资料 练习题 真题 2013考研搜集 专业课 尼科尔森微观基本原理与扩展全套资料 尼克尔森微观经济理论-基本原理与扩展 第九版答案 ch07

Chapter 7/Production Functions v 31CHAPTER 7 PRODUCTION FUNCTIONS Because the problems in this chapter do not involve optimization (cost minimization principles are not presented until Chapter 8), they tend to have a rather uninteresting focus on functional form. Computation of marginal and average productivity functions is stressed along with a few applications of Eulers theorem. Instructors may want to assign one or two of these problems for practice with specific functions, but the focus for Part 3 problems should probably be on those in Chapters 8 and 9. Comments on Problems 7.1This problem illustrates the isoquant map for fixed proportions production functions. Parts (c) and (d) show how variable proportions situations might be viewed as limiting cases of a number of fixed proportions technologies. 7.2This provides some practice with graphing isoquants and marginal productivity relationships. 7.3This problem explores a specific Cobb-Douglas case and begins to introduce some ideas about cost minimization and its relationship to marginal productivities. 7.4This is a theoretical problem that explores the concept of “local returns to scale.” The final part to the problem illustrates a rather synthetic production that exhibits variable returns to scale. 7.5This is a thorough examination of all of the properties of the general two-input Cobb-Douglas production function. 7.6This problem is an examination of the marginal productivity relations for the CES production function. 7.7This illustrates a generalized Leontief production function. Provides a two-input illustration of the general case, which is treated in the extensions. 7.8Application of Eulers theorem to analyze what are sometimes termed the “stages” of the average-marginal productivity relationship. The terms “extensive” and “intensive” margin of production might also be introduced here, although that usage appears to be archaic. 7.9Another simple application of Eulers theorem that shows in some cases cross second-order partials in production functions may have determinable signs. 7.10This is an examination of the functional definition of the elasticity of substitution. It shows that the definition can be applied to non-constant returns to scale function if returns to scale takes a simple form. Solutions7.1 a., b. function 1: use 10k, 5lfunction 2: use 8k, 8l c.Function 1: 2k + l = 8,0002.5(2k + l) = 20,0005.0k + 2.5l = 20,000 Function 2: k + l = 5,000 4(k + l) = 20,000 4k + 4l = 20,000 Thus, 9.0k, 6.5l is on the 40,000 isoquant Function 1: 3.75(2k + l) = 30,000 7.50k + 3.75l = 30,000 Function 2: 2(k + l) = 10,000 2k + 2l = 10,000 Thus, 9.5k, 5.75l is on the 40,000 isoquant 7.2 a.When k = 10, q = 10L 80 .2L2.Marginal productivity = , maximum at l = 25 The total product curve is concave.To graph this curve: When l = 20, q = 40, APl = 0 where l = 10, 40. b. See above graph. c. . d.Doubling of k and l here multiplies output by 4 (compare a and c). Hence the function exhibits increasing returns to scale. 7.3a.q = 10 if k = 100, l = 100. Total cost is 10,000. b. Setting these equal yields . Solving . So k = 3.3, l = 13.2. Total cost is 8,250.c.Because the production function is constant returns to scale, just increase all inputs and output by the ratio 10,000/8250 = 1.21. Hence, k = 4, l = 16, q = 12.1. d.Carlas ability to influence the decision depends on whether she provides a unique input for Cheers. 7.4a.If . b. c.Hence, . d. The intuitive reason for the changing scale elasticity is that this function has an upper bound of q = 1 and gains from increasing the inputs decline as q approaches this bound. 7.5a. b. . c. d.Quasiconcavity follows from the signs in part a. e.Concavity looks at: This expression is positive (and the function is concave) only if 7.6 a. Similar manipulations yield b.c. Putting these over a common denominator yields which shows constant returns to scale. d.Since = the result follows directly from part a.7.7a.If doubling k, l gives b. which are homogeneous of degree zero with respect to k and l and exhibit diminishing marginal productivities. c. which clearly varies for different values of k, l. 7.8 exhibits constant returns to scale. Thus, for any t 0,.Eulers theorem states . Here we apply the theorem for the case where t = 1: hence, . If , hence no firm would ever produce in such a range. 7.9 If , partial differentiation by l yields . Because , . That is, with only two inputs and constant returns to scale, an increase in one input must increase the marginal productivity of the other input. 7.10a.This