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The economy is a one-good two individual endowment economy in which individual $i’s$ preferences are given by $π‘ˆ_𝑖(π‘₯_𝑖)=π‘₯_𝑖$, for π‘–βˆˆ1,2, 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 $π‘ˆ_2=Ο‰βˆ’π‘ˆ_1$

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 $π‘ˆ_2=Ο‰βˆ’π‘ˆ_1$, 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.

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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.

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  • $\begingroup$ Ah you're correct. I paraphrased, but the exact wording in the question was: Consider a simple two-person, one-good endowment economy in which individual i’s preferences are given by $Ui(xi) = xi$, for $i ∈ {1,2}$, and the feasibility constraint was $x1 + x2 = Ο‰$ Show that the equation of the utility possibilities frontier for this economy is $U2 =Ο‰βˆ’U1$ $\endgroup$ – altec Mar 9 '19 at 5:14
  • $\begingroup$ Does this look like I'm on the right track here if I define alpha as above 0? imgur.com/a/57YtLis $\endgroup$ – altec Mar 9 '19 at 5:25
  • $\begingroup$ @altec: I think you are on the right track. Just sub in for the definition of $U_i$ and you're done. $\endgroup$ – Herr K. Mar 9 '19 at 16:58
  • $\begingroup$ Awesome, thanks! $\endgroup$ – altec Apr 10 '19 at 1:01

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