3145-2-prob - problems PDF

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ECO 3145 Mathematical Economics I Lecture 2 Homework Problems

1. Chiang pp.226-227 (p.239), #1 (do a couple), #2 (do a couple), #3*, #4* (* indicates more difficult) 2. Chiang p.233 (p.245), #4* 3. Chiang p.241 (p.253), #1 (do a couple), #2 4. Find the stationary points of each of the following functions. Determine whether each point is a local maximum, local minimum, or neither. a.) f ( x 1 , x 2 ) = 4x 1 − 2x 22 + x 21 + x 2 b.) f ( x 1 , x 2 ) = 8x 1 − x 21 + 14 x 2 − 7 x 22 c.) f ( x1 , x 2 ) = 4x1 − x1 + 13 x 2 − 2 x 2 d.) f ( x 1 , x 2 ) = x 1 + 3e x 2 − e x1 − e 3 x 2 e.) f ( x 1 , x 2 ) = 100 − 5x 1 + 4x 21 − 9x 2 + 5x 22 + 8x1 x2 f.) f ( x1 , x 2 ) = 13 x 31 + 3x1 x 2 + 2 x 1 − 32 x 22

5. As a consultant to the Journal of Important Stuff, you must determine the effect on sales of the number of pages devoted to economics (E) and the number of pages devoted to other subjects (A). After careful study, you have decided that the relationship between sales (S) and the allocation of pages is given by the following function: S = 100A + 310E − 12 A 2 − 2E 2 − AE

If the goal is to maximize sales, what will be your recommendation regarding the allocation of pages between economics and other subjects? Prove that you have achieved a maximum.

6. The profits of two cigarette manufacturers, Eventual Death (E) and Painful Death (P), depend upon advertising expenditures according to the following two functions. Π E = 1000A E − A 2E − A 2P Π P = 1000A P − A P A E − A 2P In these equations, Π represents profit and A represents advertising expenditures.

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a.) If each producer believes the advertising expenditure of the competitor is fixed, what will be the optimal level of expenditure and the maximum profit of each? b.) Now the two companies have decided to merge. Nonetheless, they keep the two distinct brands. The managers would like to maximize the combined profit of the two brands, i.e. Π = Π E + Π P . What levels of advertising expenditure, A E and AP, will they choose? Are the combined profits greater before or after the merger? (Note: You can deduce the answer to this last question by comparing AE and AP before and after the merger without calculating the new Π. Explain how.) c.) The managers of the merged company must decide whether to keep the distinct brands or merge them as well. In the case of a single, merged brand (Eventual Painful Death?) the profit function of the merged company is the same as in (b) except that A E = A P = A . What is the optimal level of A? Which is the best choice, separate brands or a single brand?

7. A monopolist produces the same product in two factories, A and B. Total costs are as follows: C A = Q A + Q 2A , C B = 6Q B + 1.5Q 2B , where Q represents the quantity of the product. The total quantity is Q T = Q A + Q B. The inverse demand function for the product is

P = 56 − 2Q T . What level of output should the monopolist produce in each factory in order to maximize profits?

8. Given the function f ( x 1 , x 2 ) = θx 21 + γx 22 , find the stationary point(s) and determine whether it is a local maximum, local minimum, or neither under the following conditions. a.) θ > 0, γ > 0 b.) θ < 0, γ < 0 d.) θ and γ have opposite signs

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9. 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 , x2 , x3 ) = 2x21 − 21x1 − 3x1 x2 + 3x22 − 2 x 2 x 3 + x23 2 2 2 b.) f ( x1 , x2 , x3 ) = 2 x1 x 2 − 12 x1 − 3x2 + x 2 x3 −1.5 x 3 +10 x 3

10. (Chiang and Wainwright, Exercise 11.5, #5.) The equation x 2 + y 2 = 4 describes a circle centred on (0,0) with radius of 2.

{

}

a.) Give a geometric interpretation of the set ( x, y) x 2 + y 2 ≤ 4 . b.) Is this set convex?

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

{ } b.) {( x, y) y ≥ e } c.) {( x, y) y ≤ 13 − x }

a.) ( x , y) y = e x x

2

d.) {( x, y) xy ≥ 1, x > 0 , y > 0} 12. (Chiang and Wainwright, 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]

13. 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 21 + 2x22

c.) y = 2x 12 − x 1x 2 + x 22

14. Using theorem I of concave (convex) functions, check whether the following functions are concave, convex, strictly concave, strictly convex, or neither. a.) y = − x2

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

c.) y = − x1 x 2

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15. Consider your responses to question 4. 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?

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