output (GDP)
C_path = [T+S-2,] vector, equilibrium time path of aggregate
consumption
EulErr_path = [S, T+S-1] matrix, equilibrium time path of the
Euler errors for all the savings decisions
tpi_time = scalar, number of seconds to compute TPI solution
ResDiff = [T-1,] vector, errors in the resource constraint
from period 1 to T-1. We don't use T because we are
missing one individual's consumption in that period
--------------------------------------------------------------------
'''
if TPI_solve == True:
print 'BEGIN EQUILIBRIUM TIME PATH COMPUTATION'
Gamma1 = 0.9 * b_ss
# Make sure init. period distribution is feasible in terms of K
K1, K_constr_tpi1 = ssf.get_K(Gamma1)
rparams = (A, alpha, delta)
r1 = ssf.get_r(rparams, K1, L)
BQ1 = (1 + r1) * Gamma1[-1]
if K1 <= 0 and Gamma1[-1] > 0:
print 'Initial savings distribution is not feasible because K1<=0. Some element(s) of Gamma1 must increase.'
elif K1 <= 0 and Gamma1[-1] <= 0:
print 'Initial savings distribution is not feasible because K1<=0 and b_{S+1,1}<=0. Some element(s) of Gamma1 must increase.'
elif K1 > 0 and Gamma1[-1] <= 0:
print 'Initial savings distribution is not feasible because b_{S+1,1}<=0. b_{S+1,1} must increase.'
else:
# Choose initial guess of path of aggregate capital stock
# and total bequests. Use parabola specification
# aa*x^2 + bb*x + cc
# Initial aggregate capital path
output (GDP)
C_path = [T+S-2,] vector, equilibrium time path of aggregate
consumption
EulErr_path = [S, T+S-1] matrix, equilibrium time path of the
Euler errors for all the savings decisions
tpi_time = scalar, number of seconds to compute TPI solution
ResDiff = [T-1,] vector, errors in the resource constraint
from period 1 to T-1. We don't use T because we are
missing one individual's consumption in that period
--------------------------------------------------------------------
'''
if TPI_solve == True:
print 'BEGIN EQUILIBRIUM TIME PATH COMPUTATION'
Gamma1 = 0.9 * b_ss
# Make sure init. period distribution is feasible in terms of K
K1, K_constr_tpi1 = ssf.get_K(Gamma1)
rparams = (A, alpha, delta)
r1 = ssf.get_r(rparams, K1, L)
BQ1 = (1 + r1) * Gamma1[-1]
if K1 <= 0 and Gamma1[-1] > 0:
print 'Initial savings distribution is not feasible because K1<=0. Some element(s) of Gamma1 must increase.'
elif K1 <= 0 and Gamma1[-1] <= 0:
print 'Initial savings distribution is not feasible because K1<=0 and b_{S+1,1}<=0. Some element(s) of Gamma1 must increase.'
elif K1 > 0 and Gamma1[-1] <= 0:
print 'Initial savings distribution is not feasible because b_{S+1,1}<=0. b_{S+1,1} must increase.'
else:
# Choose initial guess of path of aggregate capital stock
# and total bequests. Use parabola specification
# aa*x^2 + bb*x + cc
# Initial aggregate capital path
def TPI(params, Kpath_init, BQpath_init, Gamma1, nvec, Lpath, b_ss,
graphs):
'''
Generates steady-state time path for all endogenous objects from
initial state (K1, BQ1, Gamma1) to the steady state.
Inputs:
params = length 17 tuple, (S, T, beta, sigma, chi_b, L, A,
alpha, delta, K1, K_ss, BQ_ss, C_ss maxiter_TPI,
mindist_TPI, xi, TPI_tol)
S = integer in [3,80], number of periods an individual
lives
T = integer > S, number of time periods until steady
state
beta = scalar in [0,1), discount factor for each model
period
sigma = scalar > 0, coefficient of relative risk aversion
chi_b = scalar > 0, scale parameter on utility of bequests
L = scalar > 0, exogenous aggregate labor
A = scalar > 0, total factor productivity parameter in
firms' production function
alpha = scalar in (0,1), capital share of income
delta = scalar in [0,1], model-period depreciation rate of
capital
K1 = scalar > 0, initial period aggregate capital stock
K_ss = scalar > 0, steady-state aggregate capital stock
BQ_ss = scalar > 0, steady-state total bequests
maxiter_TPI = integer >= 1, Maximum number of iterations for TPI
mindist_TPI = scalar > 0, Convergence criterion for TPI
xi = scalar in (0,1], TPI path updating parameter
TPI_tol = scalar > 0, tolerance level for fsolve's in TPI
Kpath_init = [T+S-1,] vector, initial guess for the time path
of the aggregate capital stock
BQpath_init = [T+S-1,] vector, initial guess for the time path
of total bequests
Gamma1 = [S,] vector, initial period savings distribution
nvec = [S,] vector, exogenous labor supply n_{s,t}
Lpath = [T+S-1,] vector, exogenous time path for aggregate
labor
b_ss = [S,] vector, steady-state savings distribution
graphs = boolean, =True if want graphs of TPI objects
Functions called:
c8ssf.get_r
c8ssf.get_w
get_cbepath
c4ssf.get_K
Objects in function:
start_time = scalar, current processor time in seconds (float)
iter_TPI = integer >= 0, current iteration of TPI
dist_TPI = scalar >= 0, distance measure for fixed point
Kpath_new = [T+S-2,] vector, new path of the aggregate
capital stock implied by household and firm
optimization
BQpath_new = [T+S-1,] vector, new path of total bequests
implied by household and firm optimization
r_params = length 3 tuple, parameters passed in to get_r
w_params = length 2 tuple, parameters passed in to get_w
cbe_params = length 6 tuple. parameters passed in to
get_cbepath
rpath = [T+S-2,] vector, equilibrium time path of the
interest rate
wpath = [T+S-2,] vector, equilibrium time path of the
real wage
cpath = [S, T+S-2] matrix, equilibrium time path values
of individual consumption c_{s,t}
bpath = [S, T+S-1] matrix, equilibrium time path values
of individual savings b_{s+1,t+1}
EulErrPath = [S, T+S-1] matrix, equilibrium time path values
of Euler errors corresponding to individual
savings b_{s+1,t+1} (first column is zeros)
Kpath_constr = [T+S-2,] boolean vector, =True if K_t<=0
Kpath = [T+S-2,] vector, equilibrium time path of the
aggregate capital stock
Y_params = length 2 tuple, parameters to be passed to get_Y
Ypath = [T+S-2,] vector, equilibrium time path of
aggregate output (GDP)
Cpath = [T+S-2,] vector, equilibrium time path of
aggregate consumption
elapsed_time = scalar, time to compute TPI solution (seconds)
Returns: bpath, cpath, BQpath, wpath, rpath, Kpath, Ypath, Cpath,
EulErrpath, elapsed_time
'''
start_time = time.clock()
(S, T, beta, sigma, chi_b, L, A, alpha, delta, K1, K_ss, BQ_ss,
C_ss, maxiter_TPI, mindist_TPI, xi, TPI_tol) = params
iter_TPI = int(0)
dist_TPI = 10.
Kpath_new = Kpath_init
BQpath_new = BQpath_init
r_params = (A, alpha, delta)
w_params = (A, alpha)
cbe_params = (S, T, beta, sigma, chi_b, TPI_tol)
while (iter_TPI < maxiter_TPI) and (dist_TPI >= mindist_TPI):
iter_TPI += 1
Kpath_init = xi * Kpath_new + (1 - xi) * Kpath_init
BQpath_init = xi * BQpath_new + (1 - xi) * BQpath_init
rpath = ssf.get_r(r_params, Kpath_init, Lpath)
wpath = ssf.get_w(w_params, Kpath_init, Lpath)
cpath, bpath, EulErrPath = get_cbepath(cbe_params, nvec, rpath,
wpath, BQpath_init, Gamma1, b_ss)
Kpath_new = np.zeros(T+S-2)
Kpath_new[:T], Kpath_constr = ssf.get_K(bpath[:, :T])
Kpath_new[T:] = K_ss
Kpath_constr = np.append(Kpath_constr, np.zeros(S-1, dtype=bool))
Kpath_new[Kpath_constr] = 1
BQpath_new = np.zeros(T+S-2)
BQpath_new[:T] = ssf.get_BQ(rpath[:T], bpath[S-1, :T])
BQpath_new[T:] = BQ_ss
# Check the distance of Kpath_new and BQpath_new
Kdist_TPI = (Kpath_new[1:T] - Kpath_init[1:T]) / Kpath_init[1:T]
BQdist_TPI = (BQpath_new[1:T] - BQpath_init[1:T]) / BQpath_init[1:T]
dist_TPI = np.absolute(np.append(Kdist_TPI, BQdist_TPI)).max()
print ('iter: ', iter_TPI, ', dist: ', dist_TPI, ', max Eul err: ',
np.absolute(EulErrPath).max())
if iter_TPI == maxiter_TPI and dist_TPI > mindist_TPI:
print 'TPI reached maxiter and did not converge.'
elif iter_TPI == maxiter_TPI and dist_TPI <= mindist_TPI:
print 'TPI converged in the last iteration. Should probably increase maxiter_TPI.'
Kpath = Kpath_new
BQpath = BQpath_new
Y_params = (A, alpha)
Ypath = ssf.get_Y(Y_params, Kpath, Lpath)
Cpath = np.zeros(T+S-2)
Cpath[:T-1] = ssf.get_C(cpath[:, :T-1])
Cpath[T-1:] = C_ss
elapsed_time = time.clock() - start_time
if graphs == True:
# Plot time path of aggregate capital stock
tvec = np.linspace(1, T+S-2, T+S-2)
minorLocator = MultipleLocator(1)
fig, ax = plt.subplots()
plt.plot(tvec, Kpath)
# for the minor ticks, use no labels; default NullFormatter
ax.xaxis.set_minor_locator(minorLocator)
plt.grid(b=True, which='major', color='0.65',linestyle='-')
plt.title('Time path for aggregate capital stock')
plt.xlabel(r'Period $t$')
plt.ylabel(r'Aggregate capital $K_{t}$')
# plt.savefig('Kt_Chap8')
plt.show()
# Plot time path of total bequests
tvec = np.linspace(1, T+S-2, T+S-2)
minorLocator = MultipleLocator(1)
fig, ax = plt.subplots()
plt.plot(tvec, BQpath)
# for the minor ticks, use no labels; default NullFormatter
ax.xaxis.set_minor_locator(minorLocator)
plt.grid(b=True, which='major', color='0.65',linestyle='-')
plt.title('Time path for total bequests')
plt.xlabel(r'Period $t$')
plt.ylabel(r'Total bequests $BQ_{t}$')
# plt.savefig('BQt_Chap8')
plt.show()
# Plot time path of aggregate output (GDP)
tvec = np.linspace(1, T+S-2, T+S-2)
minorLocator = MultipleLocator(1)
fig, ax = plt.subplots()
plt.plot(tvec, Ypath)
# for the minor ticks, use no labels; default NullFormatter
ax.xaxis.set_minor_locator(minorLocator)
plt.grid(b=True, which='major', color='0.65',linestyle='-')
plt.title('Time path for aggregate output (GDP)')
plt.xlabel(r'Period $t$')
plt.ylabel(r'Aggregate output $Y_{t}$')
# plt.savefig('Yt_Chap8')
plt.show()
# Plot time path of aggregate consumption
tvec = np.linspace(1, T+S-2, T+S-2)
minorLocator = MultipleLocator(1)
fig, ax = plt.subplots()
plt.plot(tvec, Cpath)
# for the minor ticks, use no labels; default NullFormatter
ax.xaxis.set_minor_locator(minorLocator)
plt.grid(b=True, which='major', color='0.65',linestyle='-')
plt.title('Time path for aggregate consumption')
plt.xlabel(r'Period $t$')
plt.ylabel(r'Aggregate consumption $C_{t}$')
# plt.savefig('Ct_Chap8')
plt.show()
# Plot time path of real wage
tvec = np.linspace(1, T+S-2, T+S-2)
minorLocator = MultipleLocator(1)
fig, ax = plt.subplots()
plt.plot(tvec, wpath)
# for the minor ticks, use no labels; default NullFormatter
ax.xaxis.set_minor_locator(minorLocator)
plt.grid(b=True, which='major', color='0.65',linestyle='-')
plt.title('Time path for real wage')
plt.xlabel(r'Period $t$')
plt.ylabel(r'Real wage $w_{t}$')
# plt.savefig('wt_Chap8')
plt.show()
# Plot time path of real interest rate
tvec = np.linspace(1, T+S-2, T+S-2)
minorLocator = MultipleLocator(1)
fig, ax = plt.subplots()
plt.plot(tvec, rpath)
# for the minor ticks, use no labels; default NullFormatter
ax.xaxis.set_minor_locator(minorLocator)
plt.grid(b=True, which='major', color='0.65',linestyle='-')
plt.title('Time path for real interest rate')
plt.xlabel(r'Period $t$')
plt.ylabel(r'Real interest rate $r_{t}$')
# plt.savefig('rt_Chap8')
plt.show()
# Plot time path of individual savings distribution
tgrid = np.linspace(1, T, T)
sgrid = np.linspace(2, S+1, S)
tmat, smat = np.meshgrid(tgrid, sgrid)
cmap_bp = matplotlib.cm.get_cmap('summer')
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.set_xlabel(r'period-$t$')
ax.set_ylabel(r'age-$s$')
ax.set_zlabel(r'individual savings $b_{s,t}$')
strideval = max(int(1), int(round(S/10)))
ax.plot_surface(tmat, smat, bpath[:, :T], rstride=strideval,
cstride=strideval, cmap=cmap_bp)
# plt.savefig('bpath_Chap8')
plt.show()
# Plot time path of individual consumption distribution
tgrid = np.linspace(1, T-1, T-1)
sgrid = np.linspace(1, S, S)
tmat, smat = np.meshgrid(tgrid, sgrid)
cmap_cp = matplotlib.cm.get_cmap('summer')
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.set_xlabel(r'period-$t$')
ax.set_ylabel(r'age-$s$')
ax.set_zlabel(r'individual consumption $c_{s,t}$')
strideval = max(int(1), int(round(S/10)))
ax.plot_surface(tmat, smat, cpath[:, :T-1], rstride=strideval,
cstride=strideval, cmap=cmap_cp)
# plt.savefig('cpath_Chap8')
plt.show()
return (bpath, cpath, BQpath, wpath, rpath, Kpath, Ypath, Cpath,
EulErrPath, elapsed_time)