微觀經(jīng)濟理論基本原理與擴展(尼克爾森),第11版 課后習題答案microeconomic theory basic principles and extensions,student solution manual,wal

1、 The problems in this chapter are primarily mathematical. They are intended to give students some practice with the concepts introduced in Chapter 2, but the problems in themselves offer few economic insights. Conseq

2、uently, no commentary is provided. Results from some of the analytical problems are used in later chapters, however, and in those cases the student will be directed to here. Solutions 2.1 2 2 ( , ) 4 3 . U x y x y ? ?

3、a. 8 , x U x ?6 . y U y ?b. 8, x U ? 6. y U ?c. 8 6 . dU xdx ydy ? ?d. 4 . 3xyU dy xdx U y ? ? ? ?e. 2 2 4 1 3 2 16. ? ? ? ?f. 4 1 2. 3 2 3dy = = dx? ? ? ?g. The 16 U = contour line is an ellipse centered at th

4、e origin. The slope of the line at any point is given by 4 3 . dy dx x y ? ?2.2 a. Profits are given by 2 2 40 100. R C q q ? ? ? ? ? ? ?The maximum value is found by setting the derivative equal to 0: 4 40 0 d =

5、 q + dq? ? ?implies * 10 q ?and * 100.? ?CHAPTER 2: Mathematics for Microeconomics Chapter 2: Mathematics for Microeconomics 3 c. Differentiation of the original function at its optimal value yields ** 2 ( ) 0.5( )

6、. f t t g? ? ? ?Because the optimal value of t depends on , g 2 ** 2 ( ) 40 800 0.5( ) 0.5 , 2f t = t g g g? ? ? ? ? ? ? ? ? ? ? ? ?as was also shown in part c. d. If 32, g ? * 5 4. t ?Maximum height is 800 32 25.

7、?If 32.1, g ?maximum height is 800 32.1 24.92, ?a reduction of 0.08. This could have been predicted from the envelope theorem, since *2800 25 ( ) (.01) 0.08. 32 32 df t dg ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?2.6 a. Th

8、is is the volume of a rectangular solid made from a piece of metal which is x by 3x with the defined corner squares removed. b. The first order condition for maximum volume is given by 2 2 3 16 12 0. V x xt t t? ? ?

9、? ? ?Applying the quadratic formula to this expression yields 2 2 16 256 144 16 10.6 0.225 . 24 24x x x x x t x ? ? ? ? ? ?The second value given by the quadratic (1.11 ) x is obviously extraneous. c. If 0.225 , t x

10、?3 3 3 3 0.67 .04 .05 0.68 . V x x x x ? ? ? ?So volume increases without limit. d. This would require a solution using the Lagrangian method. The optimal solution requires solving three non-linear simultaneous equati

11、ons, a task not undertaken here. But it seems clear that the solution would involve a different relationship between t and x than in parts a–c. 2.7 a. Set up the Lagrangian: 1 2 1 2 5ln ( ). x x k x x ? ? ? ? ?