<p>Let <span class="math-container">$x_i(p, w_i)$</span> be consumer <span class="math-container">$i$</span>'s (Marshallian) demand and <span class="math-container">$y_j(p)$</span> be firm <span class="math-container">$j$</span>'s profit-maximizing supply at price <span class="math-container">$p$</span>.
A competitive equilibrium is a price <span class="math-container">$p$</span> such that
<span class="math-container">$$
\sum_{i=1}^I x_i(p, w_i(p, e_i)) = \sum_{j=1}^J y_j(p).
$$</span></p>
<ul>
<li>By strict concavity of <span class="math-container">$u_i$</span>'s, consumer demands are functions, rather than correspondences. Assume also that <span class="math-container">$y_j(p)$</span> are functions.</li>
<li>There can be multiple goods---e.g. consumption good/time(labor)/etc.</li>
<li>Agent's wealth <span class="math-container">$w_i$</span> may depend on price <span class="math-container">$p$</span> and endowment <span class="math-container">$e_i$</span>---e.g. in an Edgeworth box where demands are Slutsky demands at the endowments---or wealth transfer <span class="math-container">$w_i$</span> can be specified as part of equilibrium, subject to
<span class="math-container">$$
\sum_i w_i = p^T (\sum_j y_j(p) + \sum_i e_i),
$$</span>
i.e. aggregate wealth equals total value of goods.</li>
</ul>
<p><strong>Aggregating Consumers</strong></p>
<p>Under the assumption that <span class="math-container">$u_i$</span>'s are strictly concave (and differentiable, say), one can define a representative consumer via a linear social welfare function:
<span class="math-container">$$
u_{rep}(x) = \max_{\sum_{i} x_i \leq x}\sum_i \frac{1}{I} u_i(x_i).
$$</span>
Then <span class="math-container">$u_{rep}$</span> is also strictly concave. Let <span class="math-container">$u_{rep, l}$</span> denotes marginal utility with respect to good <span class="math-container">$l$</span>, then by the Envelope and Lagrange Theorems,
<span class="math-container">$$
\frac{u_{rep, l}}{u_{rep, k}} = \frac{u_{i, l}}{u_{i, k}}, \;\; \mbox{for all $i$}.
$$</span>
At a competitive equilibrium price <span class="math-container">$p$</span>, the MRS of <span class="math-container">$u_{rep}$</span> at the aggregate
demand <span class="math-container">$\sum_i x_i(p)$</span> is the ratio of prices:
<span class="math-container">$$
\frac{u_{rep, l} (\sum_i x_i(p),w_i)}{u_{rep, k}(\sum_i x_i(p), w_i)} =  \frac{p_l}{p_k}. \quad(*)
$$</span>
By concavity of <span class="math-container">$u_{rep}$</span>, the FOC <span class="math-container">$(*)$</span> implies that the aggregate demand at <span class="math-container">$p$</span> is the optimal bundle for the representative consumer <span class="math-container">$u_{rep}$</span> at <span class="math-container">$p$</span>:
<span class="math-container">$$
x_{rep}(p, \sum_i w_i) = \sum_i x_i(p, w_i).
$$</span></p>
<ul>
<li>As you already started to say in the question, this is a general construction of a (local) representative agent under concavity assumption. By the First Welfare Theorem, CE is Pareto. Under concavity of <span class="math-container">$u_i$</span>'s, Pareto allocations must arise from a linear SWF, which implies existence of a representative consumer.</li>
<li>In this specific instance, consumers receive equal social weights <span class="math-container">$\alpha_i = \frac{1}{I}$</span> because equilibrium wealth is equal across agents. In general, <span class="math-container">$\alpha_i = \frac{1}{\lambda_i}$</span> where <span class="math-container">$\lambda_i$</span> is consumer <span class="math-container">$i$</span>'s Lagrange multipler. In particular, <span class="math-container">$\alpha_i$</span> would be increasing in agent <span class="math-container">$i$</span>'s share of aggregate wealth.</li>
<li>If <span class="math-container">$u_i = u$</span> is homothetic, with possibly different levels of wealth across agents, one can simply take <span class="math-container">$u_{rep} = u$</span> trivially. (Concavity assumption not needed.)</li>
<li><em>Gorman aggregation</em>, alluded to in the comments, is a slightly different context, I believe. Gorman aggregation says that if all wealth expansion curves are parallel at all wealth distributions, then aggregate demand is a function of aggregate wealth and one has a global representative agent. The notion of Gorman aggregation is independent of equilibrium or Pareto efficiency. Here in your question, the local representative agent depends on the initial CE allocation---e.g. different endowments distributions would lead to different social weights.</li>
</ul>
<p><strong>Aggregating Firms</strong></p>
<p>As pointed out in the comments, to aggregate firms one just uses the CRS assumption and take <span class="math-container">$F_{rep} = F$</span>. For a CRS firm, doubling the input means output also doubles, with the same marginal products. In equilibrium, the firm still makes zero profit. For any one of the <span class="math-container">$J$</span> firms, aggregate production <span class="math-container">$\sum_j y_j(p)$</span> would also be optimal at the same equilibrium price <span class="math-container">$p$</span>.</p>
<p>At the given equilibrium price <span class="math-container">$p$</span>, <span class="math-container">$p = \nabla F(y_j(p))$</span>, the gradient vector of marginal products of firm <span class="math-container">$j$</span> for all <span class="math-container">$j$</span>. By the CRS/homogeneous-of-degree-one property,
<span class="math-container">$$
\nabla F_{rep}(\sum_j y_j(p)) = \nabla F(y_1(p)) = p,
$$</span>
which tells you <span class="math-container">$y_{rep}(p) = \sum_j y_j(p)$</span>.</p>
<p>Putting it together, given CE with allocation
<span class="math-container">$\{(x_i(p))_i, (y_j(p))_j \}$</span> and price <span class="math-container">$p$</span>,
the allocation <span class="math-container">$\{\sum_i x_i(p), \sum_j y_j(p) \}$</span> and price <span class="math-container">$p$</span>
is a CE for the economy with one agent <span class="math-container">$u_{rep}$</span> and one firm <span class="math-container">$F_{rep}$</span>.</p>

CC BY-SA 4.0
2021-04-09T03:22:30.103Z
2021-04-09T08:47:19.487Z
2021-04-09T08:47:19.487Z
Michael
<p>Assume the following: all firms are identical, i.e. at each time period <span class="math-container">$t$</span>,
<span class="math-container">$\forall i$</span>, <span class="math-container">$j: F_{t}^j = F_{t}^i = F_t$</span> and the technology is CRS. All consumers have the same endowment of capital <span class="math-container">$\forall i$</span>, <span class="math-container">$j : k_0^i =k_0^j = k_0$</span>. All consumers have the same time endowment in each period t, i.e. <span class="math-container">$\forall i$</span>, <span class="math-container">$j : \bar{n}_t^i = \bar{n}_t^j = n$</span> . All consumers have the same preferences, i.e. <span class="math-container">$\forall i$</span>, <span class="math-container">$j : u^i = u^j = u$</span> with <span class="math-container">$u$</span> strictly concave.</p>
<p>The question asks to state clearly and prove a theorem which says that the CE of this economy is the same as one with one firm and one consumer.</p>
<p>Now I was thinking that based on these assumptions: If all firms are identical within and across sectors, and the technology are CRS, all HH have the same <span class="math-container">$k_0^i$</span> and <span class="math-container">$U^i$</span> , and <span class="math-container">$U^i$</span> is strictly concave then, CE of the original economy is the same as one with one firm and one household. By using the First Welfare Theorem we can show that this CE is Pareto Optimal.</p>

CC BY-SA 4.0
2021-03-25T19:24:27.687Z
You may want to check out the [Gorman's aggregation condition](https://en.wikipedia.org/wiki/Gorman_polar_form).
Herr K.
CC BY-SA 4.0
2021-03-25T19:46:26.790Z
The crucial step is showing that demand is unique in the same for all consumers. With CRS, there is no difference whether there is one firm or many. The aggregate production set will be the same and no firm can make a positive profit.
Michael Greinecker
CC BY-SA 4.0
2021-03-25T19:47:12.440Z
@HerrK. Here, Gorman's theorem need not apply. The distribution of initial endowments matters.
Michael Greinecker
CC BY-SA 4.0
2021-03-25T22:21:01.517Z
@MichaelGreinecker: What could go wrong given that everything (including endowments) is so symmetric here?
Herr K.
CC BY-SA 4.0
2021-03-25T22:48:51.610Z
@HerrK. Nothing goes wrong, that's my point. Preferences need not have Gorman form for the result to work.
Michael Greinecker
<macroeconomics><pareto-efficiency><competitive-equilibrium>
Identical aggregation
Maybeline Lee
# Identical aggregation

(Asked by: username_0 on 2021-03-25)

Assume the following: all firms are identical, i.e. at each time period $t$,
$\forall i$, $j: F\_{t}^j = F\_{t}^i = F\_t$ and the technology is CRS. All consumers have the same endowment of capital $\forall i$, $j : k\_0^i =k\_0^j = k\_0$. All consumers have the same time endowment in each period t, i.e. $\forall i$, $j : \bar{n}\_t^i = \bar{n}\_t^j = n$ . All consumers have the same preferences, i.e. $\forall i$, $j : u^i = u^j = u$ with $u$ strictly concave.

The question asks to state clearly and prove a theorem which says that the CE of this economy is the same as one with one firm and one consumer.

Now I was thinking that based on these assumptions: If all firms are identical within and across sectors, and the technology are CRS, all HH have the same $k\_0^i$ and $U^i$ , and $U^i$ is strictly concave then, CE of the original economy is the same as one with one firm and one household. By using the First Welfare Theorem we can show that this CE is Pareto Optimal.

**Tags:** macroeconomics, pareto-efficiency, competitive-equilibrium
> **Comments:**
> **username_1**: You may want to check out the [Gorman's aggregation condition](https://en.wikipedia.org/wiki/Gorman_polar_form).  
> **username_2**: The crucial step is showing that demand is unique in the same for all consumers. With CRS, there is no difference whether there is one firm or many. The aggregate production set will be the same and no firm can make a positive profit.  
> **username_2**: @HerrK. Here, Gorman's theorem need not apply. The distribution of initial endowments matters.  
> **username_1**: @MichaelGreinecker: What could go wrong given that everything (including endowments) is so symmetric here?  
> **username_2**: @HerrK. Nothing goes wrong, that's my point. Preferences need not have Gorman form for the result to work.  

---

## 1 answers

### Answer by username_3 on 2021-04-09. Score: 2 | Accepted answer

Let $x\_i(p, w\_i)$ be consumer $i$'s (Marshallian) demand and $y\_j(p)$ be firm $j$'s profit-maximizing supply at price $p$.
A competitive equilibrium is a price $p$ such that
$$
\sum\_{i=1}^I x\_i(p, w\_i(p, e\_i)) = \sum\_{j=1}^J y\_j(p).
$$

* By strict concavity of $u\_i$'s, consumer demands are functions, rather than correspondences. Assume also that $y\_j(p)$ are functions.
* There can be multiple goods---e.g. consumption good/time(labor)/etc.
* Agent's wealth $w\_i$ may depend on price $p$ and endowment $e\_i$---e.g. in an Edgeworth box where demands are Slutsky demands at the endowments---or wealth transfer $w\_i$ can be specified as part of equilibrium, subject to
  $$
  \sum\_i w\_i = p^T (\sum\_j y\_j(p) + \sum\_i e\_i),
  $$
  i.e. aggregate wealth equals total value of goods.

**Aggregating Consumers**

Under the assumption that $u\_i$'s are strictly concave (and differentiable, say), one can define a representative consumer via a linear social welfare function:
$$
u\_{rep}(x) = \max\_{\sum\_{i} x\_i \leq x}\sum\_i \frac{1}{I} u\_i(x\_i).
$$
Then $u\_{rep}$ is also strictly concave. Let $u\_{rep, l}$ denotes marginal utility with respect to good $l$, then by the Envelope and Lagrange Theorems,
$$
\frac{u\_{rep, l}}{u\_{rep, k}} = \frac{u\_{i, l}}{u\_{i, k}}, \;\; \mbox{for all $i$}.
$$
At a competitive equilibrium price $p$, the MRS of $u\_{rep}$ at the aggregate
demand $\sum\_i x\_i(p)$ is the ratio of prices:
$$
\frac{u\_{rep, l} (\sum\_i x\_i(p),w\_i)}{u\_{rep, k}(\sum\_i x\_i(p), w\_i)} = \frac{p\_l}{p\_k}. \quad(\*)
$$
By concavity of $u\_{rep}$, the FOC $(\*)$ implies that the aggregate demand at $p$ is the optimal bundle for the representative consumer $u\_{rep}$ at $p$:
$$
x\_{rep}(p, \sum\_i w\_i) = \sum\_i x\_i(p, w\_i).
$$

* As you already started to say in the question, this is a general construction of a (local) representative agent under concavity assumption. By the First Welfare Theorem, CE is Pareto. Under concavity of $u\_i$'s, Pareto allocations must arise from a linear SWF, which implies existence of a representative consumer.
* In this specific instance, consumers receive equal social weights $\alpha\_i = \frac{1}{I}$ because equilibrium wealth is equal across agents. In general, $\alpha\_i = \frac{1}{\lambda\_i}$ where $\lambda\_i$ is consumer $i$'s Lagrange multipler. In particular, $\alpha\_i$ would be increasing in agent $i$'s share of aggregate wealth.
* If $u\_i = u$ is homothetic, with possibly different levels of wealth across agents, one can simply take $u\_{rep} = u$ trivially. (Concavity assumption not needed.)
* *Gorman aggregation*, alluded to in the comments, is a slightly different context, I believe. Gorman aggregation says that if all wealth expansion curves are parallel at all wealth distributions, then aggregate demand is a function of aggregate wealth and one has a global representative agent. The notion of Gorman aggregation is independent of equilibrium or Pareto efficiency. Here in your question, the local representative agent depends on the initial CE allocation---e.g. different endowments distributions would lead to different social weights.

**Aggregating Firms**

As pointed out in the comments, to aggregate firms one just uses the CRS assumption and take $F\_{rep} = F$. For a CRS firm, doubling the input means output also doubles, with the same marginal products. In equilibrium, the firm still makes zero profit. For any one of the $J$ firms, aggregate production $\sum\_j y\_j(p)$ would also be optimal at the same equilibrium price $p$.

At the given equilibrium price $p$, $p = \nabla F(y\_j(p))$, the gradient vector of marginal products of firm $j$ for all $j$. By the CRS/homogeneous-of-degree-one property,
$$
\nabla F\_{rep}(\sum\_j y\_j(p)) = \nabla F(y\_1(p)) = p,
$$
which tells you $y\_{rep}(p) = \sum\_j y\_j(p)$.

Putting it together, given CE with allocation
$\{(x\_i(p))\_i, (y\_j(p))\_j \}$ and price $p$,
the allocation $\{\sum\_i x\_i(p), \sum\_j y\_j(p) \}$ and price $p$
is a CE for the economy with one agent $u\_{rep}$ and one firm $F\_{rep}$.


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