Problem Set 3. Profit Maximization and Profit Functions ...
Problem Set 3. Profit Maximization and Profit Functions
EconS 526
1. The production function for good z is () = 100x - x2 where x is an input. The price of good z is p and the input price for x is w. a. Set up the problem for a profit maximizing firm and solve for the demand function for x. Do not forget to show the first order condition and show if the second order condition satisfies the condition for a maximum. b. What condition is required for x>0? When will the firm opt to produce x=0? c. Derive the profit function. What is the derivative of the profit function with respect to w? Show the expression and interpret it.
a.
First order condition,
max (100x - x2) -
Therefore, = 50 - 2.
(100 - 2) - = 0
The SOC is -2pw for x>0. If 100p
0.
The
determinant
of
the
Hessian
is
2 2
2 2
-
2
2
=
0.24-1.60.40.24-1.60.4
-
(0.16-0.6-0.6)2 > 0
This is a max.
Finally, the profit function is,
= (02.42)4
- 2 (02.42)5.
3. The following production function characterizes production of good y, (, ) = x + z where 0< ................
................
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