<p>The first constraint you have, <span class="math-container">$U_2(x_2)=x_2$</span>, is not really a constraint. </p>

<p>Since points on the UPF is Pareto optimal, you should instead have <span class="math-container">$U_2(x_2)\ge \bar u$</span> or <span class="math-container">$x_2\ge \bar u$</span> as a constraint, where <span class="math-container">$\bar u$</span> 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 <span class="math-container">$\bar u$</span>, how best can we make individual 1's utility as high as possible with the available resources. </p>

CC BY-SA 4.0
2019-03-09T05:14:11.310Z
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
CC BY-SA 4.0
2019-03-09T05:25:15.110Z
Does this look like I'm on the right track here if I define alpha as above 0?
https://imgur.com/a/57YtLis
altec
CC BY-SA 4.0
2019-03-09T16:58:05.807Z
@altec: I think you are on the right track. Just sub in for the definition of $U_i$ and you're done.
Herr K.
CC BY-SA 4.0
2019-03-09T04:33:03.977Z
2019-03-09T04:33:03.977Z
Herr K.
<p>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.</p>
<p><span class="math-container">$$Feasible-set : x_1+x_2=ω$$</span></p>
<p><span class="math-container">$$UPF = Feasible-set= x_1+x_2=ω$$</span></p>
<p><span class="math-container">$$U1 = x_1$$</span></p>
<p><span class="math-container">$$U2 = x_2 $$</span></p>
<p><span class="math-container">$$U1+U2 = ω$$</span></p>
<p>Let me know if something is wrong with my approach.</p>

CC BY-SA 4.0
2023-01-03T18:47:24.243Z
2023-01-03T18:47:24.243Z
Amit Ranjan
<p>The economy is a one-good two individual endowment economy in which individual <span class="math-container">$i’s$</span> preferences are given by <span class="math-container">$_(_)=_$</span>, for ∈1,2, and the feasibility constraint on the amount of x available is <span class="math-container">$_1+_2=ω$</span>.
<br></p>

<p>We need to show that the equation of the utility possibilities frontier for this economy is <span class="math-container">$_2=ω−_1$</span>
<br><br></p>

<p>Setting up the Lagrange constrained optimisation function.<br></p>

<p>Max <span class="math-container">$U_1(x_1)= x_1$</span> subject to: <br>
<span class="math-container">$$U_2(x_2) = x_2$$</span>
<span class="math-container">$$x_1 + x_2 = w$$</span></p>

<p><span class="math-container">$$L = x_1 + \lambda x_2 + \mu (w - x_1 - x_2)$$</span>
<span class="math-container">$\lambda$</span> shows how small changes to <span class="math-container">$x_2$</span> affect <span class="math-container">$U_1$</span>, and <span class="math-container">$\mu$</span> shows how small changes to the amount of x available in the economy affect <span class="math-container">$U_1$</span></p>

<p>Partial differentiation of all variables:</p>

<p><span class="math-container">$$\frac {\partial L}{\partial x_1} = 1 - \mu = 0 \rightarrow \mu = 1$$</span></p>

<p><span class="math-container">$$\frac {\partial L}{\partial x_2} =  \lambda - \mu = 0 \rightarrow \mu = \lambda$$</span></p>

<p><span class="math-container">$$\frac {\partial L}{\partial \lambda} = x_2 = 0 $$</span>
<span class="math-container">$$\frac {\partial L}{\partial \mu} = w - x_1 - x_2 = 0$$</span></p>

<p><span class="math-container">$x_0 = 0$</span>, so the Pareto Efficient point found involves the allocation where individual 1 receives all of x.</p>

<p><span class="math-container">$\mu = \lambda$</span>,so individual 1 is affected equally by changes to <span class="math-container">$x_2$</span> and by how much x is available in the economy.</p>

<p>I'm unsure how to conclude this, demonstrate that <span class="math-container">$_2=ω−_1$</span>, or if I'm even on the right track. Could anybody please advise?</p>

<p>Also any feedback on my use of LaTex is also welcome, very new to using it. </p>

<utility><pareto-efficiency>
Solving for Pareto Efficient Utility Possibility Frontier using constrained optimisation
altec
# Solving for Pareto Efficient Utility Possibility Frontier using constrained optimisation

(Asked by: username_0 on 2019-03-09)

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.

**Tags:** utility, pareto-efficiency


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## 2 answers

### Answer by username_1 on 2019-03-09. Score: 1 | Accepted answer

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.

> **Comments:**
> **username_0**: 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 $U2 =ω−U1$  
> **username_0**: Does this look like I'm on the right track here if I define alpha as above 0? https://imgur.com/a/57YtLis  
> **username_1**: @username_0: I think you are on the right track. Just sub in for the definition of $U_i$ and you're done.  

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### Answer by username_2 on 2023-01-03. Score: 0

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.


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