QUESTION $7(20$ MARKS$)$

Suppose Metnet Co$.$ has the production function of

$z=400x^{0\mathrm{.}6}{y}^{0\mathrm{.}4}$

where $x$ is the number of labor units, $y$ is the number of capital units, and $z$ is the number of production units. Suppose that labor costs RM$150$ per unit, capital costs RM$100$ per unit, and the total costs of labor and capital is equal to RM$100,000.$

a$)$ Find the number of units of labor and the number of units of capital that maximize production. Confirm your answer by constructing the bordered Hessian matrix.

$(18$ marks$)$

b$)$ Determine the maximum level of production based on your answer in $7($a$)$

$(2$ marks$)$