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Let an economy with 20 consumers, one private good $x_2$ and one public good $x_1$. The public good is produced using the private good as input with the following technology $x_1=g(z)=z^{1/2}$, where $z$ is units of private good. TheUtility function of the consumer "i" is: $$U(x_1,x_{2i})=ln(x_1)+ln(x_{2i})$$ the aggregate endowment of private good is $\sum_{i=1}^{20}w_i=20$. My problem is en the section b)

a)Obtain the pareto optimal allocation.

Here, I have the following lagrangian:

$$Max_{x_1,x_{2i},z}20ln(x_1)+\sum_{i=1}^{20}ln(x_{2i})+\lambda_1(40-\sum_{i=1}^{20}x_{2i}-z)+\lambda_2(z^{1/2}-x_1)$$

with CPO's:

$$x_1:20/x_1=\lambda_2$$ $$x_{2i}:1/x_{2i}=\lambda_1$$ $$z:-\lambda_1+\frac{1}{2}\lambda_2z^{-1/2}=0$$ $$\lambda_1: 40=\sum_{i=1}^{20}x_{2i}+z$$ $$\lambda_2:z^{1/2}=x_1$$

Where the solution $x_{2i}*=4/3$, $x_1*=\sqrt{\frac{40}{3}}$, $z*=\frac{40}{3}$ is pareto optimal.

b)Obtain the Lindahl equilibrium allocation with $·p_i=p_j \hspace{0.3cm} \forall i\neq j$. Here each consumer maximize his utility function subject to a restriction that has to be with a personalized price; and the firm which produce the public good maximize benefits. So the problem of consumer is: $$Max_{x_1,x_{2i}}ln(x_1)+ln(x_{2i})+\lambda_1(w_i-x_{2i}-p_ix_1)$$

The CPO's:

$$x_1:1/x_1=\lambda_1p_i$$ $$x_{2i}:1/x_{2i}=\lambda_1$$ $$\lambda_1:(w_i=x_{2i}+p_ix_1)$$

So, from the first and the second CPO: $x_{2i}=x_1p_i$ substituting in the third $x_{2i}=\frac{w_i}{2}$ Here is my problem. In order to obtain the Pareto allocation in a) $x_{2i}=\frac{w_i}{2}=\frac{4}{3}$ so , it must be $w_i=8/3$ for each consumer $i$ but $\sum_{i=1}^{20}w_i=20\frac{8}{3}=\frac{160}{3} \neq 40$ So What is happening?

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