Suppose an investor has utility function $U$. There are $n$ risky assets with rates of return $r_{i}, i=1,2, \ldots, n$, and one risk-free asset with rate of return $r_{f}$. The investor has initial wealth $W_{0}$. Suppose that the optimal portfolio for this investor has (random) payoff $x^{*}$. Show that

\[\mathrm{E}\left[U^{\prime}\left(x^{*}\right)\left(r_{i}-r_{f}\right)\right]=0\]

for $i=1,2, \ldots, n$.