I've been trying to figure out how to run a basic recursive problem in sage. The basic structure of the problem I have figured out, its just a matter of putting it in terms of a for/ while loop.
#Initializing values
k,k1,vk,beta,vk1=var('k,k1,vk,beta,vk1')
vk= log(k-k1)+beta*log(k1)
#I want to loop this procedure
FOC=vk.diff(k1)
k1star=solve(FOC==0,k1)
vk=(vk).subs(k1star)
vk
Any help is appreciated.
Below is what I was able to come up with. I checked the first couple iterations to make sure they were correct. After that, it looks right at a glance, but I haven't verified. This is from my old SAGE 8.4 installation (I have 9.0 on a different computer), so the print statement will be formatted differently in version 9.0.
The key is to substitute the result back into v0 before substituting back into vk.
sage: v0 = log(k)
sage: for n in range(5):
....: vk = log(k-k1) + beta*v0(k=k1)
....: FOC = vk.diff(k1)
....: k1star = solve(FOC==0, k1)
....: print(n, k1star)
....: v0 = (vk).subs(k1=k1star[0].rhs())
....:
(0, [k1 == beta*k/(beta + 1)])
(1, [k1 == (beta^2 + beta)*k/(beta^2 + beta + 1)])
(2, [k1 == (beta^3 + beta^2 + beta)*k/(beta^3 + beta^2 + beta + 1)])
(3, [k1 == (beta^4 + beta^3 + beta^2 + beta)*k/(beta^4 + beta^3 + beta^2 + beta + 1)])
(4, [k1 == (beta^5 + beta^4 + beta^3 + beta^2 + beta)*k/(beta^5 + beta^4 + beta^3 + beta^2 + beta + 1)])
Asked: 2020-07-19 19:38:18 +0100
Seen: 805 times
Last updated: Jul 20 '20
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I don't see the point : the value of
k1that nullifies $\frac{\partial vk}{\partial k1}$ is by definition an expression that does not depend onk1.At this point :
But, of course :
So there's nothing to iterate.
Aren't you trying to implement something like Newton's method of root-finding when
solvedoesn't give a explicit answer ?The context is given in Ask Sage question 52559: Cake-eating problem.
@EmmanuelCharpentier I see the issue with my approach. Though I actually made a video on solving a planners problem which is similar to this that uses value function iteration, https://www.youtube.com/watch?v=Bv_tP.... I just want to be able to code something like this.