I am trying to solve the following problem:
Consider a pure exchange economy with three commodities and two households with individual endowments:
$$e_1=(1,2,3), e_2=(3,2,1),$$
respectively, and utility functions
$$u_1(x_{11},x_{12},x_{13})=x_{11}+2x_{12}+3x_{13}$$ and $$u_2(x_{21},x_{22},x_{23})=3x_{21}+2x_{22}+x_{23}$$ respectively.
Which of the following is the Utility Possibility Frontier? Options:
A. $\displaystyle \max\left\{ u_1+\frac{u_2}{2}, u_1 + u_2, \frac{u_1}{2} +u_2 \right\} = 32$
B. $ \displaystyle \max\left\{ u_1+\frac{u_2}{3}, \frac{3}{4} u_1 + \frac{3}{4} u_2, \frac{u_1}{3} +u_2 \right\} = 24$
C. $\displaystyle \max\left\{ u_1+\frac{u_2}{3}, u_1 + u_2, \frac{u_1}{3} +u_2 \right\} = 24$
D. None of the Above
My attempt:
I tried calculating the Pareto optimal allocations by taking two commodities at a time. I found out that between good $1$ and good $2$, the Pareto efficient allocations are where: $$x_{21}= 4 \quad \text{or} \quad x_{12} = 4$$
Similarly, between good $2$ and good $3$, I find that:
$$x_{22}= 4 \quad \text{or} \quad x_{13} = 4$$ are Pareto optimal.
Similarly, between good $1$ and good $3$, I find that:
$$x_{21}= 4 \quad \text{or} \quad x_{13} = 4$$ are Pareto optimal.
Also I observe that the preferences among good $1$, good $2$ and good $3$ of individual 1 are:
$$ \text{good } 3 > \text{good } 2 > \text{good } 1 $$
And those of individual $2$ are:
$$ \text{good } 1 > \text{good } 2 > \text{good } 3 $$
Now I consider the following allocations:
$$((x_{11},x_{12},4), (x_{21},x_{22},0))$$
And the fact that $$x_{11} + x_{21} = 4$$ $$x_{12} + x_{22} = 4$$
I find that:
$$u_1 = x_{11}+2x_{12}+3x_{13} = x_{11}+2x_{12} + 12$$
$$u_2 = 3x_{21}+2x_{22}+x_{23} = 3x_{21}+2x_{22}$$
$$\implies u_1 + u_2 /3 = 16 + 2x_{12} + 2/3 x_{22}$$ $$\implies u_1 + u_2 /3 = 16 + 2/3(3x_{12} + x_{22})$$ $$\implies u_1 + u_2 /3 = 16 + 2/3(2x_{12} + 4)$$
$$\implies u_1 + u_2 /3 \le 24 $$ because $x_{12} \le 4$.
Similarly I consider the allocation:
$$((0,x_{12},x_{13}), (4,x_{22}, x_{23}))$$
And find that:
$$ u_1 /3 + u_2 \le 24 $$
I am stuck here. I am not able to see how I can take it from to the options given. Please drop hints as to how I can.
Thank you for reading this.