1. Consider the deterministic Brock-Mirman model studied in class. The discount factor is
β ∈ (0, 1), the per-period utility function is u(c) = log(c) and the production function is
f(k) = kα + (1 − δ)k, where δ = 1 and α ∈ (0, 1). Assume β = 0.99 and α = 0.3.
(a) Compute the model's steady-state k∗.
(b) Use the F and G that we derived in class by solving the Bellman equation by guess and
verify to plot the optimal decision rules for k and c in the range [0.005, ¯k], where ¯k is
the upper bound on the state space. Can you figure out such ¯k. If not, set ¯k to twice
the steady-state capital stock.
(c) Let the initial capital stock be k0 = 0.5k∗. Compute the time series for kt, t = 1, 2, ..., 200.
Plot the time series. Now do the same for k0 = 1.5k∗. Do the sequences converge to k∗?
2. Now solve the model in question (1), with the same parameter values, by discretising the state
space and iterating on the Bellman equation (use convergence criterium for the sup norm of
1e−5). For each iteration store the value of the sup norm (you may also want to print it on
the screen to see how the procedure is doing at each iteration). Use the state space you used
in (1) (b) and set the number of grid points (N + 1) to 401.
(a) Plot the optimal decision rules for k and c in the same graph as those in (1) (b). If you
have done it correctly, they should approximately agree.
(b) Plot the vector of the values of the sup norm against the the index denoting each
iteration. You should see that the sup norm declines monotonically. Would you expect
such a result? Why or why not?
(c) Now that you know your code works, solve the model again for a realistic
rate δ = 0.025 (you need to re-calculate the steady-state and adjust the state space
accordingly). In a new graph, again plot the decision rules. They should look more
linear than before as the resource constraint is more linear. The decision rules should be
especially close to linear in the neighborhood of k∗, the steady-state capital stock. (Note:
This justifies the use of LQ approximation methods we will discuss later in the module.)
Don't worry about the step-like features in the optimal consumption rule—these can be
'smoothed' out by increasing N.
3. Now solve the model with f(k) = zkα + (1 − δ)k, where z ∈ {1.1, 0.9} and the transition
matrix between these two values is
P =
0.95 0.05
0.05 0.95
.
Plot the value function and the decision rules.
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