Mathematics for Economists
InÖnite cake eating problem As it holds for anyw B = 1 +βB, )B= 1 1 β. π(w) = c(w)= w 1 + 1 ββ = w( 1 β) 1 β+β=w(^1 β). For A=ln ...
InÖnite cake eating problem βnln(cn)βnln(w)! 0. hence we can just prove that lim sup n!∞ βnV(cn)= 0 ...
InÖnite cake eating problem One can generalize the problem like we want to calculate T ∑ t= 0 βtF(t,xt,xt+ 1 )!max wherex 0 is g ...
InÖnite cake eating problem In the cake eating problem withu(x)=lnx T ∑ t= 0 βtln(xt+ 1 xt)!max,x 0 =w. Di§erentiating with resp ...
Homework (^1) Solve the consumption optimization problem withu(c)=cα,α> 0. Consider the casesα< 1 ,α= 1 ,α> 1. (^2) Sol ...
A typical example ZT 0 exp(rt)U(C(t))dt!max K^0 (t)=F(K(t))C(t)bK(t) K( (^0) )=K 0 ,K(T) 0 ,C(t) 0 is an optimal control probl ...
A typical example One should be careful as ZT 0 exp(rt)U(C(t))dt!max K^0 (t)=F(K(t))C(t)bK(t) K( 0 )=K 0 ,K(t) 0 ,C(t) 0. is m ...
A typical example One should be careful as ZT 0 exp(rt)U(C(t))dt!max K^0 (t)=F(K(t))C(t)bK(t) K( 0 )=K 0 ,C(t) 0. is much easie ...
Calculus of variations If we dropC(t)0 and reformulate as ZT 0 exp(rt)U F(K(t))bK(t)K^0 (t) dt K( 0 )=K 0 we get a calculus ...
Calculus of variations Zb a F t,x,x dt! maxmin x(a)=xa,x(b)=xb Letybe an arbitrary function, called variation, withy(a)=y(b ...
Calculus of variations 0 = ψ^0 ( 0 )$ d dλ Zb a F t,x(t)+λy(t),x(t)+λy(t) dt= = Zb a d dλ F t,x(t)+λy(t),x(t)+λy(t) ...
Calculus of variations Integrating by parts and using thaty(a)=y(b)= 0 0 = Zb a q(t)y(t)+p(t)y^0 (t)dt= = Zb a q(t)y(t)dt+ Zb a ...
Calculus of variations As it is true for any variationyone has that q(t)p^0 (t)= 0 , that is Fx^0 t,x(t),x(t) d dt Fx^0 ...
Calculus of variations φa(x(a))+φb(x(b))+ Zb a F t,x,x dt! maxmin The argument is very similar. Now letybe an arbitrary (di ...
Calculus of variations 0 =ψ^0 ( 0 )=φ^0 a(x(a))y(a)+φ^0 b(x(b))y(b)+ Zb a Fx^0 y+Fx^0 y dt. Again integrating by parts Zb a Fx ...
Calculus of variations Asyis an arbitrary function, hencey(b)andy(a)is also arbitrary, so ifx is an optimal solution then Fx^0 ...
Calculus of variations Example Solve the problem Z 2 0 ( 4 3 x^2 16 . x 4 x^2 )etdt!max x( 0 )= 8 / 3 ,x( 2 )= 1 / 3 In this ...
Calculus of variations The EulerñLagrange equation is 6 xet = d dt 16 8 x et= = et( 1 ) 16 8 x +et 8 x Sim ...
Calculus of variations The characteristic equation isλ^2 λ 3 / 4 = 0 .The roots are λ 1 = 1 / 2 ,λ 2 = 3 / 2 .Thex 8 /3 is a p ...
Calculus of variations Example On what curve can the functional Z 2 1 x 2 2 txdt, x( 1 )= 0 ,x( 2 )= 1 attain an extremum. ...
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