Applied Microeconomics: The Normative Representative Consumer and Welfare Analysis

In a previous post, I pondered some questions related to using market demand functions to make welfare statements, following broadly Microeconomic Theory by Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green (MWG).

Making welfare statements about aggregate demand revolves around a few key concepts including a positive representative consumer, a wealth distribution rule, and a social welfare function. At a high level, these concepts seem to represent the technical assumptions and characteristics that need to hold in order to make most of the basic analysis of an intermediate microeconomics course mathematically sound or tractable for applied work. Here is a shot at some high level explanations:

positive representative consumer- at a high level, a hypothetical consumer who's UMP facing society's budget constraint generates a market or economy's aggregate demand function

wealth distribution rule - for every level of aggregate wealth, assigns individual wealth.

This rule or function is what allows us to write aggregate demand as a function of prices and wealth in order to move forward with the rest of our discussion about welfare analysis.

Examples given in MWG include wealth distribution rules that are a function of shareholdings of stocks and commodities which make wealth a function of the market's price vector

social welfare function (SWF) - this assigns utility to the vector of utilities for all 'I' consumers in an economy or market. W(u1,.....,uI) or can be written in terms of indirect utilities W(v1,....vI). 

Maximizing Social Welfare and Defining the Normative Representative Consumer

The wealth distribution rule is assumed to maximize society's social welfare function subject to a given level of aggregate wealth. The optimal solution indicates a particular indirect utility function v(p,w)

Normative Representative Consumer- a positive representative consumer is a normative representative consumer relative to social welfare function W(.) if for every (p,w) the distribution of wealth maximizes W(.). v(p,w) in the optimum is the indirect utility function for the normal representative consumer. 

For v(p,w) to exist, we are assuming a SWF, and assuming it is maximized by an optimal distribution of wealth according to some specified wealth distribution rule.

An example from MWG: When v(p,w) is of the Gormon form, and the SWF is utilitarian, then an aggregate demand function can always be viewed as being generated by a normative representative consumer.