= Exponential-utility risk sharing
{title2=$\Gamma=(\sum_j\gamma_j^{-1})^{-1}$}
With positive welfare weights $a_j$, maximize $\sum_ja_j\beta_j^t[-e^{-\gamma_jc_j}/\gamma_j]$ subject to $\sum_jc_j=d$. Interior <first-order conditions> give $a_j\beta_j^te^{-\gamma_jc_j}=\nu$. Put $\Gamma=(\sum_j\gamma_j^{-1})^{-1}$, $\log\bar\beta=\Gamma\sum_j\gamma_j^{-1}\log\beta_j$ and $\log A=\Gamma\sum_j\gamma_j^{-1}\log a_j$. Then
$$
\nu=A\bar\beta^te^{-\Gamma d},\qquad
c_j=\frac\Gamma{\gamma_j}d+\frac{t}{\gamma_j}(\log\beta_j-\log\bar\beta)+\frac1{\gamma_j}(\log a_j-\log A).
$$
If <consumption> must be nonnegative, the interior formula is replaced by $c_j=\max\{0,(\log a_j+t\log\beta_j-\log\nu)/\gamma_j\}$, with $\nu$ determined by resource clearing. These are Pareto allocations; realizing them as a <competitive equilibrium with one productive asset> requires feasible financing. For common $\beta_j=\beta$ and initial ownership $\Gamma/\gamma_j$, constant holdings and $C_t^j=(\Gamma/\gamma_j)d_t$ give such an equilibrium whenever fundamental prices and the utility sums are finite.
Back to article page
