<p>Given a pure exchange economy:</p>
<p><span class="math-container">$u_A(x_A, y_A)=x_Ay_A$</span>, <span class="math-container">$u_B(x_B, y_B)=x_By_B^2$</span></p>
<p>with endowments:</p>
<p><span class="math-container">$\omega_A=(80,150)$</span> and <span class="math-container">$\omega_B=(210,180)$</span></p>
<ol>
<li>To find the equilibrium, we first find the price offer curve of <span class="math-container">$B$</span>:
<span class="math-container">\begin{eqnarray*} \max_{x_B\geq 0,y_B\geq 0} &amp; x_By_B^2 \\ \text{s.t. } &amp; p_Xx_B+y_B\leq 210p_X+180\end{eqnarray*}</span></li>
</ol>
<p>and we get <span class="math-container">$x_B = \dfrac{70p_X+60}{p_X}$</span> and <span class="math-container">$y_B =140p_X+120$</span></p>
<p>Price offer curve of <span class="math-container">$B$</span> is locus of <span class="math-container">$(x_B, y_B)$</span> pairs as found above for different prices. It is given by the blue curve in the picture below.</p>
<p><span class="math-container">$A$</span> will choose <span class="math-container">$p_X$</span> and <span class="math-container">$B$</span> will consume according to <span class="math-container">$x_B = \dfrac{70p_X+60}{p_X}$</span> and <span class="math-container">$y_B =140p_X+120$</span>, so that <span class="math-container">$A$</span> gets to consume the residual. <span class="math-container">$A$</span> chooses that point on <span class="math-container">$B$</span>'s offer curve that gives <span class="math-container">$A$</span> the highest satisfaction. In picture, it corresponds to the point of tangency of <span class="math-container">$A$</span>'s IC and <span class="math-container">$B$</span>'s offer curve. So, <span class="math-container">$A$</span> gets</p>
<p><span class="math-container">$x_A = 290 -\dfrac{70p_X+60}{p_X} = 220-\dfrac{60}{p_X}$</span> and</p>
<p><span class="math-container">$y_A =330-140p_X-120=210-140p_X$</span></p>
<p>and <span class="math-container">$A$</span> will choose <span class="math-container">$p_X$</span> by solving the following problem:
<span class="math-container">\begin{eqnarray*} \max_{p_X\geq 0} &amp; \left(220 -\dfrac{60}{p_X}\right)(210-140p_X)  \end{eqnarray*}</span>
Solving it, we get <span class="math-container">$p_X=\dfrac{3}{\sqrt{22}}$</span>.
Corresponding allocation is <span class="math-container">$\left((x_A,y_A),(x_B,y_B)\right)=\left((220-20\sqrt{22},210-420/\sqrt{22}),(70+20\sqrt{22},120+420/\sqrt{22})\right)$</span>. It is given by allocation <span class="math-container">$e$</span> in the picture below.</p>
<ol start="2">
<li>Equilibrium allocation <span class="math-container">$e$</span> is not Pareto efficient because MRS of <span class="math-container">$A$</span> is not equal to MRS of <span class="math-container">$B$</span> at this allocation.</li>
</ol>
<p><span class="math-container">$\dfrac{210-420/\sqrt{22}}{220-20\sqrt{22}} = \text{MRS}_A &gt; \text{MRS}_B = \dfrac{120+420/\sqrt{22}}{140+40\sqrt{22}}=\dfrac{3}{\sqrt{22}}=p_X$</span></p>
<ol start="3">
<li>In this problem, <span class="math-container">$A$</span> chooses the point on the <span class="math-container">$B$</span>'s IC passing through the endowment allocation, that gives <span class="math-container">$A$</span> the highest level of satisfaction.</li>
</ol>
<p><span class="math-container">$\left((x_A,y_A),(x_B,y_B)\right)$</span> satisfy:</p>
<p><span class="math-container">$x_By_B^2=210(180^2)$</span> and</p>
<p><span class="math-container">$\dfrac{y_A}{x_A} = \dfrac{y_B}{2x_B}$</span></p>
<p>and the feasibility conditions.</p>
<p>This point will be Pareto efficient and it corresponds to point <span class="math-container">$p$</span> in the picture below:</p>
<p><a href="https://i.sstatic.net/ylXzn.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ylXzn.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-09-09T15:52:03.717Z
Incredible! thank you so much Amit
Walrasian soldier
CC BY-SA 4.0
2023-09-08T05:13:44.837Z
2023-09-08T05:35:07.057Z
2023-09-08T05:35:07.057Z
Amit
<p>Not sure where you get lost. <span class="math-container">$u_B$</span> is Cobb-Douglas type, thus given a price vector <span class="math-container">$\textbf{p}$</span> and initial endowment <span class="math-container">$\textbf{w}^B$</span> it is easy to determine <span class="math-container">$x_1^B(\textbf{p}),x_2^B(\textbf{p})$</span>.</p>
<p>Once you do these, the problem is indeed</p>
<p><span class="math-container">\begin{align*}
\max_\textbf{p} \quad u_A(x^A_1,x_2^A) \\
\text{st.} \; x_1^A + x_1^B(\textbf{p}) &amp; = w_1 \\
 x_2^A + x_2^B(\textbf{p}) &amp; = w_2
\end{align*}</span></p>
<p>Pareto-optimality can be checked using the usual conditions for 2.</p>
<p>For 3., it is a similar deal to 1., but instead of a price vector, a bundle is offered by <span class="math-container">$A$</span>, thus</p>
<p><span class="math-container">\begin{align*}
\max_{x_1^B,x_2^B} \quad u_A(x^A_1,x_2^A) \\
\text{st.} \; x_1^A + x_1^B &amp; = w_1 \\
 x_2^A + x_2^B &amp; = w_2 \\
u_B(x_1^B,x_2^B) &amp; \geq u_B(w_1^B,w_2^B)
\end{align*}</span></p>

CC BY-SA 4.0
2023-09-07T08:44:48.817Z
2023-09-07T08:44:48.817Z
Giskard
<p>I'm trying to solve the following problem:</p>
<p>Consider an exchange economy with two consumers, <span class="math-container">$A$</span> and <span class="math-container">$B$</span>, whose utility functions are:
<span class="math-container">\begin{align*}
u_{A} &amp; = x_1^A x_2^A \\
u_{B} &amp; = x_1^B (x_2^B)^2
\end{align*}</span>
with endowments <span class="math-container">$\textbf{w}^A =(80 ; 150)$</span> and <span class="math-container">$\textbf{w}^B=(210; 180)$</span> respectively. Assume that consumer <span class="math-container">$A$</span> is a price setter, meaning he makes a take-it-or-leave-it price offer to consumer <span class="math-container">$B$</span>.</p>
<ol>
<li>Find the equilibrium economy in this economy.</li>
<li>Find the pareto efficient allocation (PEA) in this economy, and check if the equilibrium found in (1) is part of the PEA.</li>
<li>Now suppose that instead of offering a price vector, consumer <span class="math-container">$A$</span> makes a take-it-or-leave-it offer to consumer <span class="math-container">$B$</span> over a consuption bundle. Find this bundle. Is it the same that the one found in (1). Why? Why not? Is this solution pareto efficient?</li>
</ol>
<p>I think I should treat this problem as it was a sequential game, where player <span class="math-container">$A$</span> moves first. I know that player <span class="math-container">$B$</span> would never accept an offer that leaves him worse than in autarky, so consumer's <span class="math-container">$A$</span> offer should leave consumer <span class="math-container">$B$</span> in the same indiffence curve that goes through the endowment point. The problem is that I don't know how to solve the game mathematically and find that price vector.
I thought of solving:
<span class="math-container">\begin{align*}
\max \quad U_A(x^A_1,x_2^A) \\
\text{st.} \; x_1^A + x_1^B(\textbf{p}) &amp; = w_1 \\
 x_2^A + x_2^B(\textbf{p}) &amp; = w_2 \\
V(\textbf{p})_B = \bar{u}
\end{align*}</span>
But I'm not getting anything clear.</p>

CC BY-SA 4.0
2023-09-06T23:18:11.250Z
Is $u_A = x_1^Ax_2^B$ or $x_1^Ax_2^A$?
Amit
<microeconomics><game-theory><general-equilibrium><monopoly><pure-exchange-economy>
General equilibrium with market power
Walrasian soldier
# General equilibrium with market power

(Asked by: username_0 on 2023-09-06)

I'm trying to solve the following problem:

Consider an exchange economy with two consumers, $A$ and $B$, whose utility functions are:
\begin{align\*}
u\_{A} & = x\_1^A x\_2^A \\
u\_{B} & = x\_1^B (x\_2^B)^2
\end{align\*}
with endowments $\textbf{w}^A =(80 ; 150)$ and $\textbf{w}^B=(210; 180)$ respectively. Assume that consumer $A$ is a price setter, meaning he makes a take-it-or-leave-it price offer to consumer $B$.

1. Find the equilibrium economy in this economy.
2. Find the pareto efficient allocation (PEA) in this economy, and check if the equilibrium found in (1) is part of the PEA.
3. Now suppose that instead of offering a price vector, consumer $A$ makes a take-it-or-leave-it offer to consumer $B$ over a consuption bundle. Find this bundle. Is it the same that the one found in (1). Why? Why not? Is this solution pareto efficient?

I think I should treat this problem as it was a sequential game, where player $A$ moves first. I know that player $B$ would never accept an offer that leaves him worse than in autarky, so consumer's $A$ offer should leave consumer $B$ in the same indiffence curve that goes through the endowment point. The problem is that I don't know how to solve the game mathematically and find that price vector.
I thought of solving:
\begin{align\*}
\max \quad U\_A(x^A\_1,x\_2^A) \\
\text{st.} \; x\_1^A + x\_1^B(\textbf{p}) & = w\_1 \\
x\_2^A + x\_2^B(\textbf{p}) & = w\_2 \\
V(\textbf{p})\_B = \bar{u}
\end{align\*}
But I'm not getting anything clear.

**Tags:** microeconomics, game-theory, general-equilibrium, monopoly, pure-exchange-economy
> **Comments:**
> **username_1**: Is $u_A = x_1^Ax_2^B$ or $x_1^Ax_2^A$?  

---

## 2 answers

### Answer by username_1 on 2023-09-08. Score: 2 | Accepted answer

Given a pure exchange economy:

$u\_A(x\_A, y\_A)=x\_Ay\_A$, $u\_B(x\_B, y\_B)=x\_By\_B^2$

with endowments:

$\omega\_A=(80,150)$ and $\omega\_B=(210,180)$

1. To find the equilibrium, we first find the price offer curve of $B$:
   \begin{eqnarray\*} \max\_{x\_B\geq 0,y\_B\geq 0} & x\_By\_B^2 \\ \text{s.t. } & p\_Xx\_B+y\_B\leq 210p\_X+180\end{eqnarray\*}

and we get $x\_B = \dfrac{70p\_X+60}{p\_X}$ and $y\_B =140p\_X+120$

Price offer curve of $B$ is locus of $(x\_B, y\_B)$ pairs as found above for different prices. It is given by the blue curve in the picture below.

$A$ will choose $p\_X$ and $B$ will consume according to $x\_B = \dfrac{70p\_X+60}{p\_X}$ and $y\_B =140p\_X+120$, so that $A$ gets to consume the residual. $A$ chooses that point on $B$'s offer curve that gives $A$ the highest satisfaction. In picture, it corresponds to the point of tangency of $A$'s IC and $B$'s offer curve. So, $A$ gets

$x\_A = 290 -\dfrac{70p\_X+60}{p\_X} = 220-\dfrac{60}{p\_X}$ and

$y\_A =330-140p\_X-120=210-140p\_X$

and $A$ will choose $p\_X$ by solving the following problem:
\begin{eqnarray\*} \max\_{p\_X\geq 0} & \left(220 -\dfrac{60}{p\_X}\right)(210-140p\_X) \end{eqnarray\*}
Solving it, we get $p\_X=\dfrac{3}{\sqrt{22}}$.
Corresponding allocation is $\left((x\_A,y\_A),(x\_B,y\_B)\right)=\left((220-20\sqrt{22},210-420/\sqrt{22}),(70+20\sqrt{22},120+420/\sqrt{22})\right)$. It is given by allocation $e$ in the picture below.

2. Equilibrium allocation $e$ is not Pareto efficient because MRS of $A$ is not equal to MRS of $B$ at this allocation.

$\dfrac{210-420/\sqrt{22}}{220-20\sqrt{22}} = \text{MRS}\_A > \text{MRS}\_B = \dfrac{120+420/\sqrt{22}}{140+40\sqrt{22}}=\dfrac{3}{\sqrt{22}}=p\_X$

3. In this problem, $A$ chooses the point on the $B$'s IC passing through the endowment allocation, that gives $A$ the highest level of satisfaction.

$\left((x\_A,y\_A),(x\_B,y\_B)\right)$ satisfy:

$x\_By\_B^2=210(180^2)$ and

$\dfrac{y\_A}{x\_A} = \dfrac{y\_B}{2x\_B}$

and the feasibility conditions.

This point will be Pareto efficient and it corresponds to point $p$ in the picture below:

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

> **Comments:**
> **username_0**: Incredible! thank you so much Amit  

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### Answer by username_2 on 2023-09-07. Score: 1

Not sure where you get lost. $u\_B$ is Cobb-Douglas type, thus given a price vector $\textbf{p}$ and initial endowment $\textbf{w}^B$ it is easy to determine $x\_1^B(\textbf{p}),x\_2^B(\textbf{p})$.

Once you do these, the problem is indeed

\begin{align\*}
\max\_\textbf{p} \quad u\_A(x^A\_1,x\_2^A) \\
\text{st.} \; x\_1^A + x\_1^B(\textbf{p}) & = w\_1 \\
x\_2^A + x\_2^B(\textbf{p}) & = w\_2
\end{align\*}

Pareto-optimality can be checked using the usual conditions for 2.

For 3., it is a similar deal to 1., but instead of a price vector, a bundle is offered by $A$, thus

\begin{align\*}
\max\_{x\_1^B,x\_2^B} \quad u\_A(x^A\_1,x\_2^A) \\
\text{st.} \; x\_1^A + x\_1^B & = w\_1 \\
x\_2^A + x\_2^B & = w\_2 \\
u\_B(x\_1^B,x\_2^B) & \geq u\_B(w\_1^B,w\_2^B)
\end{align\*}


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