Solutions: Utility and Function
Answers to Exercises
Microeconomic Analysis
Third Edition
Hal R. Varian
University of California at Berkeley
W. W. Norton & Company • New York • London
Copyright c 1992, 1984, 1978 by W. W. Norton & Company, Inc.
All rights reserved Printed in the United States of America
THIRD EDITION
0-393-96282-2
W. W. Norton & Company, Inc., 500 Fifth Avenue, New York, N.Y. 10110 W. W. Norton Ltd., 10 Coptic Street, London WC1A 1PU
234567890
ANSWERS
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 = {(y, −x) : y ≤ x2 } which is certainly not convex, but the input re√ quirement set is V (y) = {x : x ≥ y} 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 so that 1 dy 1 = y dt f(x) 1.5 Substitute txi for i = 1, 2 to get f(tx1 , tx2 ) = [(tx1 )ρ + (tx2 )ρ ] ρ = t[xρ + xρ ] ρ = tf(x1 , x2 ). 1 2 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 g (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 g (0) = 0. 1.7 Let f(x) = g(h(x)) and suppose that g(h(x)) = g(h(x )). Since g is monotonic, it follows that h(x) = h(x ). Now g(h(tx)) = g(th(x)) and g(h(tx )) = g(th(x )) 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
1 1
n
i=1
∂f(x) xi , ∂xi n i=1
∂f(x) xi . ∂xi
2 ANSWERS
form
∂h g (h(x)) ∂h ∂x1 ∂x1 = . ∂h g (h(x)) ∂h ∂x2 ∂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
Microeconomic Analysis
Third Edition
Hal R. Varian
University of California at Berkeley
W. W. Norton & Company • New York • London
Copyright c 1992, 1984, 1978 by W. W. Norton & Company, Inc.
All rights reserved Printed in the United States of America
THIRD EDITION
0-393-96282-2
W. W. Norton & Company, Inc., 500 Fifth Avenue, New York, N.Y. 10110 W. W. Norton Ltd., 10 Coptic Street, London WC1A 1PU
234567890
ANSWERS
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 = {(y, −x) : y ≤ x2 } which is certainly not convex, but the input re√ quirement set is V (y) = {x : x ≥ y} 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 so that 1 dy 1 = y dt f(x) 1.5 Substitute txi for i = 1, 2 to get f(tx1 , tx2 ) = [(tx1 )ρ + (tx2 )ρ ] ρ = t[xρ + xρ ] ρ = tf(x1 , x2 ). 1 2 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 g (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 g (0) = 0. 1.7 Let f(x) = g(h(x)) and suppose that g(h(x)) = g(h(x )). Since g is monotonic, it follows that h(x) = h(x ). Now g(h(tx)) = g(th(x)) and g(h(tx )) = g(th(x )) 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
1 1
n
i=1
∂f(x) xi , ∂xi n i=1
∂f(x) xi . ∂xi
2 ANSWERS
form
∂h g (h(x)) ∂h ∂x1 ∂x1 = . ∂h g (h(x)) ∂h ∂x2 ∂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