<p>I rewrite the problem of maximization you wrote (I omit the endowments):</p>
<p><span class="math-container">$\max x_A + y_A \;\;\qquad (1)$</span></p>
<p>subject to</p>
<p><span class="math-container">$s x_B + y_B = \overline{U}\qquad (2)$</span>.</p>
<p>This problem can be seen as a <strong>problem of linear programming,</strong> and we can remember that in such problems the optimal points are, in general,  <strong>in the corners</strong> of the feasible area. After all, this is the usual solution when we have linear utility functions.</p>
<p>In our problem the corners of the constrained area are the intersection of the constraint <span class="math-container">$(2)$</span> with the coordinate axes. This already suggests that the contract curve should be on the coordinate axes.</p>
<p>I represent graphically, below, <strong>a solution through the usual Edgeworth box</strong>.</p>
<p>I show separately (to avoid visual confusion) the optimization problem from the point of view of consumer <span class="math-container">$A$</span>, as stated in <span class="math-container">$(1)-(2)$</span>, in the first picture, and the symmetric problem for consumer <span class="math-container">$B$</span> in the second picture (even if this second maximization  is not strictly necessary).</p>
<p><a href="https://i.sstatic.net/8t12M.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8t12M.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$$Fig. \;1 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$</span>
<span class="math-container">$\;$</span></p>
<p><a href="https://i.sstatic.net/MRp2y.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/MRp2y.png" alt="enter image description here" /></a></p>
<p><span class="math-container">$$Fig. \;2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$</span>
<span class="math-container">$\;$</span></p>
<p>Consider a given indifference curve of consumer <span class="math-container">$B$</span>, as <span class="math-container">$u_B= \overline U_{B1}$</span>, ending at a point <span class="math-container">$A$</span> on the <span class="math-container">$x$</span> axis <span class="math-container">$0_AC$</span>.</p>
<p>As it should be clear from  <em>Figure <span class="math-container">$1$</span></em> , the problem <span class="math-container">$(1)-(2)$</span> has its optimal solution at point <span class="math-container">$A$</span>, on the <span class="math-container">$x$</span> axis, given a level of utility of consumer <span class="math-container">$B$</span>, <span class="math-container">$u_B$</span>, equal to <span class="math-container">$\overline U_{B1}$</span>.</p>
<p>Changing the given level of <span class="math-container">$u_B$</span>, so that the blue indifference curve which is the constraint of the problem shifts,  the optimal point will go along the <span class="math-container">$x$</span> axis <span class="math-container">$0_AC$</span>, covering the whole line as the blue indifference curve shifts continuously.</p>
<p>That is, the optimal points of consumer <span class="math-container">$A$</span>, as <span class="math-container">$u_B$</span> changes assuming all possible values, coincide with the <span class="math-container">$x$</span> axis <span class="math-container">$0_AC$</span>.</p>
<p>Analogously, if we have a given indifference curve of consumer <span class="math-container">$B$</span> ending at a point <span class="math-container">$H$</span> on the <span class="math-container">$y$</span> axis <span class="math-container">$C0_B$</span>, the optimal point is <span class="math-container">$H$</span>, and shifting the given indifference curve we obtain the whole <span class="math-container">$C0_B$</span> axis.</p>
<p>We can, therefore, conclude that <strong>the contract line is the set composed of the <span class="math-container">$x$</span> axis <span class="math-container">$0_AC$</span> and the <span class="math-container">$y$</span> axis <span class="math-container">$C0_B$</span>.</strong>, the violet set in <em>Figure 1</em>.</p>
<p><span class="math-container">$\;$</span></p>
<p><strong>If we want to verify that this is actually the contract line</strong><span class="math-container">$^1$</span>, we can re-formulate the problem from the point of view of consumer <span class="math-container">$B$</span>, maximizing their utility for a given level of utility of consumer <span class="math-container">$A$</span>.</p>
<p>This is represented in <em>Figure <span class="math-container">$2$</span></em>, where the green lines are the constraints, given by fixed indifference curves of consumer <span class="math-container">$A$</span>. Following the same reasoning as for consumer <span class="math-container">$A$</span>, we see that the optimal point for consumer <span class="math-container">$B$</span> is on the <span class="math-container">$x$</span> axis at point <span class="math-container">$B$</span>, and actually it coincides with point <span class="math-container">$A$</span> of <em>Figure <span class="math-container">$1$</span></em>.</p>
<p>Changing the given level of utility of consumer <span class="math-container">$A$</span> we obtain again the <span class="math-container">$x$</span> axis <span class="math-container">$0_AC$</span>.</p>
<p>And analogously, taking as constraint indifference curves as <span class="math-container">$u_A=\overline U_{A2}$</span> we obtain the <span class="math-container">$y$</span> axis <span class="math-container">$C0_B$</span>.</p>
<hr />
<p><span class="math-container">$^1$</span> <sup> The first optimization is sufficient, but let's verify we are not mistaken. </sup></p>

CC BY-SA 4.0
2023-03-15T13:35:23.330Z
Thank you, once again I appreciate such a complete answer of yours! I just asked for help with the Walrasian Equilibria for the same functions, with s = 2 here: https://economics.stackexchange.com/questions/54741/finding-walrasian-equilibria-when-walrasian-demands-are-not-unique
Nicolas Torres
CC BY-SA 4.0
2023-03-15T19:57:50.607Z
You are welcome, you are very kind.
BakerStreet
CC BY-SA 4.0
2023-03-13T09:12:55.147Z
2023-03-26T12:26:43.927Z
2023-03-26T12:26:43.927Z
BakerStreet
<p>I'm trying to solve the following excercise:</p>
<p>Find the contract curve for an exchange economy where agents' (<span class="math-container">$A$</span> and <span class="math-container">$B$</span>) preferences and endowments are given by:</p>
<p><span class="math-container">$u_A = x_A + y_A$</span></p>
<p><span class="math-container">$u_B = s x_A + y_A$</span></p>
<p><span class="math-container">$(\omega_{1A},\omega_{2A}) = (\alpha,\beta)$</span></p>
<p><span class="math-container">$(\omega_{1B},\omega_{2B}) = (\beta,\alpha)$</span></p>
<p>Here the indifference curves for <span class="math-container">$A$</span> are lines with slope <span class="math-container">$-1$</span>, while the indifference curves for <span class="math-container">$B$</span> are lines with slope <span class="math-container">$-s$</span>, where <span class="math-container">$s$</span> is a parameter.</p>
<p>The contract curve can be found by solving</p>
<p><span class="math-container">$\max x_A + y_A$</span></p>
<p>subject to</p>
<p><span class="math-container">$s x_B + y_B = \overline{U}$</span></p>
<p><span class="math-container">$x_A + x_B = \alpha + \beta$</span></p>
<p><span class="math-container">$y_A + y_B = \alpha + \beta$</span></p>
<p>What I have tried is the usual <span class="math-container">$\frac{MU_{A,x}}{p_x} &gt; \frac{MU_{A,y}}{p_y}$</span> approach used in partial equilibrium analysis.</p>
<p>Let's set <span class="math-container">$p_x = 1$</span>.</p>
<p>Since <span class="math-container">$MU_{A,x} = MU_{A,y} = 1$</span>, the statement above corresponds to <span class="math-container">$1 &gt; \frac{1}{p_y}$</span>, equivalently, <span class="math-container">$p_y &gt; 1$</span>.</p>
<p>This implies that if <span class="math-container">$p_y &gt; 1$</span>, <span class="math-container">$A$</span> would buy as much <span class="math-container">$x$</span> as they can get by selling <span class="math-container">$y$</span>.</p>
<p>So we would get</p>
<ul>
<li><span class="math-container">$p_y &gt; 1$</span></li>
</ul>
<p><span class="math-container">$x_A = \alpha + \beta$</span></p>
<p><span class="math-container">$y_A = \beta (1 - \frac{1}{p_y})$</span></p>
<p><span class="math-container">$x_B = 0$</span></p>
<p><span class="math-container">$y_B = \alpha + \frac{\beta}{p_y}$</span></p>
<p>Conversely, if <span class="math-container">$p_y &lt; 1$</span>, <span class="math-container">$A$</span> would buy as much <span class="math-container">$y$</span> as they can get by selling <span class="math-container">$x$</span>.</p>
<p>So we would get</p>
<ul>
<li><span class="math-container">$p_y &lt; 1$</span></li>
</ul>
<p><span class="math-container">$x_A = 0$</span></p>
<p><span class="math-container">$y_A = \beta + \frac{\alpha}{p_y}$</span></p>
<p><span class="math-container">$x_B = \alpha + \beta$</span></p>
<p><span class="math-container">$y_B = \alpha (1 - \frac{1}{p_y})$</span></p>
<p>Now, if <span class="math-container">$p_y = 1$</span>, <span class="math-container">$A$</span> would be indifferent between buying or selling any amount of <span class="math-container">$x$</span> for the same amount of <span class="math-container">$y$</span>.</p>
<p>So we would get</p>
<p><span class="math-container">$x_A \in [0,\alpha + \beta]$</span></p>
<p><span class="math-container">$y_A = \alpha + \beta - x_A$</span></p>
<p><span class="math-container">$x_B = \alpha + \beta - x_A$</span></p>
<p><span class="math-container">$y_B = x_A$</span></p>
<p>We have that</p>
<p><span class="math-container">$y_B = x_A \iff \alpha + \beta - x_A = x_A \iff \alpha + \beta = 2 x_A \iff \frac{\alpha + \beta}{2} = x_A$</span></p>
<p>So the solution would be</p>
<ul>
<li><span class="math-container">$p_y = 1$</span></li>
</ul>
<p><span class="math-container">$((x_A,y_A),(x_B,y_B)) = ((\frac{\alpha + \beta}{2},\frac{\alpha + \beta}{2}),(\frac{\alpha + \beta}{2},\frac{\alpha + \beta}{2}))$</span></p>
<p>which is the midpoint of the Edgeworth box.</p>
<p>I then drew an Edgeworth box containing indifference curves for <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, for some <span class="math-container">$s &gt; 1$</span>, then plotted the segments of the 'contract curve' parametrized by <span class="math-container">$p_y$</span>, this is what I got:</p>
<p><a href="https://i.sstatic.net/IOzN2.png" rel="noreferrer"><img src="https://i.sstatic.net/IOzN2.png" alt="enter image description here" /></a></p>
<p>Red pen: <span class="math-container">$A$</span>'s indifference curves</p>
<p>Black pen: <span class="math-container">$B$</span>'s indifference curves</p>
<p>Blue pen: Exchange Area</p>
<p>Pencil: 'Contract curve' = The <span class="math-container">$y_A$</span> axis without the endpoints <span class="math-container">$\cup$</span> The segment of the <span class="math-container">$y_B$</span> axis where <span class="math-container">$\alpha &lt; y_B &lt; \alpha + \beta$</span> <span class="math-container">$\cup$</span> The midpoint of the Edgeworth box.</p>
<p>What I found weird is in General Equilibrium we don't even know the relative prices of the goods, and my method uses the relative prices.</p>
<p>Graphically, only the <span class="math-container">$y_A$</span> axis seems to be part of the contract curve, as in the rest of the 'contract curve', moving upwards along <span class="math-container">$B$</span>'s indifference curve would give <span class="math-container">$A$</span> a higher utility while keeping <span class="math-container">$B$</span>'s constant (a Pareto improvement).</p>
<p>Moreover, the endpoints of the <span class="math-container">$y_A$</span> axis seem to be part of the contract curve, as in the upper endpoint, the only possible movements along <span class="math-container">$B$</span>’s indifference curve are downwards, taking away utility from <span class="math-container">$A$</span>; while in the lower endpoint, there is nowhere even to move along either indifference curve (<span class="math-container">$A$</span> and <span class="math-container">$B$</span>’s indifference curves would consist only of that point), hence no room for Pareto improvement.</p>
<p>I also noted if I were to take <span class="math-container">$s &lt; 1$</span>, it is now the <span class="math-container">$y_B$</span> axis that seems to be part of the contract curve and none of the <span class="math-container">$y_A$</span> axis.</p>
<p>How do we actually find the contract curve?</p>

CC BY-SA 4.0
2023-03-13T07:11:11.913Z
This document might be of some help: https://piazza.com/redirect/s3?bucket=uploads&prefix=paste%2Fi5t4zjz64oq3sl%2F7a4fe44e43b0a14038982f79eb5aeaa7256bc539bc8729fbee61e11e4477f70b%2FGeneral_Equilibrium.pdf
Amit
<general-equilibrium><pareto-efficiency><pure-exchange-economy><edgeworth-box><perfect-substitutes>
How to find the contract curve when both agents have linear utilities?
Nicolas Torres
# How to find the contract curve when both agents have linear utilities?

(Asked by: username_0 on 2023-03-12)

I'm trying to solve the following excercise:

Find the contract curve for an exchange economy where agents' ($A$ and $B$) preferences and endowments are given by:

$u\_A = x\_A + y\_A$

$u\_B = s x\_A + y\_A$

$(\omega\_{1A},\omega\_{2A}) = (\alpha,\beta)$

$(\omega\_{1B},\omega\_{2B}) = (\beta,\alpha)$

Here the indifference curves for $A$ are lines with slope $-1$, while the indifference curves for $B$ are lines with slope $-s$, where $s$ is a parameter.

The contract curve can be found by solving

$\max x\_A + y\_A$

subject to

$s x\_B + y\_B = \overline{U}$

$x\_A + x\_B = \alpha + \beta$

$y\_A + y\_B = \alpha + \beta$

What I have tried is the usual $\frac{MU\_{A,x}}{p\_x} > \frac{MU\_{A,y}}{p\_y}$ approach used in partial equilibrium analysis.

Let's set $p\_x = 1$.

Since $MU\_{A,x} = MU\_{A,y} = 1$, the statement above corresponds to $1 > \frac{1}{p\_y}$, equivalently, $p\_y > 1$.

This implies that if $p\_y > 1$, $A$ would buy as much $x$ as they can get by selling $y$.

So we would get

* $p\_y > 1$

$x\_A = \alpha + \beta$

$y\_A = \beta (1 - \frac{1}{p\_y})$

$x\_B = 0$

$y\_B = \alpha + \frac{\beta}{p\_y}$

Conversely, if $p\_y < 1$, $A$ would buy as much $y$ as they can get by selling $x$.

So we would get

* $p\_y < 1$

$x\_A = 0$

$y\_A = \beta + \frac{\alpha}{p\_y}$

$x\_B = \alpha + \beta$

$y\_B = \alpha (1 - \frac{1}{p\_y})$

Now, if $p\_y = 1$, $A$ would be indifferent between buying or selling any amount of $x$ for the same amount of $y$.

So we would get

$x\_A \in [0,\alpha + \beta]$

$y\_A = \alpha + \beta - x\_A$

$x\_B = \alpha + \beta - x\_A$

$y\_B = x\_A$

We have that

$y\_B = x\_A \iff \alpha + \beta - x\_A = x\_A \iff \alpha + \beta = 2 x\_A \iff \frac{\alpha + \beta}{2} = x\_A$

So the solution would be

* $p\_y = 1$

$((x\_A,y\_A),(x\_B,y\_B)) = ((\frac{\alpha + \beta}{2},\frac{\alpha + \beta}{2}),(\frac{\alpha + \beta}{2},\frac{\alpha + \beta}{2}))$

which is the midpoint of the Edgeworth box.

I then drew an Edgeworth box containing indifference curves for $A$ and $B$, for some $s > 1$, then plotted the segments of the 'contract curve' parametrized by $p\_y$, this is what I got:

[![enter image description here](https://i.sstatic.net/IOzN2.png)](https://i.sstatic.net/IOzN2.png)

Red pen: $A$'s indifference curves

Black pen: $B$'s indifference curves

Blue pen: Exchange Area

Pencil: 'Contract curve' = The $y\_A$ axis without the endpoints $\cup$ The segment of the $y\_B$ axis where $\alpha < y\_B < \alpha + \beta$ $\cup$ The midpoint of the Edgeworth box.

What I found weird is in General Equilibrium we don't even know the relative prices of the goods, and my method uses the relative prices.

Graphically, only the $y\_A$ axis seems to be part of the contract curve, as in the rest of the 'contract curve', moving upwards along $B$'s indifference curve would give $A$ a higher utility while keeping $B$'s constant (a Pareto improvement).

Moreover, the endpoints of the $y\_A$ axis seem to be part of the contract curve, as in the upper endpoint, the only possible movements along $B$’s indifference curve are downwards, taking away utility from $A$; while in the lower endpoint, there is nowhere even to move along either indifference curve ($A$ and $B$’s indifference curves would consist only of that point), hence no room for Pareto improvement.

I also noted if I were to take $s < 1$, it is now the $y\_B$ axis that seems to be part of the contract curve and none of the $y\_A$ axis.

How do we actually find the contract curve?

**Tags:** general-equilibrium, pareto-efficiency, pure-exchange-economy, edgeworth-box, perfect-substitutes

> **Comments:**
> **username_1**: This document might be of some help: https://piazza.com/redirect/s3?bucket=uploads&prefix=paste%2Fi5t4zjz64oq3sl%2F7a4fe44e43b0a14038982f79eb5aeaa7256bc539bc8729fbee61e11e4477f70b%2FGeneral_Equilibrium.pdf  

---

## 1 answers

### Answer by username_2 on 2023-03-13. Score: 5 | Accepted answer

I rewrite the problem of maximization you wrote (I omit the endowments):

$\max x\_A + y\_A \;\;\qquad (1)$

subject to

$s x\_B + y\_B = \overline{U}\qquad (2)$.

This problem can be seen as a **problem of linear programming,** and we can remember that in such problems the optimal points are, in general, **in the corners** of the feasible area. After all, this is the usual solution when we have linear utility functions.

In our problem the corners of the constrained area are the intersection of the constraint $(2)$ with the coordinate axes. This already suggests that the contract curve should be on the coordinate axes.

I represent graphically, below, **a solution through the usual Edgeworth box**.

I show separately (to avoid visual confusion) the optimization problem from the point of view of consumer $A$, as stated in $(1)-(2)$, in the first picture, and the symmetric problem for consumer $B$ in the second picture (even if this second maximization is not strictly necessary).

[![enter image description here](https://i.sstatic.net/8t12M.png)](https://i.sstatic.net/8t12M.png)

$$Fig. \;1 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$
$\;$

[![enter image description here](https://i.sstatic.net/MRp2y.png)](https://i.sstatic.net/MRp2y.png)

$$Fig. \;2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$
$\;$

Consider a given indifference curve of consumer $B$, as $u\_B= \overline U\_{B1}$, ending at a point $A$ on the $x$ axis $0\_AC$.

As it should be clear from *Figure $1$* , the problem $(1)-(2)$ has its optimal solution at point $A$, on the $x$ axis, given a level of utility of consumer $B$, $u\_B$, equal to $\overline U\_{B1}$.

Changing the given level of $u\_B$, so that the blue indifference curve which is the constraint of the problem shifts, the optimal point will go along the $x$ axis $0\_AC$, covering the whole line as the blue indifference curve shifts continuously.

That is, the optimal points of consumer $A$, as $u\_B$ changes assuming all possible values, coincide with the $x$ axis $0\_AC$.

Analogously, if we have a given indifference curve of consumer $B$ ending at a point $H$ on the $y$ axis $C0\_B$, the optimal point is $H$, and shifting the given indifference curve we obtain the whole $C0\_B$ axis.

We can, therefore, conclude that **the contract line is the set composed of the $x$ axis $0\_AC$ and the $y$ axis $C0\_B$.**, the violet set in *Figure 1*.

$\;$

**If we want to verify that this is actually the contract line**$^1$, we can re-formulate the problem from the point of view of consumer $B$, maximizing their utility for a given level of utility of consumer $A$.

This is represented in *Figure $2$*, where the green lines are the constraints, given by fixed indifference curves of consumer $A$. Following the same reasoning as for consumer $A$, we see that the optimal point for consumer $B$ is on the $x$ axis at point $B$, and actually it coincides with point $A$ of *Figure $1$*.

Changing the given level of utility of consumer $A$ we obtain again the $x$ axis $0\_AC$.

And analogously, taking as constraint indifference curves as $u\_A=\overline U\_{A2}$ we obtain the $y$ axis $C0\_B$.

---

$^1$  The first optimization is sufficient, but let's verify we are not mistaken.

> **Comments:**
> **username_0**: Thank you, once again I appreciate such a complete answer of yours! I just asked for help with the Walrasian Equilibria for the same functions, with s = 2 here: https://economics.stackexchange.com/questions/54741/finding-walrasian-equilibria-when-walrasian-demands-are-not-unique  
> **username_2**: You are welcome, you are very kind.  

---

