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Hessian Matrix : Maximizing Profit

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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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Solution Summary

Hessian matrices are applied to calculating the maximum profit. The competitive price of the products are determined.

Solution Preview

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 = ...

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