<p>You cannot show this because it is not always true. Consider the following situation:</p>

<ul>
<li>Two consumers (Romeo (R) and Juliet (J))</li>
<li>Two goods (Apples (<span class="math-container">$A$</span>) and Chocolates (<span class="math-container">$C$</span>))</li>
<li>Romeo's endowment is <span class="math-container">$(a = 1, c = 0)$</span> and Juliet's endowment is <span class="math-container">$(a = 0, c = 1)$</span></li>
<li>Preferences are given by <span class="math-container">$u_R(a_R, c_R, a_J, c_J) = a_R + 2a_J $</span> and <span class="math-container">$u_J(a_R, c_R, a_J, c_J) = c_J + 2c_R $</span></li>
</ul>

<p>Observe that Romeo only cares about apple consumption and Juliet only cares for chocolate consumption. Even though Romeo likes apples himself but he loves to see Juliet consume his favorite fruit twice as much. Similarly, Juliet prefers to watch Romeo consume the chocolate than consuming it by herself. The only Pareto efficient allocation in this economy occurs when Romeo offers the apple to Juliet and Juliet gives her chocolate to Romeo i.e. <span class="math-container">$(a_R, c_R, a_J, c_J) = (0, 1, 1, 0)$</span>. </p>

<p>The competitive equilibrium in this economy is defined as <span class="math-container">$(p_A^*, p_C^*)$</span> and an allocation <span class="math-container">$(a_R^*, c_R^*, a_J^*, c_J^*)$</span> satisfying the property that both consumers choose their best bundle taking prices and other individual's choice as given. Since both can only choose for themselves and not for the other person they'll both end up buying what they like i.e. Romeo will buy only apples and Juliet will buy only chocolates. So endowment allocation <span class="math-container">$(a_R^*, c_R^*, a_J^*, c_J^*) = (1, 0, 0, 1)$</span> is the only competitive equilibrium and is supported by any price vector <span class="math-container">$(p_A^*, p_C^*)$</span> satisfying <span class="math-container">$p_A^* &gt; 0, p_C^*&gt; 0$</span>. Clearly it is not efficient.</p>

CC BY-SA 4.0
2019-11-10T04:30:10.203Z
Yes, I understand that there might be cases where this wouldn't hold true, but if a question asks me to find the efficiency conditions in case of externalities, how do I solve?
user708015
CC BY-SA 4.0
2019-11-10T01:10:51.223Z
2019-11-11T23:21:15.350Z
2019-11-11T23:21:15.350Z
Amit
<p>I have a 2 consumer 2 goods economy. Both the goods produce an externality for the other consumer, therefore, they are a part of the other's utility function. How do I show that despite the externality, the equilibrium is efficient? </p>

<pareto-efficiency>
Efficiency conditions for externalities
user708015
# Efficiency conditions for externalities

(Asked by: username_0 on 2019-11-09)

I have a 2 consumer 2 goods economy. Both the goods produce an externality for the other consumer, therefore, they are a part of the other's utility function. How do I show that despite the externality, the equilibrium is efficient?

**Tags:** pareto-efficiency

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## 1 answers

### Answer by username_1 on 2019-11-10. Score: 1

You cannot show this because it is not always true. Consider the following situation:

* Two consumers (Romeo (R) and Juliet (J))
* Two goods (Apples ($A$) and Chocolates ($C$))
* Romeo's endowment is $(a = 1, c = 0)$ and Juliet's endowment is $(a = 0, c = 1)$
* Preferences are given by $u\_R(a\_R, c\_R, a\_J, c\_J) = a\_R + 2a\_J $ and $u\_J(a\_R, c\_R, a\_J, c\_J) = c\_J + 2c\_R $

Observe that Romeo only cares about apple consumption and Juliet only cares for chocolate consumption. Even though Romeo likes apples himself but he loves to see Juliet consume his favorite fruit twice as much. Similarly, Juliet prefers to watch Romeo consume the chocolate than consuming it by herself. The only Pareto efficient allocation in this economy occurs when Romeo offers the apple to Juliet and Juliet gives her chocolate to Romeo i.e. $(a\_R, c\_R, a\_J, c\_J) = (0, 1, 1, 0)$.

The competitive equilibrium in this economy is defined as $(p\_A^\*, p\_C^\*)$ and an allocation $(a\_R^\*, c\_R^\*, a\_J^\*, c\_J^\*)$ satisfying the property that both consumers choose their best bundle taking prices and other individual's choice as given. Since both can only choose for themselves and not for the other person they'll both end up buying what they like i.e. Romeo will buy only apples and Juliet will buy only chocolates. So endowment allocation $(a\_R^\*, c\_R^\*, a\_J^\*, c\_J^\*) = (1, 0, 0, 1)$ is the only competitive equilibrium and is supported by any price vector $(p\_A^\*, p\_C^\*)$ satisfying $p\_A^\* > 0, p\_C^\*> 0$. Clearly it is not efficient.

> **Comments:**
> **username_0**: Yes, I understand that there might be cases where this wouldn't hold true, but if a question asks me to find the efficiency conditions in case of externalities, how do I solve?  

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