In a pure exchange economy, there are $J$ agents, and $d$ goods. Agent $j$ initially holds an endowment $x_{j} \in \mathbb{R}^{d}$ of the $d$ different goods, $j=1, \ldots, J$. Agent $j$ has preferences given by a concave utility function $U_{j}: \mathbb{R}^{d} \rightarrow \mathbb{R}$ which is strictly increasing in each of its arguments, and is twice continuously differentiable. Thus agent $j$ prefers $y \in \mathbb{R}^{d}$ to $x \in \mathbb{R}^{d}$ if and only if $U_{j}(y) \geqslant U_{j}(x)$.

The agents meet and engage in mutually beneficial trades. Thus if agent $i$ holding $z_{i}$ meets agent $j$ holding $z_{j}$, then the amounts $z_{i}^{\prime}$ held by agent $i$ and $z_{j}^{\prime}$ held by agent $j$ after trading must satisfy $U_{i}\left(z_{i}^{\prime}\right) \geqslant U_{i}\left(z_{i}\right), U_{j}\left(z_{j}^{\prime}\right) \geqslant U_{j}\left(z_{j}\right)$, and $z_{i}^{\prime}+z_{j}^{\prime}=z_{i}+z_{j}$. Meeting and trading continues until, finally, agent $j$ holds $y_{j} \in \mathbb{R}^{d}$, where

$$\sum_{j} x_{j}=\sum_{j} y_{j}$$

and there are no further mutually beneficial trades available to any pair of agents. Prove that there must exist a vector $v \in \mathbb{R}^{d}$ and positive scalars $\lambda_{1}, \ldots, \lambda_{J}$ such that

$$\nabla U_{j}\left(y_{j}\right)=\lambda_{j} v$$

for all $j$. Show that for some positive $a_{1}, \ldots, a_{J}$ the final allocations $y_{j}$ are what would be achieved by a social planner, whose objective is to obtain

$$\max \sum_{j} a_{j} U_{j}\left(y_{j}\right) \quad \text { subject to } \sum_{j} y_{j}=\sum_{j} x_{j}$$