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Unformatted text preview: 2 1 The Expenditure Function If preferences satisfy LN, the function v ( p; w ) is strictly increasing in w: We can invert that function to solve for w as a function of the level of utility. That is, given some level of utility u; we can &nd the minimal amount of income necessary to achieve utility at prices p: The function that relates income and utility in this way is the expenditure function , denoted e ( p; u ) : Example: Leontief preferences with L = 2 . The indirect utility was: v ( p; w ) == ab bp 1 + ap 2 w: Inverting we get: e ( p; u ) = bp 1 + ap 2 ab u = & p 1 a + p 2 b ¡ u: (1) The problem of expenditure minimization (EMP) for p >> and u > u (0) is to minimize px subject to u ( x ) & : The solution to EMP is h ( p; u ) and it is called the Hicksian demand correspondence . If it is singlevalued, we call it Hicksian demand function or compensated demand . Microeconomics I 3 Finally, e ( p; u ) = ph ( p; u ) ; that is e ( p; u ) is the value function of the EMP : The UMP and the EMP are related to each other: Microeconomics I 4 Proposition 3.E.1: Suppose u ( & ) is a continuous function representing preferences that are Locally Nonsatiated on X = R L + and p >> : Then: (i) If x ¡ maximizes utility for w > ; then x ¡ minimizes expenditure when the required level of utility is u ( x ¡ ) : Moreover, the minimized expenditure level is exactly w: (ii) If x ¡ minimizes expenditure when the required utility level is u then x ¡ maximizes utility when wealth equals px ¡ : Moreover the maximized utility level is exactly u: x 2 x 1 B p,w x* 1.Why LN is necessary Microeconomics I 5 The expenditure function e ( p; u ) is very interesting. We can use it to decompose the effect of a price change on the Walrasian demand into two distinct effects, called income effect and substitution effect , respectively. As e ( p; u ) is de&ned holding the utility level constant, it isolates the substitution effect. Moreover, as we will see below it plays a central role in welfare analysis. Some properties of the expenditure function. Proposition 3.E.2: Suppose u ( & ) is a continuous function representing preferences that are Locally Nonsatiated on X = R L + : The function e ( p; u ) is: (i) Homogeneous of degree 1 in p: (ii) Strictly increasing in u and nondecreasing in p l for any l: (iii) Concave in p: (iv) Continuous in p and u: Proof: (iii) Suppose x is the solution to the EMP at ( p; u ) and x is the solution at ( p ; u ) : Let p 00 = &p + (1 ¡ & ) p for any & 2 [0 ; 1] : Now we have: e ( p 00 ; u ) = p 00 x 00 = &px 00 + (1 ¡ & ) p x 00 ; (2) Microeconomics I 6 where x 00 is the solution of EMP at ( p 00 ; u ) : Since x 00 is not necessarily the solution at ( p; u ) or ( p ; u ) ; we have px 00 & e ( p; u ) and p x 00 & e ( p ; u ) : Thus: e ( p 00 ; u ) & &e ( p; u ) + (1 ¡ & ) e ( p ; u ) : (3) The intuition is the following. Suppose that initially prices are p and the bundle x is optimal in the EMP....
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This note was uploaded on 06/20/2011 for the course OPR 201 taught by Professor Pp during the Spring '11 term at Thammasat University.
 Spring '11
 PP

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