Consumption Externalities in partial equilibrium analysis 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.
Without any constraint the social planer would set $x_B \rightarrow \infty$, and not solve the &quot;first order condition&quot; 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?)