<p>I'd like to understand if I understood theory about representative agent (positive and normative): firstly it is necessary to ensure the existence of a representative household an it depends on the dependence on the total wealth of the economy regardless of the distribution of income in the economy. Secondly the representative household in order to be a positive representative household has to be the solver of a utility maximization problem under a budget constraint formally described by the following expression</p>
<p><span class="math-container">$x(p,w)= \underset{x} {\mathrm{argmax}}\ u^*(x)\\
s.t.\quad \quad \quad px\leq w $</span></p>
<p>where <span class="math-container">$w$</span> is the total wealth in the economy. There is an other requirement which is that of linear expansion path, formally written as <span class="math-container">$\dfrac{\partial x^i}{\partial w_i}=\lambda$</span> where <span class="math-container">$x^i$</span> is the individual demand. The necessary and sufficient condition that allow last one are the same which is the Gorman preference form for each household: <span class="math-container">$v_i(p,w_i)=a_i(p)+b(p)w_i$</span>. If preferences that admit a Gorman aggregate indirect utility function with <span class="math-container">$a^*(p)=\sum_i a_i(p)$</span> and <span class="math-container">$b^*(p)=b(p)$</span> then the representative household is positive. At this point I've a question: how preferences can avoid this result?.</p>
<p>When we talk about a normative representative household the difference is in the type of result of maximization problem: Pareto optimal allocation in the economy have to be the result of utility maximization. Formally two properties have to be satisfied: (1) is that given an economy <span class="math-container">$\mathcal{E}=\{\{X,p\},Y,\omega\}$</span> and two allocations <span class="math-container">$(x,y), (x'y')$</span> if <span class="math-container">$(x',y')$</span> Pareto dominates <span class="math-container">$(x,y)$</span> then the utility of the representative normative household <span class="math-container">$u^*(\sum_ix'_i)&gt;u^*(\sum_ix_i)$</span>; (2) is that there exists one more allocation <span class="math-container">$(\tilde{x},\tilde{y})$</span> for which <span class="math-container">$\sum_i\tilde{x_i}=\sum_ix_i$</span> is true and <span class="math-container">$(\tilde{x},\tilde{y})$</span> Pareto dominates <span class="math-container">$(x,y)$</span>. This second requirement is associated to the redistribution of wealth, given by the allocation <span class="math-container">$(\tilde{x},\tilde{y})$</span> if I'm not wrong, and it guarantees that its Pareto optimality is linked to the solution of the maximization of utility am I right? I'd like to know if my explanation is formally and conceptually correct, because I feel a little bit confused, especially about the formal argumentation.</p>

CC BY-SA 4.0
2022-12-15T09:36:38.343Z
Does anyone need other information in order to understand totally my question? Given the possible fact that there are doubts. I used those [slides](https://www.robertmtownsend.net/sites/default/files/files/Lecture%2020%20080719.pdf) in order to have a global vision of the theory, but I read also some stuff on Introduction to modern economic growth (2009) and Economics and consumer behavior (1980). In these two books I didn't find an helpful explanation such that of the slides, but something more articulated.
Pepus
CC BY-SA 4.0
2023-02-03T18:08:51.100Z
Does anyone can help me? I don't think is a difficult question, maybe it could be considered a common question, but I'd like to know if I'm correct.
Pepus
<microeconomics><representative-agent>
Clarity about positive and normative representative household
Pepus
# Clarity about positive and normative representative household

(Asked by: username_0 on 2022-12-13)

I'd like to understand if I understood theory about representative agent (positive and normative): firstly it is necessary to ensure the existence of a representative household an it depends on the dependence on the total wealth of the economy regardless of the distribution of income in the economy. Secondly the representative household in order to be a positive representative household has to be the solver of a utility maximization problem under a budget constraint formally described by the following expression

$x(p,w)= \underset{x} {\mathrm{argmax}}\ u^\*(x)\\
s.t.\quad \quad \quad px\leq w $

where $w$ is the total wealth in the economy. There is an other requirement which is that of linear expansion path, formally written as $\dfrac{\partial x^i}{\partial w\_i}=\lambda$ where $x^i$ is the individual demand. The necessary and sufficient condition that allow last one are the same which is the Gorman preference form for each household: $v\_i(p,w\_i)=a\_i(p)+b(p)w\_i$. If preferences that admit a Gorman aggregate indirect utility function with $a^\*(p)=\sum\_i a\_i(p)$ and $b^\*(p)=b(p)$ then the representative household is positive. At this point I've a question: how preferences can avoid this result?.

When we talk about a normative representative household the difference is in the type of result of maximization problem: Pareto optimal allocation in the economy have to be the result of utility maximization. Formally two properties have to be satisfied: (1) is that given an economy $\mathcal{E}=\{\{X,p\},Y,\omega\}$ and two allocations $(x,y), (x'y')$ if $(x',y')$ Pareto dominates $(x,y)$ then the utility of the representative normative household $u^\*(\sum\_ix'\_i)>u^\*(\sum\_ix\_i)$; (2) is that there exists one more allocation $(\tilde{x},\tilde{y})$ for which $\sum\_i\tilde{x\_i}=\sum\_ix\_i$ is true and $(\tilde{x},\tilde{y})$ Pareto dominates $(x,y)$. This second requirement is associated to the redistribution of wealth, given by the allocation $(\tilde{x},\tilde{y})$ if I'm not wrong, and it guarantees that its Pareto optimality is linked to the solution of the maximization of utility am I right? I'd like to know if my explanation is formally and conceptually correct, because I feel a little bit confused, especially about the formal argumentation.

**Tags:** microeconomics, representative-agent
> **Comments:**
> **username_0**: Does anyone need other information in order to understand totally my question? Given the possible fact that there are doubts. I used those [slides](https://www.robertmtownsend.net/sites/default/files/files/Lecture%2020%20080719.pdf) in order to have a global vision of the theory, but I read also some stuff on Introduction to modern economic growth (2009) and Economics and consumer behavior (1980). In these two books I didn't find an helpful explanation such that of the slides, but something more articulated.  
> **username_0**: Does anyone can help me? I don't think is a difficult question, maybe it could be considered a common question, but I'd like to know if I'm correct.  

---

## 0 answers

