Solving for Pareto Efficient Utility Possibility Frontier using constrained optimisation

Question

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.

Answer 1

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.

Answer 2

As we can see all points in the feasible set are pareto optimal and the UPF is the set of all pareto efficient points on the contract curve so the feasible set is the UPF in this case.

$$Feasible-set : x_1+x_2=ω$$

$$UPF = Feasible-set= x_1+x_2=ω$$

$$U1 = x_1$$

$$U2 = x_2 $$

$$U1+U2 = ω$$

Let me know if something is wrong with my approach.