<p>In the economy provided in the question, competitive equilibrium allocations is equal to the set of efficient allocations. This along with the fact that the competitive equilibrium allocations always lie in the core and core consists of efficient allocations implies that core stays the same in all replicas.</p>

CC BY-SA 4.0
2023-09-10T09:15:53.943Z
2023-09-10T09:15:53.943Z
Amit
<p>I'm trying to solve the following problem on general equilibrium:</p>
<p>Consider an economy with two individuals with utility functions <span class="math-container">$u^A(x^A,y^A) = \min \{ x^A, y^A \}$</span> and <span class="math-container">$u^B(x^B,y^B) = \min \{ x^B, y^B \}$</span>, with initial endowments <span class="math-container">$w^A = (1,0)$</span> and <span class="math-container">$w^B = (0,1)$</span>. First find the set of Pareto efficient (PE) allocations, then the set of core allocations in the unreplicated economy, <span class="math-container">$C_1$</span>, and finally in the twofold replica, <span class="math-container">$C_2$</span>.</p>
<p>My take:</p>
<p>Pareto Set = <span class="math-container">$\{ (x^A,y^A,x^B,y^B) : x^A = y^A , x^A + x^B = 1, y^A + y^B = 1 \}$</span>.</p>
<p>Because <span class="math-container">$I = 2$</span> (the number of agents is 2), then Contract Curve = Core. Here the Contract Curve is the same as the Pareto Set. Then:</p>
<p><span class="math-container">$C_1 = \{ (x^A,y^A,x^B,y^B) : x^A = y^A , x^A + x^B = 1, y^A + y^B = 1 \}$</span>.</p>
<p>In the twofold replica now I've got two agents of each type and total endowments are <span class="math-container">$w^A = (2,0)$</span>, <span class="math-container">$w^B = (0,2)$</span>. Because of equal treatment in the core, <span class="math-container">$x^A_1 = x^A_2$</span> and <span class="math-container">$y^A_1 = y^B_2$</span>, the same with the type-B individuals. So nothing change and the core is the same?</p>
<p>Shouldn't the core shrinks as the economy enlarges?</p>

<microeconomics><general-equilibrium><pareto-efficiency><pure-exchange-economy><cooperative-game-theory>
Core in a replicated economy
Walrasian soldier
# Core in a replicated economy

(Asked by: username_0 on 2023-09-10)

I'm trying to solve the following problem on general equilibrium:

Consider an economy with two individuals with utility functions $u^A(x^A,y^A) = \min \{ x^A, y^A \}$ and $u^B(x^B,y^B) = \min \{ x^B, y^B \}$, with initial endowments $w^A = (1,0)$ and $w^B = (0,1)$. First find the set of Pareto efficient (PE) allocations, then the set of core allocations in the unreplicated economy, $C\_1$, and finally in the twofold replica, $C\_2$.

My take:

Pareto Set = $\{ (x^A,y^A,x^B,y^B) : x^A = y^A , x^A + x^B = 1, y^A + y^B = 1 \}$.

Because $I = 2$ (the number of agents is 2), then Contract Curve = Core. Here the Contract Curve is the same as the Pareto Set. Then:

$C\_1 = \{ (x^A,y^A,x^B,y^B) : x^A = y^A , x^A + x^B = 1, y^A + y^B = 1 \}$.

In the twofold replica now I've got two agents of each type and total endowments are $w^A = (2,0)$, $w^B = (0,2)$. Because of equal treatment in the core, $x^A\_1 = x^A\_2$ and $y^A\_1 = y^B\_2$, the same with the type-B individuals. So nothing change and the core is the same?

Shouldn't the core shrinks as the economy enlarges?

**Tags:** microeconomics, general-equilibrium, pareto-efficiency, pure-exchange-economy, cooperative-game-theory

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## 1 answers

### Answer by username_1 on 2023-09-10. Score: 4

In the economy provided in the question, competitive equilibrium allocations is equal to the set of efficient allocations. This along with the fact that the competitive equilibrium allocations always lie in the core and core consists of efficient allocations implies that core stays the same in all replicas.


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