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 LfEulerSys(b, *objs):
'''
Generates vector of all Euler errors for a given b, which errors
characterize all optimal lifetime decisions
Inputs:
b = [u-1,] vector, remaining lifetime savings decisions
where p is the number of remaining periods
objs = length 9 tuple, (u, beta, sigma, beg_wealth, n,
r_path, w_path, p_path, p_tilde_path)
u = integer in [2,S], remaining periods in life
beta = scalar in [0,1), discount factor
sigma = scalar > 0, coefficient of relative risk aversion
beg_wealth = scalar, wealth at the beginning of first age
n = [u,] vector, remaining exogenous labor supply
r_path = [u,] vector, interest rates over remaining life
w_path = [u,] vector, wages rates over remaining life
p_c_path = [I, u] matrix, remaining lifetime
consumption good prices
p_tilde_path = [u,] vector, remaining lifetime composite
goods prices
p_tilde_path = [u,] vector of composite good prices
Functions called:
firm.get_c_tilde
firm.get_b_errors
Objects in function:
b2 = [u, ] vector, remaining savings including initial
savings
c = [u, ] vector, remaining lifetime consumption
levels implied by b2
c_constr = [u, ] boolean vector, =True if c_{s,t}<=0
b_err_params = length 2 tuple, parameters to pass into
get_b_errors (beta, sigma)
b_err_vec = [u-1,] vector, Euler errors from lifetime
consumption vector
Returns: b_err_vec
'''
(u, beta, sigma, beg_wealth, c_bar, n, r_path, w_path, p_c_path,
p_tilde_path) = objs
b2 = np.append(beg_wealth, b)
c_tilde, c_tilde_cstr = firm.get_c_tilde(c_bar, r_path, w_path, p_c_path,
p_tilde_path, n, b2)
b_err_params = (beta, sigma)
b_err_vec = firm.get_b_errors(b_err_params,
r_path[1:],
c_tilde,
c_tilde_cstr,
diff=True)
return b_err_vec
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 LfEulerSys(b, *objs):
'''
Generates vector of all Euler errors for a given b, which errors
characterize all optimal lifetime decisions
Inputs:
b = [u-1,] vector, remaining lifetime savings decisions
where p is the number of remaining periods
objs = length 9 tuple, (u, beta, sigma, beg_wealth, n,
r_path, w_path, p_path, p_tilde_path)
u = integer in [2,S], remaining periods in life
beta = scalar in [0,1), discount factor
sigma = scalar > 0, coefficient of relative risk aversion
beg_wealth = scalar, wealth at the beginning of first age
n = [u,] vector, remaining exogenous labor supply
r_path = [u,] vector, interest rates over remaining life
w_path = [u,] vector, wages rates over remaining life
p_c_path = [I, u] matrix, remaining lifetime
consumption good prices
p_tilde_path = [u,] vector, remaining lifetime composite
goods prices
p_tilde_path = [u,] vector of composite good prices
Functions called:
firm.get_c_tilde
firm.get_b_errors
Objects in function:
b2 = [u, ] vector, remaining savings including initial
savings
c = [u, ] vector, remaining lifetime consumption
levels implied by b2
c_constr = [u, ] boolean vector, =True if c_{s,t}<=0
b_err_params = length 2 tuple, parameters to pass into
get_b_errors (beta, sigma)
b_err_vec = [u-1,] vector, Euler errors from lifetime
consumption vector
Returns: b_err_vec
'''
(u, beta, sigma, beg_wealth, c_bar, n, r_path, w_path, p_c_path,
p_tilde_path) = objs
b2 = np.append(beg_wealth, b)
c_tilde, c_tilde_cstr = firm.get_c_tilde(c_bar, r_path, w_path, p_c_path, p_tilde_path,
n, b2)
b_err_params = (beta, sigma)
b_err_vec = firm.get_b_errors(b_err_params, r_path[1:], 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_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 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