def compute_paths(T, econ):
"""
Compute simulated time paths for exogenous and endogenous variables.
Parameters
===========
T: int
Length of the simulation
econ: a namedtuple of type 'Economy', containing
beta - Discount factor
Sg - Govt spending selector matrix
Sd - Exogenous endowment selector matrix
Sb - Utility parameter selector matrix
Ss - Coupon payments selector matrix
discrete - Discrete exogenous process (True or False)
proc - Stochastic process parameters
Returns
========
path: a namedtuple of type 'Path', containing
g - Govt spending
d - Endowment
b - Utility shift parameter
s - Coupon payment on existing debt
c - Consumption
l - Labor
p - Price
τ - Tax rate
rvn - Revenue
B - Govt debt
R - Risk free gross return
π - One-period risk-free interest rate
Π - Cumulative rate of return, adjusted
ξ - Adjustment factor for Π
The corresponding values are flat numpy ndarrays.
"""
# == Simplify names == #
beta, Sg, Sd, Sb, Ss = econ.beta, econ.Sg, econ.Sd, econ.Sb, econ.Ss
if econ.discrete:
P, x_vals = econ.proc
else:
A, C = econ.proc
# == Simulate the exogenous process x == #
if econ.discrete:
state = mc_sample_path(P, init=0, sample_size=T)
x = x_vals[:, state]
else:
# == Generate an initial condition x0 satisfying x0 = A x0 == #
nx, nx = A.shape
x0 = nullspace((eye(nx) - A))
x0 = -x0 if (x0[nx-1] < 0) else x0
x0 = x0 / x0[nx-1]
# == Generate a time series x of length T starting from x0 == #
nx, nw = C.shape
x = zeros((nx, T))
w = randn(nw, T)
x[:, 0] = x0.T
for t in range(1, T):
x[:, t] = dot(A, x[:, t-1]) + dot(C, w[:, t])
# == Compute exogenous variable sequences == #
g, d, b, s = (dot(S, x).flatten() for S in (Sg, Sd, Sb, Ss))
# == Solve for Lagrange multiplier in the govt budget constraint == #
# In fact we solve for ν = lambda / (1 + 2*lambda). Here ν is the
# solution to a quadratic equation a(ν**2 - ν) + b = 0 where
# a and b are expected discounted sums of quadratic forms of the state.
Sm = Sb - Sd - Ss
# == Compute a and b == #
if econ.discrete:
ns = P.shape[0]
F = scipy.linalg.inv(np.identity(ns) - beta * P)
a0 = 0.5 * dot(F, dot(Sm, x_vals).T**2)[0]
H = dot(Sb - Sd + Sg, x_vals) * dot(Sg - Ss, x_vals)
b0 = 0.5 * dot(F, H.T)[0]
a0, b0 = float(a0), float(b0)
else:
H = dot(Sm.T, Sm)
a0 = 0.5 * var_quadratic_sum(A, C, H, beta, x0)
H = dot((Sb - Sd + Sg).T, (Sg + Ss))
b0 = 0.5 * var_quadratic_sum(A, C, H, beta, x0)
# == Test that ν has a real solution before assigning == #
warning_msg = """
Hint: you probably set government spending too {}. Elect a {}
Congress and start over.
"""
disc = a0**2 - 4 * a0 * b0
if disc >= 0:
ν = 0.5 * (a0 - sqrt(disc)) / a0
else:
print("There is no Ramsey equilibrium for these parameters.")
print(warning_msg.format('high', 'Republican'))
sys.exit(0)
# == Test that the Lagrange multiplier has the right sign == #
if ν * (0.5 - ν) < 0:
print("Negative multiplier on the government budget constraint.")
print(warning_msg.format('low', 'Democratic'))
sys.exit(0)
# == Solve for the allocation given ν and x == #
Sc = 0.5 * (Sb + Sd - Sg - ν * Sm)
Sl = 0.5 * (Sb - Sd + Sg - ν * Sm)
c = dot(Sc, x).flatten()
l = dot(Sl, x).flatten()
p = dot(Sb - Sc, x).flatten() # Price without normalization
τ = 1 - l / (b - c)
rvn = l * τ
# == Compute remaining variables == #
if econ.discrete:
H = dot(Sb - Sc, x_vals) * dot(Sl - Sg, x_vals) - dot(Sl, x_vals)**2
temp = dot(F, H.T).flatten()
B = temp[state] / p
H = dot(P[state, :], dot(Sb - Sc, x_vals).T).flatten()
R = p / (beta * H)
temp = dot(P[state, :], dot(Sb - Sc, x_vals).T).flatten()
ξ = p[1:] / temp[:T-1]
else:
H = dot(Sl.T, Sl) - dot((Sb - Sc).T, Sl - Sg)
L = np.empty(T)
for t in range(T):
L[t] = var_quadratic_sum(A, C, H, beta, x[:, t])
B = L / p
Rinv = (beta * dot(dot(Sb - Sc, A), x)).flatten() / p
R = 1 / Rinv
AF1 = dot(Sb - Sc, x[:, 1:])
AF2 = dot(dot(Sb - Sc, A), x[:, :T-1])
ξ = AF1 / AF2
ξ = ξ.flatten()
π = B[1:] - R[:T-1] * B[:T-1] - rvn[:T-1] + g[:T-1]
Π = cumsum(π * ξ)
# == Prepare return values == #
path = Path(g=g,
d=d,
b=b,
s=s,
c=c,
l=l,
p=p,
τ=τ,
rvn=rvn,
B=B,
R=R,
π=π,
Π=Π,
ξ=ξ)
return path
def compute_paths(T, econ):
"""
Compute simulated time paths for exogenous and endogenous variables.
Parameters
===========
T: int
Length of the simulation
econ: a namedtuple of type 'Economy', containing
beta - Discount factor
Sg - Govt spending selector matrix
Sd - Exogenous endowment selector matrix
Sb - Utility parameter selector matrix
Ss - Coupon payments selector matrix
discrete - Discrete exogenous process (True or False)
proc - Stochastic process parameters
Returns
========
path: a namedtuple of type 'Path', containing
g - Govt spending
d - Endowment
b - Utility shift parameter
s - Coupon payment on existing debt
c - Consumption
l - Labor
p - Price
tau - Tax rate
rvn - Revenue
B - Govt debt
R - Risk free gross return
pi - One-period risk-free interest rate
Pi - Cumulative rate of return, adjusted
xi - Adjustment factor for Pi
The corresponding values are flat numpy ndarrays.
"""
# == Simplify names == #
beta, Sg, Sd, Sb, Ss = econ.beta, econ.Sg, econ.Sd, econ.Sb, econ.Ss
if econ.discrete:
P, x_vals = econ.proc
else:
A, C = econ.proc
# == Simulate the exogenous process x == #
if econ.discrete:
state = mc_sample_path(P, init=0, sample_size=T)
x = x_vals[:,state]
else:
# == Generate an initial condition x0 satisfying x0 = A x0 == #
nx, nx = A.shape
x0 = nullspace((eye(nx) - A))
x0 = -x0 if (x0[nx-1] < 0) else x0
x0 = x0 / x0[nx-1]
# == Generate a time series x of length T starting from x0 == #
nx, nw = C.shape
x = zeros((nx, T))
w = randn(nw, T)
x[:, 0] = x0.T
for t in range(1,T):
x[:, t] = dot(A, x[:, t-1]) + dot(C, w[:, t])
# == Compute exogenous variable sequences == #
g, d, b, s = (dot(S, x).flatten() for S in (Sg, Sd, Sb, Ss))
# == Solve for Lagrange multiplier in the govt budget constraint == #
## In fact we solve for nu = lambda / (1 + 2*lambda). Here nu is the
## solution to a quadratic equation a(nu**2 - nu) + b = 0 where
## a and b are expected discounted sums of quadratic forms of the state.
Sm = Sb - Sd - Ss
# == Compute a and b == #
if econ.discrete:
ns = P.shape[0]
F = scipy.linalg.inv(np.identity(ns) - beta * P)
a0 = 0.5 * dot(F, dot(Sm, x_vals).T**2)[0]
H = dot(Sb - Sd + Sg, x_vals) * dot(Sg - Ss, x_vals)
b0 = 0.5 * dot(F, H.T)[0]
a0, b0 = float(a0), float(b0)
else:
H = dot(Sm.T, Sm)
a0 = 0.5 * var_quadratic_sum(A, C, H, beta, x0)
H = dot((Sb - Sd + Sg).T, (Sg + Ss))
b0 = 0.5 * var_quadratic_sum(A, C, H, beta, x0)
# == Test that nu has a real solution before assigning == #
warning_msg = """
Hint: you probably set government spending too {}. Elect a {}
Congress and start over.
"""
disc = a0**2 - 4 * a0 * b0
if disc >= 0:
nu = 0.5 * (a0 - sqrt(disc)) / a0
else:
print "There is no Ramsey equilibrium for these parameters."
print warning_msg.format('high', 'Republican')
sys.exit(0)
# == Test that the Lagrange multiplier has the right sign == #
if nu * (0.5 - nu) < 0:
print "Negative multiplier on the government budget constraint."
print warning_msg.format('low', 'Democratic')
sys.exit(0)
# == Solve for the allocation given nu and x == #
Sc = 0.5 * (Sb + Sd - Sg - nu * Sm)
Sl = 0.5 * (Sb - Sd + Sg - nu * Sm)
c = dot(Sc, x).flatten()
l = dot(Sl, x).flatten()
p = dot(Sb - Sc, x).flatten() # Price without normalization
tau = 1 - l / (b - c)
rvn = l * tau
# == Compute remaining variables == #
if econ.discrete:
H = dot(Sb - Sc, x_vals) * dot(Sl - Sg, x_vals) - dot(Sl, x_vals)**2
temp = dot(F, H.T).flatten()
B = temp[state] / p
H = dot(P[state, :], dot(Sb - Sc, x_vals).T).flatten()
R = p / (beta * H)
temp = dot(P[state,:], dot(Sb - Sc, x_vals).T).flatten()
xi = p[1:] / temp[:T-1]
else:
H = dot(Sl.T, Sl) - dot((Sb - Sc).T, Sl - Sg)
L = np.empty(T)
for t in range(T):
L[t] = var_quadratic_sum(A, C, H, beta, x[:, t])
B = L / p
Rinv = (beta * dot(dot(Sb - Sc, A), x)).flatten() / p
R = 1 / Rinv
AF1 = dot(Sb - Sc, x[:, 1:])
AF2 = dot(dot(Sb - Sc, A), x[:, :T-1])
xi = AF1 / AF2
xi = xi.flatten()
pi = B[1:] - R[:T-1] * B[:T-1] - rvn[:T-1] + g[:T-1]
Pi = cumsum(pi * xi)
# == Prepare return values == #
path = Path(g=g,
d=d,
b=b,
s=s,
c=c,
l=l,
p=p,
tau=tau,
rvn=rvn,
B=B,
R=R,
pi=pi,
Pi=Pi,
xi=xi)
return path