Consider the following saving problem
CO
max Eti E ,81 ln ct, feirbi+Orto
subject to
et qbt+i S b, + et, and bt+i > —0. et E lethej follows a Markov chain and the transition matrix is given by
P = [P00 1 — poo 1 — Pn Pn ]'
where pm = Prob(et+i = col; = ea) and pn = Prob(et+i = eilet = el). Let 0= 2, en = 1.0, = 0.1, poi; = 0.925, p11 = 0.5, and fi = 0.96.
1, Write down the Bellman equation. 2. Assume q = fi = 0.96 and b E [-0,12.0]. Use the method of successive approximation to solve for the value function and the policy function numerically. Especially, plot graphs of the difference between the new asset and the old asset, g(b,ek) — b, for k = 0, 1. 3. Define At(b,ek) = Prob(bt = b,et = ek). Using the policy function for 6 and the transition matrix for e, compute graphs of the stationary distribution, .N(b,ek), for k = 0,1. 4. Compute the excess demand for bond z = E f bA(b,ek)db > 0 when q = 0.96. Also compute the excess supply z = E f bA(b,ek)db < 0 when q = 0.99. 5, Find the equilibrium price q that satisfies z(q) = E f bA(b,ek)db = 0. !Hint: You can use a graph of (q,z). For a more precise solution, you may want to use a optimization method, such as scipy.optimize.bisect.1