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2 No lecture on Wed February 8th Thursday 9 th Feb 14:15 - 17:00 Thursday 9 th Feb 14:15 - 17:00

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3 The Maximum Principle: A Reminder

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4 Example k F(k) capital consumption production function depreciation rate

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5 Example Solve for c Two differential equations in k,π

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6 Example Another way: differentiate

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7 Example Another way: X X

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8 Example Another way: k c No t !!!!!!

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9 Example Another way: k c k’

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10 Example Another way: k c k’ k*

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11 Example Another way: k c k*

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12 Example Another way: k c k*

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13 Example Another way: k c k*

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14 Example Another way: k c k*

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15 Example Another way: k c k* Stationary point

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16 Example Another way: k c k*

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17 Example Another way: k c k*

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18 Example k c k* k(0),c(0) ???? k(0), is given k(0) c(0), is chosen c → 0 k → 0

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19 Example k c k* k(0),c(0) ???? k(0), is given k(0) c(0), is chosen c → 0 k → 0

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20 Richard E. Bellman 1920-1983 Another approach to dynamic programming

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21 Another approach to dynamic programming For a given time τ < T define the problem:

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22 Another approach to dynamic programming Lagrange: (equating the derivative w.r.t. z t to 0 ) But: ????? ?????? ?

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23 Another approach to dynamic programming But: ????? ?????? ? The Lagrangian of the original problem:

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24 Another approach to dynamic programming But: ????? ?????? ? The Lagrangian of the original problem:

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25 Another approach to dynamic programming but this is the (first order) condition for maximizing the Hamiltonian

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26 Another approach to dynamic programming Calculating the Bellman value functions is equivalent to the maximum principle (Hamiltonian)

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27 Another approach to dynamic programming

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28 Another approach to dynamic programming Backwards Induction

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