Microeconomic Analysis - Ed 3 - Hal Varian - Manual

# Microeconomic Analysis - Ed 3 - Hal Varian - Manual -...

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Answers to Exercises Microeconomic Analysis Third Edition Ha lR .Var ian University of California at Berkeley W. W. Norton & Company New York London

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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 )= x 2 . The production set is Y = { ( y, - x ): y x 2 } 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 ( t x ). Then dy dt = n X i =1 ∂f ( x ) ∂x i x i , so that 1 y dy dt = 1 f ( x ) n X i =1 ( x ) i x i . 1.5 Substitute tx i for i =1 , 2toget f ( tx 1 ,tx 2 )=[( tx 1 ) ρ +( tx 2 ) ρ ] 1 ρ = t [ x ρ 1 + x ρ 2 ] 1 ρ = tf ( x 1 ,x 2 ) . This implies that the CES function exhibits constant returns to scale and hence has an elasticity of scale of 1. 1 .6Th i si sha l ft ru e : i f g 0 ( x ) > 0, then the function must be strictly increasing, but the converse is not true. Consider, for example, the function g ( x x 3 . This is strictly increasing, but g 0 (0) = 0. 1.7 Let f ( x g ( h ( x )) and suppose that g ( h ( x )) = g ( h ( x 0 )). Since g is monotonic, it follows that h ( x h ( x 0 ). Now g ( h ( t x )) = g ( th ( x )) and g ( h ( t x 0 )) = g ( th ( x 0 )) which gives us the required result. 1.8 A homothetic function can be written as g ( h ( x )) where h ( x )i sho - mogeneous of degree 1. Hence the TRS of a homothetic function has the

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2 ANSWERS form g 0 ( h ( x )) ∂h ∂x 1 g 0 ( h ( x )) 2 = 1 2 . That is, the TRS of a homothetic function is just the TRS of the un- derlying homogeneous function. But we already know that the TRS of a homogeneous function has the required property. 1.9 Note that we can write ( a 1 + a 2 ) 1 ρ ± a 1 a 1 + a 2 x ρ 1 + a 2 a 1 + a 2 x ρ 2 ² 1 ρ . Now simply deﬁne b = a 1 / ( a 1 + a 2 )and A =( a 1 + a 2 ) 1 ρ . 1.10 To prove convexity, we must show that for all y and y 0 in Y and 0 t 1, we must have t y +(1 - t ) y 0 in Y . But divisibility implies that t y and (1 - t ) y 0 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 s y in Y .G i v e na n y 0, let n be a nonnegative integer such that n s n - 1. By additivity, n y is in Y ;s ince s/n 1, divisibility implies ( s/n ) n y = s y 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 ( x 1 ,x 2 ) are both positive so the technology is monotonic. For the isoquant to be convex to the origin, it is suﬃcient (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 semideﬁnite. The ﬁrst principal minor of
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Microeconomic Analysis - Ed 3 - Hal Varian - Manual -...

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