<p>Set of Pareto efficient allocations (or contract curve) is the set of all feasible allocations satisfying <span class="math-container">$1 \leq x_1=y_1\leq 3$</span>. This is the line segment connecting points A and B in your graph.</p>
<p>Also, given the prices <span class="math-container">$(p_X^*, p_Y^*)=(1,1)$</span>, and endowment <span class="math-container">$((2,2),(2,2))$</span>, the budget constraint of the consumers is <span class="math-container">$p_X^*x_i+p_Y^*y_i\leq 2+2=4$</span>, and therefore, consumers will demand their bliss points i.e. both will demand the bundle <span class="math-container">$(1,1)$</span> and <span class="math-container">$((2,2),(2,2))$</span> is not a competitive equilibrium allocation.</p>

CC BY-SA 4.0
2023-04-10T06:23:16.923Z
Does the fact that the aggregate of consumers' demands doesn't equal the total endowment here imply this problem given the specific endowments, has no Walrasian Equilibrium?
Nicolas Torres
CC BY-SA 4.0
2023-04-10T06:31:16.920Z
Yes, this video has the definition of competitive (or Walrasian) equilibrium that I have used to answer this question: https://youtu.be/7U3LpSx2KCE
Amit
CC BY-SA 4.0
2023-04-10T06:36:38.353Z
I just checked and it was clear for me why the points where $x_1 < 1$ and $x_1 >  3$ are not Pareto efficient. Is the systematic way to find the CC for non-monotonic (differentiable and non-linear) utilities is to always do the usual $MRS$ procedure and then check by cases?
Nicolas Torres
CC BY-SA 4.0
2023-04-10T06:37:21.830Z
Thank you! I will check the video now and see if I got it.
Nicolas Torres
CC BY-SA 4.0
2023-04-10T06:06:05.053Z
2023-04-10T06:13:32.440Z
2023-04-10T06:13:32.440Z
Amit
<p>I just say Amit's comment on this question: <a href="https://economics.stackexchange.com/questions/15416/the-second-welfare-theorem-without-monotonicity">The second welfare theorem without monotonicity</a> so I got curious and tried to find both the contract curve for that particular problem, and the Walrasian equilibria when both agents have the endowments <span class="math-container">$(w_x,w_y)=(2,2)$</span>.</p>
<p>The problem regarding Amit's comment is</p>
<p><span class="math-container">$u_i(x_i,y_i) = - (x_i-1)^2 - (y_i-1)^2$</span></p>
<p><span class="math-container">$w_x = 4$</span></p>
<p><span class="math-container">$w_y = 4$</span></p>
<p>Since the utility function (in this case the same one for both agents) is differentiable everywhere and non linear, using <span class="math-container">$MRS_1 = MRS_2$</span> and the endowment constraints gives me the contract curve. Doing this procedure I got that the contract curve is <span class="math-container">$y_1 = x_1$</span>.</p>
<p>Clearly the allocation <span class="math-container">$((x_1,y_1),(x_2,y_2)) = ((2,2),(2,2))$</span> belongs to the contract curve.</p>
<p>However, Amit said in his comment that the alocation <span class="math-container">$((x_1,y_1),(x_2,y_2)) = ((2,2),(2,2))$</span> is not a competitive equilibrium.</p>
<p>My curiosity led me to do the experiment of trying to find the Walrasian Equilibrium for the problem above, given the endowments as the allocation <span class="math-container">$((w_{x1},w_{y1}),(w_{x2},w_{y2})) = ((2,2),(2,2))$</span>.</p>
<p>I started computing the consumers' demands as usual.</p>
<p>Since both consumers have the same utility function and endowments, solving this optimization problem would yield me the demand functions for both consumers:</p>
<p><span class="math-container">$\max -(x-1)^2-(y-1)^2$</span></p>
<p>s.t. <span class="math-container">$p_x x + p_y y = 2 p_x + 2 p_y$</span></p>
<p>Using the <span class="math-container">$MRS = \frac{p_x}{p_y}$</span> shortcut,</p>
<p><span class="math-container">$\frac{\frac{\partial u_i}{\partial x}}{\frac{\partial u_i}{\partial y}} = \frac{p_x}{p_y} \implies \frac{-2 (x-1)}{-2(y-1)} = \frac{p_x}{p_y} \implies \frac{x-1}{y-1} = \frac{p_x}{p_y} \implies x-1 = \frac{p_x}{p_y}(y-1) \implies x = \frac{p_x}{p_y} (y-1) + 1$</span></p>
<p><span class="math-container">$\implies x = \frac{p_x}{p_y} y - \frac{p_x}{p_y} + 1$</span></p>
<p>Plugging this expression for <span class="math-container">$x$</span> into the budget constraint,</p>
<p><span class="math-container">$p_x (\frac{p_x}{p_y} y - \frac{p_x}{p_y} + 1) + p_y y = 2 p_x + 2 p_y \implies \frac{{p_x}^2}{p_y} y - \frac{{p_x}^2}{p_y} + p_x + p_y y = 2 p_x + 2 p_y$</span></p>
<p><span class="math-container">$\implies (\frac{{p_x}^2}{p_y} + p_y) y = 2 p_x + 2 p_y + \frac{{p_x}^2}{p_y} - p_x \implies y = \frac{2 p_x + 2 p_y + \frac{{p_x}^2}{p_y} - p_x}{\frac{{p_x}^2}{p_y} + p_y}$</span></p>
<p>For simplicity let's take as numeraire <span class="math-container">$p_y = 1$</span></p>
<p><span class="math-container">$\implies y = \frac{2 p_x + 2 + {p_x}^2 - p_x}{{p_x}^2 + 1}$</span></p>
<p>Simplifying to get the demands for good <span class="math-container">$y$</span></p>
<p><span class="math-container">${y_1}^\star = {y_2}^\star = \frac{{p_x}^2 + p_x + 2}{{p_x}^2 + 1}$</span></p>
<p>The previous expression for <span class="math-container">$x$</span> after taking into account <span class="math-container">$p_y = 1$</span> is the numeraire becomes</p>
<p><span class="math-container">$x = p_x y - p_x + 1$</span></p>
<p>Substituting the demand for <span class="math-container">$y$</span> into this expression</p>
<p><span class="math-container">$\implies x = p_x (\frac{{p_x}^2 + p_x + 2}{{p_x}^2 + 1} - 1) + 1 \implies x = p_x \cdot \frac{{p_x}^2 + p_x + 2 - {p_x}^2 - 1}{{p_x}^2 + 1} + 1 \implies x = p_x \cdot \frac{p_x + 1}{{p_x}^2 + 1} + 1 \implies x = \frac{{p_x}^2 + p_x + {p_x}^2 + 1}{{p_x}^2 + 1} \implies {x_1}^\star = {x_2}^\star = \frac{2 {p_x}^2 + p_x + 1}{{p_x}^2 + 1}$</span></p>
<p>The equilibrium condition on good <span class="math-container">$y$</span> is</p>
<p><span class="math-container">${y_1}^\star + {y_2}^\star = 4 \implies 2 \cdot \frac{{p_x}^2 + p_x + 2}{{p_x}^2 + 1} = 4 \implies \frac{{p_x}^2 + p_x + 2}{{p_x}^2 + 1} = 2 \implies {p_x}^2 + p_x + 2 = 2 {p_x}^2 + 2 \implies p_x = {p_x}^2$</span></p>
<p><span class="math-container">$\implies {p_x}^2 - p_x = 0 \implies p_x (p_x - 1) = 0$</span></p>
<p>From here we get <span class="math-container">$p_x = 1$</span> or <span class="math-container">$p_x = 0$</span>.</p>
<p>Substituting <span class="math-container">$p_x = 0$</span> into the equilibrium condition for good <span class="math-container">$x$</span> yields a contradiction, while <span class="math-container">$p_x = 1$</span> works.</p>
<p>Therefore, <span class="math-container">${p_x}^\star = 1$</span> is the only equilibrium relative price.</p>
<p>Substituting <span class="math-container">${p_x}^\star = 1$</span> into each demand function yields the allocation <span class="math-container">$((x_1,y_1),(x_2,y_2)) = ((2,2),(2,2))$</span></p>
<p>This would imply that <span class="math-container">$((x_1,y_1),(x_2,y_2)) = ((2,2),(2,2))$</span> is the equilibrium allocation supported by the relative prices <span class="math-container">$\frac{{p_x}^\star}{{p_y}^\star} = 1$</span>, contradicting Amit's comment.</p>
<p>Is it wrong to apply any of the usual methods for non monotonic utility functions?</p>
<p>I plotted the situation on GeoGebra, which makes it seem like my results are correct:</p>
<p><a href="https://i.sstatic.net/pMoQD.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/pMoQD.jpg" alt="enter image description here" /></a></p>
<p>Geogebra's axes: Agent A's axes</p>
<p>Orange axes: Agent B's axes</p>
<p>Purple line: Contract curve</p>
<p>Blue points: Bliss point for each agent, as the labels say</p>
<p>Red circle: A's indifference curve</p>
<p>Black circle: B's indifference curve</p>
<p>Green line: Price line <span class="math-container">$\frac{{p_x}^\star}{{p_y}^\star} = 1$</span>, passing through the allocation <span class="math-container">$((x_1,y_1),(x_2,y_2)) = ((2,2),(2,2))$</span>, which is indeed tangent to both agents' indifference curves.</p>

<general-equilibrium><competitive-equilibrium><welfare-economics><pure-exchange-economy><edgeworth-box>
How to find the Walrasian equilibrium for non monotonic utility functions?
Nicolas Torres
# How to find the Walrasian equilibrium for non monotonic utility functions?

(Asked by: username_0 on 2023-04-10)

I just say username_1's comment on this question: [The second welfare theorem without monotonicity](https://economics.stackexchange.com/questions/15416/the-second-welfare-theorem-without-monotonicity) so I got curious and tried to find both the contract curve for that particular problem, and the Walrasian equilibria when both agents have the endowments $(w\_x,w\_y)=(2,2)$.

The problem regarding username_1's comment is

$u\_i(x\_i,y\_i) = - (x\_i-1)^2 - (y\_i-1)^2$

$w\_x = 4$

$w\_y = 4$

Since the utility function (in this case the same one for both agents) is differentiable everywhere and non linear, using $MRS\_1 = MRS\_2$ and the endowment constraints gives me the contract curve. Doing this procedure I got that the contract curve is $y\_1 = x\_1$.

Clearly the allocation $((x\_1,y\_1),(x\_2,y\_2)) = ((2,2),(2,2))$ belongs to the contract curve.

However, username_1 said in his comment that the alocation $((x\_1,y\_1),(x\_2,y\_2)) = ((2,2),(2,2))$ is not a competitive equilibrium.

My curiosity led me to do the experiment of trying to find the Walrasian Equilibrium for the problem above, given the endowments as the allocation $((w\_{x1},w\_{y1}),(w\_{x2},w\_{y2})) = ((2,2),(2,2))$.

I started computing the consumers' demands as usual.

Since both consumers have the same utility function and endowments, solving this optimization problem would yield me the demand functions for both consumers:

$\max -(x-1)^2-(y-1)^2$

s.t. $p\_x x + p\_y y = 2 p\_x + 2 p\_y$

Using the $MRS = \frac{p\_x}{p\_y}$ shortcut,

$\frac{\frac{\partial u\_i}{\partial x}}{\frac{\partial u\_i}{\partial y}} = \frac{p\_x}{p\_y} \implies \frac{-2 (x-1)}{-2(y-1)} = \frac{p\_x}{p\_y} \implies \frac{x-1}{y-1} = \frac{p\_x}{p\_y} \implies x-1 = \frac{p\_x}{p\_y}(y-1) \implies x = \frac{p\_x}{p\_y} (y-1) + 1$

$\implies x = \frac{p\_x}{p\_y} y - \frac{p\_x}{p\_y} + 1$

Plugging this expression for $x$ into the budget constraint,

$p\_x (\frac{p\_x}{p\_y} y - \frac{p\_x}{p\_y} + 1) + p\_y y = 2 p\_x + 2 p\_y \implies \frac{{p\_x}^2}{p\_y} y - \frac{{p\_x}^2}{p\_y} + p\_x + p\_y y = 2 p\_x + 2 p\_y$

$\implies (\frac{{p\_x}^2}{p\_y} + p\_y) y = 2 p\_x + 2 p\_y + \frac{{p\_x}^2}{p\_y} - p\_x \implies y = \frac{2 p\_x + 2 p\_y + \frac{{p\_x}^2}{p\_y} - p\_x}{\frac{{p\_x}^2}{p\_y} + p\_y}$

For simplicity let's take as numeraire $p\_y = 1$

$\implies y = \frac{2 p\_x + 2 + {p\_x}^2 - p\_x}{{p\_x}^2 + 1}$

Simplifying to get the demands for good $y$

${y\_1}^\star = {y\_2}^\star = \frac{{p\_x}^2 + p\_x + 2}{{p\_x}^2 + 1}$

The previous expression for $x$ after taking into account $p\_y = 1$ is the numeraire becomes

$x = p\_x y - p\_x + 1$

Substituting the demand for $y$ into this expression

$\implies x = p\_x (\frac{{p\_x}^2 + p\_x + 2}{{p\_x}^2 + 1} - 1) + 1 \implies x = p\_x \cdot \frac{{p\_x}^2 + p\_x + 2 - {p\_x}^2 - 1}{{p\_x}^2 + 1} + 1 \implies x = p\_x \cdot \frac{p\_x + 1}{{p\_x}^2 + 1} + 1 \implies x = \frac{{p\_x}^2 + p\_x + {p\_x}^2 + 1}{{p\_x}^2 + 1} \implies {x\_1}^\star = {x\_2}^\star = \frac{2 {p\_x}^2 + p\_x + 1}{{p\_x}^2 + 1}$

The equilibrium condition on good $y$ is

${y\_1}^\star + {y\_2}^\star = 4 \implies 2 \cdot \frac{{p\_x}^2 + p\_x + 2}{{p\_x}^2 + 1} = 4 \implies \frac{{p\_x}^2 + p\_x + 2}{{p\_x}^2 + 1} = 2 \implies {p\_x}^2 + p\_x + 2 = 2 {p\_x}^2 + 2 \implies p\_x = {p\_x}^2$

$\implies {p\_x}^2 - p\_x = 0 \implies p\_x (p\_x - 1) = 0$

From here we get $p\_x = 1$ or $p\_x = 0$.

Substituting $p\_x = 0$ into the equilibrium condition for good $x$ yields a contradiction, while $p\_x = 1$ works.

Therefore, ${p\_x}^\star = 1$ is the only equilibrium relative price.

Substituting ${p\_x}^\star = 1$ into each demand function yields the allocation $((x\_1,y\_1),(x\_2,y\_2)) = ((2,2),(2,2))$

This would imply that $((x\_1,y\_1),(x\_2,y\_2)) = ((2,2),(2,2))$ is the equilibrium allocation supported by the relative prices $\frac{{p\_x}^\star}{{p\_y}^\star} = 1$, contradicting username_1's comment.

Is it wrong to apply any of the usual methods for non monotonic utility functions?

I plotted the situation on GeoGebra, which makes it seem like my results are correct:

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

Geogebra's axes: Agent A's axes

Orange axes: Agent B's axes

Purple line: Contract curve

Blue points: Bliss point for each agent, as the labels say

Red circle: A's indifference curve

Black circle: B's indifference curve

Green line: Price line $\frac{{p\_x}^\star}{{p\_y}^\star} = 1$, passing through the allocation $((x\_1,y\_1),(x\_2,y\_2)) = ((2,2),(2,2))$, which is indeed tangent to both agents' indifference curves.

**Tags:** general-equilibrium, competitive-equilibrium, welfare-economics, pure-exchange-economy, edgeworth-box


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## 1 answers

### Answer by username_1 on 2023-04-10. Score: 2 | Accepted answer

Set of Pareto efficient allocations (or contract curve) is the set of all feasible allocations satisfying $1 \leq x\_1=y\_1\leq 3$. This is the line segment connecting points A and B in your graph.

Also, given the prices $(p\_X^\*, p\_Y^\*)=(1,1)$, and endowment $((2,2),(2,2))$, the budget constraint of the consumers is $p\_X^\*x\_i+p\_Y^\*y\_i\leq 2+2=4$, and therefore, consumers will demand their bliss points i.e. both will demand the bundle $(1,1)$ and $((2,2),(2,2))$ is not a competitive equilibrium allocation.

> **Comments:**
> **username_0**: Does the fact that the aggregate of consumers' demands doesn't equal the total endowment here imply this problem given the specific endowments, has no Walrasian Equilibrium?  
> **username_1**: Yes, this video has the definition of competitive (or Walrasian) equilibrium that I have used to answer this question: https://youtu.be/7U3LpSx2KCE  
> **username_0**: I just checked and it was clear for me why the points where $x_1 < 1$ and $x_1 >  3$ are not Pareto efficient. Is the systematic way to find the CC for non-monotonic (differentiable and non-linear) utilities is to always do the usual $MRS$ procedure and then check by cases?  
> **username_0**: Thank you! I will check the video now and see if I got it.  

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