# In modern economics, what are the least restrictive conditions shown to theoretically achieve Pareto Efficiency?

In researching this question, I've come across plenty of sources that regurgitate intro econ textbooks that say markets equilibrate to Pareto efficiency in conditions of perfect competition (where specific and minimal definitions are also hard to find). I'm sure they're correct that this happens in perfect competition, but I've also read that various economists have simplified and minimized the conditions necessary for a market to theoretically reach pareto efficiency, but I can't find any details on those.

So what are the least restrictive conditions necessary for all markets that fit those conditions to theoretically progress toward a Pareto efficient equilibrium?

• I think you may have misspoken at the end. You said what do we need for a Pareto optimal equilibrium. By the first fundamental theorem of welfare economics, any equilibrium is Pareto efficient as long as we have preferences that are locally non-satiated. Did you mean what do we need for a Pareto optimal allocation? – DornerA Mar 25 '16 at 22:25
• @DornerA I was under the impression that Pareto efficiency, Pareto optimality, and pareto optimal allocation all mean the same thing. Double checking google seems to indicate I'm right. What's the difference you're indicating? I think my question is essentially, "what are the conditions required for the first fundamental theorem of welfare economics to hold?" – B T Mar 25 '16 at 22:32
• No, I was making the distinction between an allocation and an equilibrium. There are infinitely many allocations in any economy, but there may not be an equilibrium. You asked "what are the minimal conditions necessary for a market to theoretically have a pareto efficient equilibrium?" The only thing necessary for this is the existence of said equilibrium because by the first welfare theorem, any equilibrium is pareto efficient. – DornerA Mar 25 '16 at 22:35
• @DornerA What do you mean there may not be an equilibrium? How so? I was under the impression that any given market environment has an equilibrium, and in reality the market environment changes, which then changes the equilibrium before it gets there. Is that not correct? Also, the first welfare theorem has conditions that must be met for pareto efficient equilibrium. Saying "any equilibrium is pareto efficient" just isn't correct. – B T Mar 25 '16 at 22:41
• There are many cases where an equilibrium does not exist. But this is more of an advanced topic to think about. Just assume there exists an equilibrium for simplicity. I was just noting that there is a difference between an allocation and an equilibrium. An equilibrium is an allocation which satisfies certain conditions. All equilibria are allocations, but not all allocations are equilibria. And I did mention the only necessary condition for the first welfare theorem is preferences for all agents are locally non-satiated. If this is true then "any equilibrium is pareto efficient" is 100% true. – DornerA Mar 25 '16 at 22:59

Well, when considering the minimal conditions necessary for an allocation to be Pareto optimal we must go to the primitives. First, we need the fact that all agents have rational preferences. What this means is that preferences are complete and transitive. Another thing we need is for preferences to be strictly convex.

Let $\succcurlyeq$ be the preference relation "weakly preferred to":

$\textbf{(A) Transitivity:}$ Given allocations $x,y,z \in X$, if $x\succcurlyeq y$ and $y\succcurlyeq z$ then $x\succcurlyeq z$

$\textbf{(B) Completeness:}$ Given allocations $x,y\in X$, either $x\succcurlyeq y$ or $y\succcurlyeq x$ or both $(x\sim y)$

$\textbf{(C) Strict Convexity}:\forall\, x,y \in X$ and $\forall \lambda\in[0,1]$ we have: $$\lambda f(x)+(1-\lambda) f(y)>f(\lambda x+(1-\lambda)y)$$

Also, let $X$ be the set of goods. We need $X$ to be finite.

These four axioms are sufficient for the existence of a competitive equilibrium.

Now, we must build a bare-bones theoretical market. What we need is the following:

$\textbf{(1) Complete Market:}$ Everyone has complete information and there are no transaction costs.

$\textbf{(2) Price Taking:}$ There is one price which everyone pays (and by the above statement, everyone knows this price).

$\textbf{(3) Locally Non-Satiated Preferences:}$ Basically, giving an agent more of any good will always increase their utility.

If we have $(A)-(C)$ and we have a finite number of goods, we will have a competitive equilibrium, and if we have $(1)-(2)$, we have a competitive market. Using these two and $(3)$, we can use the First Fundamental Theorem of Welfare Economics to say that our equilibrium is Pareto optimal.

• It sounds like what you're saying is that the only condition is "rational preferences" as you defined as preferences with transitivity and completeness. Are there really no other conditions needed for a market to tend toward pareto optimality? What about the conditions given in Kitsune's answer? – B T Mar 26 '16 at 2:17
• He gave two that I did not which are also necessary. We do need the market to be complete and we need price taking – DornerA Mar 26 '16 at 2:21
• Ok, well in that case, without those, your answer is incomplete. – B T Mar 26 '16 at 2:23
• Although, now that I think about it, you don't need those conditions because there are models with market power (no price taking) and imperfect information where Pareto efficiency exists – DornerA Mar 26 '16 at 2:41
• I see what you're saying, but I guess to be very specific, the question is "under what conditions would all markets satisfying those conditions tend toward a pareto efficient equilibrium", in which case I think additional criteria do apply. – B T Mar 26 '16 at 3:02

It doesn't really make sense to ask what the "minimum" conditions are for a market equilibrium to be Pareto efficient. By minimum do you mean least restrictive? Do you mean simplest conditions? You have to specify what market we're looking to put conditions on, because otherwise I can make up a really trivial market, give it some conditions, and say that it has a Pareto efficient equilibrium.

For example, I could specify a market with two goods. Person A only likes good 1 and Person B only likes good 2. They start off with 1 unit of their preferred good. Then I say the market does not allow trading. This market is now in equilibrium and is Pareto. But it has no practical interpretation really.

I'm going to interpret your question as what sort of conditions are absolutely necessary for a market equilibrium to be Pareto. In that case, just use the First Welfare Theorem:

• Complete markets: Everyone knows everything about the goods and the market, and there are no transaction costs to trade. Notice how the type of market is specified here.
• Price taking: No monopolies, no barriers to market entry/exit.
• Local non-satiated preferences: For any bundle of goods, there is an arbitrarily similar bundle that is better liked.

In this market, every competitive market is Pareto efficient.

Edit: (If you want to ensure an eqbm in the first place, you do want rational, strictly convex, strict monotone, and continuous prefs like Dorner also mentioned above, so if you want the really basic primitives, there's that. Continuity allows for continuous utility fn rep, which is nice for unique solutions to utility max, among other things)

• I do mean least restrictive, yes. – B T Mar 26 '16 at 2:17
• I don't understand local non-satiation still. If two bundles of goods are arbitrarily similar, wouldn't they also be arbitrarily similarly "liked"? – B T Mar 26 '16 at 2:21
• @denesp Yo it's 3am for me so I don't really give a diggity dang about stability tbh. I guess you'd look at excess demand function and study how they cross the x axis to check for that kinda thing. Or maybe in more complicated cases with time you look at eigenvectors. Something like that. – Kitsune Cavalry Mar 26 '16 at 8:39
• – Giskard Mar 27 '16 at 5:53
• FWT shows thatif there is an eqbm it is PO, but there very well might not be an equilibrium in the first place. To ensure an eqbm, you want some extra conditions. Rationality (complete + trans) is the obvious one, and strict convexity and strict monotonicity are others. I think continuity is one too. – Kitsune Cavalry Mar 27 '16 at 16:30