Problem_Set_2_Solution

# Problem_Set_2_Solution - Economic Growth and Development...

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Economic Growth and Development Professor Olivier de La Grandville Fall 2008 Problem Set 2 Solutions 1. (30 points) Assuming non-negative marginal productivity of labor, unstable equilibrium con- tradicts this assumption. This can be seen as follows. If sf 0 ( r * ) > n , there is a neighborhood around r * such that to the left of r * we have sf ( r ) < nr , while to the right of r * , sf ( r ) > nr , entailing disequilibrium. Furthermore, if sf 0 ( r * ) > n then sf 0 ( r * ) n > 1 and f ( r * )[1 - sf 0 ( r * ) n ] = f ( r * ) - sf ( r * ) n f 0 ( r * ) < 0. Since sf ( r * ) n = r * , we have f ( r * ) - r * f 0 ( r * ) = ∂F ∂L ( r * ) < 0 which means the non-negative marginal productivity of labor. Therefore such a point should be dismissed. 2. (30 points) The second order Taylor expansion of f ( x,y ) about the point ( a,b ) is, f ( x,y ) f ( a,b ) + f x ( a,b )( x - a ) + f y ( a,b )( y - b ) + 1 2 [ f xx ( a,b )( x - a ) 2 + 2 f xy ( a,b )( x - a )( y - b ) + f yy ( a,b )( y - b ) 2 ] If we let

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## This note was uploaded on 06/16/2010 for the course MS&E 249 taught by Professor Olivierdelagrandville during the Fall '08 term at Stanford.

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Problem_Set_2_Solution - Economic Growth and Development...

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