Understanding Duality between Individual and Collective Maximization in Macroeconomic Models

Question

I'm currently studying macroeconomic models, specifically from the book "Recursive Macroeconomic Theory." In Chapter Seven, it is mentioned that some economic models involving firms and consumers can be equivalently solved by formulating a "planner" problem, aimed at maximizing social welfare.

This approach yields the same solutions as when individual actors maximize their own objectives. I suspect this is related to the welfare theorems in economics. My prior understanding of the welfare theorem comes from MWG (Mas-Colell, Whinston, and Green), but the setups there didn’t seem as comprehensive, especially considering the complex, constraint-rich, and intertemporal aspects of macroeconomic models.

An interesting element related to this topic is the concept of a "shadow wage," which emerges from the Lagrangian formulation of these planner problems and also appears in individual optimization problems.

I'm seeking resources or explanations that delve deeper into the duality between individual and collective maximization, particularly under more general and complex conditions. Additionally, I would appreciate insights into concepts like "shadow wages" in this context.

Thank you.

Answer 1

The proof of the first welfare theorem is almost the same as the one you are familiar with from MWG. The main difference is that if you have recursive budget constraints, you have to show that you can put a total value on all consumption streams, that this value is linear in consumption streams, and that it is not infinite.

Proving a suitable version of the second welfare theorem is much more technically involved and requires some functional analysis. The problem is that the natural commodity spaces are all infinite-dimensional. One needs some additional assumptions to apply the equivalent of the separating hyperplane theorem and some further assumptions to guarantee that the resulting price systems can be written in terms of, for example, prices per period for each good. You can find some textbook treatments in "Introduction to Modern Economic Growth" by Acemoglu, Sections 5.6 and 5.7, and, on a slightly higher level, in "Recursive Methods in Economic Dynamics" by Stokey and Lucas with Prescott, Chapter 15.

There are problems for which these versions of the second welfare theorem are not enough. In particular, many stochastic models require commodity spaces in which consumption sets have an empty interior. There exist versions of the second welfare theorem that are still applicable in such cases under reasonable assumptions. One can learn more about them in the much more technical survey "Fundamental Theorems of Welfare Economics in Infinite Dimensional Commodity Spaces" by Robert A. Becker, which can be found in the book "Equilibrium Theory in Infinite Dimensional Spaces" by Khan and Yannelis.