l061013evaleconchangeWEB

l061013evaleconchangeWEB - S.Grant ECON501 1.7 Evaluation...

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S.Grant ECON501 1.7 Evaluation of Economic Change (Consumer Surplus Analysis) Ref: MWG 3.I, Varian Chapter 10 Problem At initial situation ¡ p 0 ,w 0 ¢ consumption is x 0 = x ¡ p 0 ,w 0 ¢ . A change is being considered from ¡ p 0 ,w 0 ¢ ¡ p 1 ,w 1 ¢ . We wish to evaluate it. 1.7.1 First Order Test Proposition 1.7.1 If ¡ p 1 p 0 ¢ .x 0 <w 1 w 0 then the consumer is strictly better o f under ¡ p 1 ,w 1 ¢ than under ¡ p 0 ,w 0 ¢ . 1 S.Grant ECON501 Proofs: 1. By Walras’s Law, p 0 .x 0 = w 0 . Hence, ¡ p 1 p 0 ¢ .x 0 <w 1 w 0 p 1 .x 0 <w 1 . I.e., x 0 strictly a f ordable. Thus by local non-satiation, e x ,s .t . p 1 . e x<w 1 e x  x 0 . And since x 1 % e x ,wehave x 1  x 0 . ¥ 2. (Geometry of Indirect Utility Function). Fix 2 vectors, a, b R L . Cosine of the angle, θ ,that subtends those 2 vectors is given by cos θ = a.b | a || b | and cos θ> 0 0 <θ<π/ 2 (i.e. 90 ) Also notice that from the quasi-convexity of the indirect utility function if £¡ p 1 ,w 1 ¢ ¡ p 0 ,w 0 ¢¤ ∂p 1 v ¡ p 0 ,w 0 ¢ . . . ∂p L v ¡ p 0 ,w 0 ¢ ∂w v ¡ p 0 ,w 0 ¢ > 0 then v ¡ p 1 ,w 1 ¢ >v ¡ p 0 ,w 0
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S.Grant ECON501 Now notice, that £¡ p 1 ,w 1 ¢ ¡ p 0 ,w 0 ¢¤ ∂p 1 v ¡ p 0 ,w 0 ¢ . . . ∂p L v ¡ p 0 ,w 0 ¢ ∂w v ¡ p 0 ,w 0 ¢ = ¡ p 1 p 0 ¢ . p v ¡ p 0 ,w 0 ¢ + ¡ w 1 w 0 ¢ . ∂w v ¡ p 0 ,w 0 ¢ = £ ¡ p 1 p 0 ¢ .x 0 + ¡ w 1 w 0 ¢¤ ∂w v ¡ p 0 ,w 0 ¢ > 0 if and only if ¡ w 1 w 0 ¢ > ¡ p 1 p 0 ¢ .x 0 3 S.Grant ECON501 Q. What if ¡ w 1 w 0 ¢ < ¡ p 1 p 0 ¢ .x 0 ? 1. Revealed preference: v ¡ p 1 ,w 1 ¢ ? v ¡ p 0 ,w 0 ¢ 2. Geometry of indirect utility function. θ>π/ 2 cos θ μ = a.b | a || b | < 0 There is an ¯ α (0 , 1) such that for α< ¯ α we have v ¡ (1 α ) ¡ p 0 ,w 0 ¢ + α ¡ p 1 ,w 1 ¢¢ <v ¡ p 0 ,w 0 ¢ Conclusion: If ¯ ¯ ¡ p 1 ,w 1 ¢ ¡ p 0 ,w 0 ¢¯ ¯ “small” relative to ¯ ¯ p 1 .x 0 w 1 ¯ ¯ then f rst order test is conclusive. 4
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S.Grant ECON501 1.7.2 Using the Expenditure Function Change could be evaluated by measuring “willingness to pay”. Money metric indirect utility function (indirect compensation function)
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This note was uploaded on 09/04/2010 for the course ECON 501 taught by Professor Grant during the Spring '10 term at Rice.

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l061013evaleconchangeWEB - S.Grant ECON501 1.7 Evaluation...

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