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I am asked to identify the Pareto welfare weights from the FOC of the following problem

$max_{{x_1}{x_2}} U_1(x_1)\ st\ u_2=U_2(x_2)\ and\ x_{1n}+x_{2n}=yn$

The Langrangian is: $L=U_1(x_1)+\lambda_1(u_2-U_2(x_2))+\lambda(y-x_{}+x_{2})$

and the FOC's are:

$\nabla U_1(x_1)=\lambda$

$\lambda_1\nabla U_2(x_2)=\lambda$

for the i'th variable we have

$U_{x_{1i}}(x_1)=\lambda_i$

$\lambda_1 U_{x_{2i}}(x_2)=\lambda_i$

and from the Social planner Pareto problem we have

$max_{{x_1}{x_2}} \Sigma_i^2 a_iU_i(x_i)\ st\ \ x_{1n}+x_{2n}=yn$

and the focs for this problem are:

$a_1\nabla U_1(x_1)=\lambda$

$a_2\nabla U_2(x_2)=\lambda$

for the i'th variable we have

$a_1U_{x_{1i}}(x_1)=\lambda_i$

$a_2 U_{x_{2i}}(x_2)=\lambda_i$

Therefore the pareto welfare weights are

$a_1=\frac{\lambda_i}{U_{x_{1i}}(x_1)}$ $a_2=\frac{\lambda_i}{U_{x_{2i}}(x_2)}$

But if I apply the same logic to the first problem I obtain that $a_1=1$ $a_2=\lambda_1$

But they don't add up 1.Furthermore It doesn't have sense that $a_1=1$.I think that reasoning is wrong. Please help! Thanks in advance

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