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sol_set4

# sol_set4 - Solutions to Problem Set 4 ARE 261 Question 1...

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Solutions to Problem Set 4 ARE 261 September 30, 2002 Question 1 The control is u i ,thestateis y i and the co-state variable is λ i .Th e f rst order conditions with respect to each of these variables are given by ∂L ∂u i = μ ∂F ∂u i + λ i x =0 ,i =0 , 1 ,...n 1 (1) ∂L ∂y i = ( ∂F ∂y i x + ( λ i λ i 1 ) x =0 ,i =1 , ...n 1 λ n 1 x =0 ,i = n (2) ∂L ∂λ i =( y i +1 y i ) u i x =0 i =0 , 1 , ...n 1 (3) The f rst and the third equation hold for i =0 , ..., n 1 while the second equation is di f erent for y i ( i =1 , 2 ..., n 1 )andand y n .( S i n c e y 0 is given there is no f rst order condition for this variable.) Question 2 The Lagrangian can be written as L = n 1 X i =0 ( H i x λ i ( y i +1 y i

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The f rst order conditions in terms of the Hamiltonian are ∂L ∂u i = ∂H i ∂u i x =0 = ∂H i ∂u i =0 i =0 , 1 ...n 1 (5) ∂L ∂y i = ³ ∂H ∂y i x +(( λ i λ i 1 )) ´ =0 i =1 , 2 ...n 1= ∂H ∂y i = ( λ i λ i 1 ) x (6) ∂L ∂y n = λ n 1 =0 (7) ∂L ∂λ i = ³ ( y i +1 y i )+ ∂H ∂λ i x ´ =0= y i +1 y i x = ∂H ∂λ i = u i (8) Equation (7) implies that λ n 1 =0 when y n is free.
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sol_set4 - Solutions to Problem Set 4 ARE 261 Question 1...

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