<p>Without any constraint the social planer would set <span class="math-container">$x_B \rightarrow \infty$</span>, and not solve the &quot;first order condition&quot; given by <span class="math-container">$x_B+1/x_B=0$</span> only valid for an inner solution. The centralized problem is not comparable with the budget constrained decentralized problems you solved. (By the way why should <span class="math-container">$m$</span> be the same for both consumers?)</p>

CC BY-SA 4.0
2020-12-11T07:52:05.797Z
2020-12-11T07:52:05.797Z
Bertrand
<p>I've been working on a standard partial equilibrium with externalities problem and have been having some difficulty with these (relatively) basic concepts. My issue is primarly with solving for Pareto optimality and comparing it to a competitive outcome.</p>
<p><span class="math-container">$$u^A(y_A, x_A, x_B)=y_A+\ln x_A+\frac{x_B^2}{2}$$</span>
<span class="math-container">$$u^B(y_B,x_A,x_B)=y_B+\ln x_B-x_A $$</span></p>
<p>Solving for CE is fairly straight forward using the price normalization method of <span class="math-container">$(p_x,p_y)=(p,1)$</span>
<span class="math-container">$$\max_{x_A,y_A} \mathcal{L}=y_A+\ln x_A+\frac{x_B^2}{2}+\lambda_A(m-px_A-y_A)$$</span>
<span class="math-container">$$\max_{x_B,y_B} \mathcal{L}=y_B+\ln x_B-x_A+\lambda_B(m-px_B-y_B)$$</span>
The allocation of goods in CE amounts of <span class="math-container">$x_A, x_B$</span> in this context is:
<span class="math-container">$$\{x_A^*,x_B^*\}=\left\{\frac{1}{p},\frac{1}{p}\right\}$$</span></p>
<p>Further for the pareto optimal allocation we solve:</p>
<p><span class="math-container">$$\max_{x_A,x_B,y_A,y_B} u^A(y_A, x_A, x_B)+u^B(y_B,x_A,x_B)$$</span>
<span class="math-container">$$\max_{x_A,x_B,y_A,y_B} \{y_A+\ln x_A+\frac{x_B^2}{2}+y_B+\ln x_B-x_A\}$$</span>
Solving for <span class="math-container">$x_A$</span> is fairly straight forward via the FOC, we have:
<span class="math-container">$$x_A^{**}=1$$</span>
for dealing with <span class="math-container">$x_B$</span> there is difficulty as:
<span class="math-container">$$x_B+\frac{1}{x_B}=0$$</span>
where solving gives us
<span class="math-container">$$x_B=\sqrt{-1}$$</span>
which makes no sense.</p>
<p>Im definately doing something wrong but cant figure it out.</p>

CC BY-SA 4.0
2020-12-11T07:53:34.543Z
For consumer $B$, should it not be $\max_{x_B,y_B} $ instead of $\max_{x_A,y_A} $ ?
Bertrand
CC BY-SA 4.0
2020-12-13T04:00:50.910Z
@Bertrand Noted and edited
EconJohn
<public-economics><externalities>
Consumption Externalities in partial equilibrium analysis
EconJohn
# Consumption Externalities in partial equilibrium analysis

(Asked by: username_0 on 2020-12-11)

I've been working on a standard partial equilibrium with externalities problem and have been having some difficulty with these (relatively) basic concepts. My issue is primarly with solving for Pareto optimality and comparing it to a competitive outcome.

$$u^A(y\_A, x\_A, x\_B)=y\_A+\ln x\_A+\frac{x\_B^2}{2}$$
$$u^B(y\_B,x\_A,x\_B)=y\_B+\ln x\_B-x\_A $$

Solving for CE is fairly straight forward using the price normalization method of $(p\_x,p\_y)=(p,1)$
$$\max\_{x\_A,y\_A} \mathcal{L}=y\_A+\ln x\_A+\frac{x\_B^2}{2}+\lambda\_A(m-px\_A-y\_A)$$
$$\max\_{x\_B,y\_B} \mathcal{L}=y\_B+\ln x\_B-x\_A+\lambda\_B(m-px\_B-y\_B)$$
The allocation of goods in CE amounts of $x\_A, x\_B$ in this context is:
$$\{x\_A^\*,x\_B^\*\}=\left\{\frac{1}{p},\frac{1}{p}\right\}$$

Further for the pareto optimal allocation we solve:

$$\max\_{x\_A,x\_B,y\_A,y\_B} u^A(y\_A, x\_A, x\_B)+u^B(y\_B,x\_A,x\_B)$$
$$\max\_{x\_A,x\_B,y\_A,y\_B} \{y\_A+\ln x\_A+\frac{x\_B^2}{2}+y\_B+\ln x\_B-x\_A\}$$
Solving for $x\_A$ is fairly straight forward via the FOC, we have:
$$x\_A^{\*\*}=1$$
for dealing with $x\_B$ there is difficulty as:
$$x\_B+\frac{1}{x\_B}=0$$
where solving gives us
$$x\_B=\sqrt{-1}$$
which makes no sense.

Im definately doing something wrong but cant figure it out.

**Tags:** public-economics, externalities
> **Comments:**
> **username_1**: For consumer $B$, should it not be $\max_{x_B,y_B} $ instead of $\max_{x_A,y_A} $ ?  
> **username_0**: @Bertrand Noted and edited  

---

## 1 answers

### Answer by username_1 on 2020-12-11. Score: 1

Without any constraint the social planer would set $x\_B \rightarrow \infty$, and not solve the "first order condition" given by $x\_B+1/x\_B=0$ only valid for an inner solution. The centralized problem is not comparable with the budget constrained decentralized problems you solved. (By the way why should $m$ be the same for both consumers?)


---

