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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.

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  • $\begingroup$ You may want to check out the Gorman's aggregation condition. $\endgroup$ – Herr K. Mar 25 at 19:24
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    $\begingroup$ 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. $\endgroup$ – Michael Greinecker Mar 25 at 19:46
  • $\begingroup$ @HerrK. Here, Gorman's theorem need not apply. The distribution of initial endowments matters. $\endgroup$ – Michael Greinecker Mar 25 at 19:47
  • $\begingroup$ @MichaelGreinecker: What could go wrong given that everything (including endowments) is so symmetric here? $\endgroup$ – Herr K. Mar 25 at 22:21
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    $\begingroup$ @HerrK. Nothing goes wrong, that's my point. Preferences need not have Gorman form for the result to work. $\endgroup$ – Michael Greinecker Mar 25 at 22:48
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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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