# The next step is to change the values of parameters as we want.
#
# To see all the parameters used in the model, along with their default values, see $\texttt{ConsumerParameters.py}$
#
# Parameter values are stored as attributes of the $\texttt{ConsumerType}$ the values are used for. For example, the risk-free interest rate $\texttt{Rfree}$ is stored as $\texttt{BaselineExample.Rfree}$. Because we created $\texttt{BaselineExample}$ using the default parameters values at the moment $\texttt{BaselineExample.Rfree}$ is set to the default value of $\texttt{Rfree}$ (which, at the time this demo was written, was 1.03). Therefore, to change the risk-free interest rate used in $\texttt{BaselineExample}$ to (say) 1.02, all we need to do is:
#
# -
# Change the Default Riskfree Interest Rate
BaselineExample.Rfree = 1.02
# +
## Change some parameter values
BaselineExample.Rfree = 1.02 #change the risk-free interest rate
BaselineExample.CRRA = 2. # change the coefficient of relative risk aversion
BaselineExample.BoroCnstArt = -.3 # change the artificial borrowing constraint
BaselineExample.DiscFac = .5 #chosen so that target debt-to-permanent-income_ratio is about .1
# i.e. BaselineExample.solution[0].cFunc(.9) ROUGHLY = 1.
## There is one more parameter value we need to change. This one is more complicated than the rest.
## We could solve the problem for a consumer with an infinite horizon of periods that (ex-ante)
## are all identical. We could also solve the problem for a consumer with a fininite lifecycle,
## or for a consumer who faces an infinite horizon of periods that cycle (e.g., a ski instructor
## facing an infinite series of winters, with lots of income, and summers, with very little income.)
## The way to differentiate is through the "cycles" attribute, which indicates how often the
## sequence of periods needs to be solved. The default value is 1, for a consumer with a finite
## lifecycle that is only experienced 1 time. A consumer who lived that life twice in a row, and
## then died, would have cycles = 2. But neither is what we want. Here, we need to set cycles = 0,
## to tell HARK that we are solving the model for an infinite horizon consumer.
# It plots three lines: the linear consumption function of a perfect foresight consumer, the kinked consumption
# function of a consumer who faces a constraint, and the curved consumption function of a consumer that faces risk.
# load the three agents: unconstrained perfect foresight, constrained perfect foresight, unconstrained with risk
CCC_unconstr = IndShockConsumerType(**init_lifecycle)
CCC_unconstr.delFromTimeInv('BoroCnstArt')
CCC_unconstr.addToTimeVary('BoroCnstArt')
CCC_unconstr.solve()
CCC_unconstr.unpack('cFunc')
CCC_constraint = IndShockConsumerType(**init_lifecycle)
CCC_constraint.delFromTimeInv('BoroCnstArt')
CCC_constraint.addToTimeVary('BoroCnstArt')
CCC_constraint.BoroCnstArt = [
None, -1, None, None, None, None, None, None, None, None
]
CCC_constraint.solve()
CCC_constraint.unpack('cFunc')
CCC_risk = IndShockConsumerType(**init_lifecycle_risk1)
CCC_risk.delFromTimeInv('BoroCnstArt')
CCC_risk.addToTimeVary('BoroCnstArt')
CCC_risk.solve()
CCC_risk.unpack('cFunc')
# save the data in a txt file for later plotting in Matlab
x = np.linspace(-1, 1, 500, endpoint=True)
y = CCC_unconstr.cFunc[0](x)
y2 = CCC_constraint.cFunc[0](x)
y3 = CCC_risk.cFunc[0](x)
def make_cons_func(in_BoroCnstArt, in_TranShkStd):
"""
This figure illustrates how the effect of risk is greater if there already exists a constraint.
Initialize four types: unconstrained perfect foresight, unconstrained with risk, constrained perfect foresight, and constrained with risk.
"""
WwCR_unconstr = IndShockConsumerType(**init_lifecycle)
WwCR_unconstr.delFromTimeInv("BoroCnstArt")
WwCR_unconstr.addToTimeVary("BoroCnstArt")
WwCR_unconstr.solve()
WwCR_unconstr.unpack("cFunc")
init_lifecycle_risk2["TranShkStd"] = [0, in_TranShkStd, 0, 0, 0, 0, 0, 0, 0, 0, 0]
WwCR_risk = IndShockConsumerType(**init_lifecycle_risk2)
WwCR_risk.delFromTimeInv("BoroCnstArt")
WwCR_risk.addToTimeVary("BoroCnstArt")
WwCR_risk.solve()
WwCR_risk.unpack("cFunc")
WwCR_constr = IndShockConsumerType(**init_lifecycle)
WwCR_constr.cycles = 1 # Make this consumer live a sequence of periods exactly once
WwCR_constr.delFromTimeInv("BoroCnstArt")
WwCR_constr.addToTimeVary("BoroCnstArt")
WwCR_constr.BoroCnstArt = [
None,
None,
in_BoroCnstArt,
None,
None,
None,
None,
None,
None,
None,
]
WwCR_constr.solve()
WwCR_constr.unpack("cFunc")
WwCR_constr_risk = IndShockConsumerType(**init_lifecycle_risk2)
WwCR_constr_risk.delFromTimeInv("BoroCnstArt")
WwCR_constr_risk.addToTimeVary("BoroCnstArt")
WwCR_constr_risk.BoroCnstArt = [
None,
None,
in_BoroCnstArt,
None,
None,
None,
None,
None,
None,
None,
]
WwCR_constr_risk.solve()
WwCR_constr_risk.unpack("cFunc")
x = np.linspace(-8, -4, 1000, endpoint=True)
y = WwCR_unconstr.cFunc[1](x)
y2 = WwCR_risk.cFunc[1](x)
y3 = WwCR_constr.cFunc[1](x)
y4 = WwCR_constr_risk.cFunc[1](x)
where_close = np.isclose(y, y3, atol=1e-05)
where_close_risk = np.isclose(y2, y4, atol=1e-05)
x0 = x[where_close][0]
x1 = x[where_close_risk][0]
y0 = y[where_close][0]
y1 = y2[where_close_risk][0]
# Display the figure
# print('Figure 3: Consumption Functions With and Without a Constraint and a Risk')
f = plt.figure()
plt.plot(x, y, color="black", linewidth=3, label="${c}_{t,0}$")
plt.plot(
x, y2, color="black", linestyle="--", linewidth=3, label=r"$\tilde{c}_{t,0}$"
)
plt.plot(x, y3, color="red", label="${c}_{t,1}$")
plt.plot(x, y4, color="red", linestyle="--", label=r"$\tilde{c}_{t,1}$")
plt.xlim(left=-8, right=-4.5)
plt.ylim(0, 0.30)
plt.text(-8.15, 0.305, "$c$", fontsize=14)
plt.text(-4.5, -0.02, "$w$", fontsize=14)
# plt.plot([-6.15,-6.15],[0,0.05],color="black",linestyle=":")
plt.plot([x0, x0], [0, y0], color="black", linestyle=":")
plt.plot([x1, x1], [0, y1], color="black", linestyle=":")
# plt.text(-6.2,-0.02,r"$\underline{w}_{t,1}$",fontsize=14)
plt.text(x0, -0.02, r"${w}_{t,1}$", fontsize=14)
plt.text(x1, -0.02, r"$\bar{w}_{t,1}$", fontsize=14)
plt.tick_params(
labelbottom=False,
labelleft=False,
left="off",
right="off",
bottom="off",
top="off",
)
plt.legend()
plt.show()
return None
def make_concavification_figure(in_BoroCnstArt, in_UnempProb):
"""
This figure illustrates how both risks and constraints are examples of counterclockwise concavifications.
It plots three lines: the linear consumption function of a perfect foresight consumer, the kinked consumption
function of a consumer who faces a constraint, and the curved consumption function of a consumer that faces risk.
"""
# load the three agents: unconstrained perfect foresight, constrained perfect foresight, unconstrained with risk
CCC_unconstr = IndShockConsumerType(**init_lifecycle)
CCC_unconstr.delFromTimeInv("BoroCnstArt")
CCC_unconstr.addToTimeVary("BoroCnstArt")
CCC_unconstr.solve()
CCC_unconstr.unpack("cFunc")
CCC_constraint = IndShockConsumerType(**init_lifecycle)
CCC_constraint.delFromTimeInv("BoroCnstArt")
CCC_constraint.addToTimeVary("BoroCnstArt")
CCC_constraint.BoroCnstArt = [
None,
in_BoroCnstArt,
None,
None,
None,
None,
None,
None,
None,
None,
]
CCC_constraint.solve()
CCC_constraint.unpack("cFunc")
init_lifecycle_risk1["UnempPrb"] = in_UnempProb
CCC_risk = IndShockConsumerType(**init_lifecycle_risk1)
CCC_risk.delFromTimeInv("BoroCnstArt")
CCC_risk.addToTimeVary("BoroCnstArt")
CCC_risk.solve()
CCC_risk.unpack("cFunc")
x = np.linspace(-1, 1, 500, endpoint=True)
y = CCC_unconstr.cFunc[0](x)
y2 = CCC_constraint.cFunc[0](x)
y3 = CCC_risk.cFunc[0](x)
where_close = np.isclose(y, y2, atol=1e-05)
# Display the figure
# print('Figure 1: Counterclockwise Concavifications')
f = plt.figure()
plt.plot(x, y, color="black")
plt.plot(x, y2, color="green", label="Constraint", linestyle="--")
plt.plot(x, y3, color="red", label="Risk", linestyle="--")
plt.tick_params(
labelbottom=False,
labelleft=False,
left="off",
right="off",
bottom="off",
top="off",
)
if np.any(where_close):
x0 = x[where_close][0]
y0 = y[where_close][0]
plt.text(x0 - 0.02, 0.42, "$w^{\#}$", fontsize=14)
plt.plot([x0, x0], [0.45, y0], color="black", linestyle=":", linewidth=1)
plt.text(-1.2, 1.0, "$c$", fontsize=14)
plt.text(1.12, 0.42, "$w$", fontsize=14)
plt.ylim(0.465, 1.0)
plt.legend()
plt.show()
return None
def make_future_kink(in_BoroCnstArt):
"""
This figure illustrates how a the introduction of a current constraint can hide/move a kink that was induced by a future constraint.
To construct this figure, we plot two consumption functions:
1) perfect foresight consumer that faces one constraint in period 2
2) perfect foresight consumer that faces the same constraint as above plus one more constraint in period 3
Both consumption functions first loads the parameter set of the perfect foresight lifecycle households with ten periods.
We then change the parameter "BoroCnstArt" which corresponds to big Tau in the paper, i.e. the set of future constraints ordered by time.
"""
# Make and solve the consumer with only one borrowing constraint
Bcons1 = IndShockConsumerType(**init_lifecycle)
Bcons1.delFromTimeInv("BoroCnstArt")
Bcons1.addToTimeVary("BoroCnstArt")
Bcons1.BoroCnstArt = [None, 0, None, None, None, None, None, None, None, None]
Bcons1.solve()
Bcons1.unpack("cFunc")
# Make and solve the consumer with more than one binding borrowing constraint
BCons2 = IndShockConsumerType(**init_lifecycle)
BCons2.delFromTimeInv("BoroCnstArt")
BCons2.addToTimeVary("BoroCnstArt")
BCons2.BoroCnstArt = [
None,
0,
in_BoroCnstArt,
None,
None,
None,
None,
None,
None,
None,
]
BCons2.solve()
BCons2.unpack("cFunc")
x = np.linspace(1, 1.2, 500, endpoint=True)
y_mod1 = Bcons1.cFunc[0](x)
y_mod2 = BCons2.cFunc[0](x)
where_close = np.isclose(y_mod1, y_mod2)
x0 = x[where_close][0]
y0 = y_mod1[where_close][0]
y1dd = np.diff(y_mod1, n=2)
ind1 = np.argmin(y1dd[:250]) + 1
x1 = x[ind1]
y1 = y_mod1[ind1]
y2dd = np.diff(y_mod2, n=2)
ind2 = np.argmin(y2dd[:250]) + 1
x2 = x[ind2]
y2 = y_mod2[ind2]
# Display the figure
# print('Figure 2: How a Current Constraint Can Hide a Future Kink')
f = plt.figure()
plt.plot(x, y_mod1, color="green", label="$c_{t,1}$")
plt.plot(x, y_mod2, color="red", label="$\hat{c}_{t,2}$")
# plt.text(1.15,1.01,"$\hat{c}_{t,2}$",fontsize=14)
# plt.text(1.07,1.01,"$c_{t,1}$",fontsize=14)
# plt.arrow(1.149,1.011,-0.01,0,head_width=0.001,width=0.0001,facecolor='black',length_includes_head='True')
# plt.arrow(1.085,1.011,0.01,0,head_width=0.001,width=0.0001,facecolor='black',length_includes_head='True')
plt.xlim(left=1.0, right=1.2)
plt.ylim(0.98, 1.025)
plt.tick_params(
labelbottom=False,
labelleft=False,
left="off",
right="off",
bottom="off",
top="off",
)
plt.text(0.99, 1.025, "$c$", fontsize=14)
plt.text(1.20, 0.978, "$w$", fontsize=14)
plt.text(0.988, y0, "$\hat{c}_{t,1}^{\#}$", fontsize=14)
plt.text(0.988, y1 + 0.0015, "${c}_{t,1}^{\#}$", fontsize=14)
plt.text(0.97, y2 - 0.0015, "$\hat{c}_{t,2}(w_{t,1})$", fontsize=14)
plt.annotate(
"kink that \n gets hidden",
xy=(x1, y1),
xytext=((x1 + 3) / 4, y1 + 0.005),
arrowprops=dict(facecolor="black", headwidth=4, width=1, shrink=0.15),
)
plt.text(x0 - 0.005, 0.977, "$\hat{w}_{t,1}$", fontsize=14)
plt.text(x1 - 0.005, 0.977, "$w_{t,1}$", fontsize=14)
plt.text(x2 - 0.01, 0.975, "$\hat{w}_{t,2}$", fontsize=14)
plt.plot([1, x0], [y0, y0], color="black", linestyle="--")
plt.plot([1, x1], [y1, y1], color="black", linestyle="--")
plt.plot([1, x2], [y2, y2], color="black", linestyle="--")
plt.plot([x0, x0], [0.98, y0], color="black", linestyle="--")
plt.plot([x1, x1], [0.98, y1], color="black", linestyle="--")
plt.plot([x2, x2], [0.98, y2], color="black", linestyle="--")
plt.legend()
plt.show()
return None
