Let $X \equiv\left(X_{0}, X_{1}, \ldots, X_{J}\right)^{T}$ be a zero-mean Gaussian vector, with covariance matrix $V=\left(v_{j k}\right)$. In a simple single-period economy with $J$ agents, agent $i$ will receive $X_{i}$ at time $1(i=1, \ldots, J)$. If $Y$ is a contingent claim to be paid at time 1 , define agent $i$ 's reservation bid price for $Y$, assuming his preferences are given by $E\left[U_{i}\left(X_{i}+Z\right)\right]$ for any contingent claim $Z$.

Assuming that $U_{i}(x) \equiv-\exp \left(-\gamma_{i} x\right)$ for each $i$, where $\gamma_{i}>0$, show that agent $i$ 's reservation bid price for $\lambda$ units of $X_{0}$ is

$$p_{i}(\lambda)=-\frac{1}{2} \gamma_{i}\left(\lambda^{2} v_{00}+2 \lambda v_{0 i}\right)$$

As $\lambda \rightarrow 0$, find the limit of agent $i$ 's per-unit reservation bid price for $X_{0}$, and comment on the expression you obtain.

The agents bargain, and reach an equilibrium. Assuming that the contingent claim $X_{0}$ is in zero net supply, show that the equilibrium price of $X_{0}$ will be

$$p=-\Gamma v_{0 \bullet}$$

where $\Gamma^{-1}=\sum_{i=1}^{J} \gamma_{i}^{-1}$ and $v_{0 \bullet}=\sum_{i=1}^{J} v_{0 i}$. Show that at that price agent $i$ will choose to buy

$$\theta_{i}=\left(\Gamma v_{0 \bullet}-\gamma_{i} v_{0 i}\right) /\left(\gamma_{i} v_{00}\right)$$

units of $X_{0}$.

By computing the improvement in agent $i$ 's expected utility, show that the value to agent $i$ of access to this market is equal to a fixed payment of

$$\frac{\left(\gamma_{i} v_{0 i}-\Gamma v_{0 \bullet}\right)^{2}}{2 \gamma_{i} v_{00}}$$