Stumbling Towards Purgatory

On Trials, Errors, and the Myths of Equilibrium

Abstract

This essay surveys the nuanced interplay between overlapping generations models, the formation of expectations in monetary economies, and the secular forces that have shaped our collective history of inflation—and by extension, bond pricing. Central to this discussion is Duffy’s (1994) two-period model, which refines the learning process by incorporating current inflation data and sticky prices, in contrast to Lucas’s single-period framework that effectively disregards real-time price signals. The philosophical core of the debate lies in the problem of incomplete knowledge regarding how agents form beliefs about future variables, thereby complicating any unequivocal endorsement of one learning rule over another. Against the backdrop of inflation targeting and the expansive wave of offshoring since the 1980s, bond markets have come to embrace certain pricing conventions—often couched as “real rates plus inflation expectations”—that rest precariously on specific historical circumstances. This essay underscores the importance of a multi-period framework—capable of simultaneously resolving prices and expectations across heterogeneous agents and time horizons.

Keywords: Overlapping Generations, Disequilibrium Learning, Inflation Targeting, Free Trade, Bond Pricing, Adaptive Expectations

Introduction

What if everything we believed about inflation and bond markets were propped up by assumptions so ephemeral they could crumble under the slightest breeze of new information? Such fragility owes much to the unsettling reality that we lack a definitive blueprint—a normative model, if you will—for how agents form beliefs about future economic variables.

This vulnerability is vividly illustrated by examining two influential frameworks in disequilibrium learning analysis, a theoretical approach that explores how participants converge on prices through a haze of imperfect conjectures and ever-shifting market currents. By contrasting these models, we begin to see just how tenuous our presumed certainties may be, and how easily they might unravel should the world deviate from what we expect.

Stumbling Towards “Enlightenment”

Duffy’s (1994)^⁠1 work illuminates, with notable clarity, the myriad complexities of equilibrium in an overlapping generations model that employs fiat money. Until that point, researchers and policymakers largely relied upon disequilibrium learning analysis as the principal mechanism by which agents glean the information required to satisfy the rational expectations assumption used by researchers.

This reliance traces back to Lucas (1986)^⁠2, who employed a straightforward adaptive learning rule within a single-period, Samuelson-type overlapping generations framework to demonstrate that the monetary steady state draws almost all initial price levels toward itself. Duffy, by contrast, introduced a two-period model whose informational reach is more complete—it factors in current inflation—whereas Lucas’s approach essentially ignores the inflows of newly observed inflation data. Additionally, Duffy assumed sticky prices to capture the well-documented sluggishness in aggregate price adjustments (see Sims (1989)^⁠3). The two-period horizon ensures that agents—both labor and firms—are highly attuned to intertemporal pricing effects, while price stickiness better reflects the real world’s halting response to monetary shocks.

In essence, Duffy’s model resolves a multi-period expectations equilibrium that is informationally complete, whereas Lucas confines his analysis to a single-period horizon that is informationally incomplete. This divergence in informational demands—multi-period with up-to-date inflation data versus single-period lacking it—yields starkly different outcomes regarding how adaptive learning rules converge (or fail to converge) to unique solutions in overlapping generations economies.

The philosophical significance of these competing frameworks lies in the rather unsettling fact that we remain ignorant, in any firm sense, of the precise ways in which agents form expectations about future endogenous variables. Hence, claiming that one adaptive rule is more plausible than another is fraught with difficulty. It follows that monetarists—and indeed anyone whose decisions carry significant consequences under uncertainty—must remain alert to how policy interventions, or any disruption to the current status quo anchoring expectations, can produce radically different results based on the assumptions driving agents’ beliefs. In this regard, guiding expectations via credible monetary policy remains indispensable if one hopes to steer the economy toward a particular equilibrium.

The next two sections provide a more concrete demonstration of this overarching argument. We will examine how monetary and trade policies, particularly since the 1980s, have shaped our collective experience of inflation and, by extension, moulded the belief systems that guide our assessment of bond values.

A (Very) Brief History of Treasury Pricing

Bond pricing provides a vivid illustration of how learning behaviour colours the broader market’s understanding of price dynamics—and of how, over time, our collective learned history of inflation has shaped beliefs of the main drivers of bond valuations.

I.

Since October 1979, the U.S. Federal Reserve has explicitly targeted inflation. Paul Volcker famously harnessed monetary policy to quell inflation precisely as inflation expectations were on a tear. By 1984, the Fed’s efforts had brought inflation down to around 4%.

II.

Offshoring—the relocation of production processes to lower-cost or higher-productivity regions—has a long history. For instance, U.S. semiconductor firms sent labor-intensive assembly, testing, and packaging stages to Taiwan, Hong Kong, and Malaysia in the 1960’s. Over time, more advanced production methods followed; by the 1980s, wafer fabrication had likewise migrated to Asia, notably Taiwan.

This trend broadened to encompass many more industries and gained momentum throughout the 1980s and 1990s. While it undeniably created stresses in certain labor markets—some nations and regions managed better than others—it also introduced a powerful, decades-long deflationary impulse. This secular force dovetailed neatly with the worldwide uptake of explicit inflation targets among independent central banks.

III.

Against this backdrop, policymakers and asset allocators endeavoured to construct frameworks for bond pricing. One dominant approach rests upon a “real required rate”—purportedly the level needed to clear markets—plus long-term inflation expectations, appending risk premiums for inflation, duration, and credit factors where needed. In truth, it is more of an educated guess underpinned by a dash of analysis, since none of these underlying variables is known.

Crucially, inflation expectations—and thus bond prices (indeed, the prices of most asset classes)—have been moulded by these same secular trends outlined above. Our collective learned history of inflation is very much a product of these forces: the campaign to corral inflation (I) and the global expansion of offshoring (II).

Pricing the Term Structure

Adrian, Crump, and Moench (2013)^⁠4 propose a regression-based approach that pinpoints Treasury returns’ sensitivities to the principal components of bond prices. This technique dispenses with the need to impose a normative model on Treasury yields and instead uses principal component analysis that remains agnostic regarding serial correlation in yield-pricing errors. Their method outperforms many classic affine-term-structure models, including the sort described in III.

Their statistical framework demonstrates that five principal components can price Treasuries with impressively low errors and minimal autocorrelation. Their original and subsequent work^⁠5 has shown that the first few components often correlate with broader macroeconomic factors, while higher-order components track subtler, market-microstructure effects:

1. Level (PC1) usually aligns with long-run inflation expectations (e.g., survey-based or breakevens) and can reflect slow-moving changes in the policy rate (Fed Funds or its longer-run shadow).

2. Slope (PC2) tends to move with cyclical indicators (industrial production growth, changes in unemployment, forward-looking signals) and near-term policy expectations (Fed Funds futures or OIS rates).

3. Curvature (PC3) often connects to term premia (like the ACM term premium) or medium-term macro surprises (e.g., 2–3-year growth or inflation).

4. Fourth and Fifth Factors are more elusive, though researchers frequently link them to market microstructure and supply effects, such as primary-dealer inventories, liquidity constraints (bid-ask spreads, on-/off-the-run yield differentials), or idiosyncratic issuance patterns.

These relationships—and the way markets have learned them—were shaped in an environment of monetary policy–driven constraints on inflation and the offshoring of production in areas where developed economies had scant comparative advantage. Indeed, the secular influences that served to steady inflation (and inflation expectations) also anchored the narrative investors told themselves about bond pricing.

Yet the liberal free-trade agenda and the inflation-targeting regime by an independent Federal Reserve introduced a constellation of unintended consequences. Not least, the U.S. dollar became the de facto currency in most trade and financial transactions, prompting America’s trading partners to keep buying Treasuries despite formidable fiscal deficits and soaring government debt.

Today, the implications of the way things are—rather than how some wish them to be—are not widely appreciated or, in some quarters, well understood.

Question

How would our understanding of government bond price dynamics change if either one—or both—of the primary pillars anchoring inflation expectations since the 1980s (inflation targeting and free trade) were removed?

An Observation and a (Theorist’s) Way Forward

This note attempts to show that price—and, by extension, valuation—arises from our collective learned experience of economic history, and that we compromise our ability to make objective decisions if we cling too tightly to what we assume we “know,” disregarding plausible alternative histories.

We stumble when we forget history; the best way forward is understanding that uncertainty itself is our only reliable guide.

Finally, I have quite intentionally teased out certain contradictions within standard disequilibrium learning analysis to underscore the necessity for a framework that is thoroughly multi-period, that integrates a stochastic learning process, that solves jointly for prices and expectations, and that remains flexible enough to accommodate differing expectations, time horizons, and production & consumption patterns.

Postscript

Let me add that this brisk survey of how monetary and trade policy have anchored inflation expectations can hardly do justice to the dense intricacies of our collective economic past. More comprehensive works abound; for example, Sargent’s The Conquest of American Inflation (1999) offers an excellent account of the monetary dimension, and a vast empirical literature explores the far-reaching impacts of free trade and inflation targeting. Any reader intrigued by the interplay of these forces would do well to consult these sources and others like them.

¹ Duffy, J. (1994). “On Learning and the Nonuniqueness of Equilibrium in an Overlapping Generations Model with Fiat Money.” *Journal of Economic Theory, 64*, 541–553.

² Lucas, R. E. Jr. (1986). “Adaptive Behavior and Economic Theory.” *Journal of Business, 59*, S401–S426.

³ Sims, C. A. (1989). “Models and Their Uses.” *American Journal of Agricultural Economics, 71*, 489–494.

⁴ Adrian, T., Crump, R., & Moench, E. (2012). “Pricing the Term Structure with Linear Regressions.”

⁵ See for example:

Duffee, G. R. (2013). “Forecasting Interest Rates.” In Elliott & Timmermann (Eds.), *Handbook of Economic Forecasting* (Vol. 2, Part A). Elsevier.

Adrian, Crump, & Moench (2015). *Federal Reserve Bank of New York Staff Report No. 593* (2013/2015 versions).