"""
------------------------------------------------------------------------
This script takes a Frisch elasticity parameter and then estimates the
parameters of the elliptical utility fuction that correspond to a
constant Frisch elasticity function with the input Frisch elasticity.
------------------------------------------------------------------------
"""
# Import packages
import numpy as np
import scipy.optimize as opt
[docs]
def CFE_u(theta, l_tilde, n):
"""
Disutility of labor supply from the constant Frisch elasticity
utility function.
Args:
theta (scalar): inverse of the Frisch elasticity of labor supply
l_tilde (scalar): maximum amount of labor supply
n (array_like): labor supply amount
Returns:
u (array_like): disutility of labor supply
"""
u = ((n / l_tilde) ** (1 + theta)) / (1 + theta)
return u
[docs]
def CFE_mu(theta, l_tilde, n):
"""
Marginal disutility of labor supply from the constant Frisch
elasticity utility function
Args:
theta (scalar): inverse of the Frisch elasticity of labor supply
l_tilde (scalar): maximum amount of labor supply
n (array_like): labor supply amount
Returns:
mu (array_like): marginal disutility of labor supply
"""
mu = (1.0 / l_tilde) * ((n / l_tilde) ** theta)
return mu
[docs]
def elliptical_u(b, k, upsilon, l_tilde, n):
"""
Disutility of labor supply from the elliptical utility function
Args:
b (scalar): scale parameter of elliptical utility function
k (scalar): shift parameter of elliptical utility function
upsilon (scalar): curvature parameter of elliptical utility
function
l_tilde (scalar): maximum amount of labor supply
n (array_like): labor supply amount
Returns:
u (array_like): disutility of labor supply
"""
u = b * ((1 - ((n / l_tilde) ** upsilon)) ** (1 / upsilon)) + k
return u
[docs]
def elliptical_mu(b, upsilon, l_tilde, n):
"""
Marginal disutility of labor supply from the elliptical utility
function
Args:
b (scalar): scale parameter of elliptical utility function
upsilon (scalar): curvature parameter of elliptical utility
function
l_tilde (scalar): maximum amount of labor supply
n (array_like): labor supply amount
Returns:
mu (array_like): marginal disutility of labor supply
"""
mu = (
b
* (1.0 / l_tilde)
* ((1.0 - (n / l_tilde) ** upsilon) ** ((1.0 / upsilon) - 1.0))
* (n / l_tilde) ** (upsilon - 1.0)
)
return mu
[docs]
def sumsq(params, *objs):
"""
This function generates the sum of squared deviations between the
constant Frisch elasticity function and the elliptical utility
function.
Args:
params (tuple): parameters to estimate, (b, k, upsilon)
objs (tuple): other parameters of utility function,
(theta, l_tilde, n_grid)
Returns:
ssqdev (scalar): sum of squared errors
"""
theta, l_tilde, n_grid = objs
b, k, upsilon = params
CFE = CFE_u(theta, l_tilde, n_grid)
ellipse = elliptical_u(b, k, upsilon, l_tilde, n_grid)
errors = CFE - ellipse
ssqdev = (errors**2).sum()
return ssqdev
[docs]
def sumsq_MU(params, *objs):
"""
This function generates the sum of squared deviations between the
marginals of the constant Frisch elasticity function and the
elliptical utility function
Args:
params (tuple): parameters to estimate, (b, k, upsilon)
objs (tuple): other parameters of utility function,
(theta, l_tilde, n_grid)
Returns:
ssqdev (scalar): sum of squared errors
"""
theta, l_tilde, n_grid = objs
b, upsilon = params
CFE_MU = CFE_mu(theta, l_tilde, n_grid)
ellipse_MU = elliptical_mu(b, upsilon, l_tilde, n_grid)
# CFE_MU = (1.0 / l_tilde) * ((n_grid / l_tilde) ** theta)
# ellipse_MU = (b * (1.0 / l_tilde) * ((1.0 - (n_grid / l_tilde) **
# upsilon) **
# ((1.0 / upsilon) - 1.0)) *
# (n_grid / l_tilde) ** (upsilon - 1.0))
errors = CFE_MU - ellipse_MU
ssqdev = (errors**2).sum()
return ssqdev
[docs]
def estimation(frisch, l_tilde):
"""
This function estimates the parameters of an elliptical utility
function that fits a constant frisch elasticity function.
Args:
frisch (scalar): Frisch elasticity of labor supply
l_tilde (scalar): maximum amount of labor supply
Returns:
b_MU_til (scalar): estimated b from elliptical utility function
upsilon_MU_til (scalar): estimated upsilon from elliptical
utility function
"""
# Set parameters
theta = 1 / frisch
N = 101
# Estimate parameters of elliptical utility function
# Initial guesses
b_init = 0.6701
# k_init = -.6548
upsilon_init = 2.3499
# don't estimate near edge of range of labor supply
n_grid = np.linspace(0.01, 0.8, num=N)
# Estimate params using marginal utilities
ellipse_MU_params_init = np.array([b_init, upsilon_init])
ellipse_MU_objs = (theta, l_tilde, n_grid)
bnds_MU = ((None, None), (None, None))
ellipse_MU_params_til = opt.minimize(
sumsq_MU,
ellipse_MU_params_init,
args=(ellipse_MU_objs),
method="L-BFGS-B",
bounds=bnds_MU,
tol=1e-15,
)
(b_MU_til, upsilon_MU_til) = ellipse_MU_params_til.x
return b_MU_til, upsilon_MU_til
