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Unformatted text preview: Walrasian demand for u (x1 ; x2 ) = x1 x2 u (x1 ; x2 )
x1 w p1 p1 x1 + p2 x2 w
w + 1 =0 0; x1 and
0; x2 (p1 x1 + p2 x2 one must solve: (1 ) 0 w ) = 0, u (x1 ; x2 )
1 x1 We must have strictly positive consumption (why?), hence
Moreover the budget constraint must bind (why?), hence
Therefore the FOC are
u (x1 ; x2 ) = w p 1 x1 and (1 0; w = 0, and w p2 2 x2 2 =0 1 0; 2 0 =0 =
w + 2 =0 ) u (x1 ; x2 ) = w p 2 x2 Summing both sides and using Walras’Law we get
u (x1 ; x2 ) = w ( p 1 x1 + p 2 x2 ) = ww Some algebra yields
x1 (p ; w ) = w
p1 and x2 (p ; w ) = (1 ) w
p2 and w = 1 1
p1 p2 Luca’ Rough Guide to Convex Optimization
Where does the recipe come from? Roee told you, but here is a quick summary.
Let f : Rn ! R be a continuous, increasing, and quasi-concave function, and let
C Rn be a convex set. We want to …nd a solution to the following problem:
max f (x )
x 2C where
C = fx 2 Rn : gi (x ) 0 with i = 1; :::; N g This is the most general way to state a maximization problem. Example
1 If C = Rn , we have an unconstrained problem. 2 If you have constraints like h (x ) 3 If you have constraints like h (x ) b , de…ne gj (x ) = [h (x ) b ].
If you have constraints like h (x ) = b , de…ne gj (x ) = h (x ) b and
gk (x ) = [h (x ) b ] and rewrite as fgj (x ) 0 and gk (x ) 0g. 4 b , de…ne gj (x ) = h (x ) b. Geometry at Work A level curve for some function f : Rn ! R is given by f (x ) = c for some c 2 R.
better than’set is
% (x ) = fy 2 Rn : u (y ) u (x )g : Draw C and some level curves (when are better than sets convex?).
At a maximum, level curves and constraint set are ‘
Tangent to C : a plane through the point does not intersect the interior of C .
Tangent to the level curves: a plane through the point does not intersect the
interior of % (x ). Geometry at Work
An hyperplane is H = fx 2 Rn : (x y ) = 0g . An hyperplane H supports C at a point x if C is a strict subset of
H = fx 2 Rn : (x y) 0g : Take x on the boundary of C . The tangent to C...
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This note was uploaded on 05/13/2013 for the course ECON 2100 taught by Professor Board,o during the Fall '08 term at Pittsburgh.
- Fall '08