def compute_sequence(self, x0, ts_length=None, Pay=None):
""" Simulate quantities and prices for our economy """
lq = LQ(self.Q, self.R, self.A, self.B, self.C, N=self.W, beta=self.beta)
xp, up, wp = lq.compute_sequence(x0, ts_length)
self.h = self.Sh.dot(xp)
self.k = self.Sk.dot(xp)
self.i = self.Si.dot(xp)
self.b = self.Sb.dot(xp)
self.d = self.Sd.dot(xp)
self.c = self.Sc.dot(xp)
self.g = self.Sg.dot(xp)
self.s = self.Ss.dot(xp)
# === Value of J-period risk-free bonds === #
# === See p.145: Equation (7.11.2) === #
e1 = np.zeros((1,self.nc))
e1[0,0] = 1
self.R1_Price = np.empty((ts_length+1,1))
self.R2_Price = np.empty((ts_length+1,1))
self.R5_Price = np.empty((ts_length+1,1))
for i in range(ts_length+1):
self.R1_Price[i,0] = self.beta*e1.dot(self.Mc).dot(np.linalg.matrix_power(self.A0,1)).dot(xp[:,i])/e1.dot(self.Mc).dot(xp[:,i])
self.R2_Price[i,0] = self.beta**2*e1.dot(self.Mc).dot(np.linalg.matrix_power(self.A0,2)).dot(xp[:,i])/e1.dot(self.Mc).dot(xp[:,i])
self.R5_Price[i,0] = self.beta**5*e1.dot(self.Mc).dot(np.linalg.matrix_power(self.A0,5)).dot(xp[:,i])/e1.dot(self.Mc).dot(xp[:,i])
# === Gross rates of return on 1-period risk-free bonds === #
self.R1_Gross = 1/self.R1_Price
# === Net rates of return on J-period risk-free bonds === #
# === See p.148: log of gross rate of return, divided by j === #
self.R1_Net = np.log(1/self.R1_Price)/1
self.R2_Net = np.log(1/self.R2_Price)/2
self.R5_Net = np.log(1/self.R5_Price)/5
# === Value of asset whose payout vector is Pay*xt === #
# See p.145: Equation (7.11.1)
if isinstance(Pay,np.ndarray) == True:
self.Za = Pay.T.dot(self.Mc)
self.Q = solve_discrete_lyapunov(self.A0.T*self.beta**0.5,self.Za)
self.q = self.beta/(1-self.beta)*np.trace(self.C.T.dot(self.Q).dot(self.C))
self.Pay_Price = np.empty((ts_length+1,1))
self.Pay_Gross = np.empty((ts_length+1,1))
self.Pay_Gross[0,0] = np.nan
for i in range(ts_length+1):
self.Pay_Price[i,0] = (xp[:,i].T.dot(self.Q).dot(xp[:,i]) + self.q)/e1.dot(self.Mc).dot(xp[:,i])
for i in range(ts_length):
self.Pay_Gross[i+1,0] = self.Pay_Price[i+1,0]/(self.Pay_Price[i,0] - Pay.dot(xp[:,i]))
return
# == Formulate as an LQ problem == #
Q = 1
R = np.zeros((2, 2))
Rf = np.zeros((2, 2))
Rf[0, 0] = q
A = [[1 + r, -c_bar + mu],
[0, 1]]
B = [[-1],
[0]]
C = [[sigma],
[0]]
# == Compute solutions and simulate == #
lq = LQ(Q, R, A, B, C, beta=beta, T=T, Rf=Rf)
x0 = (0, 1)
xp, up, wp = lq.compute_sequence(x0)
# == Convert back to assets, consumption and income == #
assets = xp[0, :] # a_t
c = up.flatten() + c_bar # c_t
income = wp[0, 1:] + mu # y_t
# == Plot results == #
n_rows = 2
fig, axes = plt.subplots(n_rows, 1, figsize=(12, 10))
plt.subplots_adjust(hspace=0.5)
for i in range(n_rows):
axes[i].grid()
axes[i].set_xlabel(r'Time')
bbox = (0., 1.02, 1., .102)
