<p>In the proof of the existence of the Walrasian equilibrium, there is a map defined using the Max map. Why this map is defined in this way?I am reading " Microeconomics; Gravelle, Rees". On page 257 the authors define the mapping in the picture. The reason is that they want to use the Brouwer's Fixed Point Theorem. First, they define a mapping from the set of normalized price vectors to the set of excess demand vectors (Z). Then they have to define another mapping from the set of excess demand vectors to the set of normalized price vectors, and they have defined it as the mapping in the attached image. I wonder how the fixed point of the composition of these two functions is the equilibrium?<a href="https://i.sstatic.net/I9Ot2.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/I9Ot2.jpg" alt="enter image description here"></a></p>

CC BY-SA 4.0
2019-01-17T11:39:41.640Z
Could you please elaborate. What do you mean by "why"? Do you understand the rest of the proof?
Giskard
CC BY-SA 4.0
2019-01-17T11:45:07.047Z
I am reading " Microeconomics; Gravelle, Rees". On page 257 the authors define the mapping in the picture. The reason is that they want to use the Brouwer's Fixed Point Theorem. First, they define a mapping from the set of normalized price vectors to the set of excess demand vectors (Z). Then they have to define another mapping from the set of excess demand vectors to the set of normalized price vectors, and they have defined it as the mapping in the attached image. I wonder how the fixed point of the composition of these two functions is the equilibrium?
Richardlion
CC BY-SA 4.0
2019-01-17T12:02:57.043Z
Thanks! This makes it much clearer. I recommend that you click the edit button and add the info in your comment to the body of your question.
Giskard
<mathematical-economics>
Existence of the general equilibrium
Richardlion
# Existence of the general equilibrium

(Asked by: username_0 on 2019-01-17)

In the proof of the existence of the Walrasian equilibrium, there is a map defined using the Max map. Why this map is defined in this way?I am reading " Microeconomics; Gravelle, Rees". On page 257 the authors define the mapping in the picture. The reason is that they want to use the Brouwer's Fixed Point Theorem. First, they define a mapping from the set of normalized price vectors to the set of excess demand vectors (Z). Then they have to define another mapping from the set of excess demand vectors to the set of normalized price vectors, and they have defined it as the mapping in the attached image. I wonder how the fixed point of the composition of these two functions is the equilibrium?[![enter image description here](https://i.sstatic.net/I9Ot2.jpg)](https://i.sstatic.net/I9Ot2.jpg)

**Tags:** mathematical-economics
> **Comments:**
> **username_1**: Could you please elaborate. What do you mean by "why"? Do you understand the rest of the proof?  
> **username_0**: I am reading " Microeconomics; Gravelle, Rees". On page 257 the authors define the mapping in the picture. The reason is that they want to use the Brouwer's Fixed Point Theorem. First, they define a mapping from the set of normalized price vectors to the set of excess demand vectors (Z). Then they have to define another mapping from the set of excess demand vectors to the set of normalized price vectors, and they have defined it as the mapping in the attached image. I wonder how the fixed point of the composition of these two functions is the equilibrium?  
> **username_1**: Thanks! This makes it much clearer. I recommend that you click the edit button and add the info in your comment to the body of your question.  

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## 0 answers

