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Saturday, March 28, 2009

Mathematical Formalism in Economics

Roman Frydman responds to the response to the Anatole Kaletsky article, Goodbye, homo economicus:

Your posting of Kaletsky’s article has led to a much overdue discussion of the usefulness of mathematical formalism for understanding market outcomes. This is particularly important as the recent discussions of failures of economic models have focused on specific assumptions, such as incompleteness of markets, contracts or nonlinearities (Willem Buiter), or neo-Keynesian versus new classical approaches (Paul Krugman).

The attached note, which draws heavily on my recent book and subsequent papers with Michael Goldberg, argues that the question of whether and what type of mathematical formalism can help us understand market outcomes in modern capitalism is more subtle than Kaletsky’s critics might have realized.

Here's the note:

What type of mathematical formalism can help us understand market outcomes in modern capitalism?

Mark Thoma reports that the article by Anatole Kaletsky Goodbye, homo economicus, calling for an intellectual revolution in economics, “did not get the best reception here and elsewhere, and there were also protests that arrived by email.”

What really irked Kaletsky’s antagonists was his attack on the use of mathematical formalism in economics. But what has gone completely unnoticed in the subsequent discussion is that Kaletsky’s attack on mathematical formalism focused not on the use of mathematics in economics as such, but on the portrayal by contemporary models of the “market economy as a mechanical system.”

This characterization of contemporary macro and finance models seems uncontroversial. Regardless of whether these models are based on REH or behavioral considerations, they represent the causal mechanisms that supposedly underpin change on individual and aggregate levels through mechanical rules. Thus, they ignore the key feature of modern economies: the fact that individuals and companies engage in innovative activities, discovering new ways of using existing physical and human capital and technology, as well as new technologies and new capital in which to invest.

Moreover, the institutions and the broader social context within which this entrepreneurial activity takes place also change in novel ways. Innovation in turn influences future returns from economic activity in ways that no one, including economists, and market participants, can fully foresee, and thus that do not conform to any rule that can be prespecified in advance.

In our recent book, Imperfect Knowledge Economics (IKE), Michael Goldberg and I trace the empirical failures and fundamental epistemological flaws within the contemporary models of “rational” or “irrational” behavior to a common source: in modeling aggregate outcomes, contemporary economists fully prespecify the causal mechanism that underpins change in real-world markets.

To remedy this flaw, IKE jettisons mechanical models of change and attempts to construct economic models of individual behavior and aggregate outcomes on the basis of qualitative regularities that can be formalized with mathematical conditions. An aggregate model based on such micro-foundations generates only qualitative predictions of market outcomes.

This brings us back to the key question: whether, and if so, some mathematical formalism might be useful in our quest to understand individual behavior and market outcomes.

In our recent paper, Macroeconomic Theory for a World of Imperfect Knowledge, Goldberg and I show that the answer to this question may lie in the non-standard use of probabilistic formalism.

Our article has an extensive formal analysis of what this might entail and what it implies both theoretically and empirically. However, for the reader who is interested in a quick overview, it might be useful to reproduce one section (4.1.1.) of our article that does so more informally. The paragraphs that follow discuss how IKE explores the middle ground between Knight’s and Keynes’s arguments against the use of standard probability theory in economic analysis and contemporary reliance on models that generate “sharp predictions”: one “overarching” probability distribution, which is presumed to adequately capture market outcomes, past and future.

Contemporary models represent outcomes at each point in time -- and thus how they unfold over time -- with a single "overarching" conditional probability distribution. The relationships between the moments of this distribution and the set of causal variables constitute the model's empirical content that can be confronted with the time-series data.

By contrast, early modern economists argued that standard probabilistic representations cannot adequately represent change. Indeed, both Frank Knight and John Maynard Keynes emphasized that economic decisions and institutional and policy changes are fraught with radical uncertainty; the complete set of outcomes and their associated probabilities can neither be inferred from past data nor known in advance.

Radical uncertainty is often thought of as a situation in which no economic theory is possible: neither economists (nor market participants) are able to represent mathematically any aspects of the causal mechanism underpinning change. IKE adopts an intermediate position between radical uncertainty and the contemporary presumption that models that fully prespecify change are not only within reach of economic analysis, but anything less is not worthy of scientific status.

Of course, if economic decisions stem only from erratic "animal spirits," no economic theory is possible. As Edmund Phelps recently put it, "animal spirits can't be modelled." Although animal spirits may play a role, IKE explores the possibility that individual decision-making displays some qualitative regularity that can be represented with a mathematical model.

Departing from the position of Knight and Keynes, IKE makes use of probabilistic formalism. This facilitates the formalization of conditions that specify the microfoundations of IKE models and the mathematical derivation of their qualitative implications. However, IKE recognizes the importance of early modern arguments that market participants, let alone economists, have access to only imperfect knowledge of which causal factors may be useful for understanding outcomes and how they influence those outcomes.

Like extant approaches, IKE represents revisions of market participants' forecasting strategies, and more broadly change in how individuals make decisions, with transitions across probability distributions. But IKE constrains these revisions with qualitative conditions only. Consequently, it does not follow extant approaches in presuming that individual decision making and market outcomes can be adequately represented with a single overarching probability distribution. At the same time, IKE does not adopt the other extreme position that uncertainty is so radical as to preclude economists from saying anything useful and empirically relevant about how market outcomes unfold over time.

Because its restrictions on change are qualitative, IKE models represent outcomes at every point in time with myriad probability distributions. Nevertheless, the qualitative restrictions of IKE models constrain all transitions across probability distributions to share one or more qualitative features. These common features, which are embodied in what we call partially predetermined probability distributions, enable economists to model mathematically some aspects of the causal mechanism that underpin individual decision making and market outcomes. Such probabilistic representations constitute the empirical content of IKE models.

Although IKE acknowledges the limits to knowledge, it constrains its models sufficiently to distinguish empirically among alternative explanations of aggregate outcomes. In our book, we develop several alternative IKE models and show that their qualitative predictions enable us to reject some in favor of others on the basis of time-series data. Jettisoning sharp predictions may appear to lower the "scientific standard" that economists have self-imposed on their models. But as Friedrich Hayek anticipated, replacing the "pretense of exact knowledge" with imperfect knowledge as the foundation for economic analysis is crucial for understanding markets. Remarkably, stopping short of sharp predictions is also necessary to escape the epistemological flaws of extant fully predetermined models.”

In our book, we show how IKE models shed new light on the salient features of the empirical record on exchange rates, which have confounded international macroeconomists for decades. In part III of Macroeconomics for a World of Imperfect Knowledge, Michael and I sketch our methodology and show how it can be applied to study price movements in other asset markets. Although these results are promising, it is much too early to claim broader usefulness for IKE in macroeconomic and policy modeling.

Moreover, our discussion reveals that the question as to whether mathematical formalism may help us understand economic phenomena is more subtle than Kaletsky’s critics might have realized. As Kaletsky points out, early modern economists relied on a largely narrative mode of analysis. Although imprecise by contemporary standards, narrative accounts had the important advantage of leaving economists relatively free to explore the complexity and opaqueness of the interdependence between individual rationality, the social context of decision-making, and market outcomes. (Of course, a narrative mode of analysis also constrains argument, but this constraint is relatively weak compared to the rigor of mathematical language.)

Indeed, the giants of early modern economics uncovered remarkably powerful and durable insights, such as Hayek's prescient prediction that socialist planning is bound in principle to fail; Knight's assertion that standard probabilistic uncertainty cannot adequately characterize business decisions; and Keynes's closely related arguments concerning the importance of radical uncertainty, the social context, and conventions for forecasting returns and risk on investment in real and financial assets. These insights point to the fundamental flaw in the research program of contemporary economists: the causal mechanism that underpins change in capitalist economies is not completely intelligible to anyone, including market participants, economists, policy officials, or social planners.

In contrast to the conventional approach, which seeks to understand economic decisions with universal mechanistic rules, the constraints of IKE models are qualitative and context dependent. If qualitative regularities can be established in contexts other than asset markets, IKE can show how they can be incorporated into mathematical models. But in contexts in which change cannot be adequately characterized with reasonably long-lasting qualitative conditions, empirically relevant models of the observed time-series may be beyond the reach of economic analysis. In this sense, IKE provides the boundary to what modern macroeconomic theory --- which aims to explain empirical regularities in aggregate outcomes with models that are based on mathematical microfoundations --- can deliver.

    Posted by on Saturday, March 28, 2009 at 08:46 AM in Economics, Methodology | Permalink  TrackBack (0)  Comments (25)

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