Pablo Cuba Borda

Is Bolivia’s total fertility rate below the U.S. rate? In a new data release, the National Institute of Statistics of Bolivia reports the country’s total fertility rate at 1.56 in 2026. The U.S. will be around 1.57. This is NOT my estimate. I am just copying and pasting a table from nimbus.ine.gob.bo/index.php/… (Appendix 1, page 68). Can I say it again? This is the official data. If I had been told 10 years ago that, by 2026, Bolivia’s total fertility rate would be below the U.S. rate, I would not have believed it. I would have happily bet $1 against $1 million that it would not happen. In fact, I look at the table, and I still need to pinch myself. Bolivia was around 6.5 in the mid-1980s and around 3 in 2010 or so. I don’t know. Sometimes I wish I were a better writer, so I could express what I am feeling right now. More than anything, the sense that ten years ago I missed a fundamental change in humanity. What else can I be missing now? Thanks to @LibCarlos for the link and the information..

A guide for students of economics: Ten statements that demonstrate that someone does not understand modern economics or what an equilibrium is, and that you can safely ignore everything else they say. 1. “Equilibrium means the economy is stable or at rest.” Many assume that an equilibrium is a peaceful state with no forces at play. Instead, an equilibrium is just an arrangement of actions and expectations over time that are mutually consistent. It can be locally unstable, explosive, or fragile. Nothing in the definition of equilibrium implies stability. 2. “Equilibrium implies optimality or social efficiency.” Equilibrium is often conflated with efficiency, but equilibrium merely reflects decentralized consistency, not welfare maximization. Market power, externalities, incomplete markets, nominal rigidities, and frictions routinely produce inefficient equilibria. I often teach a first-year macro graduate course, and not a single one of the equilibria I define is efficient. 3. “Equilibrium is a unique outcome.” Many often expect models to have one equilibrium. In reality, multiple equilibria arise naturally in dynamic, strategic, and incomplete-market environments. Models of coordination failures, self-fulfilling expectations, bubbles, overlapping generations, and liquidity traps all hinge on the existence of equilibrium multiplicity. 4. “Equilibrium requires perfect foresight or perfect information.” Equilibrium does not assume agents know the future. In fact, equilibria are often stochastic. The definition of equilibrium only requires that beliefs are consistent with the (perceived) stochastic laws of motion implied by the model. Bayesian learning, noisy signals, ambiguity, and subjective uncertainty all fit well within an equilibrium framework, provided beliefs converge to an internally consistent (but possibly incorrect) distribution. Bonus point: equilibria are compatible with agents having diverging beliefs that never converge to a single Dirac distribution. 5. “Real economies are rarely in equilibrium, so the concept is unrealistic.” Equilibrium is not meant to describe the daily state of the world. It is a conceptual device used to understand the outcome of our models under the assumptions we make. Also, see point 1 above. 6. “Equilibrium requires agents to be fully rational in a psychological sense.” Equilibrium only assumes internal consistency: agents optimize given preferences and constraints. It does not assume realism about human cognition. We can and do define equilibria in models with behavioral biases, bounded rationality, inattention, or rule-of-thumb behavior. We only need to ensure that the resulting actions and beliefs are mutually compatible. 7. “Equilibrium eliminates dynamics or learning.” Equilibrium is sometimes misinterpreted as a static state in which nothing evolves. In fact, many equilibria are sequences of probability distributions over states driven by shocks, policy rules, and endogenous responses. Learning dynamics (Bayesian updating, adaptive rules, experience-based expectations) can occur within equilibrium if the evolution of beliefs is self-consistent. 8. “Equilibrium renders expectations unimportant.” A common misconception is that equilibrium mechanically determines outcomes. In reality, expectations are often central: they determine investment, consumption, asset prices, and policy responses. Many equilibria differ only in their expectations. This is why communication, credibility, and forward guidance matter even in fully rational models. 9. “Equilibrium excludes policy intervention.” Some interpret equilibrium as a laissez-faire concept. In fact, equilibrium analysis is the foundation of modern policy evaluation. Fiscal, monetary, and regulatory interventions work through equilibrium responses (prices, wages, interest rates, quantities) and must satisfy equilibrium conditions to be credible. Equilibrium is a tool for policy design, not a barrier to it. 10. “Equilibriums…” Aequilibrium is a Latin neuter noun of the second declension, which forms a nominative plural in “a”. It is composed of aequus (equal; the same root as equality or equity) and libra (balance or scales or the name of several currencies over history). A final thought: “equilibrium” is a term of art. Its meaning in economics differs from its use in the natural sciences or in everyday language. Terms of art are ubiquitous across academic disciplines, and the first act of intellectual diligence when one starts studying a discipline is to learn what they mean.

This week I’ll be lecturing at the ECB on deep learning. More than ever, I plan to emphasize the importance of soft inductive bias, double descent, and benign overparameterization—topics I’ve discussed in many previous threads. These three closely related phenomena compel us to reconsider much of what we thought we knew about the model classes used in function approximation and their generalization properties. They also demonstrate why intuition derived from traditional statistics can fail when we transition into the high-dimensional, overparameterized regimes characteristic of modern machine learning. Much of the new material in my lectures comes from a superb recent paper by Andrew Gordon Wilson, Deep Learning is Not So Mysterious or Different. Wilson argues that “rather than restricting the hypothesis space to avoid overfitting, we should embrace a flexible hypothesis space, with a soft preference for simpler solutions that are consistent with the data.” Several aspects of the paper struck me in particular: 1️⃣ The deep links with PAC-Bayes theory, a connection especially relevant to me, given my many papers in Bayesian econometrics. 2️⃣ The insistence that soft inductive bias, double descent, and benign overparameterization are not unique to deep learning but arise in a wide family of model classes. 3️⃣ The move away from attributing the soft inductive bias primarily to optimizers and instead focusing on the structure of the hypothesis space itself. Wilson also discusses mode connectivity, a phenomenon that still puzzles me deeply. I must confess I have almost no intuition for why distinct minima in the loss landscape should be connected by paths of low loss. If anyone has a convincing intuition or geometric explanation, I would be very grateful to hear it. In any case, read the paper—or even better, watch this wonderful interview with the author on Machine Learning Street Talk: 🔗 youtube.com/watch?v=M-jTeBCE…📷

A few days ago, Charlie Plosser passed away, another sad loss in a year that has not been kind to the profession. Charlie was one of the great macroeconomists of his generation, and it is difficult to single out just one of his contributions. His 1983 paper with John Long in the JPE, “Real Business Cycles” people.bu.edu/rking/EC702/lp… not only popularized the term RBC but also developed a network structure for the economy that has gained renewed relevance as we seek to integrate disaggregated microdata into business cycle models. His 1982 JME paper with Charles Nelson, “Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications” hedibert.org/wp-content/uplo… showed that many macroeconomic series did not revert to a linear trend after a shock. The insight that thinking of the economy as fluctuating around a fixed linear trend could be highly misleading has shaped much of my own research and how I interpret both U.S. and Spanish economic history. But perhaps the work that influenced me most was his 1988 JME paper with Robert King and Sergio Rebelo, “Production, Growth and Business Cycles I: The Basic Neoclassical Model”, and especially its technical appendix: people.bu.edu/rking/EC702/kp… This paper synthesized and extended the program of understanding business cycles through the stochastic neoclassical growth model. Even today’s heterogeneous-agent New Keynesian (HANK) models, which represent the state of the art, are variations of this framework, enriched with agent heterogeneity and nominal (and sometimes real) rigidities. The technical appendix trained a generation of students, myself included, in how to loglinearize dynamic equilibrium models and derive their quantitative representation. I still recall, as an undergraduate in 1996, reading the 1988 paper and writing to Sergio Rebelo (then at Rochester) to request a copy of the appendix. A few weeks later, I received a thick yellow folder in the mail. I remember the excitement of opening it and working through the math. I still teach much of that material in my second-year graduate course on computational economics, now reframed as a perturbation problem and with updated notation: sas.upenn.edu/~jesusfv/Lectu… Charlie later closed his career as President and CEO of the Federal Reserve Bank of Philadelphia. As a visiting scholar there, I often spoke with him. His views on monetary policy—the importance of systematic rules, and of focusing on price stability rather than pursuing disparate goals—are more relevant today than ever. Charlie’s example as a researcher, long-time editor of the JME, and dedicated public servant leaves a lasting legacy.