Hessian Matrix : Maximizing Profit
Question: The profit maximizing input choice
A competitive firm's profit function can be written as
π := p * q - w * L - r * k
where p is the competitive price of the product and w and r the per unit cost of the two inputs labour (L) and capital (k).
the firm takes p,w, and r as given and chooses L and k to maximize profits.
If the relationship between inputs and output is given by
q := Lα * kα
the profit function takes the form
π := p * Lα * kα - w * L - r * k
What restriction must be placed on the parameter α to ensure that the second order conditions for an extreme value of π are satisfied.
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