Mathematics for Economists

Optimal control, Bolza problem Asx( 0 )= 1 C 1 +C 2 = 1 .By the transversality condition 0 =Fx^0 ( 1 )= 2 x( 1 ), x( 1 )= 0 , ...

Homework (^1) Solve the problems in calculus of variations with the method of optimal control. (^2) Show with the method of opti ...

Stability DeÖnition An equilibrium pointxof a dynamic systemXis called attractive if for any initial valuex 0 the solutionX(t,x ...

Stability Example The solutiony=0 of the system ay^00 +by^0 +cy= 0 is asymptotically stable if the real part of the characteris ...

Stability Example The solutiony=0 of the system yt+ 2 +ayt+ 1 +byt= 0 is asymptotically stable if the absolute value of the cha ...

Stability Example There is an example of a non-linear system which is attractive but not locally stable. x ^1 x 2 = x 12 (x 2 ...

Phase diagram in on dimension. Let x^0 =x( 1 x). -0.5 0 0.5 1 1.5 0.25 0 -0.25 -0.5 -0.75 x y x y It has two equilibriumx=0 and ...

Phase diagram in two dimension A saddle point. x^0 =  1 1 4 1  x x ' = A x + B yy ' = C x + D y A = 1C = 4B = 1D = 1 -5 -4 -3 ...

Phase diagram A node x^0 =  3 1 1 2  x x ' = A x + B yy ' = C x + D y A = - 3C = 1 B = 1D = - 2 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 ...

Phase diagram An improper node x^0 =  1 1 1 2  x x ' = A x + B yy ' = C x + D y A = 1C = 1 B = - 1D = 3 -5 -4 -3 -2 -1 0 1 2 ...

Phase diagram A focus x^0 =  1 /2 1 1 1 / 2  x x ' = A x + B yy ' = C x + D y A = - 1/2C = - 1 B = 1D = - 1/2 -5 -4 -3 -2 - ...

Phase diagram (^1) Identify the four regions given byx 10 =0 andx 20 = 0 .This curves are called nullclines or demarcation lines ...