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The profit maximization assumption implies $$\text{if } x_i \succ x_i^* \text{ then } p_ix_i > p_i w_i$$

Okay so this just says if the agent is utility maximizing / rational, then if he doesn't choose a bundle strictly preferable to his bundle then it must not be affordable.

Why is the local non-satiation assumption needed to then say

$$\text{if } x_i \succeq x_i^* \text{ then } p_ix_i \geq p_i w_i$$

Why isn't this just automatic from the profit maximization assumption? If we know $x_i \succ x_i^* \implies p_ix_i > p_i w_i$, isn't it obvious that $x_i = x_i^* \implies p_ix_i = p_i w_i$ and so $$\text{if } x_i \succeq x_i^* \text{ then } p_ix_i \geq p_i w_i$$

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Ok I think I might understand now why local nonsatiation is important for tending toward a pareto optimal market allocation. Consider the following picture, where all the circles represent possible allocations, and their position on the graph represents the utility received by each person in a simple two-person market:

Satiated pareto efficiency graph

In this case, X, Y, Z, and D all give person 1 the same utility. In such a situation, X, Y, and Z are all possible equilibria given complete markets and price taking behavior even though they're not pareto-optimal.

In a situation with local nonsatiation, this situation couldn't exist, and thus a pareto optimal equilibrium is ensured.

Weak pareto optimality doesn't require local non-satiation.

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