Notes on Representative Agents

Author

Affiliation

Hirotaka Fukui

Kobe University, Graduate School of Economics

Modified

November 24, 2025

Important

All rights reserved. All errors in this document are the responsibility of the author. Although every effort has been made to ensure the accuracy of this material, I cannot guarantee that it is free of errors or misprints. Use at your own risk. If you find any errors, please get in touch with me.

Aggregation and Social Welfare

Macroeconomics often assumes a representative individual to describe household behavior and aggregate welfare. This simplification builds on the Second Fundamental Theorem of Welfare Economics, which shows that a social planner can achieve a Pareto-efficient allocation of resources. By assigning proper welfare weights to each household, a centralized economy can replicate the outcomes of competitive markets.

However, assuming that one individual can represent an entire population is a strong abstraction. It ignores differences in income, preferences, and opportunities across people. Recent studies have moved beyond this idea, developing models that incorporate household heterogeneity and inequality.

Even so, the representative individual remains central to modern macroeconomics. Critics, following the Debreu–Sonnenschein–Mantel theorem, argue that aggregating individual behavior into a single agent is theoretically inconsistent. Others see it as a useful simplification that can approximate aggregate outcomes under certain assumptions.

Debreu-Mantel-Sonnenschein Theorem

When studying how prices and income affect demand, economists usually work with market-level data rather than household-level observations. For businesses and policymakers, what matters is the overall demand curve, not each household’s choices. Ideally, a few parameters could summarize how total demand responds to changes in prices and income.

This raises a key question: if we only observe aggregate data, what kind of utility function should we assume for individuals? Microeconomic theory provides clear conditions—such as convex, continuous, and transitive preferences—under which individual demand satisfies symmetry, homogeneity, and the weak axiom of revealed preference.

Yet these properties do not necessarily hold when we aggregate across many households. Market demand may behave quite differently from individual demand. Even if individual preferences are non-convex, the overall demand function can still appear smooth and continuous. This insight lies at the heart of the Debreu–Sonnenschein–Mantel theorem, which shows that aggregate demand can take almost any shape consistent with Walras’ law and continuity, regardless of individual preferences.

Theorem 1 (Debreu-Mantel-Sonnenschein Theorem) Let $\epsilon >0$ and $N \in \mathbb{N}$. Consider a set of prices $\textbf{P}_{\epsilon}=\{p \in \mathbb{R}^{N}_{+}: p_{j}/p_{j'} \geq \epsilon,~\forall j~\text{and}~ j' \}$ and any continuous function $\textbf{x}: \textbf{P}_{\epsilon} \rightarrow \mathbb{R}^{N}_{+}$ that satisfies Walras’s Law and is homogeneous of degree 0. Then there exists an exchange economy with $N$ commodities and $H<\infty$ households, where the aggregate excess demand is given by $\textbf{x}(p)$ over the set $\textbf{P}_{\epsilon}$

--- title: Notes on Representative Agents engine: julia format: html: code-tools: true --- ::: {.callout-important} All rights reserved. All errors in this document are the responsibility of the author. Although every effort has been made to ensure the accuracy of this material, I cannot guarantee that it is free of errors or misprints. Use at your own risk. If you find any errors, please get in touch with me. ::: # Aggregation and Social Welfare Macroeconomics often assumes a representative individual to describe household behavior and aggregate welfare. This simplification builds on the Second Fundamental Theorem of Welfare Economics, which shows that a social planner can achieve a Pareto-efficient allocation of resources. By assigning proper welfare weights to each household, a centralized economy can replicate the outcomes of competitive markets. However, assuming that one individual can represent an entire population is a strong abstraction. It ignores differences in income, preferences, and opportunities across people. Recent studies have moved beyond this idea, developing models that incorporate household heterogeneity and inequality. Even so, the representative individual remains central to modern macroeconomics. Critics, following the Debreu–Sonnenschein–Mantel theorem, argue that aggregating individual behavior into a single agent is theoretically inconsistent. Others see it as a useful simplification that can approximate aggregate outcomes under certain assumptions. ## Debreu-Mantel-Sonnenschein Theorem When studying how prices and income affect demand, economists usually work with market-level data rather than household-level observations. For businesses and policymakers, what matters is the overall demand curve, not each household’s choices. Ideally, a few parameters could summarize how total demand responds to changes in prices and income. This raises a key question: if we only observe aggregate data, what kind of utility function should we assume for individuals? Microeconomic theory provides clear conditions---such as convex, continuous, and transitive preferences---under which individual demand satisfies symmetry, homogeneity, and the weak axiom of revealed preference. Yet these properties do not necessarily hold when we aggregate across many households. Market demand may behave quite differently from individual demand. Even if individual preferences are non-convex, the overall demand function can still appear smooth and continuous. This insight lies at the heart of the Debreu–Sonnenschein–Mantel theorem, which shows that aggregate demand can take almost any shape consistent with Walras’ law and continuity, regardless of individual preferences. ::: {#thm-DMS} ## Debreu-Mantel-Sonnenschein Theorem Let $\epsilon >0$ and $N \in \mathbb{N}$. Consider a set of prices $\textbf{P}_{\epsilon}=\{p \in \mathbb{R}^{N}_{+}: p_{j}/p_{j'} \geq \epsilon,~\forall j~\text{and}~ j' \}$ and any continuous function $\textbf{x}: \textbf{P}_{\epsilon} \rightarrow \mathbb{R}^{N}_{+}$ that satisfies Walras's Law and is homogeneous of degree 0. Then there exists an exchange economy with $N$ commodities and $H<\infty$ households, where the aggregate excess demand is given by $\textbf{x}(p)$ over the set $\textbf{P}_{\epsilon}$ :::