In a manufacturing firm, the total number of units produced is given by the Cobb-Douglas production function below:

P (x, y, z) = 60 x^1/6 y^1/3 z^1/2

where x is the number of units of labour, y is the number of units of capital and z is the number of units of equipment.

(a) Show that a doubling of the number of units for labour, capital and equipment will result in a
doubling of production P.

(b) If the firm has a total budget of B dollars and the company can buy x, y and z at $1600, $240

and $200 per unit respectively, how should it distribute its spending among the three categories in order to maximize production?