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?
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.