# 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 assumptions are different. First one states that if a bundle is better than the optimal one the consumer cannot afford it. The second one states that if a bundle is as good (not necessarily better) than the optimal one it has to cost as least as much, not less.

Consider a space with just one type of good, $x$, and a utility function $U(x) = 0$. Let the endowment of the consumer be $w = 1$. While $$\text{if } x_i \succ x_i^* \text{ then } p_ix_i > p_i w_i$$ is still true, $$\text{if } x_i \succeq x_i^* \text{ then } p_ix_i \geq p_i w_i$$ isn't, because $x^* = 0$ is both optimal and feasible, thus $$x^* \succeq x^* \text{ AND } p x^* < p w.$$ More complicated examples (multiple goods, global non-satiation fulfilled) may also be constructed.

• The math isn't helping me understand this. How would local satiation prevent a market from reaching a pareto optimal state? – B T Mar 27 '16 at 3:28
• @BT You have posted your own question about that, haven't you? – Giskard Mar 27 '16 at 5:50
• Well, I posted a question about the conditions under which the equilibrium is guaranteed to be pareto optimal. The answer involves local satiation as a condition, but the question didn't explicitly ask why it was a condition, and this one does. – B T Mar 27 '16 at 6:04
• I think the problem is this example doesn't explain what local nonsatiation is or why its needed. Rather, it instead provides an example for why utility maximizing doesn't imply the second claim is necessarily true. I would have preferred something explaining exactly what local nonsatiation provides that resolves the situation...which is what the title asks. But it is nonetheless a very clever example – Stan Shunpike Mar 27 '16 at 6:54
• @StanShunpike It is unfortunate that - if I understand you correctly now - the title and the body of the question are very different. It is also unfortunate that you did not comment when I answered the question 5 months ago... – Giskard Mar 27 '16 at 7:49

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.