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tags: micro
---
# Question 3 of 2020 May Micro
## Question:
Construct a two-consumer, two-good pure exchange economy in which every non-wasteful feasible allocation is strongly Pareto optimal but no competitive equilibrium exists. For simplicity, assume each consumer owns exactly one unit of each good.

## Suggested Answer:

### Setting
Let $b_1=(x_1, y_1)$ and $b_2=(x_2, y_2)$ denote the endowment of consumer 1 and 2, respectively. Since it is a pure exchange economy, $x_1+x_2 =2$ and $y_1+y_2 =2$, we can rewrite the endowment as $b_1=(x_1, y_1)$ and $b_2=(2- x_1, 2- y_1)$.

Both consumers have a lexicographic preference, that is, for consumer $i$

$$
x_i>x_i' \implies (x_i, y_i) \succ_i (x_i', y_i')
$$

$$
x_i=x_i' \text{ and } y_i>y_i' \implies (x_i, y_i) \succ_i (x_i', y_i')
$$

### Strongly Pareto Optimal

Ginve any allocation $b_1=(x_1, y_1)$ and $b_2=(2- x_1, 2- y_1)$, if we want to make consumer 1 better off, we need to either increase $x_1$ or increase $y_1$ and remain $x_1$ as the same. If we increase $x_1$, then $2-x_1$ will decrease and consumer 2 will be worse off. If we increase $y_1$ and remain $x_1$ as the same, $2-x_2$ reamin the same but $2-y_2$ will decrease, so consumer 2 will also be worse off.

By symmetry, we cannot make any consumer better off and do not make another worse off. Hence, any feasible allocation is strongly Pareto optimal.

### Competitive Equilibrium
Let $P_x$ and $P_y$ denote the possible price of good $x$ and good $y$, respectively. Ginve an allocation $b_1=(x_1, y_1)$ and $b_2=(2- x_1, 2- y_1)$, without loss of generality, we assume $y_1 >0$.

First, if $P_x=0$, two consumers will want to buy an infinity amount of good $x$, so it could not constitute a competitive equilibrium.

Second, suppose $P_x>0$ and $P_y>0$. Consumer 1 will want to sell all of her good $y$ and use that money to buy $x$. However, consumer 2 will not be willing to sell his $x$ to consumer 1. Hence, it could not constitute a competitive equilibrium.

Third, consider $P_x>0$ and $P_y=0$. Both consumers will want to buy an infinity amount of good $y$, so it could not constitute a competitive equilibrium.

Therefore, any allocation could not constitute a competitive equilibrium.