In a two-period model, two agents enter a negotiation at time 0 . Agent $j$ knows that he will receive a random payment $X_{j}$ at time $1(j=1,2)$, where the joint distribution of $\left(X_{1}, X_{2}\right)$ is known to both agents, and $X_{1}+X_{2}>0$. At the outcome of the negotiation, there will be an agreed risk transfer random variable $Y$ which agent 1 will pay to agent 2 at time 1 . The objective of agent 1 is to maximize $E U_{1}\left(X_{1}-Y\right)$, and the objective of agent 2 is to maximize $E U_{2}\left(X_{2}+Y\right)$, where the functions $U_{j}$ are strictly increasing, strictly concave, $C^{2}$, and have the properties that

$$\lim _{x \downarrow 0} U_{j}^{\prime}(x)=+\infty, \quad \lim _{x \uparrow \infty} U_{j}^{\prime}(x)=0$$

Show that, unless there exists some $\lambda \in(0, \infty)$ such that

$$\frac{U_{1}^{\prime}\left(X_{1}-Y\right)}{U_{2}^{\prime}\left(X_{2}+Y\right)}=\lambda \quad \text { almost surely }$$

the risk transfer $Y$ could be altered to the benefit of both agents, and so would not be the conclusion of the negotiation.

Show that, for given $\lambda>0$, the relation $(*)$ determines a unique risk transfer $Y=Y_{\lambda}$, and that $X_{2}+Y_{\lambda}$ is a function of $X_{1}+X_{2}$.