# Uncertainty and Pareto efficient policies

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

• I am somewhat confused. Are there any resource constraints at all? – Giskard May 19 at 17:07

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.