Economic Theory; Macroeconomics Sequence
Economic Theory; Macroeconomics Sequence ECON 714
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This 3 page Class Notes was uploaded by April Jerde on Thursday September 17, 2015. The Class Notes belongs to ECON 714 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 7 views. For similar materials see /class/205158/econ-714-university-of-wisconsin-madison in Economcs at University of Wisconsin - Madison.
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Date Created: 09/17/15
Econ 714 Simple Dynamic Programming Sang Yoon Tim Lee In a Brock Mirman economy the planner chooses Ch kt1f 0 to maximize f V 1mm t0 Ctkt1 S 14k where A gt 0 and 1x 8 E 0 1 There are many ways to solve this model Note that the inequality will always be binding since the one period utility function and production function are increasing 1 N Euler Equations This is what we39re already used to Check the concavity of the functions and Slater 39s condition and verify that the foc39s of the Lagrangean are sufficient for a saddle point Then the foc39s wrt Ct and kt At Ct At BMHIXAk ll These can be combined to give Ct1 0471 7 Ak Ct 5 H1 How much more can we do from here If you remember from Rody39s class given kg and a transversality condition we can solve for the optimal path iteratively But this looks tedious and impossiblethat39s why in the Rody39s class we had to assort to graphs and usually focused only on the steady state Iterating the Bellman Equation This is what we would do to solve for the value function in a computer in this case it39s even possible to solve by hand Practically one wouldn39t get too involved with the background theory we39re learning now but 1 03 just to be straight note that even this simple economy does not apply to the theory we39ve learned up to now since the log function is unbounded We could set an upper bound for the return function by setting an upper bound for k but the log function is still unbounded below Refer to SLP page 92 for when it is still valid to use the Bellman equation for now take it for granted that it is Set vn1k E Tvnk mafkm lmAk Id BungH k e 0 Again note that we can39t verify that T is a contraction mapping using Blackwell39s condition because we39re not in the space of bounded functions this was an assump tion for applying Blackwell But still Thm 414 on SNL page 92 ensures that iter ations works Set 120k 0 to obtain that k 0 Then 121k ln Aka The next iteration yields i 8 k 7 1 u and we could get 122k by plugging this in to 020 Ak 1 3ka 1 AW ngagkn H n 7 i k e and continuing the iteration we would get gk E k Ball vac 17 11nlA1 7 M 1 f2 13 Although solvable by hand in this case it is not recommended unless you are using ln k lnAth a computer However it is critical that you understand the algorithm so that you are able to feed it to a computer Guess and Verify This is the most conventional and maybe the only way to solve for the value function by hand In many cases it is obvious what the value function will look like in other cases it is not In this case we kind of see that the value function should look like vk E Flnk where E and F are undetermined constants Now plug this in the functional equa tion E F lnk max lnAk 7 k IBE F Ink k e0Ak The foc for the RHS yields M k 7 18FAka Plug this in the FE to get Aka M a E Flnk i 1HW ME Fln1 IBPAk and matching coefficients we obtain x P 7 178 1 BDC E 178 lnA1ilth lnA m 178 and plugging this in FE we get the same answer as above References Recursive Macroeconomic Theory Ljungqvist L and T Sargent Recursive Methods in Economic Dynamics Stokey N and RE Lucas Jr with E Prescott
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