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Checking su¢ ciency for Optimization Problems - Some Results Concavity/Convexity of Lagrangian Suppose Lagrangian L(x) is concave (convex) in x; then x , that solves …rst order conditions obtained from lagrangian function, solves maximization (minimization) problem. Thus to check if demand functions, obtained from di¤erentiating Lagrangian with respect x1 , x2 ; and ; indeed solve maximization problem of consumer, one could examine concavity of Lagrangian function. While doing this one has to remeber that sum of concave functions is concave and sum of convex functions is convex; i.e. say f and g are concave functions, then we know that: f +g is also concave. Also if f is concave, then f is convex. (Note that since linear function is concave and convex at the same time, one can ignore budget constraint while checking concavity/covexity of Lagrangian). De…nition of concave/convex two-variable functions: 00 00 00 00 00 Theorem 1 (a) f is concave () f11 6 0; f22 6 0 and f11 f22 (f12 )2 > 0: 00 00 00 00 00 (b) f is convex () f11 > 0; f22 > 0 and f11 f22 (f12 )2 > 0: Example: Problem Set 1, Question 2. Consumer solves following maximization problem: Problem 2 max x1 x2 s.t. p1 x1 + p2 x2 6 M We get following Lagrangian: L(x1 ; x2 ) = x1 x2 ( p1 x1 + p2 x2 M) By showing that Lagrangian function is concave, one shows that demand functions (derived in class) are indeed solution of maximization problem. For this it is su¢ cient to check concavity of utility function: x1 x2 . f1 = x1 f11 = ( 1 x2 1) x1 2 x2 6 0: () 61 6 0: () 61 f2 = x1 x2 1 f22 = ( 1) x1 x2 f12 = x1 1 x2 1 00 00 00 f11 f22 (f12 )2 = ( 2 1)( x21 1) 1 2 2 x2 2 ( )2 x21 2 2 x2 2 >0 () ( 1)( 1) () + 6 1: >0 (General Result applied to 2 variables, 1 constraint) Su¢ cient Condition for Optimization problems with two variables and one constraint If consumer wants to solve following maximization problem: Max x1 x2 subject to the constraint: p1 x1 + p2 x2 6 M One could use Lagrange’s method to …nd solution. Lagrangian would be: L(x1 ; x2 ) = x1 x2 (p1 x1 + p2 x2 M ) First order necessary conditions for maximum are: 1. @L x1 = x1 1 x2 p1 = 0 2. @L x2 = x1 x2 1 p2 = 0 3. @L = (p1 x1 + p2 x2 M) = 0 To check second order conditions, let’s de…ne following matrix, called Bordered Hessian: 0 De…nition 3 Bordered Hessian: g1 g2 g1 L11 L21 g2 L12 L22 Where g1 for example stands for partial derivative of constraint and L11 respectively for second-partial of Lagrangian with respect to …rst variable. De…ne determinant of the Bordered Hessian matrix: D 0 g1 g2 g1 L11 L21 g2 L12 L22 = [L11 (g2 )2 2L12 g1 g2 + L22 (g1 )2 ] Theorem 4 If (x1 ; x2 ; ) solve …rst order conditions given by equations (1,2,3) and if D> 0 (< 0) when evaluated at (x1 ; x2 ; ); then (x1 ; x2 ) is a local maximum (minimum) of f (x1 ; x2 ) subject to the constraint of g(x1 ; x2 ) = 0: 2 Calculating all the derivatives of the Bordered Hessian matrix: L11 = ( 1) x1 2 x2 L22 = ( 1) x1 x2 2 L12 = x1 g1 = p1 g2 = p2 1 x2 1 To calculate Determinant of Bordered Hessian in our example, one should evaluate determinant at (x1 ; x2 ; ); i.e. at the demand functions as found solving …rst order conditions and also but as one can see determinant does not depend on in this case. 3