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ECO 3145 Mathematical Economics I

Module 3 – Static Optimization III – unconstrained – sufficient conditions for global optima

Practice Problems

1.   (C&W, Exercise 11.5, #5.) The equation x 2  + y2  = 4 describes a circle centred on (0,0) with radius of 2.

a.)  Give a geometric interpretation of the set (x, y) x 2  + y2    4 .

b.) Is this set convex?

2.   Sketch the graph of each of the following sets and indicate whether it is convex.

a.) (x, y) y = ex 

b.) (x, y) y  e x 

c.) (x, y) y  13 − x 2

d.) (x, y) xy  1 , x  0 , y  0

3.   (C&W, Exercise 11.5, #7.) Given the two vectors u = 10  6 and v = 4  8 , which of the following are convex combinations of u and v?

a.) 7  7                b.) 5.2   7.6               c.) 6.2   8.2

4.   Using the definition of concave (convex) functions, check whether the following functions are concave, convex, strictly concave, strictly convex, or neither.

a.) y = x 2                        b.) y = x 1(2)  + 2x2(2)

5.   Using theorem I of concave (convex) functions, check whether the following functions are concave, convex, strictly concave, strictly convex, or neither.

a.) y = −x 2                    b.) y = (x1  + x 2 )2             c.) y = −x1 x 2

6.   Consider your responses to question II.2 in the Module 1 problems.

a.) Using theorem II of concave (convex) functions, note the cases in which it is possible, by

means of the second order conditions, to determine whether the function is concave, convex, strictly concave, strictly convex, or neither.

b.) If the function is concave or convex, what does that imply about the local minimum or maximum?