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09-17-08 - Econ 302 Intermediate Microeconomic Theory...

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Econ 302: Intermediate Microeconomic Theory Andres Elberg University of Illinois at Urbana-Champaign September 17, 2008 Lecture 7: Demand September 17, 2008 1 / 49
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Optimal Choice: Analytical Solution Formally, the consumer problem can be stated as the following constrained maximization problem: max x 1 , x 2 u ( x 1 , x 2 ) subject to p 1 x 1 + p 2 x 2 = m Lecture 7: Demand September 17, 2008 2 / 49
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Lagrangian Function To solve this problem we form the Lagrangian: L ( x 1 , x 2 , λ ) = u ( x 1 , x 2 ) + λ ( m & p 1 x 1 & p 2 x 2 ) And obtain the &rst order conditions: ( 1 ) L x 1 ( x 1 , x 2 , λ ) = u ( x 1 , x 2 ) x 1 & λ p 1 = 0 ( 2 ) L x 2 ( x 1 , x 2 , λ ) = u ( x 1 , x 2 ) x 2 & λ p 2 = 0 ( 3 ) L ∂λ ( x 1 , x 2 , λ ) = m & p 1 x 1 & p 2 x 2 = 0 Lecture 7: Demand September 17, 2008 3 / 49
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Optimal Choice: Analytical Solution From equations (1) and (2) we get: u ( x 1 , x 2 ) x 1 u ( x 1 , x 2 ) x 2 = p 1 p 2 Which is the same condition that we obtained before: At an (interior) optimum, the Marginal Rate of Substitution must be equal to the price ratio. Lecture 7: Demand September 17, 2008 4 / 49
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Optimal Choice: Examples We will obtain the demand functions for di/erent type of preferences: 1 Cobb-Douglas 2 Perfect Complements 3 Perfect Substitutes Lecture 7: Demand September 17, 2008 5 ± 49
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Optimal Choice: Examples Cobb-Douglas Preferences Recall the Cobb-Douglas utility function: u ( x 1 , x 2 ) = x a 1 x 1 & a 2 Cobb-Douglas preferences are well-behaved so we& ll have an interior optimum. Let&s set up the Lagrangian for the problem: L ( x 1 , x 2 , λ ) = x a 1 x 1 & a 2 + λ ( m & p 1 x 1 & p 2 x 2 ) Lecture 7: Demand September 17, 2008 6 / 49
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Optimal Choice: Examples Cobb-Douglas Preferences The &rst order conditions are: ( 1 ) L x 1 ( x 1 , x 2 , λ ) = ax a & 1 1 x 1 & a 2 & λ p 1 = 0 ( 2 ) L x 2 ( x 1 , x 2 , λ ) = ( 1 & a ) x a 1 x & a 2 & λ p 2 = 0 ( 3 ) L ∂λ ( x 1 , x 2 , λ ) = m & p 1 x 1 & p 2 x 2 = 0 Lecture 7: Demand September 17, 2008 7 / 49
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Optimal Choice: Examples Cobb-Douglas Preferences Using equations (1) and (2) we &nd that the following condition must hold at an optimum: a 1 & a & x 2 x 1 ± = p 1 p 2 To solve for the demand functions we use the other condition that must hold with monotonic preferences: p 1 x 1 + p 2 x 2 = m Lecture 7: Demand September 17, 2008 8 / 49
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Optimal Choice: Examples Cobb-Douglas Preferences We &nd: x 1 ( p 1 , p 2 , m ) = a & m p 1 ± x 2 ( p 1 , p 2 , m ) = ( 1 & a ) & m p 2 ± Lecture 7: Demand September 17, 2008 9 / 49
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Optimal Choice: Examples Perfect Complements Suppose the two goods are perceived by the consumer as being perfect complements The consumer preferences can be described by the utility function: u ( x 1 , x 2 ) = min f x 1 , x 2 g We know the consumer will want to consume both goods in &xed proportions In particular, x 1 = x 2 Lecture 7: Demand September 17, 2008 10 / 49
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Optimal Choice: Examples Perfect Complements We have then two conditions that the optimal bundle must satisfy: 1 x 1 = x 2 2 p 1 x 1 + p 2 x 2 = m (i.e. the consumer spends all her income at an optimum) Using (1) and (2) we &nd: x & i = m p 1 + p 2 , i = 1 , 2 Lecture 7: Demand September 17, 2008 11 / 49
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Optimal Choice: Examples Perfect Substitutes Suppose now that the consumer perceives both goods as being perfect substitutes Her utility function is u ( x 1 , x 2 ) = x 1 + x 2
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09-17-08 - Econ 302 Intermediate Microeconomic Theory...

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