# Identify the Pareto welfare weights

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