Mathematics for Economists
Dynamic programming DeÖnition A strategy is a Markovian strategy if the strategy is dependent only on the present state, that is ...
Dynamic programming The conditions of the above theorem are quite reasonable. They are (just) guaranteeing the existence of the ...
Dynamic programming DeÖnition A set-valued mapping, that is a correspondence,Φis called (^1) upper semi continuous if for everyx ...
Dynamic programming The main advantage of the compact valued and continuous correspondences is that the parametric optimization ...
Dynamic programming First let us Önd the solution of the problem att=T.In this case the state variable is the wealthw the corre ...
Dynamic programming Now we move to time periodT 1 .By Bellmanís principle one should solve the problem VT 1 (w) = max c 2 [ 0 , ...
Dynamic programming In this case the value of the goal function is r w k+ 1 + s k w w k+ 1 = r w k+ 1 + s k w(k+ (^1) )w k ...
Dynamic programming It gives us the induction hypothesis: VTt = p 1 +k+k^2 +.. .+kt p w σTt = w 1 +k+k^2 +.. .+kt The only thi ...
Dynamic programming Calculating the derivatives 1 2 p c = p 1 +k+k^2 +.. .+kt k 2 p k(wc) p c=p^1 1 +k+k^2 +.. .+kt p wpc k kc= ...
Dynamic programming The value of the goal function is VT(t+ 1 )(w)= r w ∑tn+=^10 kn + s t ∑ n= 0 kn s k w w ∑tn+=^10 kn = = ...
Cake eating problem We have a cake of sizew 1 .We want to eat it inT periods. Our utility for the consumption plan(ct)Tt= 1 is T ...
Cake eating problem (^1) The utility function is twice continuously di§erentiable onx> 0. (^2) u^0 (c)> 0 .Increasing util ...
Cake eating problem As the set of feasible solutions is compact and the utility function is strictly concave there is a unique s ...
Cake eating problem As the constraints are linear Slaterís condition holds. The Lagrange function is L= T ∑ t= 1 βt^1 u(ct)+λ w ...
Cake eating problem DeÖnition The relation u^0 (ct)=βu^0 (ct+ 1 ) is called Euler equation. ...
Cake eating problem Example Solve the cake eating problem for theu(x)$lnxfunction. Observe that in this formulation one can use ...
Cake eating problem It is easy to see that c 1 = w 1 1 +β+.. .+βT^1 , c 2 = βw^1 1 +β+.. .+βT^1 , .. . cT = β T (^1) w 1 1 +β+.. ...
Cake eating problem We can solve the problem as a dynamic programming problem. With utility functionrt(ct)$βt^1 u(ct)and transit ...
Cake eating problem With backwards iteration for timet=T 2 βT^3 u^0 (ct)VT^0 1 (wtct) = βT^3 u^0 (ct)VT^0 1 (wT 1 )= 0 βT^3 u^ ...
Cake eating problem VT 1 (w) = max c 2 [ 0 ,w] βT^2 u(c)+VT(wc) = = max c 2 [ 0 ,w] βT^2 u(c)+βT^1 u(wc) = = βT^2 u(c(w) ...
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