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HW_03_Solutions - ECONOMICS 11 WINTER 2010 HW ASSIGNMENT#3...

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E CONOMICS 11, W INTER 2010: HW A SSIGNMENT #3 - S OLUTIONS Question 1 The Lagrangian can be written as follows: ( 𝑥𝑥 , 𝑦𝑦 , 𝜆𝜆 ) = 𝑥𝑥 + 𝑦𝑦 1 2 + 𝜆𝜆 ( 𝑀𝑀 − 𝑥𝑥𝑝𝑝 𝑥𝑥 − 𝑦𝑦𝑝𝑝 𝑦𝑦 ) The first order conditions are: 1 − 𝜆𝜆𝑝𝑝 𝑥𝑥 = 0 1 2 𝑦𝑦 1 2 − 𝜆𝜆𝑝𝑝 𝑦𝑦 = 0 𝑀𝑀 − 𝑥𝑥𝑝𝑝 𝑥𝑥 − 𝑦𝑦𝑝𝑝 𝑦𝑦 = 0 Solving the first two equations for λ and setting them equal yields: 1 𝑝𝑝 𝑥𝑥 = 1 2 𝑦𝑦 1 2 𝑝𝑝 𝑦𝑦 Solve for y to get: 𝑦𝑦 = 𝑝𝑝 𝑥𝑥 2 𝑝𝑝 𝑦𝑦 2 Plugging this expression into the first order condition for λ and solving for x yields: 𝑀𝑀 − 𝑥𝑥𝑝𝑝 𝑥𝑥 − � 𝑝𝑝 𝑥𝑥 2 𝑝𝑝 𝑦𝑦 2 𝑝𝑝 𝑦𝑦 = 0 𝑥𝑥 = 𝑀𝑀 𝑝𝑝 𝑥𝑥 𝑝𝑝 𝑥𝑥 4 𝑝𝑝 𝑦𝑦 Note that x must be greater than or equal to zero. If x > 0, then we have an interior solution and the demand functions are given by the solutions to the maximization problem above. If, on the other hand, x = 0, then we have a corner solution where the individual spends all of their income on y. Hence, the demand functions can be expressed as follows:
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𝑋𝑋�𝑝𝑝 𝑥𝑥 , 𝑝𝑝 𝑦𝑦 , 𝑀𝑀� = 𝑀𝑀 𝑝𝑝 𝑥𝑥 𝑝𝑝 𝑥𝑥 4 𝑝𝑝 𝑦𝑦 𝑖𝑖𝑖𝑖 𝑀𝑀 𝑝𝑝 𝑥𝑥 𝑝𝑝 𝑥𝑥 4 𝑝𝑝 𝑦𝑦 > 0 0 𝑖𝑖𝑖𝑖 𝑀𝑀 𝑝𝑝 𝑥𝑥 𝑝𝑝 𝑥𝑥 4 𝑝𝑝 𝑦𝑦 0 𝑌𝑌�𝑝𝑝 𝑥𝑥 , 𝑝𝑝 𝑦𝑦 , 𝑀𝑀� = 𝑝𝑝 𝑥𝑥 2 𝑝𝑝 𝑦𝑦 2 𝑖𝑖𝑖𝑖 𝑀𝑀 𝑝𝑝 𝑥𝑥 𝑝𝑝 𝑥𝑥 4 𝑝𝑝 𝑦𝑦 > 0 𝑀𝑀 𝑝𝑝 𝑦𝑦 𝑖𝑖𝑖𝑖 𝑀𝑀 𝑝𝑝 𝑥𝑥 𝑝𝑝 𝑥𝑥 4 𝑝𝑝 𝑦𝑦 0 Question 2 Part (a) – The Lagrangian can be written as follows: ( 𝑥𝑥 , 𝑦𝑦 , 𝜆𝜆 ) = 𝑥𝑥𝑦𝑦 3 + 𝜆𝜆 ( 𝑀𝑀 − 𝑥𝑥𝑝𝑝 𝑥𝑥 − 𝑦𝑦𝑝𝑝 𝑦𝑦 ) The first order conditions are: 𝑦𝑦 3 − 𝜆𝜆𝑝𝑝 𝑥𝑥 = 0 3 𝑥𝑥𝑦𝑦 2 − 𝜆𝜆𝑝𝑝 𝑦𝑦 = 0 𝑀𝑀 − 𝑥𝑥𝑝𝑝 𝑥𝑥 − 𝑦𝑦𝑝𝑝 𝑦𝑦 = 0 Solving the first two equations for λ and setting them equal yields: 𝑦𝑦 3 𝑝𝑝 𝑥𝑥 = 3 𝑥𝑥𝑦𝑦 2 𝑝𝑝 𝑦𝑦 Note that this equation is identical to that which you would get if you used the marginal utility per dollar method. Solving this equation for y as a function of x yields: 𝑦𝑦 = 3 𝑥𝑥𝑝𝑝 𝑥𝑥 𝑝𝑝 𝑦𝑦 Plugging this expression into the first order condition for λ
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