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duality

# duality - Duality Theory for Utility Maximization John Rust...

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Duality Theory for Utility Maximization John Rust, Juan Diaz, Sung Jin Cho University of Maryland <http://gemini.econ.umd.edu/jrust/econ306/ch5_lec.pdf> February 24, 2003 1

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Indirect Utility Function Definition: The Indirect Utility Function V ( p , y ) is equal to the maximized value of the utility function when prices are p and income is y : V ( p , y ) = max x u ( x ) subject to: p 0 x y = u ( x ( p , y )) , (1) where x ( p , y ) is the demand function, i.e. the utility maximizing consumption bundle at price vector p and income y . Properties of the Indirect Utility Function: 1. V ( p , y ) is homogeneous of degree 0 in ( p , y ) 2. V ( p , y ) is strictly increasing in y . 3. V ( p , y ) is decreasing in p 4. V ( p , y ) is quasiconvex in ( p , y ) 5. Roy’s Identity holds: x i ( p , y ) = - V ( p , y ) / p i V ( p , y ) / y , i = 1 ,..., n . (2) 2
Some Definitions Homogeneous of Degree 1 A function V ( p , y ) is homogeneous of degree k in ( p , y ) if for all λ > 0 we have V ( λ p , λ y ) = λ k V ( p , y ) . (3) Thus, a function that is homogeneous of degree 0 satisfies: V ( λ p , λ y ) = λ 0 V ( p , y ) = 1 V ( p , y ) = V ( p , y ) . (4) Why is the Indirect Utility Function homogeneous of degree 0? By definition we have: V ( λ p , λ y ) = max x u ( x ) subject to: λ p 0 x λ y (5) Notice that { x | λ p 0 x λ y } = { x | p 0 x y } , i.e. multiplying all prices and incomes by the same factor λ does not change the person’s budget set! 3

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Why is V ( p , y ) Homogeneous of Degree 0? (continued) If the budget set does not change, then the utility maximizing bundle cannot change, i.e. x ( λ p , λ y ) = x ( p , y ) (6) Since the demand function is homogeneous of degree 0 (by the equation above), it follows that indirect utility function is also: V ( λ p , λ y ) = u ( x ( λ p , λ y )) = u ( x ( p , y )) = V ( p , y ) . (7) However this last equation tells us that V ( p , y ) is homogeneous of degree 0 in ( p , y ) . A practical way of saying this is that rational consumers have no money illusion. That is, the consumer does not care if we measure prices and income in pennies ( λ = 100 ) , or in dollars ( λ = 1 ) . Consumer behavior is predicted to be completely invariant to what units prices and incomes are measured in. 4
Why is V ( p , y ) increasing in y ? The answer to this is easy: if the consumer has more income, then the consumer can afford to consume more, but this must strictly increase the consumer’s utility level. Why is V ( p , y ) decreasing in p ? The answer to this is also easy: if the consumer faces higher prices, then the consumer can’t afford to consume as much as he/she did on the same income y when prices were lower. But if the consumer consumes less, then the maximized utility level, V ( p , y ) , must decrease. There is one exception to this: if a consumer has a utility function that doesn’t care about some good x j (i.e. x j does not enter the consumer’s utility function u ( x 1 ,..., x n ) ), then an increase in p j has no effect on the consumer’s utilty level. This is why utility is not strictly decreasing in p .

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