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ECO 3145 Mathematical Economics I

Module 2  Static Optimization II  unconstrained  sufficient conditions for local optima

Practice Problems

1.   For question II. 1 in the Module 1 problems, derive the second differential. Put the latter in matrix form.

2.   For question II.2 in the Module 1 problems, determine whether each stationary point is a local maximum, local minimum, or neither.

3.   Consider the function given in part (b.) of question II. 1 in the Module 1 problems.

a.) How many leading principal sub-matrices does the Hessian of the function have? Write

them out.

b.) What are the values of the leading principal minors?

c.) What shape is the function positive definite, negative definite, or neither?

4.   For question II.3 in the Module 1 problems, prove that your recommended allocation of pages achieves a maximum of sales.

5.   A monopolist produces the same product in two factories, A and B. Total costs are as follows:

CA  = QA  + QA(2)   ,

CB  = 6QB  +1.5QB(2)   ,

where Q represents the quantity of the product. The total quantity is QT  = QA  + QB . The

inverse demand function for the product is

P = 56 − 2QT  .

What level of output should the monopolist produce in each factory in order to maximize profits? Prove that the solution you have found is actually a local maximum.

6.   Given the function f(x1 , x 2 ) = x1(2)  + x2(2) , find the stationary point(s) and determine whether it is a local maximum, local minimum, or neither under the following conditions.

a.)   0,   0

b.)   0,   0

a.)   and  have opposite signs

7.   Find the stationary points of the following functions. Use the second order conditions to determine whether the point is a local maximum or a local minimum.

a.) f(x1 , x 2 , x 3 ) = 2x1(2)  − 21x1  − 3x1 x 2  + 3x2(2)  − 2x 2 x 3  + x3(2)

b.) f(x1 , x 2 , x 3 ) = 2x1 x 2  −  x1(2)  − 3x2(2)  + x 2 x 3  −1.5x3(2)  +10x3