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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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Please see attached file.

We start from the first order condition for the firm to maximize its profit:
F.O.C:
dπ / dL = pαLα-1 kα - w = 0
dπ / dk = pαLα kα-1 - r = 0
Rearrange:
(1) pαLα-1 kα = w
(2) pαLα kα-1 = r
equation (1) / (2):
k/L = w/r
or, k = wL / r
substitute into equation (1)
pαLα-1 (wL / r)α = ...

Solution Summary

Hessian matrices are applied to calculating the maximum profit.

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