def makeTargetMfig(Rfree, DiscFac, CRRA, PermShkStd, TranShkStd):
inf_hor = IndShockConsumerType(quietly=True, **base_params)
inf_hor.Rfree = Rfree
inf_hor.DiscFac = DiscFac
inf_hor.CRRA = CRRA
inf_hor.PermShkStd = [PermShkStd]
inf_hor.TranShkStd = [TranShkStd]
inf_hor.update_income_process()
mPlotMin = 0
mPlotMax = 250
inf_hor.aXtraMax = mPlotMax
inf_hor.solve(quietly=True, messaging_level=logging.CRITICAL)
soln = inf_hor.solution[0]
Bilt, cFunc = soln.Bilt, soln.cFunc
cPlotMin = 0, cFunc(mPlotMax)
if Bilt.GICMod: # tattle
soln.check_GICMod(soln, quietly=False, messaging_level=logging.WARNING)
mBelwStE = np.linspace(mPlotMin, mPlotMax, 1000)
EPermGroFac = inf_hor.PermGroFac[0]
def EmDelEq0(mVec):
return (EPermGroFac / Rfree) + (1.0 - EPermGroFac / Rfree) * mVec
cBelwStE_Best = cFunc(mBelwStE) # "best" = optimal c
cBelwStE_Sstn = EmDelEq0(mBelwStE) # "sustainable" c
mBalLvl = Bilt.mBalLvl
plt.figure(figsize=(12, 8))
plt.plot(mBelwStE, cBelwStE_Best, label="$c(m_{t})$")
plt.plot(mBelwStE,
cBelwStE_Sstn,
label="$\mathsf{E}_{t}[\Delta m_{t+1}] = 0$")
plt.xlim(mPlotMin, mPlotMax)
plt.ylim(cPlotMin, cFunc(mPlotMax))
plt.plot(
[mBalLvl, mBalLvl],
[0, 2.5],
color="black",
linestyle="--",
)
plt.tick_params(
# labelbottom=False,
# labelleft=False,
# left="off",
right="off",
# bottom="off",
top="off",
)
plt.text(0, 1.47, r"$c$", fontsize=26)
plt.text(3.02, 0, r"$m$", fontsize=26)
plt.text(mBalLvl - 0.05, -0.1, "m̌", fontsize=26)
plt.legend(fontsize='x-large')
plt.show()
return None
def makeTargetMfig(Rfree, DiscFac, CRRA, PermShkStd, TranShkStd):
baseAgent_Inf = IndShockConsumerType(verbose=0, cycles=0, **base_params)
baseAgent_Inf.Rfree = Rfree
baseAgent_Inf.DiscFac = DiscFac
baseAgent_Inf.CRRA = CRRA
baseAgent_Inf.PermShkStd = [PermShkStd]
baseAgent_Inf.TranShkStd = [TranShkStd]
baseAgent_Inf.updateIncomeProcess()
baseAgent_Inf.checkConditions()
mPlotMin = 0
mPlotMax = 250
baseAgent_Inf.aXtraMax = mPlotMax
baseAgent_Inf.solve()
baseAgent_Inf.unpack('cFunc')
cPlotMin = 0
cPlotMax = baseAgent_Inf.cFunc[0](mPlotMax)
if (baseAgent_Inf.GPFInd >= 1):
baseAgent_Inf.checkGICInd(verbose=3)
mBelwTrg = np.linspace(mPlotMin, mPlotMax, 1000)
EPermGroFac = baseAgent_Inf.PermGroFac[0]
EmDelEq0 = lambda m: (EPermGroFac / Rfree) + (1.0 - EPermGroFac / Rfree
) * m
cBelwTrg_Best = baseAgent_Inf.cFunc[0](mBelwTrg) # "best" = optimal c
cBelwTrg_Sstn = EmDelEq0(mBelwTrg) # "sustainable" c
mNrmTrg = baseAgent_Inf.solution[0].mNrmSS
plt.figure(figsize=(12, 8))
plt.plot(mBelwTrg, cBelwTrg_Best, label="$c(m_{t})$")
plt.plot(mBelwTrg,
cBelwTrg_Sstn,
label="$\mathsf{E}_{t}[\Delta m_{t+1}] = 0$")
plt.xlim(mPlotMin, mPlotMax)
plt.ylim(cPlotMin, cPlotMax)
plt.plot(
[mNrmTrg, mNrmTrg],
[0, 2.5],
color="black",
linestyle="--",
)
plt.tick_params(
# labelbottom=False,
# labelleft=False,
# left="off",
right="off",
# bottom="off",
top="off",
)
plt.text(0, 1.47, r"$c$", fontsize=26)
plt.text(3.02, 0, r"$m$", fontsize=26)
plt.text(mNrmTrg - 0.05, -0.1, "m̌", fontsize=26)
plt.legend(fontsize='x-large')
plt.show()
return None
def makeConvergencePlot(DiscFac, CRRA, Rfree, permShkStd):
# Construct finite horizon agent with baseline parameters
baseAgent_Fin = IndShockConsumerType(quietly=True,
messaging_level=logging.CRITICAL,
**base_params)
baseAgent_Fin.DiscFac = DiscFac
baseAgent_Fin.CRRA = CRRA
baseAgent_Fin.Rfree = Rfree
baseAgent_Fin.permShkStd = [permShkStd]
baseAgent_Fin.cycles = 100
baseAgent_Fin.solve(quietly=True, messaging_level=logging.CRITICAL)
baseAgent_Fin.unpack('cFunc')
# figure plot limits
mMin, mMax, cPlotMin, cPlotMax = 0, 7, 0, 7
mPlotMin = 0
mLocCLabels = 5.6 # Defines horizontal limit of figure
mPlotTop = 6.5 # 3.5 # 6.5 # Defines maximum m value where functions are plotted
mPts = 100 # Number of points at which functions are evaluated
plt.figure(figsize=(12, 8))
plt.ylim([cPlotMin, cPlotMax])
plt.xlim([mMin, mMax])
mBelwLabels = np.linspace(mPlotMin, mLocCLabels - 0.1,
mPts) # Range of m below loc of labels
m_FullRange = np.linspace(mPlotMin, mPlotTop, mPts) # Full plot range
# c_Tm0 defines the last period consumption rule (c=m)
c_Tm0 = m_FullRange
# c_Tm1 defines the second-to-last period consumption rule
c_Tm1 = baseAgent_Fin.cFunc[-2](mBelwLabels)
c_Tm5 = baseAgent_Fin.cFunc[-6](
mBelwLabels) # c_Tm5 defines the T-5 period consumption rule
# c_Tm10 defines the T-10 period consumption rule
c_Tm10 = baseAgent_Fin.cFunc[-11](mBelwLabels)
# c_Limt defines limiting infinite-horizon consumption rule
c_Limt = baseAgent_Fin.cFunc[0](mBelwLabels)
plt.plot(mBelwLabels, c_Limt, label="$c(m)$")
plt.plot(mBelwLabels, c_Tm1, label="$c_{T-1}(m)$")
plt.plot(mBelwLabels, c_Tm5, label="$c_{T-5}(m)$")
plt.plot(mBelwLabels, c_Tm10, label="$c_{T-10}(m)$")
plt.plot(m_FullRange, c_Tm0, label="$c_{T}(m) = 45$ degree line")
plt.legend(fontsize='x-large')
plt.tick_params(labelbottom=False,
labelleft=False,
left="off",
right="off",
bottom="off",
top="off")
plt.show()
return None
def makeBoundsFigure(UnempPrb, PermShkStd, TranShkStd, DiscFac, CRRA):
baseAgent_Inf = IndShockConsumerType(verbose=0, cycles=0, **base_params)
baseAgent_Inf.UnempPrb = UnempPrb
baseAgent_Inf.PermShkStd = [PermShkStd]
baseAgent_Inf.TranShkStd = [TranShkStd]
baseAgent_Inf.DiscFac = DiscFac
baseAgent_Inf.CRRA = CRRA
baseAgent_Inf.updateIncomeProcess()
baseAgent_Inf.checkConditions()
baseAgent_Inf.solve()
baseAgent_Inf.unpack('cFunc')
# Retrieve parameters (makes code readable)
Rfree = baseAgent_Inf.Rfree
CRRA = baseAgent_Inf.CRRA
EPermGroFac = baseAgent_Inf.PermGroFac[0]
mNrmTrg = baseAgent_Inf.solution[0].mNrmSS
UnempPrb = baseAgent_Inf.UnempPrb
κ_Min = 1.0 - (Rfree**(-1.0)) * (Rfree * DiscFac)**(1.0 / CRRA)
h_inf = (1.0 / (1.0 - EPermGroFac / Rfree))
cFunc_Uncnst = lambda m: (h_inf - 1) * κ_Min + κ_Min * m
cFunc_TopBnd = lambda m: (1 - UnempPrb**(1.0 / CRRA) *
(Rfree * DiscFac)**(1.0 / CRRA) / Rfree) * m
cFunc_BotBnd = lambda m: (1 - (Rfree * DiscFac)**(1.0 / CRRA) / Rfree) * m
# Plot the consumption function and its bounds
cMaxLabel = r"c̅$(m) = (m-1+h)κ̲$" # Use unicode kludge
cMinLabel = r"c̲$(m)= (1-\Phi_{R})m = κ̲ m$"
mPlotMax = 25
mPlotMin = 0
# mKnk is point where the two upper bounds meet
mKnk = ((h_inf - 1) * κ_Min) / (
(1 - UnempPrb**(1.0 / CRRA) *
(Rfree * DiscFac)**(1.0 / CRRA) / Rfree) - κ_Min)
mBelwKnkPts = 300
mAbveKnkPts = 700
mBelwKnk = np.linspace(mPlotMin, mKnk, mBelwKnkPts)
mAbveKnk = np.linspace(mKnk, mPlotMax, mAbveKnkPts)
mFullPts = np.linspace(mPlotMin, mPlotMax, mBelwKnkPts + mAbveKnkPts)
plt.figure(figsize=(12, 8))
plt.plot(mFullPts, baseAgent_Inf.cFunc[0](mFullPts), label=r'$c(m)$')
plt.plot(mBelwKnk, cFunc_Uncnst(mBelwKnk), label=cMaxLabel, linestyle="--")
plt.plot(
mAbveKnk,
cFunc_Uncnst(mAbveKnk),
label=
r'Upper Bound $ = $ Min $[\overline{\overline{c}}(m),\overline{c}(m)]$',
linewidth=2.5,
color='black')
plt.plot(mBelwKnk, cFunc_TopBnd(mBelwKnk), linewidth=2.5, color='black')
plt.plot(mAbveKnk,
cFunc_TopBnd(mAbveKnk),
linestyle="--",
label=r"$\overline{\overline{c}}(m) = κ̅m = (1 - ℘^{1/ρ}Φᵣ)m$")
plt.plot(mBelwKnk, cFunc_BotBnd(mBelwKnk), color='red', linewidth=2.5)
plt.plot(mAbveKnk,
cFunc_BotBnd(mAbveKnk),
color='red',
label=cMinLabel,
linewidth=2.5)
plt.tick_params(labelbottom=False,
labelleft=False,
left='off',
right='off',
bottom='off',
top='off')
plt.xlim(mPlotMin, mPlotMax)
plt.ylim(mPlotMin, 1.12 * cFunc_Uncnst(mPlotMax))
plt.text(mPlotMin,
1.12 * cFunc_Uncnst(mPlotMax) + 0.05,
"$c$",
fontsize=22)
plt.text(mPlotMax + 0.1, mPlotMin, "$m$", fontsize=22)
plt.legend(fontsize='x-large')
plt.show()
return None
def makeGrowthplot(PermGroFac, DiscFac):
# cycles=0 tells the solver to find the infinite horizon solution
baseAgent_Inf = IndShockConsumerType(verbose=0, cycles=0, **base_params)
baseAgent_Inf.PermGroFac = [PermGroFac]
baseAgent_Inf.DiscFac = DiscFac
baseAgent_Inf.updateIncomeProcess()
baseAgent_Inf.checkConditions()
baseAgent_Inf.solve()
baseAgent_Inf.unpack('cFunc')
if (baseAgent_Inf.GPFInd >= 1):
baseAgent_Inf.checkGICInd(verbose=3)
elif baseAgent_Inf.solution[0].mNrmSS > 3.5:
print('Solution exists but is outside the plot range.')
else:
def EcLev_tp1_Over_p_t(a):
'''
Taking end-of-period assets a as input, return ratio of expectation
of next period's consumption to this period's permanent income
Inputs:
a: end-of-period assets
Returns:
EcLev_tp1_Over_p_{t}: next period's expected c level / current p
'''
# Extract parameter values to make code more readable
permShkVals = baseAgent_Inf.PermShkDstn[0].X
tranShkVals = baseAgent_Inf.TranShkDstn[0].X
permShkPrbs = baseAgent_Inf.PermShkDstn[0].pmf
tranShkPrbs = baseAgent_Inf.TranShkDstn[0].pmf
Rfree = baseAgent_Inf.Rfree
EPermGroFac = baseAgent_Inf.PermGroFac[0]
PermGrowFac_tp1 = EPermGroFac * permShkVals # Nonstochastic growth times idiosyncratic permShk
RNrmFac_tp1 = Rfree / PermGrowFac_tp1 # Growth-normalized interest factor
# 'bank balances' b = end-of-last-period assets times normalized return factor
b_tp1 = RNrmFac_tp1 * a
# expand dims of b_tp1 and use broadcasted sum of a column and a row vector
# to obtain a matrix of possible market resources next period
# because matrix mult is much much faster than looping to calc E
m_tp1_GivenTranAndPermShks = np.expand_dims(b_tp1,
axis=1) + tranShkVals
# List of possible values of $\mathbf{c}_{t+1}$ (Transposed by .T)
cRat_tp1_GivenTranAndPermShks = baseAgent_Inf.cFunc[0](
m_tp1_GivenTranAndPermShks).T
cLev_tp1_GivenTranAndPermShks = cRat_tp1_GivenTranAndPermShks * PermGrowFac_tp1
# compute expectation over perm shocks by right multiplying with probs
EOverPShks_cLev_tp1_GivenTranShkShks = np.dot(
cLev_tp1_GivenTranAndPermShks, permShkPrbs)
# finish expectation over trans shocks by right multiplying with probs
EcLev_tp1_Over_p_t = np.dot(EOverPShks_cLev_tp1_GivenTranShkShks,
tranShkPrbs)
# return expected consumption
return EcLev_tp1_Over_p_t
# Calculate the expected consumption growth factor
# mBelwTrg defines the plot range on the left of target m value (e.g. m <= target m)
mNrmTrg = baseAgent_Inf.solution[0].mNrmSS
mBelwTrg = np.linspace(1, mNrmTrg, 50)
c_For_mBelwTrg = baseAgent_Inf.cFunc[0](mBelwTrg)
a_For_mBelwTrg = mBelwTrg - c_For_mBelwTrg
EcLev_tp1_Over_p_t_For_mBelwTrg = [
EcLev_tp1_Over_p_t(i) for i in a_For_mBelwTrg
]
# mAbveTrg defines the plot range on the right of target m value (e.g. m >= target m)
mAbveTrg = np.linspace(mNrmTrg, 3.5, 50)
# EcGro_For_mAbveTrg: E [consumption growth factor] when m_{t} is below target m
EcGro_For_mBelwTrg = np.array(
EcLev_tp1_Over_p_t_For_mBelwTrg) / c_For_mBelwTrg
c_For_mAbveTrg = baseAgent_Inf.cFunc[0](mAbveTrg)
a_For_mAbveTrg = mAbveTrg - c_For_mAbveTrg
EcLev_tp1_Over_p_t_For_mAbveTrg = [
EcLev_tp1_Over_p_t(i) for i in a_For_mAbveTrg
]
# EcGro_For_mAbveTrg: E [consumption growth factor] when m_{t} is bigger than target m_{t}
EcGro_For_mAbveTrg = np.array(
EcLev_tp1_Over_p_t_For_mAbveTrg) / c_For_mAbveTrg
Rfree = 1.0
EPermGroFac = 1.0
mNrmTrg = baseAgent_Inf.solution[0].mNrmSS
# Calculate Absolute Patience Factor Phi = lower bound of consumption growth factor
APF = (Rfree * DiscFac)**(1.0 / CRRA)
plt.figure(figsize=(12, 8))
# Plot the Absolute Patience Factor line
plt.plot([0, 3.5], [APF, APF],
label="\u03A6 = [(\u03B2 R)^(1/ \u03C1)]/R")
# Plot the Permanent Income Growth Factor line
plt.plot([0, 3.5], [EPermGroFac, EPermGroFac], label="\u0393")
# Plot the expected consumption growth factor on the left side of target m
plt.plot(mBelwTrg, EcGro_For_mBelwTrg, color="black")
# Plot the expected consumption growth factor on the right side of target m
plt.plot(mAbveTrg,
EcGro_For_mAbveTrg,
color="black",
label="$\mathsf{E}_{t}[c_{t+1}/c_{t}]$")
# Plot the target m
plt.plot(
[mNrmTrg, mNrmTrg],
[0, 3.5],
color="black",
linestyle="--",
label="",
)
plt.xlim(1, 3.5)
plt.ylim(0.94, 1.10)
plt.text(2.105, 0.930, "$m_{t}$", fontsize=26, fontweight="bold")
plt.text(
mNrmTrg - 0.02,
0.930,
"m̌",
fontsize=26,
fontweight="bold",
)
plt.tick_params(
labelbottom=False,
labelleft=False,
left="off",
right="off",
bottom="off",
top="off",
)
plt.legend(fontsize='x-large')
plt.show()
return None
def makeGICFailExample(DiscFac, PermShkStd, UnempPrb):
# Construct the "GIC fails" example.
GIC_fails_dictionary = dict(base_params)
GIC_fails_dictionary['Rfree'] = 1.04
GIC_fails_dictionary['PermGroFac'] = [1.00]
GICFailsExample = IndShockConsumerType(
verbose=0,
cycles=0, # cycles=0 makes this an infinite horizon consumer
**GIC_fails_dictionary)
GICFailsExample.DiscFac = DiscFac
GICFailsExample.PermShkStd = [PermShkStd]
GICFailsExample.UnempPrb = UnempPrb
GICFailsExample.updateIncomeProcess()
GICFailsExample.checkConditions()
# Get calibrated parameters to make code more readable
LivPrb = GICFailsExample.LivPrb[0]
Rfree = GICFailsExample.Rfree
DiscFac = GICFailsExample.DiscFac
CRRA = GICFailsExample.CRRA
permShkPrbs = GICFailsExample.PermShkDstn[0].pmf
permShkVals = GICFailsExample.PermShkDstn[0].X
EPermGroFac = GICFailsExample.PermGroFac[0]
# np.dot multiplies vectors; probability times value for each outcome is expectation
EpermShkInv = np.dot(permShkPrbs,
permShkVals**(-1)) # $ \Ex[\permShk^{-1}] $
InvEpermShkInv = (EpermShkInv)**(-1) # $ (\Ex[\permShk^{-1}])^{-1}$
PermGroFac = EPermGroFac * InvEpermShkInv # Uncertainty-adjusted permanent growth factor
ERNrmFac = Rfree / PermGroFac # Interest factor normalized by uncertainty-adjusted growth
ErNrmRte = ERNrmFac - 1 # Interest rate is interest factor - 1
# "sustainable" C = P + (discounted) interest income
# "sustainable" c = 1 + (discounted, normalized) interest income
EmDelEq0 = lambda m: 1 + (m - 1) * (ErNrmRte / ERNrmFac
) # "sustainable" c where E[Δ m] = 0
GICFailsExample.solve(
) # Above, we set up the problem but did not solve it
GICFailsExample.unpack(
'cFunc'
) # Make the consumption function easily accessible for plotting
mPlotMin = 0
mPts = 1000
m = np.linspace(mPlotMin, 5, mPts)
c_Limt = GICFailsExample.cFunc[0](m)
c_Sstn = EmDelEq0(m) # "sustainable" consumption
plt.figure(figsize=(12, 8))
plt.plot(m, c_Limt, label="$c(m_{t})$")
plt.plot(m, c_Sstn, label="$\mathsf{E}_{t}[\Delta m_{t+1}] = 0$")
plt.xlim(0, 5.5)
plt.ylim(0, 1.6)
plt.tick_params(
labelbottom=False,
labelleft=False,
left="off",
right="off",
bottom="off",
top="off",
)
plt.legend(fontsize='x-large')
plt.show()
print(f'Current Growth Impatience Factor is {GICFailsExample.GPFInd}')
return None
#
# 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.
## Note that another complication with the cycles attribute is that it does not come from
def makeBoundsFigure(UnempPrb, permShkStd, TranShkStd, DiscFac, CRRA):
inf_hor = IndShockConsumerType(quietly=True,
messaging_level=logging.CRITICAL,
**base_params)
inf_hor.UnempPrb = UnempPrb
inf_hor.permShkStd = [permShkStd]
inf_hor.TranShkStd = [TranShkStd]
inf_hor.DiscFac = DiscFac
inf_hor.CRRA = CRRA
inf_hor.update_income_process()
inf_hor.solve(quietly=True, messaging_level=logging.CRITICAL)
soln = inf_hor.solution[0]
Bilt, Pars = soln.Bilt, soln.Pars
cFunc = soln.cFunc
mPlotMin, mPlotMax = 0, 25
inf_hor.aXtraMax = mPlotMax
# Retrieve parameters (makes code more readable)
Rfree, EPermGroFac = Pars.Rfree, Pars.PermGroFac
κ_Min = 1.0 - (Rfree**(-1.0)) * (Rfree * DiscFac)**(1.0 / CRRA)
h_inf = (1.0 / (1.0 - EPermGroFac / Rfree))
def cFunc_Uncnst(mVec):
return (h_inf - 1) * κ_Min + κ_Min * mVec
def cFunc_TopBnd(mVec):
return (1 - UnempPrb**(1.0 / CRRA) *
(Rfree * DiscFac)**(1.0 / CRRA) / Rfree) * mVec
def cFunc_BotBnd(mVec):
return (1 - (Rfree * DiscFac)**(1.0 / CRRA) / Rfree) * mVec
# Plot the consumption function and its bounds
cPlotMaxLabel = r"c̅$(m) = (m-1+h)κ̲$" # Use unicode kludge
cPlotMinLabel = r"c̲$(m)= (1-\Phi_{R})m = κ̲ m$"
# mKnk is point where the two upper bounds meet
mKnk = ((h_inf - 1) * κ_Min) / (
(1 - UnempPrb**(1.0 / CRRA) *
(Rfree * DiscFac)**(1.0 / CRRA) / Rfree) - κ_Min)
mBelwKnkPts, mAbveKnkPts = 50, 100
mBelwKnk = np.linspace(mPlotMin, mKnk, mBelwKnkPts)
mAbveKnk = np.linspace(mKnk, mPlotMax, mAbveKnkPts)
mFullPts = np.linspace(mPlotMin, mPlotMax, mBelwKnkPts + mAbveKnkPts)
plt.figure(figsize=(12, 8))
plt.plot(mFullPts, cFunc(mFullPts), label=r'$c(m)$')
plt.plot(mBelwKnk,
cFunc_Uncnst(mBelwKnk),
label=cPlotMaxLabel,
linestyle="--")
plt.plot(
mAbveKnk,
cFunc_Uncnst(mAbveKnk),
label=
r'Upper Bound $ = $ Min $[\overline{\overline{c}}(m),\overline{c}(m)]$',
linewidth=2.5,
color='black')
plt.plot(mBelwKnk, cFunc_TopBnd(mBelwKnk), linewidth=2.5, color='black')
plt.plot(mAbveKnk,
cFunc_TopBnd(mAbveKnk),
linestyle="--",
label=r"$\overline{\overline{c}}(m) = κ̅m = (1 - ℘^{1/ρ}Φᵣ)m$")
plt.plot(mBelwKnk, cFunc_BotBnd(mBelwKnk), color='red', linewidth=2.5)
plt.plot(mAbveKnk,
cFunc_BotBnd(mAbveKnk),
color='red',
label=cPlotMinLabel,
linewidth=2.5)
plt.tick_params(labelbottom=False,
labelleft=False,
left='off',
right='off',
bottom='off',
top='off')
plt.xlim(mPlotMin, mPlotMax)
plt.ylim(mPlotMin, 1.12 * cFunc_Uncnst(mPlotMax))
plt.text(mPlotMin,
1.12 * cFunc_Uncnst(mPlotMax) + 0.05,
"$c$",
fontsize=22)
plt.text(mPlotMax + 0.1, mPlotMin, "$m$", fontsize=22)
plt.legend(fontsize='x-large')
plt.show()
return None
