Consider the case where a firm uses two inputs: capital and Labor, The profit function is

Where P is the output price and w and r are the wage rate and rental rate, respectively.

a.Assume the Cobb-Douglas function of the form

b.Find the optimal values of labor and Capital for the profit function.

c.Derive the conditions under which profit is a maximum.

d.Find the maximum value function

Numerical version of this model can be discussed but first solve for general model.