<p>Since <span class="math-container">$X_A$</span> denotes the number of goods <span class="math-container">$X$</span> consumed by <span class="math-container">$A$</span>, whereas <span class="math-container">$X_B$</span> denotes the number of goods <span class="math-container">$X$</span> consumed by <span class="math-container">$B$</span>, and the economy has a total of 20 goods, it makes sense that
<span class="math-container">$$
X_A + X_B = 20.
$$</span>
(Consuming less would be feasible but suboptimal.) We can rearrange this to
<span class="math-container">$$
X_A = 20 - X_B.
$$</span>
From the equality of the MRSs you have <span class="math-container">$X_A = X_B$</span>, hence
<span class="math-container">$$
X_A = 20 - X_A.
$$</span></p>

CC BY-SA 4.0
2022-10-08T17:22:36.383Z
2022-10-08T17:22:36.383Z
Giskard
<p>There is a mistake in the answer you posted. When <span class="math-container">$u_i = x_i^2 + y_i$</span> for <span class="math-container">$i\in \{A, B\}$</span>, set of efficient allocations (along the contract curve) does not satisfy MRS<span class="math-container">$_A$</span> = MRS<span class="math-container">$_B$</span>. This is because the utility functions are strictly quasi-convex. For example: An allocation <span class="math-container">$\left((x_A, y_A), (x_B, y_B)\right) = \left((10,10), (10, 10)\right)$</span> is not Pareto efficient because <span class="math-container">$\left((x_A', y_A'), (x_B', y_B')\right) = \left((\sqrt{110},0), (20-\sqrt{110}, 20)\right)$</span> is Pareto superior to it. Efficient allocations will be a subset of the boundaries of the Edgeworth box. As an exercise try and find them.</p>

CC BY-SA 4.0
2022-10-09T13:25:47.997Z
2022-10-09T13:25:47.997Z
Amit
<p>I do not know the step Xa=20-xa
The economy has 20Xa, so what does this step mean?
In my view, replacing Xb should be 40-20Xa</p>
<p><a href="https://i.sstatic.net/b5NUe.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/b5NUe.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-10-08T17:20:55.323Z
Hi! Please avoid unnecessary tags (like finance here). Also provide text that enables future visitors to find the question with the right keywords, making the question useful. Finally, please use MathJax to type in your equations. There is a cheatsheet  for it in the meta section of our site.
Giskard
<microeconomics>
Meaning of equation in pure exchange economy
Gavin
# Meaning of equation in pure exchange economy

(Asked by: username_0 on 2022-10-08)

I do not know the step Xa=20-xa
The economy has 20Xa, so what does this step mean?
In my view, replacing Xb should be 40-20Xa

[![enter image description here](https://i.sstatic.net/b5NUe.png)](https://i.sstatic.net/b5NUe.png)

**Tags:** microeconomics

> **Comments:**
> **username_1**: Hi! Please avoid unnecessary tags (like finance here). Also provide text that enables future visitors to find the question with the right keywords, making the question useful. Finally, please use MathJax to type in your equations. There is a cheatsheet  for it in the meta section of our site.  

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## 2 answers

### Answer by username_1 on 2022-10-08. Score: 1 | Accepted answer

Since $X\_A$ denotes the number of goods $X$ consumed by $A$, whereas $X\_B$ denotes the number of goods $X$ consumed by $B$, and the economy has a total of 20 goods, it makes sense that
$$
X\_A + X\_B = 20.
$$
(Consuming less would be feasible but suboptimal.) We can rearrange this to
$$
X\_A = 20 - X\_B.
$$
From the equality of the MRSs you have $X\_A = X\_B$, hence
$$
X\_A = 20 - X\_A.
$$


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### Answer by username_2 on 2022-10-09. Score: 1

There is a mistake in the answer you posted. When $u\_i = x\_i^2 + y\_i$ for $i\in \{A, B\}$, set of efficient allocations (along the contract curve) does not satisfy MRS$\_A$ = MRS$\_B$. This is because the utility functions are strictly quasi-convex. For example: An allocation $\left((x\_A, y\_A), (x\_B, y\_B)\right) = \left((10,10), (10, 10)\right)$ is not Pareto efficient because $\left((x\_A', y\_A'), (x\_B', y\_B')\right) = \left((\sqrt{110},0), (20-\sqrt{110}, 20)\right)$ is Pareto superior to it. Efficient allocations will be a subset of the boundaries of the Edgeworth box. As an exercise try and find them.


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