 Solution

Use the same Kakutani fixed-point theorem pair as in part (b). If any coordinate of $p^{\ast }$ vanished, the additional boundary condition would force some coordinate of every member of $z(p^{\ast })$, including $v^{\ast }$, to be positive. This contradicts $v^{\ast }\le 0$. Hence $p_i^{\ast }>0$ for every $i$.

Now Walras law reads ${\sum }_i{p}_i^{\ast }{v}_i^{\ast }=0$, a sum of nonpositive terms with strictly positive coefficients. Every term must vanish, soThe boundary hypothesis first excludes boundary prices; strict positivity of all prices then turns nonpositive excess demand into exact market clearing.