# 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 $$𝑈_𝑖(𝑥_𝑖)=𝑥_𝑖$$, for 𝑖∈1,2, and the feasibility constraint on the amount of x available is $$𝑥_1+𝑥_2=ω$$.

We need to show that the equation of the utility possibilities frontier for this economy is $$𝑈_2=ω−𝑈_1$$

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 $$𝑈_2=ω−𝑈_1$$, 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.
• 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 $x1 + x2 = ω$ Show that the equation of the utility possibilities frontier for this economy is $U2 =ω−U1$ – altec Mar 9 at 5:14
• @altec: I think you are on the right track. Just sub in for the definition of $U_i$ and you're done. – Herr K. Mar 9 at 16:58