= Solution
For a <predictable> position $\pi$, the current net wealth increment $\pi\Delta S_t+Y_t-y$ is conditionally <Gaussian> with mean $\pi\mu+a-y$ and <variance> $\pi^2\sigma^2+2\pi\rho b\sigma+b^2$. Its exponential-loss multiplier is
$$
\exp\left[-\gamma(\pi\mu+a-y)+\tfrac12\gamma^2(\pi^2\sigma^2+2\pi\rho b\sigma+b^2)\right].
$$
Differentiate its quadratic exponent. The unique minimizing position is
$$
\boxed{\pi_t^{\mathrm{swap}}=\frac\mu{\gamma\sigma^2}-\frac{\rho b}{\sigma},\quad1\leq t\leq T.}
$$
The first term is the speculative demand; the second offsets the part of the income correlated with the tradable price increment. Independent period vectors and <backward induction> justify using this same position at every date. Completing the square gives minimal exponent $-\kappa-\gamma d(y)$, where
$$
d(y)=a-y-\frac{\mu\rho b}{\sigma}-\frac\gamma2b^2(1-\rho^2).
$$
Therefore <hedging a Gaussian income stream with exponential utility> has value
$$
\boxed{V_0^{\mathrm{swap}}(x;y)=-\exp[-\gamma x-T\kappa-\gamma T d(y)].}
$$
The residual <variance> is $b^2(1-\rho^2)$; it disappears for perfectly correlated income.
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