Solving for Pareto Efficient Utility Possibility Frontier using constrained optimisation The economy is a one-good two individual endowment economy in which individual $i’s$ preferences are given by $_(_)=_$, for ∈1,2, and the feasibility constraint on the amount of x available is $_1+_2=ω$.


We need to show that the equation of the utility possibilities frontier for this economy is $_2=ω−_1$


Setting up the Lagrange constrained optimisation function.

Max $U_1(x_1)= x_1$ subject to: 
$$U_2(x_2) = x_2$$
$$x_1 + x_2 = w$$

$$L = x_1 + \lambda x_2 + \mu (w - x_1 - x_2)$$
$\lambda$ shows how small changes to $x_2$ affect $U_1$, and $\mu$ shows how small changes to the amount of x available in the economy affect $U_1$

Partial differentiation of all variables:

$$\frac {\partial L}{\partial x_1} = 1 - \mu = 0 \rightarrow \mu = 1$$

$$\frac {\partial L}{\partial x_2} =  \lambda - \mu = 0 \rightarrow \mu = \lambda$$

$$\frac {\partial L}{\partial \lambda} = x_2 = 0 $$
$$\frac {\partial L}{\partial \mu} = w - x_1 - x_2 = 0$$

$x_0 = 0$, so the Pareto Efficient point found involves the allocation where individual 1 receives all of x.

$\mu = \lambda$,so individual 1 is affected equally by changes to $x_2$ and by how much x is available in the economy.

I'm unsure how to conclude this, demonstrate that $_2=ω−_1$, or if I'm even on the right track. Could anybody please advise?

Also any feedback on my use of LaTex is also welcome, very new to using it.
The first constraint you have, $U_2(x_2)=x_2$, is not really a constraint. 

Since points on the UPF is Pareto optimal, you should instead have $U_2(x_2)\ge \bar u$ or $x_2\ge \bar u$ as a constraint, where $\bar u$ is some arbitrary level of utility for individual 2 on the UPF. In other words, given that we're keeping individual 2's utility to be no lower than $\bar u$, how best can we make individual 1's utility as high as possible with the available resources.