lecture_05_handout

0 and 2 x2 0 0 0 and 2 x2 0 1 0 0 the recipe

Info iconThis preview shows page 1. Sign up to view the full content.

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

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 ) x2 and 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 = >0 1 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 s 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. + The ‘ 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’ . 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...
View Full Document

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.

Ask a homework question - tutors are online