Mathematics for Economists
Cake eating problem Observe that the partial derivative is by the parameter of the goal function so with g(p,x) = βT^2 u(x)+βT^1 ...
Cake eating problem In the general case using the Fermat principle for the max βTt^1 u^0 (cTt)=VT^0 t+ 1 (wTt+ 1 ) By the Bellma ...
Cake eating problem Example Solve the cake eating problem for theu(x)$lnxfunction. IfT=2 then the Euler equation is 1 c 1 =β^1 c ...
Cake eating problem The value function is V 1 (w 1 ) = lnc 1 +βlnc 2 =ln 1 1 +βw^1 +βln β 1 +βw^1 = = ln^1 1 +β +lnw 1 +βlnw 1 + ...
Cake eating problem IfT=3 then by the Euler equations 1 c 1 =β 1 c 2 =β 21 c 3 and the resource equation is c 1 +c 2 +c 3 =w 1 T ...
Production saving model Example Solve the problem T ∑ t= 0 βtU(ct)! max ct+kt+ 1 = f(kt). wherek 0 is given. ...
Production saving model The simplest way is to turn it to an unconstrained optimization problem that is T ∑ t= 0 βtU(f(kt)kt+ 1 ...
Production saving model Consider the terms withkt+ 1 witht<T.It is in two terms of the sum βtU(f(kt)kt+ 1 )+βt+^1 U(f(kt+ 1 ) ...
Production saving model We can get the same equation with dynamic programing. By the Bellman equation Vt(kt) = max ct 2 [ 0 ,kt] ...
Production saving model To calculateVt^0 + 1 (kt+ 1 )we want to use the envelope theorem. The parametric goal function is g(kt+ ...
Production saving model Again we rewrite the value function and introduce a new control parameterkt+ 2 =f(kt+ 1 )ct+ 1 Vt+ 1 (kt ...
Production saving model Hence βtU^0 (f(kt)kt+ 1 )=βtU^0 (ct)=Vt^0 + 1 (kt+ 1 )=βt+^1 U^0 (f(kt+ 1 )kt+ 2 )f^0 (kt+ 1 ). Hence U^ ...
Production saving model We can also consider the problem as a KuhnñTucker problem T ∑ t= 0 βtU(ct)! max ct+kt+ 1 f(kt) 0 , ct, ...
Production saving model ∂L ∂ct =β tU (^0) (ct)+λtμ t=^0. By the Inada conditionλt> 0 .hence ct+kt+ 1 f(kt)= 0. ...
Production saving model ∂L ∂kt =λt 1 λtf^0 (kt)νtkt= 0. If we exclude corner solutions thenνt=μt=0 and then λt 1 = λtf^0 (kt) λt ...
Production saving model βt^1 U^0 (ct 1 ) = βtU^0 (ct)f^0 (kt) U^0 (ct 1 ) = βU^0 (ct)f^0 (kt) U^0 (f(kt 1 )kt) = βU^0 (f(kt)kt+ ...
Production saving model The Euler equation of the production saving model is U^0 (ct)=βU^0 (ct+ 1 )f^0 (kt+ 1 ), for the cake ea ...
Production saving model Example Solve the production saving model withU(c)=lnc,f(k)=kα. Asf should be concave henceα 1 .Of cour ...
Production saving model Letst$kt+ 1 /ktα=kt+ 1 /f(kt)be the saving rate. ct=f(kt)kt+ 1 hence kt+ 1 ct = kt+ 1 f(kt)kt+ 1 = kt+ 1 ...
Production saving model Hence the Euler equation is st 1 st = αβ 1 1 st+ 1 1 st st =^1 αβ ( 1 st+ 1 ) αβ 1 st 1 = 1 st+ 1 s ...
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