2008-Final-Solutions

# 2008-Final-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 Final Exam Solutions Winter 2008 Monday, March 18, 2008 Problem 1 (35 Points) (i) The set P of Pareto-optimal allocations (i.e., the contract curve) is determined by the (closure of the) set of all y ( ) which satisfy y ( ) = ( y 1 ( ) , y 2 ( )) arg max y [0 , 1 ] [0 , 2 ] (1- ) u 1 ( y ) + u 2 ( - y ) (1) for (0 , 1), where = ( 1 , 2 ) is the total endowment vector. Since consumer 2 has a Cobb- Douglas utility function, it needs to be true that - y ( ) &gt; 0, so that y i ( ) &lt; i for all i { 1 , 2 } and all (0 , 1). Hence, the Lagrangean for the optimization problem (1) is given by L ( y ; , ) = (1- ) y 1 y 2 + (ln( 1- y 1 ) + 2 ln( 2- y 2 )) + ( y 1 , y 2 ) , where = ( 1 , 2 ) 0 is the Lagrange-multiplier associated with the nonnegativity constraint. The optimality conditions are L y 1 = (1- ) y 2- 1- y 1 + 1 ! = 0 , (2) L y 2 = (1- ) y 1- 2 2- y 2 + 2 ! = 0 , (3) and 1 y 1 ! = 2 y 2 ! = 0 . (4) Multiplying (2) and (3) by y 1 and y 2 respectively, we obtain by virtue of (4) that (1- ) y 1 y 2- y 1 1- y 1 = (1- ) y 1 y 2- 2 y 2 2- y 2 = 0 , so that y 1 = 2 1 y 2 2 + y 2 . (5) The last inequality implies that either y = 0 or y 0. In the interesting case when y 0, we have that 2(1- ) 1 y 2 2 2 + y 2 + 2 y 2 2- y 2 = 0 , 1 or, equivalently, y 2 ( ) = (1- ) 1 2 - 1 r (1- ) 1 2 2- 6 (1- ) 1 2 + 1 2 (1- ) 1 2 2 A 2 [0 , 2 ] , where A [0 , 1] as (0 , 1). Hence, by (5) it is y 1 ( ) = 2 A 1 1 + A [0 , 1 ] Note that y = 0 can be excluded from consideration at the very outset, since consumer 1s prefer- ences can be equivalently represented by the Cobb-Douglas utility function u 1 ( x 1 ) = ln( u 1 ( x 1 )) = ln x 1 1 + ln x 1 2 . (ii) Let p = ( p 1 , p 2 ) be a price vector. Consumer 1s Walrasian demand (offer curve) is x 1 ( p, 1 ) = p 1 2 p 1 , p 1 2 p 2 , and consumer 2s Walrasian demand (offer curve) is x 2 ( p, 2 ) = p 2 3 p 1 , 2 p 2 3 p 2 . Since supply equals demand for good 1, we have that x 1 1 ( p, 1 ) + x 2 1 ( p, 2 ) = p 1 2 p 1 + p 2 3 p 1 = 18 p 1 + 4 p 2 2 p 1 + 3 p 1 + 6 p 2 3 p 1 = 1 1 + 2 1 = 18 + 3 = 21 . The last equation can be rewritten in the form 10 + 4( p 2 /p 1 ) = 21, so that p 2 p 1 = 11 / 4 = 2 . 75 ....
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## 2008-Final-Solutions - Stanford University Department of...

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