<p>Consider a 2-consumer, 2-goods pure-exchange economy:
<span class="math-container">$&lt;(u_A:\mathbb{R}^2_+\rightarrow\mathbb{R},(\omega^X_A,\omega^Y_A)), (u_B:\mathbb{R}^2_+\rightarrow\mathbb{R}, (\omega^X_B,\omega^Y_B))&gt;$</span></p>
<p>Set of feasible allocations is
<span class="math-container">$\mathcal{F} = \{((x_A,y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=\omega_A^X+\omega_B^X \ \wedge \ y_A+y_B=\omega_A^Y+\omega_B^Y\}$</span></p>
<p>We can define utility possibility set as:</p>
<p><span class="math-container">$\text{UPS} = \{(\mu_A,\mu_B)\in\mathbb{R}^2|\mu_A\leq u_A(x_A,y_A) \ \wedge \mu_B \leq u_B(x_B,y_B) \text{ for some } ((x_A,y_A),(x_B,y_B))\in\mathcal{F}\}$</span></p>
<p>Utility Possibility Frontier is defined as
<span class="math-container">$\text{UPF} = \{(\mu_A,\mu_B)\in\mathbb{R}^2|(\mu_A,\mu_B) \text{ is the limit point of UPS and there exists no } (\mu_A',\mu_B')\in\text{UPS satisfying } \mu_A'&gt;\mu_A, \mu_B'&gt;\mu_B  \}$</span></p>
<p>Consider the exchange economy <span class="math-container">$u_A(x_A,y_A) = \max(x_A,y_A)$</span>, <span class="math-container">$u_B(x_B,y_B) = \max(x_B,y_B)$</span>. Total endowment of X and Y is <span class="math-container">$1$</span> unit each.</p>
<p>Set of Pareto efficient allocations in this economy consists of two allocations
<span class="math-container">$\{((1,0),(0,1)),((0,1),(1,0))\}$</span></p>
<p>Utility possibility set is:</p>
<p><span class="math-container">$\text{UPS}=\{(\mu_A,\mu_B)\in\mathbb{R}^2|\max(\mu_A,\mu_B)\leq 1\}$</span></p>
<p>and utility possibility frontier is:</p>
<p><span class="math-container">$\text{UPF}=\{(\mu_A,\mu_B)\in\mathbb{R}^2|\max(\mu_A,\mu_B)= 1\}$</span></p>
<p>Here is the picture:
<a href="https://i.sstatic.net/yu2hg.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yu2hg.png" alt="enter image description here" /></a></p>
<p>Observe that the only utility pair in the above example that corresponds to Pareto efficient allocation(s) is <span class="math-container">$(\mu_A,\mu_B)=(1,1)$</span>. If we consider <span class="math-container">$(\mu_A,\mu_B)=(1,0)$</span> on the frontier, this utility pair can be achieved by an inefficient allocation such as <span class="math-container">$((1,1),(0,0))$</span>. This example demonstrates that a utility pair on the frontier may not always correspond to a Pareto-efficient allocation.</p>

CC BY-SA 4.0
2023-10-11T11:41:08.807Z
Thank you so much for the explanation! It really helped.
Ananya
CC BY-SA 4.0
2023-10-11T08:10:42.413Z
2023-10-11T08:46:29.590Z
2023-10-11T08:46:29.590Z
Amit
<p>Are all allocations on UPF pareto optimal?
Was studying this in context of Public and Private goods. If someone can help.</p>

CC BY-SA 4.0
2023-10-10T19:44:36.423Z
Please specify all acronyms, such as UPF. Maybe also add a bit more detail, context, and your own thoughts/attempts to think through this question.
BB King
CC BY-SA 4.0
2023-10-10T20:14:30.190Z
UPF stands for Utility Possibility Frontier. This was a question posed in my public economics exam for the unit -Public goods as 1 marker
Ananya
CC BY-SA 4.0
2023-10-11T11:40:40.477Z
There are different nonequivalent definitions of both Pareto optimality and the UPF. In the current form, the question cannot be answered.
Michael Greinecker
<public-economics><public-goods>
Are all allocations on UPF pareto optimal?
Ananya
# Are all allocations on UPF pareto optimal?

(Asked by: username_0 on 2023-10-10)

Are all allocations on UPF pareto optimal?
Was studying this in context of Public and Private goods. If someone can help.

**Tags:** public-economics, public-goods

> **Comments:**
> **username_1**: Please specify all acronyms, such as UPF. Maybe also add a bit more detail, context, and your own thoughts/attempts to think through this question.  
> **username_0**: UPF stands for Utility Possibility Frontier. This was a question posed in my public economics exam for the unit -Public goods as 1 marker  
> **username_2**: There are different nonequivalent definitions of both Pareto optimality and the UPF. In the current form, the question cannot be answered.  

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## 1 answers

### Answer by username_3 on 2023-10-11. Score: 2 | Accepted answer

Consider a 2-consumer, 2-goods pure-exchange economy:
$<(u\_A:\mathbb{R}^2\_+\rightarrow\mathbb{R},(\omega^X\_A,\omega^Y\_A)), (u\_B:\mathbb{R}^2\_+\rightarrow\mathbb{R}, (\omega^X\_B,\omega^Y\_B))>$

Set of feasible allocations is
$\mathcal{F} = \{((x\_A,y\_A),(x\_B,y\_B))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_A+x\_B=\omega\_A^X+\omega\_B^X \ \wedge \ y\_A+y\_B=\omega\_A^Y+\omega\_B^Y\}$

We can define utility possibility set as:

$\text{UPS} = \{(\mu\_A,\mu\_B)\in\mathbb{R}^2|\mu\_A\leq u\_A(x\_A,y\_A) \ \wedge \mu\_B \leq u\_B(x\_B,y\_B) \text{ for some } ((x\_A,y\_A),(x\_B,y\_B))\in\mathcal{F}\}$

Utility Possibility Frontier is defined as
$\text{UPF} = \{(\mu\_A,\mu\_B)\in\mathbb{R}^2|(\mu\_A,\mu\_B) \text{ is the limit point of UPS and there exists no } (\mu\_A',\mu\_B')\in\text{UPS satisfying } \mu\_A'>\mu\_A, \mu\_B'>\mu\_B \}$

Consider the exchange economy $u\_A(x\_A,y\_A) = \max(x\_A,y\_A)$, $u\_B(x\_B,y\_B) = \max(x\_B,y\_B)$. Total endowment of X and Y is $1$ unit each.

Set of Pareto efficient allocations in this economy consists of two allocations
$\{((1,0),(0,1)),((0,1),(1,0))\}$

Utility possibility set is:

$\text{UPS}=\{(\mu\_A,\mu\_B)\in\mathbb{R}^2|\max(\mu\_A,\mu\_B)\leq 1\}$

and utility possibility frontier is:

$\text{UPF}=\{(\mu\_A,\mu\_B)\in\mathbb{R}^2|\max(\mu\_A,\mu\_B)= 1\}$

Here is the picture:
[![enter image description here](https://i.sstatic.net/yu2hg.png)](https://i.sstatic.net/yu2hg.png)

Observe that the only utility pair in the above example that corresponds to Pareto efficient allocation(s) is $(\mu\_A,\mu\_B)=(1,1)$. If we consider $(\mu\_A,\mu\_B)=(1,0)$ on the frontier, this utility pair can be achieved by an inefficient allocation such as $((1,1),(0,0))$. This example demonstrates that a utility pair on the frontier may not always correspond to a Pareto-efficient allocation.

> **Comments:**
> **username_0**: Thank you so much for the explanation! It really helped.  

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