Mathematics for Economists
Constrained positive deÖnite matrixes Example Solvex 1 x 2 !min,x 12 +x 22 = 1. The Lagrangian is L(x 1 ,x 2 ,λ)=x 1 x 2 +λ x 1 ...
Constrained positive deÖnite matrixes The Hessian of the Lagrangian is A= 2 λ 1 1 2λ ,B=( 2 x 1 , 2 x 2 ). For the Örst root ...
Constrained positive deÖnite matrixes For the third root p^12 ,p^12 ,λ= 1 / 2 H= 0 @ 0 2 / p 2 2/ p 2 2 / p 2 1 1 2 / p 2 ...
Constrained positive deÖnite matrixes Example Solve the problemx 12 +x 22 +x 32 !min,x 1 +x 2 +x 3 = 1. The Lagrangian is L(x 1 ...
Constrained positive deÖnite matrixes But we must calculate 3 1 =2 determinants: Deleting the last row and the last column det 0 ...
Constrained positive deÖnite matrixes Example Solve the problemx 1 x 2 x 3 !max,x 12 +x 22 +x 32 = 1 The Lagrangianx 1 x 2 x 3 + ...
Constrained positive deÖnite matrixes A maximal solution is whenx 1 =x 2 =x 3 = 1 / p 3 in this case λ=x 21 xx^2 3 =^1 2 p 3 H= ...
Constrained positive deÖnite matrixes But we should also check another determinant asnp=2. Deleting the last row and column det ...
Constrained positive deÖnite matrixes Now check thex 1 =x 2 =x 3 = 1 / p 3 which is the minimal solution. λ=x^1 x^2 2 x 3 =^1 2 ...
Constrained positive deÖnite matrixes But we should also check another determinant asnp=2. Deleting the last row and last column ...
Constrained positive deÖnite matrixes Observe thatx 2 =x 3 = 0 ,x 1 = 1 ,λ=0 is also a solution of the equations x 2 x 3 + 2 λx ...
Constrained positive deÖnite matrixes Example Solvex 12 x 22 x 32 !max,x 12 +x 22 +x 32 = 1. The Lagrangian x 12 x 22 x 32 +λ x ...
Constrained positive deÖnite matrixes The Hessian A= 0 @ 2 + 2 λ 0 0 0 2 + 2 λ 0 0 0 2 + 2 λ 1 A,B= 2 x 1 2 x 2 2 x 3 . Ifx 1 ...
Constrained positive deÖnite matrixes But we must also check (^) (^) (^) 0 2 0 2 0 0 0 0 4 (^) (^) =^16 :Hence the determinants ...
Constrained positive deÖnite matrixes Now letx 1 = 0 ,x 2 = 1 ,x 3 =0 andλ= 1 ,which solves the equations. H= 0 BB @ 0 0 2 0 0 4 ...
Constrained positive deÖnite matrixes Example Solvex 32 x 12 x 22 !max,x 12 +x 22 +x 32 = 1. The solution isx 3 = 1 ,x 1 =x 2 = ...
Dynamic programming Example Consumer saving problem: (^1) The planning period is Önitet= 1 , 2 ,.. .,T. (^2) At every periodtthe ...
Dynamic programming Problem We want to maximize the aggregate reward T ∑ t= 1 rt(st,at)!max, under the conditions that s 1 2 S s ...
Dynamic programming DeÖnition (^1) Sis called the state space. (^2) Ais called the action space. (^3) rt(s,a)is the reward funct ...
Dynamic programming DeÖnition (^1) The historyhtat timetis the sequenceht=(s 1 ,a 1 ,.. .at 1 ,st). (Observe that there is noatb ...
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