《高级微观经济学(第三版) 答案》由会员分享，可在线阅读，更多相关《高级微观经济学(第三版) 答案（53页珍藏版）》请在金锄头文库上搜索。
1、Answers to Exercises Microeconomic Analysis Third Edition Hal R. Varian University of California at Berkeley W. W. Norton since s/n 1, divisibility implies (s/n)ny = sy is in Y . 1.11.a This is closed and nonempty for all y 0 (if we allow inputs to be negative). The isoquants look just like the Leon
2、tief technology except we are measuring output in units of logy rather than y. Hence, the shape of the isoquants will be the same. It follows that the technology is monotonic and convex. 1.11.b This is nonempty but not closed. It is monotonic and convex. 1.11.c This is regular. The derivatives of f(
3、x1,x2) are both positive so the technology is monotonic. For the isoquant to be convex to the origin, it is suffi cient (but not necessary) that the production function is concave. To check this, form a matrix using the second derivatives of the production function, and see if it is negative semidef
4、i nite. The fi rst principal minor of the Hessian must have a negative determinant, and the second principal minor must have a nonnegative determinant. 2f(x) x2 1 = 1 4x 3 2 1 x 1 2 2 2f(x) x1x2 = 1 4x 1 2 1 x 1 2 2 2f(x) x2 2 = 1 4x 1 2 1x 3 2 2 Ch. 2 PROFIT MAXIMIZATION3 Hessian = “ 1 4x 3/2 1 x1/
5、2 2 1 4x 1/2 1 x1/2 2 1 4x 1/2 1 x1/2 2 1 4x 1/2 1 x3/2 2 # D1= 1 4x 3/2 1 x1/2 2 1. It is monotonic and weakly convex. 1.11.f This is regular. To check monotonicity, write down the production function f(x) = ax1 x 1x2+ bx2and compute f(x) x1 = a 1 2x 1/2 1 x1/2 2 . This is positive only if a 1 2 q
6、x2 x1, thus the input requirement set is not always monotonic. Looking at the Hessian of f, its determinant is zero, and the determinant of the fi rst principal minor is positive. Therefore f is not concave. This alone is not suffi cient to show that the input requirement sets are not convex. But we
7、 can say even more: f is convex; therefore, all sets of the form x1,x2:ax1 x 1x2+ bx2 y for all choices of y are convex. Except for the border points this is just the complement of the input requirement sets we are interested in (the inequality sign goes in the wrong direction). As complements of co
8、nvex sets (such that the border line is not a straight line) our input requirement sets can therefore not be themselves convex. 1.11.g This function is the successive application of a linear and a Leontief function, so it has all of the properties possessed by these two types of functions, including
9、 being regular, monotonic, and convex. Chapter 2. Profit Maximization 4 ANSWERS 2.1 For profi t maximization, the Kuhn-Tucker theorem requires the follow- ing three inequalities to hold ? pf(x ) xj wj ? x j = 0, pf(x ) xj wj 0, x j 0. Note that if x j 0, then we must have wj/p = f(x)/xj. 2.2 Suppose
10、 that x0 is a profi t-maximizing bundle with positive profi ts (x0) 0. Since f(tx0) tf(x0), for t 1, we have (tx0) = pf(tx0) twx0 t(pf(x0) wx0) t(x0) (x0). Therefore, x0 could not possibly be a profi t-maximizing bundle. 2.3 In the text the supply function and the factor demands were computed for th
11、is technology. Using those results, the profi t function is given by (p,w) = p ? w ap ?a a1 w ? w ap ?1 a1 . To prove homogeneity, note that (tp,tw) = tp ? w ap ?a a1 tw ? w ap ?1 a1 = t(p,w), which implies that (p,w) is a homogeneous function of degree 1. Before computing the Hessian matrix, factor
12、 the profi t function in the following way: (p,w) = p 1 1aw a a1 ? a a 1a a 1 1a ? = p 1 1aw a a1(a), where (a) is strictly positive for 0 0 and x j = 0, the above conditions imply f(x) xi f(x) xj wi wj . 8 ANSWERS This means that it would decrease cost to substitute xifor xj, but since there is no
13、xjused, this is not possible. If we have interior solutions for both xiand xj, equality must hold. 4.3 Following the logic of the previous exercise, we equate marginal costs to fi nd y1= 1. We also know y1+ y2= y, so we can combine these two equations to get y2= y1. It appears that the cost function
14、 is c(y) = 1/2+y1 = y1/2. However, on refl ection this cant be right: it is obviously better to produce everything in plant 1 if y1 1/2, we have x1= x2= y/3. The cost function is then c(w1,w2,y) = minw1,w2,(w1+ w2)/3y. 5.5 The input requirement set is not convex.Since y = maxx1,x2, the fi rm will us
15、e whichever factor is cheaper; hence the cost function is c(w1,w2,y) = minw1,w2y. The factor demand function for factor 1 has the form x1= ( yif w1 w2 . 5.6 We have a = 1/2 and c = 1/2 by homogeneity, and b = 3 since x1/w2= x2/w1. 5.7 Set up the minimization problem min x1+ x2 x1x2= y. Substitute to
16、 get the unconstrained minimization problem min x1+ y/x1. The fi rst-order condition is 1 y/x2 1, which implies x1= y. By symmetry, x2= y. We are given that 2y = 4, so y = 2, from which it follows that y = 4. Ch. 5 COST FUNCTION11 5.8 If p = 2, the fi rm will produce 1 unit of output. If p = 1, the fi rst- order condition suggests y = 1/2, but this yields negative profi ts.The fi r
