<p>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 <em>what market we're looking to put conditions on</em>, because otherwise I can make up a really trivial market, give it some conditions, and say that it has a Pareto efficient equilibrium.</p>

<p>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.</p>

<p>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:</p>

<blockquote>
  <ul>
  <li><strong>Complete markets</strong>: 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.</li>
  <li><strong>Price taking</strong>: No monopolies, no barriers to market entry/exit.</li>
  <li><strong>Local non-satiated preferences</strong>: For any bundle of goods, there is an arbitrarily similar bundle that is better liked.</li>
  </ul>
</blockquote>

<p>In this market, every competitive market is Pareto efficient.</p>

<p><em>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)</em></p>

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2016-03-26T02:17:46.640Z
I do mean least restrictive, yes.
B T
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2016-03-26T02:21:52.867Z
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
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2016-03-26T02:24:38.100Z
Do one of these points cover externalities? Does "no transaction costs" also imply that property rights are fully defined and therefore there are no externalities?
B T
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2016-03-26T02:32:07.263Z
You're taking the word "arbitrary" too literally. I mean that the difference between two bundles being arbitrarily small, but still with one having a clear preference over the other. No transaction costs means Coase Theorem holds, and all externalities will be internalized.
Kitsune Cavalry
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2016-03-26T02:55:10.677Z
Ah I see about transaction costs, gotcha. But for the nonsatiation condition, for a given market actor wouldn't there be one best "bundle" for which there is no better bundle?
B T
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2016-03-26T03:13:11.763Z
And when you say "in this market, ever competitive market is pareto efficient", is "competitive market" an additional constraint or do the 3 quoted conditions define a "competitive market"?
B T
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2016-03-26T08:28:10.907Z
@KitsuneCavalry From the question: "for all markets that fit those conditions to theoretically progress toward a Pareto efficient equilibrium" You have shown that an equilibrium exists, but is it an attractor? (Is it a state towards which a system in inequilibrium will progress?)
Giskard
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2016-03-26T08:36:02.243Z
@BT Local non-satiation just means something like more stuff is better. Obviously not true for small periods of time (1 cookie vs 1000) but in the long run holds true. I also use competitive market and Walrasian eqbm interchangeably (a bit lazily I guess)
Kitsune Cavalry
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2016-03-26T08:39:51.970Z
@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
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2016-03-26T22:56:17.093Z
Ah I see about local nonsatiation, with a little further reading. It seems monotone preferences essentially means "more is better", but local nonsatiation is more general in that it just means that any market actor wants something more/better and that thing exists to motivate them to trade. I'm a bit confused why that would be a requirement, since surely at the pareto optimal states, someone might be at their bliss point.
B T
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2016-03-26T23:00:33.337Z
@denesp I don't understand your contention about equilibrium. If an equilibrium isn't "an attractor" then it isn't going to be an equilibrium.
B T
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2016-03-27T00:40:35.600Z
@BT What if there are multiple equilibria? Which one would the market move to? That depends on stability. You can def. have an equilibrium that isn't stable, so any sudden changes in the market could move you away from the equilibrium.
Kitsune Cavalry
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2016-03-27T00:45:46.650Z
Does equilibrium means something different than it does in physics? Wikipedia doesn't seem to think so. The definition of an equilibrium is a stable point, so an unstable equilibrium is a contradiction. Yes there can be multiple equilibrium, and this makes me realize I was wrong that an equilibrium must be an attractor. But yes, obviously changes in the market environment can change the set of equilibria that exist. That's not the same thing as an equilibrium being unstable.
B T
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2016-03-27T02:54:21.937Z
If you are at equilibrium, you will stay there, but if you don't start there, you might approach a different equilibrium.
Kitsune Cavalry
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2016-03-27T05:53:03.380Z
@BT http://www.merriam-webster.com/dictionary/unstable%20equilibrium
Giskard
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2016-03-27T06:10:00.317Z
Sounds like we agree, we're just using different terms.
B T
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2016-03-27T06:36:51.420Z
@KitsuneCavalry I'm ready to accept one of these answers, and I'm inclined to chose yours since its more understandable to me, but I'm wondering if the answer does in fact have to include the transitivity, completeness, and strict convexity dorner wrote about. Does it?
B T
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2016-03-27T16:30:48.253Z
FWT shows that*if* 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
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2016-03-26T02:08:22.350Z
2016-03-27T16:37:51.687Z
2016-03-27T16:37:51.687Z
Kitsune Cavalry
<p>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.</p>

<p>Let $\succcurlyeq$ be the preference relation "weakly preferred to":</p>

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

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

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

<p>Also, let $X$ be the set of goods. We need $X$ to be finite.</p>

<p>These four axioms are sufficient for the existence of a competitive equilibrium.</p>

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

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

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

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

<p>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. </p>

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2016-03-26T02:17:44.797Z
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
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2016-03-26T02:21:29.130Z
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
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2016-03-26T02:23:07.840Z
Ok, well in that case, without those, your answer is incomplete.
B T
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2016-03-26T02:41:46.273Z
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
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2016-03-26T03:02:32.283Z
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
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2016-03-26T15:23:22.060Z
@BT I have edited my answer to be as rigorous as possible. These are all of the assumptions necessary to achieve Pareto optimality. However, there are Pareto optimal points which are not competitive equilibria, but that's not what your question is asking.
DornerA
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2016-03-26T02:05:49.730Z
2016-03-26T15:21:58.867Z
2016-03-26T15:21:58.867Z
DornerA
<p>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.</p>

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

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2016-03-25T22:25:46.367Z
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
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2016-03-25T22:32:31.200Z
@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
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2016-03-25T22:35:32.180Z
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
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2016-03-25T22:41:28.390Z
@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
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2016-03-25T22:59:37.470Z
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
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2016-03-26T00:36:07.197Z
Oh ok, I see what you mean about allocation. I still don't understand the confusion about my last sentence. What does "locally non-satiated" mean?
B T
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2016-03-26T01:18:13.040Z
I'm just saying that the way that it's stated presumes that the allocation in question is an equilibrium. So in that case, the only assumption we would need is local non-satiation (which is basically a mathematical way of saying "more is better." Any strictly increasing utility function will satisfy this). Therefore as it stands, that is the answer to your question. However if you said "what is necessary for an ALLOCATION to be considered Pareto efficient?" we would be dealing with a different question. I can write this all out (answering both questions) as an answer if you would like
DornerA
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2016-03-26T01:23:04.460Z
An answer would be great. I changed my last sentence to be a little clearer as to what I meant. I'm not asking about a particular allocation, but under what conditions the eventual equilibrium will be pareto efficient.
B T
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2016-03-26T08:26:47.360Z
"Least restrictive" is still poorly defined in this question. There are several sets of conditions that will show a competitive equilibrium exists. How is someone to know which are more restrictive?
Giskard
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2016-03-26T22:38:29.327Z
Its possible that there are multiple equally minimally restrictive sets of conditions. Comparing two sets, if neither set can be simplified into the other set, then I'm going to say they're equally least restrictive. In fact, I'd be very interested if there were multiple sets of conditions that are all minimally restrictive (can't be simplified).
B T
<pareto-efficiency>
In modern economics, what are the least restrictive conditions shown to theoretically achieve Pareto Efficiency?
B T
# In modern economics, what are the least restrictive conditions shown to theoretically achieve Pareto Efficiency?

(Asked by: username_0 on 2016-03-25)

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?

**Tags:** pareto-efficiency
> **Comments:**
> **username_1**: 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?  
> **username_0**: @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?"  
> **username_1**: 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.  
> **username_0**: @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.  
> **username_1**: 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.  
> **username_0**: Oh ok, I see what you mean about allocation. I still don't understand the confusion about my last sentence. What does "locally non-satiated" mean?  
> **username_1**: I'm just saying that the way that it's stated presumes that the allocation in question is an equilibrium. So in that case, the only assumption we would need is local non-satiation (which is basically a mathematical way of saying "more is better." Any strictly increasing utility function will satisfy this). Therefore as it stands, that is the answer to your question. However if you said "what is necessary for an ALLOCATION to be considered Pareto efficient?" we would be dealing with a different question. I can write this all out (answering both questions) as an answer if you would like  
> **username_0**: An answer would be great. I changed my last sentence to be a little clearer as to what I meant. I'm not asking about a particular allocation, but under what conditions the eventual equilibrium will be pareto efficient.  
> **username_2**: "Least restrictive" is still poorly defined in this question. There are several sets of conditions that will show a competitive equilibrium exists. How is someone to know which are more restrictive?  
> **username_0**: Its possible that there are multiple equally minimally restrictive sets of conditions. Comparing two sets, if neither set can be simplified into the other set, then I'm going to say they're equally least restrictive. In fact, I'd be very interested if there were multiple sets of conditions that are all minimally restrictive (can't be simplified).  

---

## 2 answers

### Answer by username_1 on 2016-03-26. Score: 4 | Accepted answer

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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: He gave two that I did not which are also necessary. We do need the market to be complete and we need price taking  
> **username_0**: Ok, well in that case, without those, your answer is incomplete.  
> **username_1**: 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  
> **username_0**: 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.  
> **username_1**: @BT I have edited my answer to be as rigorous as possible. These are all of the assumptions necessary to achieve Pareto optimality. However, there are Pareto optimal points which are not competitive equilibria, but that's not what your question is asking.  

---

### Answer by username_3 on 2016-03-26. Score: 3

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)*

> **Comments:**
> **username_0**: I do mean least restrictive, yes.  
> **username_0**: I don't understand local non-satiation still. If two bundles of goods are arbitrarily similar, wouldn't they also be arbitrarily similarly "liked"?  
> **username_0**: Do one of these points cover externalities? Does "no transaction costs" also imply that property rights are fully defined and therefore there are no externalities?  
> **username_3**: You're taking the word "arbitrary" too literally. I mean that the difference between two bundles being arbitrarily small, but still with one having a clear preference over the other. No transaction costs means Coase Theorem holds, and all externalities will be internalized.  
> **username_0**: Ah I see about transaction costs, gotcha. But for the nonsatiation condition, for a given market actor wouldn't there be one best "bundle" for which there is no better bundle?  
> **username_0**: And when you say "in this market, ever competitive market is pareto efficient", is "competitive market" an additional constraint or do the 3 quoted conditions define a "competitive market"?  
> **username_2**: @KitsuneCavalry From the question: "for all markets that fit those conditions to theoretically progress toward a Pareto efficient equilibrium" You have shown that an equilibrium exists, but is it an attractor? (Is it a state towards which a system in inequilibrium will progress?)  
> **username_3**: @BT Local non-satiation just means something like more stuff is better. Obviously not true for small periods of time (1 cookie vs 1000) but in the long run holds true. I also use competitive market and Walrasian eqbm interchangeably (a bit lazily I guess)  
> **username_3**: @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.  
> **username_0**: Ah I see about local nonsatiation, with a little further reading. It seems monotone preferences essentially means "more is better", but local nonsatiation is more general in that it just means that any market actor wants something more/better and that thing exists to motivate them to trade. I'm a bit confused why that would be a requirement, since surely at the pareto optimal states, someone might be at their bliss point.  
> **username_0**: @denesp I don't understand your contention about equilibrium. If an equilibrium isn't "an attractor" then it isn't going to be an equilibrium.  
> **username_3**: @BT What if there are multiple equilibria? Which one would the market move to? That depends on stability. You can def. have an equilibrium that isn't stable, so any sudden changes in the market could move you away from the equilibrium.  
> **username_0**: Does equilibrium means something different than it does in physics? Wikipedia doesn't seem to think so. The definition of an equilibrium is a stable point, so an unstable equilibrium is a contradiction. Yes there can be multiple equilibrium, and this makes me realize I was wrong that an equilibrium must be an attractor. But yes, obviously changes in the market environment can change the set of equilibria that exist. That's not the same thing as an equilibrium being unstable.  
> **username_3**: If you are at equilibrium, you will stay there, but if you don't start there, you might approach a different equilibrium.  
> **username_2**: @BT http://www.merriam-webster.com/dictionary/unstable%20equilibrium  
> **username_0**: Sounds like we agree, we're just using different terms.  
> **username_0**: @KitsuneCavalry I'm ready to accept one of these answers, and I'm inclined to chose yours since its more understandable to me, but I'm wondering if the answer does in fact have to include the transitivity, completeness, and strict convexity dorner wrote about. Does it?  
> **username_3**: FWT shows that*if* 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.  

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