def EulerSys_b(b, *objs):
'''
Generates vector of all Euler errors for a given b, which errors
characterize all optimal lifetime savings decisions
Inputs:
b = [S-1,] vector, lifetime savings decisions
objs = length 9 tuple,
(alpha, beta, sigma, r, w, p, p, c_bar, n)
alpha = [I,] vector, expenditure shares on each good
beta = scalar in [0,1), discount factor for each model
period
sigma = scalar > 0, coefficient of relative risk aversion
r = scalar > 0, interest rate
w = scalar > 0, real wage
p_tilde = scalar > 0, composite good price
p_c = [I,] vector, price for each consumption good
c_bar = [I,] vector, minimum consumption values for all goods
n = [S,] vector, exogenous labor supply n_{s}
Functions called:
firm.get_c_tilde
firm.get_c
firm.get_b_errors
Objects in function:
c_params = length 3 tuple, parameters for get_c (r, w, p)
c_tilde = [S,] vector, remaining lifetime consumption
levels implied by b
c_tilde_cstr = [S, ] boolean vector, =True if c_{s,t}<=0
c_params = length 2 tuple, parameters for get_c
(alpha, p)
c = [I,S] matrix, consumption values for each good
and age c_{i,s}
c_cstr = [I,S] boolean matrix, =True if c_{i,s}<=0
b_err_params = length 2 tuple, parameters to pass into
get_b_errors (beta, sigma)
b_err_vec = [S-1,] vector, Euler errors from lifetime
consumption vector
Returns: b_err_vec
'''
alpha, beta, sigma, r, w, p_tilde, p_c, c_bar, I, S, n = objs
c_tilde, c_tilde_cstr = firm.get_c_tilde(c_bar, r, w, p_c, p_tilde, n,
np.append([0], b))
c, c_cstr = firm.get_c(np.tile(np.reshape(alpha, (I, 1)), (1, S)),
np.tile(np.reshape(c_bar, (I, 1)), (1, S)),
np.tile(c_tilde, (I, 1)),
np.tile(np.reshape(p_c, (I, 1)), (1, S)), p_tilde)
b_err_params = (beta, sigma)
b_err_vec = firm.get_b_errors(b_err_params,
r,
c_tilde,
c_tilde_cstr,
diff=True)
return b_err_vec
def get_cbess(params, b_guess, p_c, c_bar, I, S, n):
'''
Generates vectors for individual savings, composite consumption,
industry-specific consumption, constraint vectors, and Euler errors
for a given set of prices (r, w, p, p).
Inputs:
params = length 7 tuple, (alpha, beta, sigma, r, w, p ss_tol)
alpha = [I,] vector, expenditure shares on each good
beta = scalar in [0,1), discount factor for each model
period
sigma = scalar > 0, coefficient of relative risk aversion
r = scalar > 0, interest rate
w = scalar > 0, real wage
p_tilde = scalar > 0, composite good price
ss_tol = scalar > 0, tolerance level for steady-state fsolve
b_guess = [S-1,] vector, initial guess for savings vector
p_c = [I,] vector, prices in each industry
c_bar = [I,] vector, minimum consumption values for all goods
n = [S,] vector, exogenous labor supply n_{s}
Functions called:
EulerSys_b
firm.get_c_tilde
firm.get_c
firm.get_b_errors
Objects in function:
eulb_objs = length 9 tuple, objects to be passed in to
EulerSys_b: (alpha, beta, sigma, r, w, p, p,
c_bar, n)
b = [S-1,] vector, optimal lifetime savings decisions
c_tidle_params = length 3 tuple, parameters for get_c_tilde(r, w, p_tilde)
c_tilde = [S,] vector, optimal lifetime consumption
c_tilde_cstr = [S,] boolean vector, =True if c_s<=0
c_params = length 2 tuple, parameters for get_c (alpha, p)
c = [I,S] matrix, optimal consumption of each good
given prices
c_cstr = [I,S] boolean matrix, =True if c_{i,s}<=0 for given
c_s
eul_params = length 2 tuple, parameters for get_b_errors
(beta, sigma)
euler_errors = [S-1,] vector, Euler errors from savings decisions
Returns: b, c, c_cstr, c, c_cstr, euler_errors
'''
alpha, beta, sigma, r, w, p_tilde, ss_tol = params
eulb_objs = (alpha, beta, sigma, r, w, p_tilde, p_c, c_bar, I, S, n)
b = opt.fsolve(EulerSys_b, b_guess, args=(eulb_objs), xtol=ss_tol)
c_tilde, c_tilde_cstr = firm.get_c_tilde(c_bar, r, w, p_c, p_tilde, n, np.append([0], b))
c_params = (alpha, p_tilde)
c, c_cstr = firm.get_c(np.tile(np.reshape(alpha,(I,1)),(1,S)), np.tile(np.reshape(c_bar,(I,1)),(1,S)),
np.tile(c_tilde,(I,1)), np.tile(np.reshape(p_c,(I,1)),(1,S)) , p_tilde)
eul_params = (beta, sigma)
euler_errors = firm.get_b_errors(eul_params, r, c_tilde, c_tilde_cstr, diff=True)
return b, c, c_cstr, c, c_cstr, euler_errors
def EulerSys_b(b, *objs):
'''
Generates vector of all Euler errors for a given b, which errors
characterize all optimal lifetime savings decisions
Inputs:
b = [S-1,] vector, lifetime savings decisions
objs = length 9 tuple,
(alpha, beta, sigma, r, w, p, p, c_bar, n)
alpha = [I,] vector, expenditure shares on each good
beta = scalar in [0,1), discount factor for each model
period
sigma = scalar > 0, coefficient of relative risk aversion
r = scalar > 0, interest rate
w = scalar > 0, real wage
p_tilde = scalar > 0, composite good price
p_c = [I,] vector, price for each consumption good
c_bar = [I,] vector, minimum consumption values for all goods
n = [S,] vector, exogenous labor supply n_{s}
Functions called:
firm.get_c_tilde
firm.get_c
firm.get_b_errors
Objects in function:
c_params = length 3 tuple, parameters for get_c (r, w, p)
c_tilde = [S,] vector, remaining lifetime consumption
levels implied by b
c_tilde_cstr = [S, ] boolean vector, =True if c_{s,t}<=0
c_params = length 2 tuple, parameters for get_c
(alpha, p)
c = [I,S] matrix, consumption values for each good
and age c_{i,s}
c_cstr = [I,S] boolean matrix, =True if c_{i,s}<=0
b_err_params = length 2 tuple, parameters to pass into
get_b_errors (beta, sigma)
b_err_vec = [S-1,] vector, Euler errors from lifetime
consumption vector
Returns: b_err_vec
'''
alpha, beta, sigma, r, w, p_tilde, p_c, c_bar, I, S, n = objs
c_tilde, c_tilde_cstr = firm.get_c_tilde(c_bar, r, w, p_c, p_tilde, n, np.append([0], b))
c, c_cstr = firm.get_c(np.tile(np.reshape(alpha,(I,1)),(1,S)), np.tile(np.reshape(c_bar,(I,1)),(1,S)),
np.tile(c_tilde,(I,1)), np.tile(np.reshape(p_c,(I,1)),(1,S)) , p_tilde)
b_err_params = (beta, sigma)
b_err_vec = firm.get_b_errors(b_err_params, r, c_tilde, c_tilde_cstr, diff=True)
return b_err_vec
def paths_life(params, beg_age, beg_wealth, c_bar, n, r_path,
w_path, p_c_path, p_tilde_path, b_init):
'''
Solve for the remaining lifetime savings decisions of an individual
who enters the model at age beg_age, with corresponding initial
wealth beg_wealth.
Inputs:
params = length 5 tuple, (S, alpha, beta, sigma, tp_tol)
S = integer in [3,80], number of periods an individual
lives
alpha = [I,S-beg_age+1], expenditure share on good for remaing lifetime
beta = scalar in [0,1), discount factor for each model
period
sigma = scalar > 0, coefficient of relative risk aversion
tp_tol = scalar > 0, tolerance level for fsolve's in TPI
beg_age = integer in [1,S-1], beginning age of remaining life
beg_wealth = scalar, beginning wealth at beginning age
n = [S-beg_age+1,] vector, remaining exogenous labor
supplies
r_path = [S-beg_age+1,] vector, remaining lifetime interest
rates
w_path = [S-beg_age+1,] vector, remaining lifetime wages
p_c_path = [I, S-beg_age+1] matrix, remaining lifetime
consumption good prices
p_tilde_path = [S-beg_age+1,] vector, remaining lifetime composite
goods prices
b_init = [S-beg_age,] vector, initial guess for remaining
lifetime savings
Functions called:
LfEulerSys
firm.get_c_tilde
firm.get_c
firm.get_b_errors
Objects in function:
u = integer in [2,S], remaining periods in life
b_guess = [u-1,] vector, initial guess for lifetime savings
decisions
eullf_objs = length 9 tuple, objects to be passed in to
LfEulerSys: (p, beta, sigma, beg_wealth, n,
r_path, w_path, p_path, p_tilde_path)
b_path = [u-1,] vector, optimal remaining lifetime savings
decisions
c_tilde_path = [u,] vector, optimal remaining lifetime
consumption decisions
c_path = [u,I] martrix, remaining lifetime consumption
decisions by consumption good
c_constr = [u,] boolean vector, =True if c_{u}<=0,
b_err_params = length 2 tuple, parameters to pass into
firm.get_b_errors (beta, sigma)
b_err_vec = [u-1,] vector, Euler errors associated with
optimal savings decisions
Returns: b_path, c_tilde_path, c_path, b_err_vec
'''
S, alpha, beta, sigma, tp_tol = params
u = int(S - beg_age + 1)
if beg_age == 1 and beg_wealth != 0:
sys.exit("Beginning wealth is nonzero for age s=1.")
if len(r_path) != u:
#print len(r_path), S-beg_age+1
sys.exit("Beginning age and length of r_path do not match.")
if len(w_path) != u:
sys.exit("Beginning age and length of w_path do not match.")
if len(n) != u:
sys.exit("Beginning age and length of n do not match.")
b_guess = 1.01 * b_init
eullf_objs = (u, beta, sigma, beg_wealth, c_bar, n, r_path,
w_path, p_c_path, p_tilde_path)
b_path = opt.fsolve(LfEulerSys, b_guess, args=(eullf_objs),
xtol=tp_tol)
c_tilde_path, c_tilde_cstr = firm.get_c_tilde(c_bar, r_path, w_path, p_c_path, p_tilde_path,
n, np.append(beg_wealth, b_path))
c_path, c_cstr = firm.get_c(alpha[:,:u], c_bar, c_tilde_path, p_c_path, p_tilde_path)
b_err_params = (beta, sigma)
b_err_vec = firm.get_b_errors(b_err_params, r_path[1:], c_tilde_path,
c_tilde_cstr, diff=True)
return b_path, c_tilde_path, c_path, b_err_vec
def get_cbepath(params, Gamma1, r_path, w_path, p_c_path, p_tilde_path,
c_bar, I, n):
'''
Generates matrices for the time path of the distribution of
individual savings, individual composite consumption, individual
consumption of each type of good, and the Euler errors associated
with the savings decisions.
Inputs:
params = length 6 tuple, (S, T, alpha, beta, sigma, tp_tol)
S = integer in [3,80], number of periods an individual
lives
T = integer > S, number of time periods until steady
state
I = integer, number unique consumption goods
alpha = [I,T+S-1] matrix, expenditure shares on each good along time path
beta = scalar in [0,1), discount factor for each model
period
sigma = scalar > 0, coefficient of relative risk aversion
tp_tol = scalar > 0, tolerance level for fsolve's in TPI
Gamma1 = [S-1,] vector, initial period savings distribution
r_path = [T+S-1,] vector, the time path of
the interest rate
w_path = [T+S-1,] vector, the time path of
the wage
p_c_path = [I, T+S-1] matrix, time path of consumption goods prices
p_tilde_path = [T+S-1] vector, time path of composite price
c_bar = [I,T+S-1] matrix, minimum consumption values for all
goods along time path
n = [S,] vector, exogenous labor supply n_{s}
Functions called:
paths_life
Objects in function:
b_path = [S-1, T+S-1] matrix, distribution of savings along time path
c_tilde_path = [S, T+S-1] matrix, distribution of composite consumption along time path
c_path = [S, T+S-1, I] array, distribution of consumption of each cons good along time path
eulerr_path = [S-1, T+S-1] matrix, Euler equation errors along the time path
pl_params = length 4 tuple, parameters to pass into paths_life
(S, beta, sigma, TP_tol)
u = integer >= 2, represents number of periods
remaining in a lifetime, used to solve incomplete
lifetimes
b_guess = [u-1,] vector, initial guess for remaining lifetime
savings, taken from previous cohort's choices
b_lf = [u-1,] vector, optimal remaining lifetime savings
decisions
c_tilde_lf = [u,] vector, optimal remaining lifetime composite consumption
decisions
c_lf = [u,I] matrix, optimal remaining lifetime invididual consumption decisions
b_err_vec_lf = [u-1,] vector, Euler errors associated with
optimal remaining lifetime savings decisions
DiagMaskb = [u-1, u-1] boolean identity matrix
DiagMaskc = [u, u] boolean identity matrix
Returns: b_path, c_tilde_path, c_path, eulerr_path
'''
S, T, alpha, beta, sigma, tp_tol = params
b_path = np.append(Gamma1.reshape((S-1,1)), np.zeros((S-1, T+S-2)),
axis=1)
c_tilde_path = np.zeros((S, T+S-1))
c_path = np.zeros((S, T+S-1, I))
eulerr_path = np.zeros((S-1, T+S-1))
# Solve the incomplete remaining lifetime decisions of agents alive
# in period t=1 but not born in period t=1
c_tilde_path[S-1, 0], c_tilde_cstr = firm.get_c_tilde(c_bar[:,0],r_path[0],w_path[0],
p_c_path[:,0], p_tilde_path[0],n[S-1],Gamma1[S-2])
c_path[S-1, 0, :], c_cstr = firm.get_c(alpha[:,0],c_bar[:,0],c_tilde_path[S-1,0],
p_c_path[:,0],p_tilde_path[0])
for u in xrange(2, S):
# b_guess = b_ss[-u+1:]
b_guess = np.diagonal(b_path[S-u:, :u-1])
pl_params = (S, alpha[:,:u], beta, sigma, tp_tol)
b_lf, c_tilde_lf, c_lf, b_err_vec_lf = paths_life(pl_params,
S-u+1, Gamma1[S-u-1], c_bar[:,:u], n[-u:], r_path[:u],
w_path[:u], p_c_path[:, :u], p_tilde_path[:u], b_guess)
# Insert the vector lifetime solutions diagonally (twist donut)
# into the c_tilde_path, b_path, and EulErrPath matrices
DiagMaskb = np.eye(u-1, dtype=bool)
DiagMaskc = np.eye(u, dtype=bool)
b_path[S-u:, 1:u] = DiagMaskb * b_lf + b_path[S-u:, 1:u]
c_tilde_path[S-u:, :u] = DiagMaskc * c_tilde_lf + c_tilde_path[S-u:, :u]
DiagMaskc_tiled = np.tile(np.expand_dims(np.eye(u, dtype=bool),axis=2),(1,1,I))
c_lf = np.tile(np.expand_dims(c_lf.transpose(),axis=0),((c_lf.shape)[1],1,1))
c_path[S-u:, :u, :] = (DiagMaskc_tiled * c_lf +
c_path[S-u:, :u, :])
eulerr_path[S-u:, 1:u] = (DiagMaskb * b_err_vec_lf +
eulerr_path[S-u:, 1:u])
# Solve for complete lifetime decisions of agents born in periods
# 1 to T and insert the vector lifetime solutions diagonally (twist
# donut) into the c_tilde_path, b_path, and EulErrPath matrices
DiagMaskb = np.eye(S-1, dtype=bool)
DiagMaskc = np.eye(S, dtype=bool)
DiagMaskc_tiled = np.tile(np.expand_dims(np.eye(S, dtype=bool),axis=2),(1,1,I))
for t in xrange(1, T+1): # Go from periods 1 to T
# b_guess = b_ss
b_guess = np.diagonal(b_path[:, t-1:t+S-2])
pl_params = (S, alpha[:,t-1:t+S-1], beta, sigma, tp_tol)
b_lf, c_lf, c_i_lf, b_err_vec_lf = paths_life(pl_params, 1,
0, c_bar[:,t-1:t+S-1], n, r_path[t-1:t+S-1],
w_path[t-1:t+S-1], p_c_path[:, t-1:t+S-1],
p_tilde_path[t-1:t+S-1], b_guess)
c_i_lf = np.tile(np.expand_dims(c_i_lf.transpose(),axis=0),((c_i_lf.shape)[1],1,1))
# Insert the vector lifetime solutions diagonally (twist donut)
# into the c_tilde_path, b_path, and EulErrPath matrices
b_path[:, t:t+S-1] = DiagMaskb * b_lf + b_path[:, t:t+S-1]
c_tilde_path[:, t-1:t+S-1] = DiagMaskc * c_lf + c_tilde_path[:, t-1:t+S-1]
c_path[:, t-1:t+S-1, :] = (DiagMaskc_tiled * c_i_lf +
c_path[:, t-1:t+S-1, :])
eulerr_path[:, t:t+S-1] = (DiagMaskb * b_err_vec_lf +
eulerr_path[:, t:t+S-1])
return b_path, c_tilde_path, c_path, eulerr_path
def get_cbess(params, b_guess, p_c, c_bar, I, S, n):
'''
Generates vectors for individual savings, composite consumption,
industry-specific consumption, constraint vectors, and Euler errors
for a given set of prices (r, w, p, p).
Inputs:
params = length 7 tuple, (alpha, beta, sigma, r, w, p ss_tol)
alpha = [I,] vector, expenditure shares on each good
beta = scalar in [0,1), discount factor for each model
period
sigma = scalar > 0, coefficient of relative risk aversion
r = scalar > 0, interest rate
w = scalar > 0, real wage
p_tilde = scalar > 0, composite good price
ss_tol = scalar > 0, tolerance level for steady-state fsolve
b_guess = [S-1,] vector, initial guess for savings vector
p_c = [I,] vector, prices in each industry
c_bar = [I,] vector, minimum consumption values for all goods
n = [S,] vector, exogenous labor supply n_{s}
Functions called:
EulerSys_b
firm.get_c_tilde
firm.get_c
firm.get_b_errors
Objects in function:
eulb_objs = length 9 tuple, objects to be passed in to
EulerSys_b: (alpha, beta, sigma, r, w, p, p,
c_bar, n)
b = [S-1,] vector, optimal lifetime savings decisions
c_tidle_params = length 3 tuple, parameters for get_c_tilde(r, w, p_tilde)
c_tilde = [S,] vector, optimal lifetime consumption
c_tilde_cstr = [S,] boolean vector, =True if c_s<=0
c_params = length 2 tuple, parameters for get_c (alpha, p)
c = [I,S] matrix, optimal consumption of each good
given prices
c_cstr = [I,S] boolean matrix, =True if c_{i,s}<=0 for given
c_s
eul_params = length 2 tuple, parameters for get_b_errors
(beta, sigma)
euler_errors = [S-1,] vector, Euler errors from savings decisions
Returns: b, c, c_cstr, c, c_cstr, euler_errors
'''
alpha, beta, sigma, r, w, p_tilde, ss_tol = params
eulb_objs = (alpha, beta, sigma, r, w, p_tilde, p_c, c_bar, I, S, n)
b = opt.fsolve(EulerSys_b, b_guess, args=(eulb_objs), xtol=ss_tol)
c_tilde, c_tilde_cstr = firm.get_c_tilde(c_bar, r, w, p_c, p_tilde, n,
np.append([0], b))
c_params = (alpha, p_tilde)
c, c_cstr = firm.get_c(np.tile(np.reshape(alpha, (I, 1)), (1, S)),
np.tile(np.reshape(c_bar, (I, 1)), (1, S)),
np.tile(c_tilde, (I, 1)),
np.tile(np.reshape(p_c, (I, 1)), (1, S)), p_tilde)
eul_params = (beta, sigma)
euler_errors = firm.get_b_errors(eul_params,
r,
c_tilde,
c_tilde_cstr,
diff=True)
return b, c, c_cstr, c, c_cstr, euler_errors