Exam (elaborations) TEST BANK FOR Microeconomic Analysis 3rd Edition By Hal R. Varian (Solution Manual)

Chapter 1. Technology 1.1 False. There are many counterexamples. Consider the technology generated by a production function f(x) = x2. The production set is Y = f(y;−x) : y  x2g which is certainly not convex, but the input requirement set is V (y) = fx : x p yg which is a convex set. 1.2 It doesn't change. 1.3 1 = a and 2 = b. 1.4 Let y(t) = f(tx). Then dy dt = Xn i=1 @f(x) @xi xi; so that 1 y dy dt = 1 f(x) Xn i=1 @f(x) @xi xi: 1.5 Substitute txi for i = 1;2 to get f(tx1; tx2) = [(tx1) + (tx2)] 1  = t[x 1 + x 2] 1  = tf(x1; x2): This implies that the CES function exhibits constant returns to scale and hence has an elasticity of scale of 1. 1.6 This is half true: if g0(x) 0, then the function must be strictly increasing, but the converse is not true. Consider, for example, the function g(x) = x3. This is strictly increasing, but g0(0) = 0. 1.7 Let f(x) = g(h(x)) and suppose that g(h(x)) = g(h(x0)). Since g is monotonic, it follows that h(x) = h(x0). Now g(h(tx)) = g(th(x)) and g(h(tx0)) = g(th(x0)) which gives us the required result. 1.8 A homothetic function can be written as g(h(x)) where h(x) is homogeneous of degree 1. Hence the TRS of a homothetic function has the 2 ANSWERS form g0(h(x)) @h @x1 g0(h(x)) @h @x2 = @h @x1 @h @x2 : That is, the TRS of a homothetic function is just the TRS of the underlying homogeneous function. But we already know that the TRS of a homogeneous function has the required property. 1.9 Note that we can write (a1 + a2) 1   a1 a1 + a2 x 1 + a2 a1 + a2 x 2 1  : Now simply dene b = a1=(a1 + a2) and A = (a1 +a2)1  . 1.10 To prove convexity, we must show that for all y and y0 in Y and 0  t  1, we must have ty + (1−t)y0 in Y . But divisibility implies that ty and (1 − t)y0 are in Y , and additivity implies that their sum is in Y . To show constant returns to scale, we must show that if y is in Y , and s 0, we must have sy in Y. Given any s 0, let n be a nonnegative integer such that n  s  n − 1. By additivity, ny is in Y ; 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 Leontief technology except we are measuring output in units of log y 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(x1; x2) are both positive so the technology is monotonic. For the isoquant to be convex to the origin, it is sucient (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 semidenite. The rst principal minor of the Hessian must have a negative determinant, and the second principal minor must have a nonnegative determinant. @2f(x) @x21 = −1 4x −32 1 x 12 2 @2f(x) @x1@x2 = 1 4x −1 2 1 x −12 2 @2f(x) @x22 = −1 4x 12 1 x −3 2 2