<p>The Pareto optimal set should always pass through the origin because it's one of the point that makes it impossible to better off a consumer without harming the other.</p>

CC BY-SA 4.0
2018-12-17T17:24:48.023Z
This doesn't answer OP's question, which asks whether strict monotonicity is sufficient to guarantee the existence of a "generalized contract curve".
Herr K.
CC BY-SA 4.0
2018-12-17T15:12:27.697Z
2018-12-17T15:12:27.697Z
Becky
<p>General equilibrium framework; two individuals, two goods represented in an Edgeworth box.</p>

<p>Is it true that if the preference are both strongly monotonic the Pareto set will go from the origin of an individual's axis to the one of the other's axis? By origin, I mean the points of the box where an individual has everything and the other nothing. The reason would be that the allocation of all goods to one of the single individuals is Pareto optimal. </p>

CC BY-SA 4.0
2018-12-12T18:04:29.750Z
Could you please define what exactly you mean by "go from one origin to the other"? E.g. if all the points in the box are in the Pareto set, does the set go from one origin to the other?
Giskard
CC BY-SA 4.0
2018-12-12T18:52:10.610Z
@denesp edited.
PostDocing
CC BY-SA 4.0
2018-12-12T19:17:02.623Z
I know what the word origin means, please answer my question: "...if all the points in the box are in the Pareto set, does the set go from one origin to the other?"
Giskard
CC BY-SA 4.0
2018-12-12T19:39:06.860Z
It is plain English to me, downvoting b/c you are not getting the question is a bit childish. I mean is the Pareto set supposed to contained both the origins of the axis of the two guys when preferences are strongly mononic?
PostDocing
CC BY-SA 4.0
2018-12-12T23:53:16.677Z
By saying that the Pareto set goes from one origin to the other, you are presupposing that the set can be represented by a curve (e.g. the [contract curve](https://en.wikipedia.org/wiki/Contract_curve)). But a set is a more general concept than a curve. While it's intuitive to describe a curve that connects the two origins as "going from one to the other", it's less intuitive to use the same language on a [(connected) set](https://en.wikipedia.org/wiki/Connected_space) that contains the two origins. This is perhaps why @denesp was having trouble understanding your question.
Herr K.
CC BY-SA 4.0
2018-12-13T01:08:27.647Z
It would help if you could clarify what you mean by "the Pareto set going from one origin to the other". If you mean "the **contract curve** going from one origin to the other", then the answer is no: you'd at least need strict convexity to ensure that the contract curve exists. If you mean "**the Pareto set is i) connected and ii) contains the two origins**", then the answer is probably yes. Strict monotonicity does imply that the two origins are Pareto efficient; however, the connectedness of the set is not immediately obvious to me.
Herr K.
CC BY-SA 4.0
2018-12-13T04:20:59.613Z
Alessandro, I was indeed asking for clarifications because these details matter when you make mathematical claims. Please believe me, I am not trying to be bothersome. Multiple times you got upset with me on this site for the same reason. Also, downvote was not by me.
Giskard
CC BY-SA 4.0
2018-12-13T04:22:27.270Z
It seems to me that if continuity is not assumed then we do not have connectedness.
Giskard
<general-equilibrium><pareto-efficiency>
Pareto set and Edgeworth box
PostDocing
# Pareto set and Edgeworth box

(Asked by: username_0 on 2018-12-12)

General equilibrium framework; two individuals, two goods represented in an Edgeworth box.

Is it true that if the preference are both strongly monotonic the Pareto set will go from the origin of an individual's axis to the one of the other's axis? By origin, I mean the points of the box where an individual has everything and the other nothing. The reason would be that the allocation of all goods to one of the single individuals is Pareto optimal.

**Tags:** general-equilibrium, pareto-efficiency
> **Comments:**
> **username_1**: Could you please define what exactly you mean by "go from one origin to the other"? E.g. if all the points in the box are in the Pareto set, does the set go from one origin to the other?  
> **username_0**: @denesp edited.  
> **username_1**: I know what the word origin means, please answer my question: "...if all the points in the box are in the Pareto set, does the set go from one origin to the other?"  
> **username_0**: It is plain English to me, downvoting b/c you are not getting the question is a bit childish. I mean is the Pareto set supposed to contained both the origins of the axis of the two guys when preferences are strongly mononic?  
> **username_2**: By saying that the Pareto set goes from one origin to the other, you are presupposing that the set can be represented by a curve (e.g. the [contract curve](https://en.wikipedia.org/wiki/Contract_curve)). But a set is a more general concept than a curve. While it's intuitive to describe a curve that connects the two origins as "going from one to the other", it's less intuitive to use the same language on a [(connected) set](https://en.wikipedia.org/wiki/Connected_space) that contains the two origins. This is perhaps why @denesp was having trouble understanding your question.  
> **username_2**: It would help if you could clarify what you mean by "the Pareto set going from one origin to the other". If you mean "the **contract curve** going from one origin to the other", then the answer is no: you'd at least need strict convexity to ensure that the contract curve exists. If you mean "**the Pareto set is i) connected and ii) contains the two origins**", then the answer is probably yes. Strict monotonicity does imply that the two origins are Pareto efficient; however, the connectedness of the set is not immediately obvious to me.  
> **username_1**: Alessandro, I was indeed asking for clarifications because these details matter when you make mathematical claims. Please believe me, I am not trying to be bothersome. Multiple times you got upset with me on this site for the same reason. Also, downvote was not by me.  
> **username_1**: It seems to me that if continuity is not assumed then we do not have connectedness.  

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## 1 answers

### Answer by username_3 on 2018-12-17. Score: 0 | Accepted answer

The Pareto optimal set should always pass through the origin because it's one of the point that makes it impossible to better off a consumer without harming the other.

> **Comments:**
> **username_2**: This doesn't answer OP's question, which asks whether strict monotonicity is sufficient to guarantee the existence of a "generalized contract curve".  

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