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Exam 1 - Practice Answers - Midterm Exam Eco 701 Fall 2009...

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Midterm Exam Eco 701 - Fall 2009 Practice General Instructions: Please answer all of the questions in your blue book. You have the entire class time (approximately 2 hours) to complete the exam. Be sure to mark your answers clearly and show all work. If you have to cross out mistakes or superfluous material, please do so in such a way so that I do not get confused in grading your exam. You may answer the questions in any order you choose, but once again, please take care in indicating which question you are answering. Please hand in the exam sheet with the exam. 1 Consumer Theory A consumer has the utility function u = x ( α - βx ) + y , where α and β are parameters. 1. Find the regular demand functions for each good (in any order you choose). Answer: Setting up the Lagrangian gives the first two first- order conditions as: ∂L ∂x = α - 2 βx - λp x = 0 ∂L ∂y = 1 - λp y = 0 If we solve the second equation for λ , we find that λ = 1 p y . Note that this is a little different than the usual case where we cannot solve for λ explicitly in terms of parameters this early in the problem. This is due to the quasi-linearity of the setup. In any case, plugging λ = 1 p y into the first equation gives: x * = α - p x p y 2 β But we do not have an equation for y . What to do? Well, the budget constraint still applies, so if we plug x * into the budget constraint 1
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I = p x x + p y y , we get: y * = I p y - 1 2 αp x βp y + 1 2 p 2 x βp 2 y 2. Find the compensated demand functions for each good. Answer: Set- ting up the Lagrangian “in reverse” and differentiating gives: ∂L ∂x = p x + λ ( α - 2 βx ) = 0 ∂L ∂y = p y - λ = 0 Solving these equations in the same way outlined above gives the same exact answers as before.
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