What is the purpose of the local non-satiation assumption in the first welfare theorem?

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

• The consumer problem is not about profit maximization. Commented Jul 20, 2022 at 15:15

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:

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.

Local non-satiation assumption is required to prove the first welfare theorem.

Consider the following pure exchange economy consisting of two consumers - $$1$$ and $$2$$, and two Goods - $$X$$ and $$Y$$:

• Preferences of $$1$$ and $$2$$ are given by the following utility functions: $$u_1(x_1, y_1) = 0$$ and $$u_2(x_2, y_2) = x_2+y_2$$
• Endowments of $$1$$ and $$2$$ are given by the following: $$\omega_1 = (10,0)$$ and $$\omega_2 = (0,20)$$

Observe that the endowment allocation is also the competitive equilibrium of this economy i.e. $$(x_1^*, y_1^*) = (10, 0)$$ and $$(x_2^*, y_2^*) = (0, 20)$$ supported by any pair of prices $$p = (p_X, p_Y)$$ satisfying the condition $$0 < p_Y \leq p_X$$. However, this allocation is not Pareto efficient. This is because an allocation where $$2$$ consumes all of both the goods is Pareto superior to it.

In this example, it is clear that $$1$$'s preferences does not satisfy LNS. Also we can see that this condition fails to hold:

$$(x_1, y_1) \succsim (x_1^*, y_1^*) = (10,0)$$ implies $$p_Xx_1+p_Yy_1 \geq p\cdot\omega_1 = 10p_X$$

To see this, consider $$(x_1, y_1) = (0,0)$$.

When preferences satisfy LNS, we can prove that the following proposition is true:

Proposition. If $$x_i^*$$ is the demand of consumer $$i$$ when price is $$p$$ and endowment is $$\omega$$, then $$x_i\succsim x_i^* \ \Rightarrow \ p\cdot x_i \geq p \cdot \omega_i$$.

Proof. Suppose $$x_i\succsim x_i^*$$ but $$p\cdot x_i < p \cdot \omega_i$$. Then by LNS, there exists a bundle $$x_i'$$ sufficiently close to $$x_i$$, so that $$x_i' \succ x_i$$ and $$p\cdot x_i' holds. This implies that there exists a bundle $$x_i'\succ x_i^*$$ which is also affordable, contradicting that $$x_i^*$$ is the demand.