The economy is a one-good two individual endowment economy in which individual $iβs$ preferences are given by $π_π(π₯_π)=π₯_π$, for πβ1,2, and the feasibility constraint on the amount of x available is $π₯_1+π₯_2=Ο$.
We need to show that the equation of the utility possibilities frontier for this economy is $π_2=Οβπ_1$
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 $π_2=Οβπ_1$, 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.