Assume, as in Problem $12\mathrm{.}2,$ that prices are completely unresponsive to unanticipated monetary shocks for one period and completely flexible thereafter. Assume also that y $=$ c $$ ar and m $$ p $=$ b $+$ hy $$ ki hold each period. Suppose, however, that the money supply follows a random walk: mt $=$ mt$1+$ ut$,$ where ut is a mean$-$zero, serially uncorrelated disturbance.

$($a$)$ Let Et denote expectations as of period t$.$ Explain why, for any t$,$ Et $[$Et$+1[$pt$+2]$ pt$+1]=0,$ and thus why Etmt$+1$ Etpt$+1=$ b $+$ hyn $$ krn$,$ where yn and rn are the flexible$-$price levels of y and r$.$

$($b$)$ Use the result in part $($a$)$ to solve for yt$,$ pt$,$ it$,$ and rt in terms of mt$1$ and ut$.$