11hw7 - Homework Assignment #7 18.4 Find the general...

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Unformatted text preview: Homework Assignment #7 18.4 Find the general expression (in terms of all the parameters) for the commodity bundle ( x 1 ,x 2 ) which maximizes the Cobb-Douglas utility function U ( x 1 ,x 2 ) = kx a 1 x 1- a 2 on the budget set p 1 x 1 + p 2 x 2 = I . Answer: We assume p and I > 0, so the problem has a non-trivial solution. We will also assume that 0 < a < 1. The constraint has derivative ( p 1 ,p 2 ) 6 = , implying that the NDCQ condition is satisfied. The Lagrangian is L = kx a 1 x 1- a 2- ( p 1 x 1 + p 2 x 2- I ). This yields first-order conditions 0 = akx a- 1 1 x 1- a 2- p 1 and (1- a ) kx a 1 x- a 2- p 2 . Collecting the terms on the left-hand side and dividing, we obtain a 1- a x 2 x 1 = p 2 p 1 . Rearranging, we find p 1 x 1 = ap 2 x 2 / (1- a ). We now use the budget constraint to find (1+ a/ (1- a )) p 2 x 2 = I , so x 2 = (1- a ) I/p 2 and x 1 = aI/p 1 . The second-order conditions should also be checked, but Chapter 18 did not cover that. In this case we know that Cobb-Douglas utility is concave (as shown in class), so we have a global maximum (Chapter 21). 18.9 Maximize x 2 y 2 z 2 subject to x 2 + y 2 + z 2 = c 2 , where c is some fixed positive constant. What is the maximum value of the objective function on the constraint set? Show that for all x , y , z ....
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11hw7 - Homework Assignment #7 18.4 Find the general...

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