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I have always wondered why such a negative result such as Sonnenschein–Mantel–Debreu theorem (SMD) theorem is usually ignored or disregarded by practitioners, specially in macro.

The result roughly says that there is no deep empirical implications of the rationality assumptions when aggregated because the aggregate demand itself only inherits Walras' law and homogeneity of degree zero. The lack of empirical implications can be translated to the lack of identification and so we cannot hope to recover the underlying preferences from an aggregate demand setup as much of the cross-country empirical macro or calibration macro seems to try. For additional discussion of the importance of the SMD theorem, see this piece.

In sum what are the implications of the SMD theorem for macroeconomics.

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    $\begingroup$ I recently though about this. It depends whom you speak to. Someone like Taleb (for instance) would say SMD is ignored due to selection bias, and for rhetorical effect. An interesting account, of a more general nature, of why this sort of work perpetuates (models built upon models with false premise) is presented by Gilboa. Furthermore, it is not only practitioners that ignore such negative results.... $\endgroup$ – Rusan Kax Jan 16 '15 at 19:37
  • $\begingroup$ You confuse aggregate demand and excess demand. If one has data on endowments, the theory has observable implications. $\endgroup$ – Michael Greinecker Jan 17 '15 at 13:10
  • $\begingroup$ Yes, the writing is confusing, but SMD referes to excess demand. However, observing several endowments is very difficult. In a lab yes, in an economy, specially when we think of it dynamically the endowment is Ko or the initial capital. How many initial capitals can you observe? $\endgroup$ – user157623 Jan 17 '15 at 13:14
  • $\begingroup$ @user157623 I don't, the IRS does. $\endgroup$ – Michael Greinecker Jan 19 '15 at 5:18
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This is somewhat broad and argumentative question, but I'll try to provide one possible answer (disclaimer: I'm not a GE theorist). SMD theorem states that imposing only the abstract structure of general equilibrium has few empirical implications. Fair enough. But it's incorrect to say that

we cannot hope to recover the underlying preferences from an aggregate demand setup as much of the cross-country empirical macro or calibration macro seems to try.

because once we impose additional assumptions about preferences, technologies and endowments, we do obtain additional empirical predictions (and of course, different sets of assumptions will lead to different predictions). This is precisely what economists are doing most of the time! It's not 1950's anymore - these days, there is very little research that tries to arrive at general results from abstract axioms. Most theoretical work in macroeconomics will present specific model, derive its qualitative and quantitative implications and often will include an empirical component.

Another often-heard argument, present e.g. in the linked piece, is that SMD theorem shows GE models are in general unstable or indeterminate, and thus we must resort to unrealistic restrictions, such as representative agent assumption, to guarantee that our models are well-behaved. But this confuses necessary and sufficient condition: yes, assuming representative agent is sufficient to obtain well-behaved aggregate demand function, but that doesn't mean all, or even most, models with heterogeneity are somehow automatically unstable. And a logical possibility of unstable equilibrium says nothing about whether such situation is empirically relevant (maybe it is, but critics usually provide no evidence for such claims).

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    $\begingroup$ "Most theoretical work in macroeconomics will present specific model, derive its qualitative and quantitative implications and often will include an empirical component". I suspect the OPs question can be rephrased as this - I'm not making a statement about if I agree/disagree with that article. Just suggesting this is the thrust of the OPs question. $\endgroup$ – Rusan Kax Jan 16 '15 at 22:16
  • $\begingroup$ But isn't then then the case that the output of the estimation depends solely on the additional assumptions made? $\endgroup$ – user157623 Jan 16 '15 at 22:36
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    $\begingroup$ @user157623 But that's always the case. If I wanted to test Newton's theory of gravity by calculating whether the predicted motion of planets matches observations, I won't get anywhere with just Newton's laws - I'd need additional "assumptions" in the form of initial positions, velocities and masses of planets too. $\endgroup$ – ivansml Jan 17 '15 at 14:13
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The SMD results deprived us of the certainty that if the individual level is as we model it, then the market and macro level will be just like gigantic individuals.

So we have to turn to the real world, and record for how many markets, in how many countries, and in how many instances, did the markets and the economies appear to have behaved in a way that negates this convenient projection so much that it makes it really misleading. And judge whether the number of cases we found justifies the abandonment of models that use this projection, compared to the number of cases that they are modeled adequately through it.

The cases are not many. I am not a fan of "representative consumer" modeling -I agree with the link in the question that it is an evasion. But this is the only point of that polemic with which I agree -because it is a clever evasion, one that permitted macroeconomics to produce results that did have relevance to the real-world (see also this post about the representative-individual framework).

Moreover, the main purpose of macroeconomic models is not to uncover the underlying preferences -it is to provide relations between macro-economic variables that stand the test of data.

I suspect that somebody (a bit old) will think "but what about the Oil Crisis of the 70's". And somebody else (younger) will say "but what about the recent financial crisis?" Etc. Well, the clear inadequacy of the Economics to predict crises, has nothing to do with whether the market demand schedule is downward sloping or not. It has to do with the fact that Economics has concentrated on equilibrium analysis and of fluctuations around it in an otherwise rather stable underlying structure.

So there is an open road ahead which we are gradually taking -and in this perspective I would say that the SMD results liberate us from internal myopic theoretical constraints, by essentially proving that the aggregate can be much more/different than "the sum of its parts", as the saying goes. Given it, we don't have to argue that individuals are self-destructive in order to produce disastrous behavior on aggregate, for example.

(PS: This issue of crying "but what about the crises?" every time we discuss the usefulness of macroeconomics, reminds me of a symphonic concert: 120 minutes of perfect harmony -and we will remember much more forcefully the one second that a violin went awry. In between crises, macroeconomic models do a pretty good job -but when things work, much fewer people notice that they do work).

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  • $\begingroup$ Do you have a good reference for a rigorous proof of the Sonnenschein–Mantel–Debreu theorem? $\endgroup$ – Hans Mar 11 '17 at 8:36
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    $\begingroup$ @Hans The exposition in MasColell, Whinston and Green' book "Microeconomic Theory"ch. 17) is rigorous and useful enough, and contains the references to strict rigorous proofs (including the three original papers) $\endgroup$ – Alecos Papadopoulos Mar 11 '17 at 11:16
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In current mainstream macro practice, the representative-agent (RA) assumption (of DSGE models and nearly identified with them) side-steps the SMD:

The theorems of Hugo Sonnenschein, Rolf Mantel and Gerard Debreu in the early 1970s established that the restrictions that generate well-behaved individual demand functions do not constrain aggregate demand functions to exhibit the same properties [...]. The new classicals sidestepped the problem of aggregation either by imagining an economy composed of identical individuals or by assuming that there is one individual who represents the whole economy, so that the solution to the optimization problem of this representative agent gives the aggregate relationships in that economy. In fact, they adopted the representative-agent model from the optimal-growth literature of the 1960s.

Using such models, Lucas and others developed the characteristic conclusions of the new classical school, such as the ineffectiveness of monetary policy with respect to the real economy (see Hoover 1988). Policy ineffectiveness was widely regarded by Keynesians as a politically conservative conclusion. Initially, it was interpreted as a direct consequence of the rational-expectations hypothesis, which was then regarded as politically suspect. Later, economists came to see that the assumptions of flexible prices and perfect competition were the critical factors in the policy ineffectiveness proposition. Once a wedge had been driven between policy ineffectiveness and the assumption of rational expectations, the rational expectations hypothesis was accepted by a wider spectrum of macroeconomists [...]. New Keynesians found that rational expectations did not rule out an important role for the government in stabilizing the economy.

[...]

The representative-agent program elevates the claims of microeconomics in some version or other to the utmost importance, while at the same time not acknowledging that the very microeconomic theory it privileges undermines, in the guise of the Sonnenschein–Debreu–Mantel theorem, the likelihood that the utility function of the representative agent will be any direct analogue of a plausible utility function for an individual agent. Kirman’s (1992) survey article on the representative agent, which highlights the lack of analogy, is well-cited; yet, it is striking that almost all of the citations are by critics of the representative-agent program; there is little evidence that advocates have even noticed the argument against their approach. [...]

All the different views mainstream macroeconomists have about the state of their field and about possible areas of improvement should not diminish the degree to which they converged methodologically in studying fluctuations. They all analyse such phenomena usually through a dynamic stochastic general equilibrium model with a representative agent, firmly grounded on microeconomic principles. Moreover, several of them agree with Chari (2010: 2) that “any interesting model must be a dynamic stochastic general equilibrium model. From this perspective, there is no other game in town.” Therefore, he continues, “a useful aphorism in macroeconomics is: ‘If you have an interesting and coherent story to tell, you can tell it in a DSGE model. If you cannot, your story is incoherent.’”

But RA/DGSE has brought its own problems/critics... especially post-2008:

The representative-agent (RA) assumption prevent DSGE models to address distributional issues, which are one of the major cause of the Great Recession and they are fundamental for studying the effects of policies. [...]

The RA assumption coupled with the implicit presence of a Walrasian auctioneer, which sets prices before exchanges take place, rule out almost by definition the possibility of interactions carried out by heterogeneous individuals. This prevents DSGE model to accurately study the dynamics of credit and financial markets. Indeed, the assumption that the representative agent always satisfies the transversality condition, removes the default risk from DSGE models (Goodhart, 2009). As a consequence, agents face the same interest rate (no risk premia) and all transactions can be undertaken in capital markets without the need of banks. The abstraction from default risks does not allow DSGE models to contemplate the conflict between price and financial stability that Central Banks always face (Howitt, 2011): they just care about the nth-order distortions caused by price misallignments which can eventually result in inflation without considering the huge costs of financial crisis (Stiglitz, 2011, 2015). No surprise that DSGE models work fine in normal time but they are unequipped not only to forecast but also to explain the current crisis (Goodhart, 2009; Krugman, 2011)

There is a bit more subtlety to RA than what I've covered here; RA can include some sources of parametric heterogeneity but assumes a sort of "structural" homogeneity, if I understand that correctly.

This RA-solution to SMD has become the main theoretical attack surface against DSGE models, as far as I can tell. It's also found in the ACE/ABM (computational/agent-based modelling) literature, e.g. Fagiolo and Roventini basically reiterate the above:

First, the well-known Sonnenschein (1972), Mantel (1974), Debreu (1974) theorems prove that the uniqueness and stability of the general equilibrium cannot be attained even if one places stringent and unrealistic assumptions about agents. Moreover, Saari and Simon (1978) show that an inffnite amount of information is required to reach the equilibrium for any initial price vector. Given such nihilist conclusions, neoclassical economists took the short-cut of the representative agent (RA) to obtain stable and unique equilibrium. Indeed, if the choices of heterogeneous agents collapse to the RA ones, one can circumvent all the aggregation problems and develop GE macroeconomic models with rigorous Walrasian micro-foundations grounded on rationality and constrained optimization. However, the RA assumption is far from being innocent: there are (at least) four reasons for which it cannot be defended (Kirman, 1992) [...]

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