2008-Final-Solutions

2008-Final-Solutions - Stanford University Department of...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Stanford University Department of Management Science and Engineering MS&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 ( ) > 0, so that y i ( ) < 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 ....
View Full Document

Page1 / 7

2008-Final-Solutions - Stanford University Department of...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online