options = dict(print=True,
algorithm='newton',
maxit=253)
S = growth_model.solve(vtrue, ktrue, print=True, algorithm='newton', maxit=120)
v, pr = growth_model.Value(S.Wealth, order)
k = growth_model.Policy(S.Wealth)
# resid = growth_model.residuals(s)
# ============ Simulate Model
T = 20
data = growth_model.simulate(T, np.atleast_2d(smin))
# ============ Compute Linear-Quadratic Approximation
growth_model.lqapprox(sstar, kstar)
vlq, plq = growth_model.Value(S.Wealth, order)
klq = growth_model.Policy(S.Wealth)
# ============ Compute Analytic Solution
vtrue = vstar + b * (np.log(S.Wealth) - np.log(sstar))
# ============== Make plots:
Wealth = S.Wealth.T
# Plot Optimal Policy
demo.figure('Optimal Investment Policy', 'Wealth', 'Investment')
plt.plot(Wealth, np.c_[k, klq])
demo.annotate(sstar, kstar,'$s^*$ = %.2f\n$k^*$ = %.2f' % (sstar, kstar), 'bo', (10, -7))
model = DPmodel(basis, reward, transition, bounds,
x=['released'], discount=delta, e=e, w=w)
# Deterministic Steady-State
xstar = 1 # deterministic steady-state action
sstar = 1 + (a[0] *(1-delta)/b[0]) ** (1 / b[1]) # deterministic steady-state stock
# Check Model Derivatives
# dpcheck(model,sstar,xstar)
## SOLUTION
# Compute Linear-Quadratic Approximation at Collocation Nodes
model.lqapprox(sstar, xstar)
# Solve Bellman Equation
model.solve() # no need to pass LQ to model, it's already there
resid, s, v, x = model.solution()
# Plot Optimal Policy
demo.figure('Optimal Irrigation Policy', 'Reservoir Level', 'Irrigation')
plt.plot(s, x.T)
# Plot Value Function
demo.figure('Value Function', 'Reservoir Level', 'Value')
plt.plot(s, v.T)
# Plot Shadow Price Function
demo.figure('Shadow Price Function', 'Reservoir Level', 'Shadow Price')
discount=delta,
e=e,
w=w)
# Deterministic Steady-State
xstar = 1 # deterministic steady-state action
sstar = 1 + (a[0] * (1 - delta) / b[0])**(1 / b[1]
) # deterministic steady-state stock
# Check Model Derivatives
# dpcheck(model,sstar,xstar)
## SOLUTION
# Compute Linear-Quadratic Approximation at Collocation Nodes
model.lqapprox(sstar, xstar)
# Solve Bellman Equation
model.solve() # no need to pass LQ to model, it's already there
resid, s, v, x = model.solution()
# Plot Optimal Policy
demo.figure('Optimal Irrigation Policy', 'Reservoir Level', 'Irrigation')
plt.plot(s, x.T)
# Plot Value Function
demo.figure('Value Function', 'Reservoir Level', 'Value')
plt.plot(s, v.T)
# Plot Shadow Price Function
demo.figure('Shadow Price Function', 'Reservoir Level', 'Shadow Price')