# 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.

See attached file for full problem description.

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