WENDNER: INTERMEDIATE MICROECONOMICS Second Order Conditions 1. Notation and Basic Concepts N = Number of Choice Variables, M

WENDNER: INTERMEDIATE MICROECONOMICS Second Order Conditions

1. Notation and Basic Concepts n = number of choice variables, m = number of constraints, f(x1, ..., xn) = i objective function, g (x1, ..., xn) = i-th (non)linear constraint.

Hessian Matrix of f(x1, ..., xn):   f11 f12 ··· f1n  f f ··· f  H =  21 22 2n  (1)  ············  fn1 fn2 ··· fnn

Leading principal minors of the Hessian Matrix of f(x1, ..., xn): ¯ ¯ ¯ f f ¯ |H | = |f | = f , |H | = ¯ 11 12 ¯ ··· , 1 11 11 2 ¯ f f ¯ ¯ ¯ 21 22 ¯ ¯ ¯ f11 f12 ··· f1k ¯ ¯ ¯ ¯ f21 f22 ··· f2k ¯ |Hk| = ¯ ¯ , ··· , |Hn| = |H| , ¯ ············ ¯ ¯ ¯ fk1 fk2 ··· fkk where |Hi| denotes the i−th leading principal minor. Bordered Hessian Matrix of the Lagrange-function L: 1 2 m L = f(x1, ..., xn)+ λ1 g (x1, ..., xn)+λ2 g (x1, ..., xn)+···+λm g (x1, ..., xn):   Lλ1λ1 ···Lλ1λm Lλ1x1 ···Lλ1xn    ··················   L ···L L ···L   λmλ1 λmλm λmx1 λmxn  Hb =    Lx1λ1 ···Lx1λm Lx1x1 ···Lx1xn   ·················· 

Lxnλ1 ···Lxnλm Lxnx1 ···Lxnxn   1 1 0 ··· 0 g1 ··· gn    ··················   m m   0 ··· 0 Pg1 ··· Pgn  =  1 m m i m i  . (2)  g1 ··· g1 f11 + i=1 λi g11 ··· f1n + i=1 λi g1n   ··················  1 m Pm i Pm i gn ··· gn fn1 + i=1 λi gn1 ··· fnn + i=1 λi gnn

Ronald Wendner Notes V-1 v1.1 If the constraints are linear — as gij = 0, i = 1, ..., n, j = 1, ..., n — the bordered Hessian becomes:   1 1 0 ··· 0 g1 ··· gn    ········· ·········   m m   0 ··· 0 g1 ··· gn  Hb =  1 m  . (3)  g1 ··· g1 f11 ··· f1n   ········· ·········  1 m gn ··· gn fn1 ··· fnn Notice that quadrants 1 to 3 represent the “border,” and quadrant 4 equals the (un-bordered) Hessian matrix. Leading principal minors of the bordered Hessian. As the second quadrant of bordered Hessian matrix (=first m rows and columns) is a nullmatrix, the first leading principal minor of interest concerns the square matrix of the first m + 2 rows and columns! I.e., the first leading principal minor we are looking at is: |Hbm+2|, and NOT |Hbm+1|. ¯ ¯ ¯ 1 1 ¯ ¯ 0 ··· 0 g1 g2 ¯ ¯ ¯ ¯ ··············· ¯ ¯ m m ¯ |Hbm+2| = ¯ 0 ··· 0 g1 g2 ¯ ¯ 1 m ¯ ¯ g1 ··· g1 f11 f12 ¯ ¯ g1 ··· gm f f ¯ ¯ 2 2 21 22 ¯ ¯ 1 1 1 ¯ ¯ 0 ··· 0 g1 g2 g3 ¯ ¯ ¯ ¯ ·················· ¯ ¯ m m m ¯ ¯ 0 ··· 0 g1 g2 g3 ¯ |Hbm+3| = ¯ 1 m ¯ , ··· , |Hbm+n| = |Hb| . ¯ g1 ··· g1 f11 f12 f13 ¯ ¯ 1 m ¯ ¯ g2 ··· g2 f21 f22 f23 ¯ ¯ 1 m ¯ g3 ··· g3 f31 f32 f33 2. Concavity and Negative (Semi)Definiteness of H

Suppose, m = 0. Concavity of f(x1, ..., xn) is a sufficient (second order) condition for the first order conditions to actually represent a maximum for unconstrained maximization problems. Concavity of f(x1, ..., xn) implies (and is implied by) negative semi-definiteness of the Hessian matrix. The Hessian matrix is negative semi-definite if (i) the sign of the first leading principal minor is non-positive, i.e., f11 ≤ 0, and (ii) the signs of the further i leading principal minors alternate. I.e., sign |Hi| = sign (−1) or |Hi| = 0. If all inequalities hold strictly, the Hessian matrix is (strictly) negative definite, f(x1, ..., xn) is strictly concave, and the maximizer (if it exists) is unique.

Ronald Wendner Notes V-2 v1.1 3. Negative (Semi)Definiteness of Hb Suppose, m > 0. Negative (semi)definiteness of Hb is a sufficient (second order) condition for the first order conditions to actually represent a maximum for constrained maximization problems. The bordered Hessian matrix is negative semi-definite if (i) the sign of the “first” leading principal minor m+1 |Hbm+2| equals the sign (−1) (or equals zero), and (ii) the signs of the fur- m+i−1 ther leading principal minors alternate. I.e., sign |Hbm+i| = sign (−1) (or equal zero). If all inequalities hold strictly, the bordered Hessian matrix is (strictly) negative definite, and the maximizer (if it exists) is unique.

4. Example: n=2, m=1, linear constraint The bordered Hessian matrix is:   1 1 0 g1 g2  1  Hb = g1 f11 f12 , (4) 1 g2 f21 f22 and the only leading principal minor we are interest in is |Hb1+2| = |Hb|. I.e., ¯ ¯ ¯ 1 1 ¯ ¯ 0 g1 g2 ¯ ¯ 1 ¯ |Hb| = ¯ g1 f11 f12 ¯ (5) ¯ 1 ¯ g2 f21 f22 If we are looking for a (unique) maximum, we need to verify that Hb is negative deﬁnite, i.e., sign|Hb| = sign(−1)1+1, which is positive: |Hb| > 0. Notice: For this linear constraint — considering that f1/(−g1) = f2/(−g2) — we exactly get our condition for quasiconcavity (with n = 2, m = 1).

Good luck for your Final!

Ronald Wendner Notes V-3 v1.1