E308 6 and 7

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

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
= + + = + = () ( ) (, ,) 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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 12

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online