Uncertainty and Pareto efficient policies

Question

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, ${x_1^*,x_2^*}?$

Answer 1

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.