<p>Pareto efficient allocations can be found by maximizing a weighted average of the utilities of the agents. Let <span class="math-container">$\lambda$</span> be the <em>Pareto weight</em> for agent 1 and <span class="math-container">$1 - \lambda$</span> the weight for agent 2 (where <span class="math-container">$\lambda \in [0,1]$</span>). This then gives the following problem:
<span class="math-container">$$
\max_{x_1, x_2} -\lambda \left[\pi_1(x_1 - b_{11})^2 + \pi_2(x_2 - b_{12})^2\right] - (1-\lambda)\left[ \pi_1 (x_1 - b_{21})^2 + \pi_2 (x_2 - b_{22})^2\right].
$$</span>
The first order conditions with respect to <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> give:
<span class="math-container">$$
\lambda \pi_1 2(x_1 - b_{11}) + (1-\lambda) \pi_1 2(x_1 - b_{21}) = 0\\
\lambda \pi_2 2(x_2 - b_{21}) + (1-\lambda) \pi_2 2(x_2 - b_{22}) = 0.
$$</span></p>
<p>Simplifying gives:
<span class="math-container">$$
x_1 = \lambda b_{11} + (1-\lambda) b_{21},\\
x_2 = \lambda b_{21} + (1-\lambda) b_{22}.
$$</span>
So <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are weighted averages of the bliss points.</p>

CC BY-SA 4.0
2021-05-19T16:28:44.120Z
2021-05-19T17:02:35.480Z
2021-05-19T17:02:35.480Z
tdm
<p>There are two economic agents <span class="math-container">$i\in \{1,2\}$</span> with state dependent utility <span class="math-container">$u_{is}=-(x-b_{is})^2$</span> where <span class="math-container">$x\in R$</span> and <span class="math-container">$b_{is}\in R$</span> is bliss point of <span class="math-container">$i$</span> in state <span class="math-container">$s\in\{1,2\}$</span>. Assume <span class="math-container">$b_{1s}\lt b_{2s}$</span> for <span class="math-container">$\forall s\in\{1,2\}$</span>. State <span class="math-container">$s\in\{1,2\}$</span> occurs with probability <span class="math-container">$\pi_{s}$</span>. We denote <span class="math-container">$U_i(x_1,x_2)=\pi_1u_{i1}(x_1)+\pi_2u_{i2}(x_2)$</span>.</p>
<p>Ex-post Pareto efficient policies in state 1 and 2, <span class="math-container">$x_1^*$</span> and <span class="math-container">$x_2^*$</span> respectively, I found them to be as follows:</p>
<p>Any policy in <span class="math-container">$[b_{11},b_{21}]$</span> is ex- post Pareto efficient in state 1 and any policy not in <span class="math-container">$[b_{11},b_{21}]$</span> is ex-post Pareto inefficient in state 1. Hence <span class="math-container">$x_1^* = z$</span>, where <span class="math-container">$z$</span> is any <span class="math-container">$z\in [b_{11}, b_{21}].$</span> By similar arguments <span class="math-container">$x_2^* = z$</span>, where <span class="math-container">$z$</span> is any <span class="math-container">$z\in[b_{12}, b_{22}].$</span></p>
<p>But how would we characterize ex-ante Pareto efficient policy pairs, <span class="math-container">${x_1^*,x_2^*}?$</span></p>

CC BY-SA 4.0
2021-05-19T17:07:18.647Z
I am somewhat confused. Are there any resource constraints at all?
Giskard
<microeconomics><pareto-efficiency><nonexpert><uncertainty>
Uncertainty and Pareto efficient policies
Maybeline Lee
# Uncertainty and Pareto efficient policies

(Asked by: username_0 on 2021-03-14)

There are two economic agents $i\in \{1,2\}$ with state dependent utility $u\_{is}=-(x-b\_{is})^2$ where $x\in R$ and $b\_{is}\in R$ is bliss point of $i$ in state $s\in\{1,2\}$. Assume $b\_{1s}\lt b\_{2s}$ for $\forall s\in\{1,2\}$. State $s\in\{1,2\}$ occurs with probability $\pi\_{s}$. We denote $U\_i(x\_1,x\_2)=\pi\_1u\_{i1}(x\_1)+\pi\_2u\_{i2}(x\_2)$.

Ex-post Pareto efficient policies in state 1 and 2, $x\_1^\*$ and $x\_2^\*$ respectively, I found them to be as follows:

Any policy in $[b\_{11},b\_{21}]$ is ex- post Pareto efficient in state 1 and any policy not in $[b\_{11},b\_{21}]$ is ex-post Pareto inefficient in state 1. Hence $x\_1^\* = z$, where $z$ is any $z\in [b\_{11}, b\_{21}].$ By similar arguments $x\_2^\* = z$, where $z$ is any $z\in[b\_{12}, b\_{22}].$

But how would we characterize ex-ante Pareto efficient policy pairs, ${x\_1^\*,x\_2^\*}?$

**Tags:** microeconomics, pareto-efficiency, nonexpert, uncertainty
> **Comments:**
> **username_1**: I am somewhat confused. Are there any resource constraints at all?  

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## 1 answers

### Answer by username_2 on 2021-05-19. Score: 3

Pareto efficient allocations can be found by maximizing a weighted average of the utilities of the agents. Let $\lambda$ be the *Pareto weight* for agent 1 and $1 - \lambda$ the weight for agent 2 (where $\lambda \in [0,1]$). This then gives the following problem:
$$
\max\_{x\_1, x\_2} -\lambda \left[\pi\_1(x\_1 - b\_{11})^2 + \pi\_2(x\_2 - b\_{12})^2\right] - (1-\lambda)\left[ \pi\_1 (x\_1 - b\_{21})^2 + \pi\_2 (x\_2 - b\_{22})^2\right].
$$
The first order conditions with respect to $x\_1$ and $x\_2$ give:
$$
\lambda \pi\_1 2(x\_1 - b\_{11}) + (1-\lambda) \pi\_1 2(x\_1 - b\_{21}) = 0\\
\lambda \pi\_2 2(x\_2 - b\_{21}) + (1-\lambda) \pi\_2 2(x\_2 - b\_{22}) = 0.
$$

Simplifying gives:
$$
x\_1 = \lambda b\_{11} + (1-\lambda) b\_{21},\\
x\_2 = \lambda b\_{21} + (1-\lambda) b\_{22}.
$$
So $x\_1$ and $x\_2$ are weighted averages of the bliss points.


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