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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.

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

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