2008-midterm-Solutions

# 2008-midterm-Solutions - Stanford University Department of...

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Unformatted text preview: Stanford University Department of Management Science and Engineering MS&amp;E 241 Economic Analysis Midterm Exam Solutions Winter 2008 Thursday, February 14, 2008 Problem 1 (35 Points) (i) A set of indifference curves of the consumers demand function u ( x ) is given in Figure 1. For any point x in the interior of the choice set X , the upper contour set U x = { y X : u ( x ) u ( y ) } is nonconvex. (ii) Given the price vector p = ( p 1 ,p 2 ) 0, the consumers disposable income w &gt; 0, and her resulting budget set B ( p,w ) = { x X : p x w } , we would observe one of the unique consumption bundles x 1 ( p,w ) = w p 1 , (1- ) w p 2 arg max x B ( p,w ) u 1 ( x ) and x 2 ( p,w ) = (1- ) w p 1 , w p 2 arg max x B ( p,w ) u 2 ( x ) , if the consumers utility function was either u 1 or u 2 . Given a price vector p = ( p 1 ,p 2 ) 0 determine the consumers Walrasian demand vector x * ( p,w, ). The Walrasian demand correspondence x ( p,w ), which is the solution to the consumers actual utility maximization problem, has to contain x 1 ( p,w ), x 2 ( p,w ), or both, i.e., x ( p,w ) = arg max x B ( p,w ) u ( x ) x 1 ( p,w ) ,x 2 ( p,w ) . Since both x 1 ( p,w ) and x 2 ( p,w ) are feasible (i.e., they both lie in B ( p,w )) and the consumers preferences are locally nonsatiated (since the utility function is strictly increasing on X ), we have by virtue of the Weak Axiom of Revealed Preference (WARP) that the Walrasian demand is equal to { x 1 ( p,w ) ,x 2 ( p,w ) } if and only if u 1 ( x 1 ( p,w )) = u 2 ( x 2 ( p,w )), which is satisfied if and only if p 1 = p 2 . When the prices for the two commodities are different, one can easily verify that ( p 1- p 2 ) ( u ( x 1 ( p,w ))- u ( x 2 ( p,w )) ) &lt; . Hence, the consumers Walrasian demand correspondence x : R 3 R is given by x ( p,w ) = { x 1 ( p,w ) } , if p 1 &lt; p 2 , { x 1 ( p,w ) ,x 2 ( p,w ) } , if p 1 = p 2 , { x 2 ( p,w ) } , otherwise. 1 Figure 1: Consumer Choice with Nonconvex Preferences (Problem 1, parts (i) and (ii)). (iii) The consumers Walrasian demand correspondence, obtained in part (ii), is depicted in Figure 2. As predicted by Berges maximum theorem, the Walrasian demand correspondence is compact- valued and upper semi-continuous. Thus, along the price path p ( s ) = (1 ,s ) it is continuous almost everywhere (whenever it is single-valued), for all s R ++ \ { 1 } . Note also that (0 , 0) &lt; D s x ( p ( s ) ,w ) dx 1 ( p ( s ) ,w ) ds , dx 2 ( p ( s ) ,w ) ds = ,- (1- ) w s 2 , ,- w s 2 for all s (0 , ) \ { 1 } . In other words, good 2 is a non-Giffen good, for the demand for it is decreasing in price. A demand for good 2 in the interval ((1- ) w/p 1 ,w/p 1 ) can never be observed, as a consequence of the nonconvexity of the consumers preferences....
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## 2008-midterm-Solutions - Stanford University Department of...

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