the proof of the theorem that makes the theorem nonconstructive.

Forty – or so – years after he first proved the fix-point theorem nonconstructively, Brouwer gave an impeccable intuitionistic (constructive) proof of his celebrated theorem.

This paper, and other papers, on some of the other mathematical infelicities of CGE claims on the constructivity of Scarf’s algorithm (and other so-called constructive and computable algorithms) are available in the forthcoming Elgar Companion to Computable Economics (edited by Velupillai, Zambelli & Kinsella).

In passing I may wonder, I think, what meaning (mathematical, computable or economic) could be attached to a ‘true fixed point’ which is provably nonconstructive and uncomputable?

KVV

Dear ‘Agentmodelling’,

I did try to leave a response to your kind and informative posting of the relevant paragraphs from Scarf’s book. Somehow, it does not seem to have succeeded in getting ‘posted’.

So, here’s another try!

First of all, thank you for making available these important paragraphs from Scarf’s book.

Secondly, it may be apposite to point out that even the Gods, sometimes, ‘nod’!

It is not ‘the passage to the limit’ that ‘is the nonconstructive aspect of Brouwer’s theorem’ – but a particular kind of ‘passage to the limit’: that which is associated with the invoking of the Bolzano-Weierstrass Theorem.

Moreover, it is not ONLY ‘the passage to the limit’ that ‘is the nonconstructive aspect of Brouwer’s theorem’. The appeal to various nonconstructive logical laws – such as the ‘tertium non datur (law of the excluded middle) – are also responsible for the ‘nonconstructive aspects of Brwouer’s theorem’.

Finally, I suppose it is reasonable to wonder what meaning – mathematical, computable or economic – could be attached to the notion of a ‘true fixed point’, which is provable nonconstructive and uncomputable?

Many thanks.

Best wishes,

K. Vela Velupillai

This is an utterly pointless paper, followed by a pointless and incoherent discussion by the author.