<p>(a) To find the competitive equilibrium, we first find the demands. Utility maximisation problem of consumer <span class="math-container">$i$</span> is
<span class="math-container">\begin{eqnarray*} \max_{x_i\geq 0, y_i\geq 0} &amp; \sqrt{x_i} + y_i\\ \text{s.t. } &amp; px_i+y_i\leq 8p+10\end{eqnarray*}</span>
Solving this problem, we get <span class="math-container">$i$</span>'s demand for commodity <span class="math-container">$x_i$</span> as:
<span class="math-container">\begin{eqnarray*} x_i(p) = \min\left\{\dfrac{1}{4p^2},\dfrac{8p+10}{p}\right\} \end{eqnarray*}</span>
We can now solve for the competitive equilibrium price <span class="math-container">$p$</span> by equating demand for <span class="math-container">$X$</span> and with supply of <span class="math-container">$X$</span> i.e. using <span class="math-container">$x_A(p)+x_B(p)+x_C(p)=24$</span>, and we get <span class="math-container">$p = \dfrac{1}{4\sqrt{2}}$</span>. Corresponding competitive equilibrium allocation is the endowment itself i.e. each consumer <span class="math-container">$i$</span> consumes <span class="math-container">$(x_i,y_i)=(8,10)$</span>. This equilibrium is also Pareto efficient because the environment in the problem satisfies the conditions for the First Welfare Theorem to hold.</p>
<p>(b) In this part we have an exchange economy with externalities. Competitive equilibrium in this case requires consumers to solve the following problems:
Utility maximisation problem of consumer <span class="math-container">$A$</span> is
<span class="math-container">\begin{eqnarray*} \max_{x_A\geq 0, y_A\geq 0} &amp; \sqrt{x_A} + \sqrt{x_B}+ y_A\\ \text{s.t. } &amp; px_A+y_A\leq 8p+10\end{eqnarray*}</span>
Utility maximisation problem of consumer <span class="math-container">$B$</span> is
<span class="math-container">\begin{eqnarray*} \max_{x_B\geq 0, y_B\geq 0} &amp; \sqrt{x_A} + \sqrt{x_B}+ y_B\\ \text{s.t. } &amp; px_B+y_B\leq 8p+10\end{eqnarray*}</span>
Utility maximisation problem of consumer <span class="math-container">$C$</span> is
<span class="math-container">\begin{eqnarray*} \max_{x_C\geq 0, y_C\geq 0} &amp; \sqrt{x_C}+ y_C\\ \text{s.t. } &amp; px_C+y_C\leq 8p+10\end{eqnarray*}</span>
Observe that each consumer <span class="math-container">$i$</span>'s demand for <span class="math-container">$x_i$</span> is same as before i.e.
<span class="math-container">\begin{eqnarray*} x_i(p) = \min\left\{\dfrac{1}{4p^2},\dfrac{8p+10}{p}\right\} \end{eqnarray*}</span>
Using demand equal supply, we get the same equilibrium as before i.e. <span class="math-container">$p = \dfrac{1}{4\sqrt{2}}$</span> and the corresponding competitive equilibrium allocation is also the endowment itself i.e. each consumer <span class="math-container">$i$</span> consumes <span class="math-container">$(x_i,y_i)=(8,10)$</span>. However, competitive equilibrium in this case is not Pareto efficient. This is because allocation <span class="math-container">$(x_A,y_A)=(x_B,y_B)=\left(\frac{32}{3},9.4\right)$</span> <span class="math-container">$(x_C,y_C)=\left(\frac{8}{3},11.2\right)$</span> is Pareto superior to it.</p>
<p>(c) Set of interior Pareto efficient allocations are all feasible allocations that satisfy <span class="math-container">$x_A=x_B=\frac{32}{3}$</span>, <span class="math-container">$x_C=\frac{8}{3}$</span>.</p>
<p>There are also corner Pareto efficient allocations such as allocations where <span class="math-container">$y_C=0$</span>. These are the feasible allocations where <span class="math-container">$x_C\leq\frac{8}{3}$</span>, and <span class="math-container">$x_A=x_B=\frac{24-x_C}{2}$</span>.</p>
<p>Likewise, there are corner allocations where <span class="math-container">$y_A=0$</span> and/or <span class="math-container">$y_B=0$</span>. Try to find those.</p>

CC BY-SA 4.0
2023-10-01T18:49:28.347Z
2023-10-01T18:49:28.347Z
Amit
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<p>How do you solve this exchange economy problem?</p>

<microeconomics>
How do you solve this problem
John Travolta
# How do you solve this problem

(Asked by: username_0 on 2023-09-28)

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How do you solve this exchange economy problem?

**Tags:** microeconomics


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## 1 answers

### Answer by username_1 on 2023-10-01. Score: 2

(a) To find the competitive equilibrium, we first find the demands. Utility maximisation problem of consumer $i$ is
\begin{eqnarray\*} \max\_{x\_i\geq 0, y\_i\geq 0} & \sqrt{x\_i} + y\_i\\ \text{s.t. } & px\_i+y\_i\leq 8p+10\end{eqnarray\*}
Solving this problem, we get $i$'s demand for commodity $x\_i$ as:
\begin{eqnarray\*} x\_i(p) = \min\left\{\dfrac{1}{4p^2},\dfrac{8p+10}{p}\right\} \end{eqnarray\*}
We can now solve for the competitive equilibrium price $p$ by equating demand for $X$ and with supply of $X$ i.e. using $x\_A(p)+x\_B(p)+x\_C(p)=24$, and we get $p = \dfrac{1}{4\sqrt{2}}$. Corresponding competitive equilibrium allocation is the endowment itself i.e. each consumer $i$ consumes $(x\_i,y\_i)=(8,10)$. This equilibrium is also Pareto efficient because the environment in the problem satisfies the conditions for the First Welfare Theorem to hold.

(b) In this part we have an exchange economy with externalities. Competitive equilibrium in this case requires consumers to solve the following problems:
Utility maximisation problem of consumer $A$ is
\begin{eqnarray\*} \max\_{x\_A\geq 0, y\_A\geq 0} & \sqrt{x\_A} + \sqrt{x\_B}+ y\_A\\ \text{s.t. } & px\_A+y\_A\leq 8p+10\end{eqnarray\*}
Utility maximisation problem of consumer $B$ is
\begin{eqnarray\*} \max\_{x\_B\geq 0, y\_B\geq 0} & \sqrt{x\_A} + \sqrt{x\_B}+ y\_B\\ \text{s.t. } & px\_B+y\_B\leq 8p+10\end{eqnarray\*}
Utility maximisation problem of consumer $C$ is
\begin{eqnarray\*} \max\_{x\_C\geq 0, y\_C\geq 0} & \sqrt{x\_C}+ y\_C\\ \text{s.t. } & px\_C+y\_C\leq 8p+10\end{eqnarray\*}
Observe that each consumer $i$'s demand for $x\_i$ is same as before i.e.
\begin{eqnarray\*} x\_i(p) = \min\left\{\dfrac{1}{4p^2},\dfrac{8p+10}{p}\right\} \end{eqnarray\*}
Using demand equal supply, we get the same equilibrium as before i.e. $p = \dfrac{1}{4\sqrt{2}}$ and the corresponding competitive equilibrium allocation is also the endowment itself i.e. each consumer $i$ consumes $(x\_i,y\_i)=(8,10)$. However, competitive equilibrium in this case is not Pareto efficient. This is because allocation $(x\_A,y\_A)=(x\_B,y\_B)=\left(\frac{32}{3},9.4\right)$ $(x\_C,y\_C)=\left(\frac{8}{3},11.2\right)$ is Pareto superior to it.

(c) Set of interior Pareto efficient allocations are all feasible allocations that satisfy $x\_A=x\_B=\frac{32}{3}$, $x\_C=\frac{8}{3}$.

There are also corner Pareto efficient allocations such as allocations where $y\_C=0$. These are the feasible allocations where $x\_C\leq\frac{8}{3}$, and $x\_A=x\_B=\frac{24-x\_C}{2}$.

Likewise, there are corner allocations where $y\_A=0$ and/or $y\_B=0$. Try to find those.


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