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# Hessian Matrix : Maximizing Profit

Question: The profit maximizing input choice

A competitive firm's profit function can be written as
&#960; := 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 := L&#945; * k&#945;
the profit function takes the form
&#960; := p * L&#945; * k&#945; - w * L - r * k

What restriction must be placed on the parameter &#945; to ensure that the second order conditions for an extreme value of &#960; are satisfied.

See attached file for full problem description.

#### Solution Preview

We start from the first order condition for the firm to maximize its profit:
F.O.C:
d&#960; / dL = p&#945;L&#945;-1 k&#945; - w = 0
d&#960; / dk = p&#945;L&#945; k&#945;-1 - r = 0
Rearrange:
(1) p&#945;L&#945;-1 k&#945; = w
(2) p&#945;L&#945; k&#945;-1 = r
equation (1) / (2):
k/L = w/r
or, k = wL / r
substitute into equation (1)
p&#945;L&#945;-1 (wL / r)&#945; = ...

#### Solution Summary

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

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