Lake Model Solutions

Excercise 1

We begin by initializing the variables and import the necessary modules



In [1]:

    
%pylab inline
import LakeModel

alpha = 0.012
lamb = 0.2486
b = 0.001808
d = 0.0008333
g = b-d
N0 = 100.
e0 = 0.92
u0 = 1-e0
T = 50









    



Populating the interactive namespace from numpy and matplotlib

Now construct the class containing the initial conditions of the problem



In [2]:

    
LM0 = LakeModel.LakeModel(lamb,alpha,b,d)
x0 = LM0.find_steady_state()# initial conditions

print "Initial Steady State: ", x0









    



Initial Steady State:  [ 0.94737184  0.05262816]

New legislation changes $\lambda$ to $0.2$



In [3]:

    
LM1 = LakeModel.LakeModel(0.2,alpha,b,d)



In [4]:

    
xbar = LM1.find_steady_state() # new steady state
X_path = vstack(LM1.simulate_stock_path(x0*N0,T)) # simulate stocks
x_path = vstack(LM1.simulate_rate_path(x0,T)) # simulate rates
print "New Steady State: ", xbar









    



New Steady State:  [ 0.93540871  0.06459129]

Now plot stocks



In [5]:

    
figure(figsize=[10,9])
subplot(3,1,1)
plot(X_path[:,0])
title(r'Employment')
subplot(3,1,2)
plot(X_path[:,1])
title(r'Unemployment')
subplot(3,1,3)
plot(X_path.sum(1))
title(r'Labor Force')









    Out[5]:





<matplotlib.text.Text at 0x10a459dd0>

And how the rates evolve:



In [6]:

    
figure(figsize=[10,6])
subplot(2,1,1)
plot(x_path[:,0])
hlines(xbar[0],0,T,'r','--')
title(r'Employment Rate')
subplot(2,1,2)
plot(x_path[:,1])
hlines(xbar[1],0,T,'r','--')
title(r'Unemployment Rate')









    Out[6]:





<matplotlib.text.Text at 0x10a963c90>

We see that it takes 20 periods for the economy to converge to it's new steady state levels

Exercise 2

This next exercise has the economy expriencing a boom in entrances to the labor market and then later returning to the original levels. For 20 periods the economy has a new entry rate into the labor market



In [7]:

    
bhat = 0.003
T_hat = 20
LM1 = LakeModel.LakeModel(lamb,alpha,bhat,d)

We simulate for 20 periods at the new parameters



In [8]:

    
X_path1 = vstack(LM1.simulate_stock_path(x0*N0,T_hat)) # simulate stocks
x_path1 = vstack(LM1.simulate_rate_path(x0,T_hat)) # simulate rates

Now using the state after 20 periods for the new initial conditions we simulate for the additional 30 periods



In [9]:

    
X_path2 = vstack(LM0.simulate_stock_path(X_path1[-1,:2],T-T_hat+1)) # simulate stocks
x_path2 = vstack(LM0.simulate_rate_path(x_path1[-1,:2],T-T_hat+1)) # simulate rates

Finally we combine these two paths and plot



In [10]:

    
x_path = vstack([x_path1,x_path2[1:]]) # note [1:] to avoid doubling period 20
X_path = vstack([X_path1,X_path2[1:]]) # note [1:] to avoid doubling period 20



In [11]:

    
figure(figsize=[10,9])
subplot(3,1,1)
plot(X_path[:,0])
title(r'Employment')
subplot(3,1,2)
plot(X_path[:,1])
title(r'Unemployment')
subplot(3,1,3)
plot(X_path.sum(1))
title(r'Labor Force')









    Out[11]:





<matplotlib.text.Text at 0x10aa8ee50>

And the rates:



In [12]:

    
figure(figsize=[10,6])
subplot(2,1,1)
plot(x_path[:,0])
hlines(x0[0],0,T,'r','--')
title(r'Employment Rate')
subplot(2,1,2)
plot(x_path[:,1])
hlines(x0[1],0,T,'r','--')
title(r'Unemployment Rate')









    Out[12]:





<matplotlib.text.Text at 0x10ae15a90>



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