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.