# 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.

• For consumer $B$, should it not be $\max_{x_B,y_B}$ instead of $\max_{x_A,y_A}$ ? – Bertrand Dec 11 '20 at 7:53
• @Bertrand Noted and edited – EconJohn Dec 13 '20 at 4:00

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?)