I have 2,001 states, and say the transition matrix of Markov chain is given. Then there is a variable x(s) attached to each state s \in \{1,...,2001\}, and I have to solve the following equation for any x(s):
\frac{a}{f(x(s))}+x(s)=b+\mathbb{E}_{s}[\frac{1}{f(x(s'))}] .
where the function f is non-linear.
I understood this problem as solving a system of non-linear equations, so I used nlsolver package.
I could solve for 50 states, but for 2,001 states, I think I will never get my result as I have to simulate this problem 10,000 times.
This is a replication exercise, and the author mentions that this is a fairly simple numerical exercise. Would there be another approach to solve this problem more time efficiently?