Tran_b = 0.99 # and the bad state
# The HARK framework allows permanent shocks
Perm_g = Perm_b = 1.0 # KS assume there are no aggregate permanent shocks
# Aggregate productivity shock distribution by state.
# First element is probabilities of different outcomes, given the state you are in.
# Second element is agg permanent shocks (here we don't have any, so just they are just 1.).
# Third element is agg transitory shocks, which are calibrated the same as in Krusell Smith.
KSAggShkDstn = [
[np.array([1.0]),np.array([Perm_g]),np.array([Tran_g])], # Aggregate good
[np.array([1.0]),np.array([Perm_b]),np.array([Tran_b])] # Aggregate bad
]
KSEconomy.AggShkDstn = KSAggShkDstn
# %% [markdown]
# #### Summing Up
#
# The combined idiosyncratic and aggregate assumptions can be summarized mathematically as follows.
#
# $\forall \{s,s'\}=\{g,b\}\times\{g,b\}$, the following two conditions hold:
#
# $$\underbrace{\pi_{ss'01}}_{p(s \rightarrow s',u \rightarrow e)}+\underbrace{\pi_{ss'00}}_{p(s \rightarrow s', u \rightarrow u)} = \underbrace{\pi_{ss'11}}_{p(s\rightarrow s', e \rightarrow e) } + \underbrace{\pi_{ss'10}}_{p(s \rightarrow s', e \rightarrow u)} = \underbrace{\pi_{ss'}}_{p(s\rightarrow s')}$$
#
# $$u_s \frac{\pi_{ss'00}}{\pi_{ss'}}+ (1-u_s) \frac{\pi_{ss'10}}{\pi_{ss'}} = u_{s'}$$
# %% [markdown]
# ### Solving the Model
# Now, we have fully defined all of the elements of the macroeconomy, and we are in postion to construct an object that represents the economy and to construct a rational expectations equilibrium.
def consExample(filename):
KSAgentDictionary = {
"CRRA" : 3.0, # Coefficient of relative risk aversion
"DiscFac" : 0.99, # Intertemporal discount factor
"LivPrb" : [1.0], # Survival probability
"AgentCount" : 1000, # Number of agents of this type (only matters for simulation)
"aNrmInitMean" : 0.0, # Mean of log initial assets (only matters for simulation)
"aNrmInitStd" : 0.0, # Standard deviation of log initial assets (only for simulation)
"pLvlInitMean" : 0.0, # Mean of log initial permanent income (only matters for simulation)
"pLvlInitStd" : 0.0, # Standard deviation of log initial permanent income (only matters for simulation)
"PermGroFacAgg" : 1.0, # Aggregate permanent income growth factor (only matters for simulation)
"T_age" : None, # Age after which simulated agents are automatically killed
"T_cycle" : 1, # Number of periods in the cycle for this agent type
#
# Parameters for constructing the "assets above minimum" grid
"aXtraMin" : 0.001, # Minimum end-of-period "assets above minimum" value
"aXtraMax" : 20, # Maximum end-of-period "assets above minimum" value
"aXtraExtra" : [None], # Some other value of "assets above minimum" to add to the grid
"aXtraNestFac" : 3, # Exponential nesting factor when constructing "assets above minimum" grid
"aXtraCount" : 24, # Number of points in the grid of "assets above minimum"
#
# Parameters describing the income process
"PermShkCount" : 1, # Number of points in discrete approximation to permanent income shocks - no shocks of this kind!
"TranShkCount" : 1, # Number of points in discrete approximation to transitory income shocks - no shocks of this kind!
"PermShkStd" : [0.], # Standard deviation of log permanent income shocks - no shocks of this kind!
"TranShkStd" : [0.], # Standard deviation of log transitory income shocks - no shocks of this kind!
"UnempPrb" : 0.0, # Probability of unemployment while working - no shocks of this kind!
"UnempPrbRet" : 0.0, # Probability of "unemployment" while retired - no shocks of this kind!
"IncUnemp" : 0.0, # Unemployment benefits replacement rate
"IncUnempRet" : 0.0, # "Unemployment" benefits when retired
"tax_rate" : 0.0, # Flat income tax rate
"T_retire" : 0, # Period of retirement (0 --> no retirement)
"BoroCnstArt" : 0.0, # Artificial borrowing constraint; imposed minimum level of end-of period assets
"cycles" : 0, # Consumer is infinitely lived
"PermGroFac" : [1.0], # Permanent income growth factor
#
# New Parameters that we need now
'MgridBase': np.array([0.1, 0.3, 0.6,
0.8, 0.9, 0.98,
1.0, 1.02, 1.1,
1.2, 1.6, 2.0,
3.0]), # Grid of capital-to-labor-ratios (factors)
'MrkvArray': np.array([[0.875, 0.125],
[0.125, 0.875]]),# Transition probabilities for macroecon. [i,j] is probability of being in state j next
# period conditional on being in state i this period.
'PermShkAggStd' : [0.0, 0.0], # Standard deviation of log aggregate permanent shocks by state. No continous shocks in a state.
'TranShkAggStd' : [0.0, 0.0], # Standard deviation of log aggregate transitory shocks by state. No continuous shocks in a state.
'PermGroFacAgg' : 1.0
}
# Economy Dictionary
KSEconomyDictionary = {
'PermShkAggCount': 1,
'TranShkAggCount': 1,
'PermShkAggStd' : [0.0, 0.0],
'TranShkAggStd' : [0.0, 0.0],
'DeprFac' : 0.025, # Depreciation factor
'CapShare' : 0.36, # Share of capital income in cobb-douglas production function
'DiscFac' : 0.99,
'CRRA' : 3.0,
'PermGroFacAgg' : [1.0, 1.0],
'AggregateL' : 1.0, # Fix aggregate labor supply at 1.0 - makes interpretation of z easier
'act_T' : 1200, # Number of periods for economy to run in simulation
'intercept_prev' : [0.0, 0.0], # Make some initial guesses at linear savings rule intercepts for each state
'slope_prev' : [1.0, 1.0], # Make some initial guesses at linear savings rule slopes for each state
'MrkvArray' : np.array([ [0.875, 0.125],
[0.125, 0.875]]), # Transition probabilities
'MrkvNow_init' : 0 # Pick a state to start in (we pick the first state)
}
### Representative Agent
# Create the Krusell-Smith agent as an instance of AggShockMarkovConsumerType
KSAgent = AggShockMarkovConsumerType(**KSAgentDictionary)
# Construct the income distribution for the Krusell-Smith agent
prb_eg = 0.96 # Probability of employment in the good state
prb_ug = 1 - prb_eg # Probability of unemployment in the good state
prb_eb = 0.90 # Probability of employment in the bad state
prb_ub = 1 - prb_eb # Probability of unemployment in the bad state
p_ind = 1 # Persistent component of income is always 1
ell_ug = ell_ub = 0 # Labor supply is zero for unemployed consumers in either agg state
ell_eg = 1.0/prb_eg # Labor supply for employed consumer in good state
ell_eb = 1.0/prb_eb # 1=pe_g*ell_ge+pu_b*ell_gu=pe_b*ell_be+pu_b*ell_gu
# IncomeDstn is a list of lists, one for each aggregate Markov state
# Each contains three arrays of floats, representing a discrete approximation to the income process.
# Order:
# state probabilities
# idiosyncratic persistent income level by state (KS have no persistent shocks p_ind is always 1.0)
# idiosyncratic transitory income level by state
KSAgent.IncomeDstn[0] = [
[np.array([prb_eg,prb_ug]), np.array([p_ind,p_ind]), np.array([ell_eg,ell_ug])], # Agg state good
[np.array([prb_eb,prb_ub]), np.array([p_ind,p_ind]), np.array([ell_eb,ell_ub])] # Agg state bad
]
KSEconomy = CobbDouglasMarkovEconomy(agents = [KSAgent], **KSEconomyDictionary) # Combine production and consumption sides into an "Economy"
# Calibrate the magnitude of the aggregate shocks
Tran_g = 1.01 # Productivity z in the good aggregate state
Tran_b = 0.99 # and the bad state
# Aggregate productivity shock distribution by state.
# First element is probabilities of different outcomes, given the state you are in.
# Second element is agg permanent shocks (here we don't have any, so just they are just 1.).
# Third element is agg transitory shocks, which are calibrated the same as in Krusell Smith.
KSAggShkDstn = [
[np.array([1.0]),np.array([1.0]),np.array([Tran_g])], # Aggregate good
[np.array([1.0]),np.array([1.0]),np.array([Tran_b])] # Aggregate bad
]
KSEconomy.AggShkDstn = KSAggShkDstn
# Construct the economy, make an initial history, then solve
KSAgent.getEconomyData(KSEconomy) # Makes attributes of the economy, attributes of the agent
KSEconomy.makeAggShkHist() # Make a simulated history of the economy
# Tolerance level
KSEconomy.tolerance = 0.01
# Solve the economy using the market method.
KSEconomy.solve()
### Plots
print('Consumption function at each aggregate market resources gridpoint (in general equilibrium):')
m_grid = np.linspace(0,20,200)
c_m = KSAgent.solution[0].cFunc[0](m_grid,10.0*np.ones_like(m_grid))
f = plt.figure()
plt.plot(m_grid,c_m, color = '#4a69bd', linewidth = 2.5)
plt.ylabel(r'$c(y)$', labelpad = 15, rotation = 0)
plt.ylim([-0.15, 2.15])
plt.yticks([0 + 0.25 * k for k in range(9)], [r'${:.1f}$'.format(0 + 0.25 * k) if k % 2 == 0 else '' for k in range(9)])
plt.xlabel(r'$y$')
plt.xlim([-1, 21])
plt.xticks([0 + 2.5 * k for k in range(9)], [r'${:.0f}$'.format(0 + 2.5 * k) if k % 2 == 0 else '' for k in range(9)])
plt.show()
f.savefig(filename, bbox_inches='tight')
