$.$ The following question is based on the international factor movement problem

$($Lecture $9).$ Suppose the domestic production function is Y$=10$K$0\mathrm{.}5$L$0\mathrm{.}5.$

$(1)$ We first assume a closed economy. The domestic market has $50$ units of labor and $200$

units of capital. When oversea capital movement is permitted, $150$ units of capital move

from home to abroad, until domestic and foreign rental rates are equalized. Please

calculate the domestic rental rate r after the capital movement. $($Hint: you need to write

down the expression of MPK$)]$