There are two persons and two goods in an exchange economy. Initial endowment is $$ \omega = (\omega_1,\omega_2) =\left((1,0),(0,1)\right)$$
The utilities of two agents are given by:
$$\begin{cases}u_1&=&x_{11}x_{12}
\\u_2&=&2x_{21}+x_{22}\end{cases}$$
where $x$ is the allocation such that $x=\left((x_{11},x_{12}),(x_{21},x_{22})\right)$
and $x_{11}+x_{21}=1$ and $x_{12}+x_{22}=1$ i.e. non wasteful allocation.
What will be the Pareto optimal allocation?
I tried to find Pareto Optimal allocation using hit and trial but it is not coming. Is there any general way to find Pareto Optimal allocation for utilities of above kind?
The 2 conditions for person 1 are:
$\Large{\frac{\frac{\partial U_1}{\partial x_{11}}}{\frac{\partial U_2}{\partial x_{12}}}=\normalsize\frac{p_1}{p_2}}\quad \normalsize(1)$
$p_1x_{11}+p_2x_{12}=1\cdot p_1+0\cdot p_2\quad (2)$
At the beginning $x_{11}=1$ and $x_{12}=0$. That is the reason why the RHS looks like this.
$\frac{x_{12}}{x_{11}}=\frac{p_1}{p_2}\Rightarrow p_1x_{11}=p_2x_{12}$
Inserting the expression for $p_2x_{12}$ in (2):
$p_1x_{11}+p_1x_{11}=1\cdot p_1+0\cdot p_2$
$2p_1x_{11}=1p_1\Rightarrow x_{11}=\frac{1}{2}$
Now you can use the given equation $x_{11}+x_{21}=1$.
It is obvious that $x_{21}=\frac{1}{2}$
The 2 conditions for person 2 are:
$\Large{\frac{\frac{\partial U_2}{\partial x_{21}}}{\frac{\partial U_2}{\partial x_{22}}}=\normalsize\frac{p_1}{p_2}}\quad \normalsize(3)$
$p_1x_{21}+p_2x_{22}=0\cdot p_1+1\cdot p_2\quad (4)$
The calculations are similar to the calculations above.