def __init__(self, Rfree=1.001, CRRA=2):
PFaux = PerfForesightConsumerType(
) # set up a consumer type and use default parameteres
PFaux.cycles = 0 # Make this type have an infinite horizon
PFaux.DiscFac = 1 / Rfree
PFaux.Rfree = Rfree
PFaux.LivPrb = [1.0]
PFaux.PermGroFac = [1.0]
PFaux.CRRA = CRRA
PFaux.solve() # solve the consumer's problem
PFaux.unpackcFunc() # unpack the consumption function
self.cFunc = PFaux.solution[0].cFunc
def plot1(Epsilon, DiscFac, PopGrowth, YearsPerGeneration, Initialk):
'''Inputs:
Epsilon: Elasticity of output with respect to capital/labour ratio
DiscFac: One period discount factor
YearPerGeneration: No. of years per generation
PopGrowth: Gross growth rate of population in one period'''
#
kMax = 0.1
# Define some parameters
Beta = DiscFac**YearsPerGeneration
xi = PopGrowth**YearsPerGeneration
Q = (1-Epsilon)*(Beta/(1+Beta))/xi
kBar = Q**(1/(1-Epsilon))
# Create an agent that will solve the consumption problem
PFagent = PerfForesightConsumerType(**init_perfect_foresight)
PFagent.cycles = 1 # let the agent live the cycle of periods just once
PFagent.T_cycle = 2 # Number of periods in the cycle
PFagent.assign_parameters(PermGroFac = [0.000001]) # Income only in the first period
PFagent.LivPrb = [1.]
PFagent.DiscFac = Beta
# Hark seems to have trouble with log-utility
# so pass rho = 1 + something small.
PFagent.CRRA = 1.001
PFagent.solve()
# Plot the OLG capital accumulation curve and 45 deg line
plt.figure(figsize=(9,6))
kt_range = np.linspace(0, kMax, 300)
# Analitical solution plot
ktp1 = Q*kt_range**Epsilon
plt.plot(kt_range, ktp1, 'b-', label='Capital accumulation curve')
plt.plot(kt_range, kt_range, 'k-', label='45 Degree line')
# Plot the path
kt_ar = Initialk
ktp1_ar = 0.
for i in range(3):
# Compute Next Period's capital using HARK
wage = (1-Epsilon)*kt_ar**Epsilon
c = PFagent.solution[0].cFunc(wage)
a = wage - c
k1 = a/xi
plt.arrow(kt_ar, ktp1_ar, 0., k1-ktp1_ar,
length_includes_head=True,
lw=0.01,
width=0.0005,
color='black',
edgecolor=None)
plt.arrow(kt_ar, k1, k1-kt_ar , 0.,
length_includes_head=True,
lw=0.01,
width=0.0005,
color='black',
edgecolor=None)
# Update arrow
kt_ar = k1
ktp1_ar = kt_ar
# Plot kbar and initial k
plt.plot(kBar, kBar, 'ro', label=r'$\bar{k}$')
plt.plot(Initialk, 0.0005, 'co', label = '$k_0$')
plt.legend()
plt.xlim(0 ,kMax)
plt.ylim(0, kMax)
plt.xlabel('$k_t$')
plt.ylabel('$k_{t+1}$')
plt.show()
return None
def FisherPlot(Y1, Y2, B1, R, c1Max, c2Max):
# Basic setup of perfect foresight consumer
# We first create an instance of the class
# PerfForesightConsumerType, with its standard parameters.
PFexample = PerfForesightConsumerType()
PFexample.cycles = 1 # let the agent live the cycle of periods just once
PFexample.T_cycle = 2 # Number of periods in the cycle
PFexample.PermGroFac = [1.] # No automatic growth in income across periods
PFexample.LivPrb = [1.] # No chance of dying before the second period
PFexample.aNrmInitStd = 0.
PFexample.AgentCount = 1
CRRA = PFexample.CRRA
Beta = PFexample.DiscFac
# Set interest rate and bank balances from input.
PFexample.Rfree = R
PFexample.aNrmInitMean = B1
# Solve the model: this generates the optimal consumption function.
# Try-except blocks "try" to execute the code in the try block. If an
# error occurs, the except block is executed and the application does
# not halt
try:
PFexample.solve()
except:
print(
'Those parameter values violate a condition required for solution!'
)
# Create the figure
c1Min = 0.
c2Min = 0.
plt.figure(figsize=(8, 8))
plt.xlabel('Period 1 Consumption $C_1$')
plt.ylabel('Period 2 Consumption $C_2$')
plt.ylim([c2Min, c2Max])
plt.xlim([c1Min, c1Max])
# Plot the budget constraint
C1_bc = np.linspace(c1Min, B1 + Y1 + Y2 / R, 10, endpoint=True)
C2_bc = (Y1 + B1 - C1_bc) * R + Y2
plt.plot(C1_bc, C2_bc, 'k-', label='Budget Constraint')
# Plot the optimal consumption bundle
C1 = PFexample.solution[0].cFunc(B1 + Y1 + Y2 / R)
C2 = PFexample.solution[1].cFunc((Y1 + B1 - C1) * R + Y2)
plt.plot(C1, C2, 'ro', label='Optimal Consumption')
# Plot the indifference curve
V = C1**(1 - CRRA) / (1 - CRRA) + Beta * C2**(1 - CRRA) / (
1 - CRRA) # Get max utility
C1_V = np.linspace(((1 - CRRA) * V)**(1 / (1 - CRRA)) + 0.5, c1Max, 1000)
C2_V = (((1 - CRRA) * V - C1_V**(1 - CRRA)) / Beta)**(1 / (1 - CRRA))
plt.plot(C1_V, C2_V, 'b-', label='Indiferrence Curve')
# Add a legend and display the plot
plt.legend()
plt.show()
return None
def FisherPlot2(Y1, Y2, B1, RHi, RLo, c1Max, c2Max):
# Basic setup of perfect foresight consumer
PFexample = PerfForesightConsumerType(
) # set up a consumer type and use default parameteres
PFexample.cycles = 1 # let the agent live the cycle of periods just once
PFexample.T_cycle = 2 # Number of periods in the cycle
PFexample.PermGroFac = [1.] # No automatic growth in income across periods
PFexample.LivPrb = [1.] # No chance of dying before the second period
PFexample.aNrmInitStd = 0.
PFexample.AgentCount = 1
CRRA = 2.
Beta = PFexample.DiscFac
# Set the parameters we enter
PFexample.aNrmInitMean = B1
# Create two models, one for RfreeHigh and one for RfreeLow
PFexampleRHi = deepcopy(PFexample)
PFexampleRHi.Rfree = RHi
PFexampleRLo = deepcopy(PFexample)
PFexampleRLo.Rfree = RLo
c1Min = 0.
c2Min = 0.
# Solve the model for RfreeHigh
try:
PFexampleRHi.solve()
PFexampleRLo.solve()
except:
print(
'Those parameter values violate a condition required for solution!'
)
# Plot the chart
plt.figure(figsize=(8, 8))
plt.xlabel('Period 1 Consumption $C_1$')
plt.ylabel('Period 2 Consumption $C_2$')
plt.ylim([c2Min, c2Max])
plt.xlim([c1Min, c1Max])
# Plot the budget constraints
C1_bc_RLo = np.linspace(c1Min, B1 + Y1 + Y2 / RLo, 10, endpoint=True)
C2_bc_RLo = (Y1 + B1 - C1_bc_RLo) * RLo + Y2
plt.plot(C1_bc_RLo, C2_bc_RLo, 'k-', label='Budget Constraint R Low')
C1_bc_RHi = np.linspace(c1Min, B1 + Y1 + Y2 / RHi, 10, endpoint=True)
C2_bc_RHi = (Y1 + B1 - C1_bc_RHi) * RHi + Y2
plt.plot(C1_bc_RHi, C2_bc_RHi, 'k--', label='Budget Constraint R High')
# The optimal consumption bundles
C1_opt_RLo = PFexampleRLo.solution[0].cFunc(B1 + Y1 + Y2 / RLo)
C2_opt_RLo = PFexampleRLo.solution[1].cFunc((Y1 + B1 - C1_opt_RLo) * RLo +
Y2)
C1_opt_RHi = PFexampleRHi.solution[0].cFunc(B1 + Y1 + Y2 / RHi)
C2_opt_RHi = PFexampleRHi.solution[1].cFunc((Y1 + B1 - C1_opt_RHi) * RHi +
Y2)
# Plot the indifference curves
V_RLo = C1_opt_RLo**(1 - CRRA) / (1 - CRRA) + Beta * C2_opt_RLo**(
1 - CRRA) / (1 - CRRA) # Get max utility
C1_V_RLo = np.linspace(((1 - CRRA) * V_RLo)**(1 / (1 - CRRA)) + 0.5, c1Max,
1000)
C2_V_RLo = (((1 - CRRA) * V_RLo - C1_V_RLo**(1 - CRRA)) /
Beta)**(1 / (1 - CRRA))
plt.plot(C1_V_RLo, C2_V_RLo, 'b-', label='Indiferrence Curve R Low')
V_RHi = C1_opt_RHi**(1 - CRRA) / (1 - CRRA) + Beta * C2_opt_RHi**(
1 - CRRA) / (1 - CRRA) # Get max utility
C1_V_RHi = np.linspace(((1 - CRRA) * V_RHi)**(1 / (1 - CRRA)) + 0.5, c1Max,
1000)
C2_V_RHi = (((1 - CRRA) * V_RHi - C1_V_RHi**(1 - CRRA)) /
Beta)**(1 / (1 - CRRA))
plt.plot(C1_V_RHi, C2_V_RHi, 'b--', label='Indiferrence Curve R High')
# The substitution effect
C1_Subs = (V_RHi * (1 - CRRA) /
(1 + Beta * (RLo * Beta)**((1 - CRRA) / CRRA)))**(1 /
(1 - CRRA))
C2_Subs = C1_Subs * (RLo * Beta)**(1 / CRRA)
# The Human wealth effect
Y2HW = Y2 * RLo / RHi
C1HW = Y2HW / (RLo + (RLo)**(1 / CRRA))
C2HW = C1HW * (RLo * Beta)**(1 / CRRA)
C1_bc_HW = np.linspace(c1Min, B1 + Y1 + Y2HW / RLo, 10, endpoint=True)
C2_bc_HW = (Y1 + B1 - C1_bc_HW) * RLo + Y2HW
plt.plot(C1_bc_HW, C2_bc_HW, 'k:')
VHW = (C1HW**(1 - CRRA)) / (1 - CRRA) + (Beta * C2HW**
(1 - CRRA)) / (1 - CRRA)
C1_V_HW = np.linspace(((1 - CRRA) * VHW)**(1 / (1 - CRRA)) + 0.5, c1Max,
1000)
C2_V_HW = (((1 - CRRA) * VHW - C1_V_HW**(1 - CRRA)) / Beta)**(1 /
(1 - CRRA))
plt.plot(C1_V_HW, C2_V_HW, 'b:')
# Plot the points of interest
plt.plot(C1_opt_RLo,
C2_opt_RLo,
'ro',
label='A: Optimal Consumption R Low')
plt.plot(C1_Subs,
C2_Subs,
'go',
label='B: Income effect DB \n Substitution effect BC')
plt.plot(C1_opt_RHi,
C2_opt_RHi,
'mo',
label='C: Optimal Consumption R High')
plt.plot(C1HW, C2HW, 'co', label='D: HW effect AD')
plt.legend()
plt.show()
return None
from HARK.ConsumptionSaving.ConsIndShockModel import PerfForesightConsumerType # Import the consumer type import HARK.ConsumptionSaving.ConsumerParameters as Params # Import default parameters # Now extract the default values of the parameters of interest CRRA = Params.CRRA Rfree = Params.Rfree DiscFac = Params.DiscFac PermGroFac = Params.PermGroFac rfree = Rfree-1 # %% # Now create a perfect foresight consumer example, PFagent = PerfForesightConsumerType(**Params.init_perfect_foresight) PFagent.cycles = 0 # We need the consumer to be infinitely lived PFagent.LivPrb = [1.0] # Suppress the possibility of dying # Solve the agent's problem PFagent.solve() # %% # Plot the consumption function # Remember, after invoking .solve(), the consumption function is stored as PFagent.solution[0].cFunc # Set out some range of market resources that we want to plot consumption for mMin = 0 mMax = 20 numPoints = 100 m_range = np.linspace(mMin, mMax, numPoints) # This creates an array of points in the given range