= Solution
Let
$$
\boxed{\overline y=a-\frac{\mu\rho b}{\sigma}-\frac\gamma2b^2(1-\rho^2).}
$$
Part (c) has $d(y)=\overline y-y$. Comparing the optimized values with and without the contract, $V_0^{\mathrm{swap}}=-e^{-\gamma x-T\kappa-\gamma T(\overline y-y)}$ and $V_0=-e^{-\gamma x-T\kappa}$, shows that $y>\overline y$ makes the swap value strictly more negative. Thus \b[The investor prefers not to enter when $y>\overline y$.] He is indifferent at equality and prefers entry for a smaller payment.
The <periodic utility indifference payment for Gaussian income> equals the mean income less the cost of hedging its correlated component and the exponential-utility penalty for the residual unhedgeable <variance>. Comparing unoptimized strategies would not justify this price threshold.
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