E308 6 and 7

E308 6 and 7 - DUALITY IN CONSUMPTION I We have considered...

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DUALITY IN CONSUMPTION I We have considered the utility maximization problem: max U=U(x 1 , x 2 ) s . t . p 1 x 1 +p 2 x 2 =m The first order condition are solved to derive the demand functions: x 1 *=x 1 (p 1 , p 2 , m) and x 2 *=x 2 (p 1 , p 2 , m). For any given pair of prices (p 1 , p 2 ) and income m, (x 1 * ,x 2 *) will be the bundle that maximize the consumer’s utility subject to the budget constraint. The corresponding utility level would be V=U[x 1 *(p 1 , p 2 , m), x 2 (p 1 , p 2 , m)]. Note that any change in the parameters (p 1 , p 2 , m) will lead to a change in (x 1 * ,x 2 *) and therefore the maximally attained level of utility V will also change. We may express U[x 1 *(p 1 , p 2 , m), x 2 (p 1 , p 2 , m)] directly as a function of (p 1 , p 2 , m): V=U[x 1 *(p 1 , p 2 , m), x 2 (p 1 , p 2 , m)]= V(p 1 , p 2 , m). This is because as the Indirect Utility Function , the utility function ranks consumption bundles according to the consumer’s preferences. The indirect utility function ranks price income combinations (p 1 , p 2 , m). For every combination of (p 1 , p 2 , m), there is a specific budget line and the preferred bundle on that line. If the preferred bundle on a different budget line lies on a higher indifference curve, the consumer in better off with the latter budget line. The budget line A 0 B 0 (Figure 1) corresponds to (p 1 0 , p 2 0 , m 0 ). The best point on A 0 B 0 is K. The line A 1 B 1 corresponds to (p 1 1 , p 2 1 , m 1 ) and the best point on A 1 B 1 is L. Because L is on a higher indifference curve, the consumer can reach a higher level of utility given (p 1 1 , p 2 1 , m 1 ) than from (p 1 0 , p 2 0 , m 0 ). Figure 1 1
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x 2 m 0 /p 2 0 A 0 m 1 /p 2 1 A 1 K L O B 0 m 0 /p 1 0 B 1 m 1 /p 1 1 x 1 Hence the combination (p 1 1 , p 2 1 , m 1 ) makes the consumer better off and V ( p 1 1 , p 2 1 , m 1 )>V (p 1 0 , p 2 0 , m 0 ) Example of Indirect Utility Function: (1) Ux x = 12 x m p 1 1 2 * = , x m p 2 2 2 * = Ux x x x m p m p m pp Vp p m (* , * ) * * (, ,) 1 2 22 2 =⋅ = = = (2) x =+ x pm pp p 1 2 * () = + 11 2 , x 2 1 * = + 21 2 x x x m p p p p ) * * ( ) ( ) 1 2 2 1 1 2 =+= + ⋅+ 2
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= + + = + = () ( ) (, ,) m pp mp p Vp p m 12 The indirect utility function is homogeneous of degree 0 in (p 1 , p 2 , m). If both price and income are demand, V will not be affected. We have seen before that if p 1 , p 2 and m are all changed proportionately, x 1 *(p 1 , p 2 , m) and x 2 *(p 1 , p 2 , m) will not change. Hence
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E308 6 and 7 - DUALITY IN CONSUMPTION I We have considered...

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