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In [1]:

    
import matplotlib.pyplot as plt
import numpy as np
%matplotlib inline  

# number of time steps
N = 300

# relative size of rUK to Scotland - this is basically used to define the relative government spends
scaling = 9.0

# fixed spending rate of rUK
G_UK = 50

# fiscal transfer to Scotland
# taken as a fraction of the rUK spend (based on relative size factor) 
# with another fraction for the proportion devolved (with the remainder provided by Sottish tax and spend) 
FT_S = 0.5*(G_UK/(scaling))

# tax rates
theta_UK = 0.25
theta_S  = 0.25

# savings rates (actually spending rates!)
alpha_UK = 0.95
alpha_S  = 0.95

# Scottish (rUK) export rate (relative to GDP) - approximate value from table 6.11 of Active Citizen
EX_S  = 0.31
# Scottish (rUK) import rate (relative to GDP) - approximate value from table 6.11 of Active Citizen
IM_S  = 0.41
# UK import rate (relative to GDP) - simply Scottish export rate scaled by relative sizes of economies
IM_UK = EX_S/scaling

# containers - rUK
C_UK   = np.zeros(N) # consumption
Y_UK   = np.zeros(N) # income
Y_d_UK = np.zeros(N) # disposable income
T_UK   = np.zeros(N) # tax revenue
H_h_UK = np.zeros(N) # private savings
H_g_UK = np.zeros(N) # government debt
CA_UK  = np.zeros(N) # current account

# containers - Scotland
C_S   = np.zeros(N) # consumption
Y_S   = np.zeros(N) # income
Y_d_S = np.zeros(N) # disposable income
T_S   = np.zeros(N) # tax revenue
H_h_S = np.zeros(N) # private savings
H_g_S = np.zeros(N) # government debt
CA_S  = np.zeros(N) # current account
# Scottish G is not a constant
G_S = np.zeros(N) # Scottish government spending


for t in range(1, N):
    
    ### UK ###
    
    # calculate consumer spending
    C_UK[t]   = alpha_UK*Y_d_UK[t-1] 
    
    CA_UK[t] = IM_S*Y_S[t-1] - IM_UK*Y_UK[t-1]
    
    # calculate total income (consumer spending plus constant government spending plus CA)
    Y_UK[t]   = G_UK + C_UK[t] + CA_UK[t]
    
    # calculate the tax take
    T_UK[t] = theta_UK * Y_UK[t]
    
    # calculate disposable income
    Y_d_UK[t] = Y_UK[t] - T_UK[t]
    
    # calculate the change in private savings
    H_h_UK[t] = H_h_UK[t-1] + (1-alpha_UK)*Y_d_UK[t-1] 
    
    # calculate the change in government debt from rUK operations
    H_g_UK[t] = H_g_UK[t-1] + T_UK[t]- G_UK
    
    ### SCOTLAND ###
    
    # calculate consumer spending
    C_S[t]   = alpha_S*Y_d_S[t-1] 
    
    # calculate government spending (tax take plus fiscal transfer)
    G_S[t] = T_S[t-1] + FT_S
    
    # calculate CA
    CA_S[t] = - IM_S*Y_S[t-1] + IM_UK*Y_UK[t-1]
    
    # calculate total income (consumer spending plus constant government spending + CA)
    Y_S[t]   = G_S[t] + C_S[t] + CA_S[t]
    
    # calculate the tax take
    T_S[t] = theta_S * Y_S[t]
    
    # calculate disposable income
    Y_d_S[t] = Y_S[t] - T_S[t]
    
    # calculate the change in private savings
    H_h_S[t] = H_h_S[t-1] + (1-alpha_S)*Y_d_S[t-1] 
    
    # calculate the change in government debt from Scottish operations
    # this has already been modified for rUK operations - just add on Scottish value
    H_g_UK[t] = H_g_UK[t] + T_S[t]- G_S[t]



In [2]:

    
fig = plt.figure(figsize=(14, 10))

consumption_plot = fig.add_subplot(341, xlim=(0, N), ylim=(0, np.max([np.max(Y_UK),np.max(Y_S)])*1.1))
consumption_plot.plot(range(N), C_UK, lw=3)
consumption_plot.plot(range(N), C_S, lw=3)
consumption_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('consumption')

gov_plot = fig.add_subplot(342, xlim=(0, N), ylim=(0, np.max([np.max(Y_UK), np.max(Y_S)])*1.1))
gov_plot.plot(range(N), np.repeat(G_UK,N), lw=3)
gov_plot.plot(range(N), G_S, lw=3)
gov_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('government spending')

ca_plot = fig.add_subplot(343, xlim=(0, N), ylim=(np.min([np.min(CA_UK),np.min(CA_S)])*1.1, np.max([np.max(CA_UK),np.max(CA_S)])*1.1))
ca_plot.plot(range(N), CA_UK, lw=3)
ca_plot.plot(range(N), CA_S, lw=3)
ca_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('trade balance')

income_plot = fig.add_subplot(344, xlim=(0, N), ylim=(0, np.max([np.max(Y_UK),np.max(Y_S)])*1.1))
income_plot.plot(range(N), Y_UK, lw=3)
income_plot.plot(range(N), Y_S, lw=3)
income_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('income')

tax_plot = fig.add_subplot(345, xlim=(0, N), ylim=(0, np.max([np.max(T_UK),np.max(T_S)])*1.1))
tax_plot.plot(range(N), T_UK, lw=3)
tax_plot.plot(range(N), T_S, lw=3)
tax_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('tax revenue')

tax_gdp_plot = fig.add_subplot(346, xlim=(0, N), ylim=(0, 1))
tax_gdp_plot.plot(range(N), T_UK/Y_UK, lw=3)
tax_gdp_plot.plot(range(N), T_S/Y_S, lw=3)
tax_gdp_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('tax revenue (% GDP)')

deficit_plot = fig.add_subplot(347, xlim=(0, N), ylim=(np.min([np.min(T_UK-G_UK),np.min(T_S-G_S)])*1.1, np.max([np.max(T_UK-G_UK),np.max(T_S-G_S)])*1.1))
deficit_plot.plot(range(N), T_UK-np.repeat(G_UK,N), lw=3)
deficit_plot.plot(range(N), T_S-G_S, lw=3)
deficit_plot.plot(range(N), ((T_UK-np.repeat(G_UK,N))+(T_S-G_S)), lw=3)
deficit_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('government budget')

debt_plot = fig.add_subplot(348, xlim=(0, N), ylim=(np.min(H_g_UK)*1.1,0))
debt_plot.plot(range(N), H_g_UK, lw=3)
debt_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('government debt')

gov_balance_gdp_plot = fig.add_subplot(3,4,9, xlim=(0, N), ylim=(-0.2, 0.2))
gov_balance_gdp_plot.plot(range(N-1), (T_UK[1:]-np.repeat(G_UK,N-1))/Y_UK[1:], lw=3)
gov_balance_gdp_plot.plot(range(N-1), (T_S[1:]-G_S[1:])/Y_S[1:], lw=3)
gov_balance_gdp_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('government balance (% GDP)')

private_balance_gdp_plot = fig.add_subplot(3,4,10, xlim=(0, N), ylim=(-0.2,0.2))
private_balance_gdp_plot.plot(range(N-1), np.diff(H_h_UK)/Y_UK[1:], lw=3)
private_balance_gdp_plot.plot(range(N-1), np.diff(H_h_S)/Y_S[1:], lw=3)
private_balance_gdp_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('private balance (% GDP)')

ca_gdp_plot = fig.add_subplot(3,4,11, xlim=(0, N), ylim=(-0.2,0.2))
ca_gdp_plot.plot(range(N-1), np.divide(CA_UK[1:],Y_UK[1:]), lw=3)
ca_gdp_plot.plot(range(N-1), np.divide(CA_S[1:], Y_S[1:]), lw=3)
ca_gdp_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('trade balance (% GDP)')

# space subplots neatly
plt.tight_layout()









    



-c:44: RuntimeWarning: invalid value encountered in divide
-c:45: RuntimeWarning: invalid value encountered in divide

Some thoughts...

rUK = blue, Scotland = green, aggregate = red
top row of plots shows income identity, first three plots sum to last one
middle row is government
bottom row is sectoral balances as percent of GDP
government has deficit in both regions if private sector are saving (i.e. alpha < 1)
in absolute terms rUK has bigger deficit, but relative to GDP it is much smaller
government balance equals private balance + trade balance for each regions (bottom row of plots)
if alpha = 1, private balance obviously goes to zero (no savings) and trade deficit perfectly balances government deficit (in this case we see the government have a balanced budget on aggregate (red) - the surplus from the rUK exactly matches the fiscal transfer to Scotland)
any absolute trade deficit needs to be less than G at the outset otherwise it explodes (i.e. more money being spent than exists, G is only source of funds)
basically economies grow until amount flowing out (IM and savings; both function of Y) equals amount flowing in (EX, G and FT)
absolute trade balances are obsiously equal and opposite but relative to GDP Scotlands deficit is much larger than rUKs surplus (by factor of 9, again, obviously)

Would be good to

add investment
use realistic figures for rUK government spend and Scottish block grant (very crude implementation here), spending propensities
could add spending out of wealth (Modigliani consumption function) but probably guessing parameters
could handle Scottish tax differently (i.e. Scottish rate)
could add bonds (and interest rate and liquidity preference) but possibly overkill
if modelling as distinct countries need to central banks with reserves (e.g. gold)



In [3]:

    
### NO SAVING ###

# savings rates (actually spending rates!)
alpha_UK = 1.0
alpha_S  = 1.0


for t in range(1, N):
    
    ### UK ###
    
    # calculate consumer spending
    C_UK[t]   = alpha_UK*Y_d_UK[t-1] 
    
    CA_UK[t] = IM_S*Y_S[t-1] - IM_UK*Y_UK[t-1]
    
    # calculate total income (consumer spending plus constant government spending plus CA)
    Y_UK[t]   = G_UK + C_UK[t] + CA_UK[t]
    
    # calculate the tax take
    T_UK[t] = theta_UK * Y_UK[t]
    
    # calculate disposable income
    Y_d_UK[t] = Y_UK[t] - T_UK[t]
    
    # calculate the change in private savings
    H_h_UK[t] = H_h_UK[t-1] + (1-alpha_UK)*Y_d_UK[t-1] 
    
    # calculate the change in government debt from rUK operations
    H_g_UK[t] = H_g_UK[t-1] + T_UK[t]- G_UK
    
    ### SCOTLAND ###
    
    # calculate consumer spending
    C_S[t]   = alpha_S*Y_d_S[t-1] 
    
    # calculate government spending (tax take plus fiscal transfer)
    G_S[t] = T_S[t-1] + FT_S
    
    # calculate CA
    CA_S[t] = - IM_S*Y_S[t-1] + IM_UK*Y_UK[t-1]
    
    # calculate total income (consumer spending plus constant government spending + CA)
    Y_S[t]   = G_S[t] + C_S[t] + CA_S[t]
    
    # calculate the tax take
    T_S[t] = theta_S * Y_S[t]
    
    # calculate disposable income
    Y_d_S[t] = Y_S[t] - T_S[t]
    
    # calculate the change in private savings
    H_h_S[t] = H_h_S[t-1] + (1-alpha_S)*Y_d_S[t-1] 
    
    # calculate the change in government debt from Scottish operations
    # this has already been modified for rUK operations - just add on Scottish value
    H_g_UK[t] = H_g_UK[t] + T_S[t]- G_S[t]
   

fig = plt.figure(figsize=(14, 10))

consumption_plot = fig.add_subplot(341, xlim=(0, N), ylim=(0, np.max([np.max(Y_UK),np.max(Y_S)])*1.1))
consumption_plot.plot(range(N), C_UK, lw=3)
consumption_plot.plot(range(N), C_S, lw=3)
consumption_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('consumption')

gov_plot = fig.add_subplot(342, xlim=(0, N), ylim=(0, np.max([np.max(Y_UK), np.max(Y_S)])*1.1))
gov_plot.plot(range(N), np.repeat(G_UK,N), lw=3)
gov_plot.plot(range(N), G_S, lw=3)
gov_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('government spending')

ca_plot = fig.add_subplot(343, xlim=(0, N), ylim=(np.min([np.min(CA_UK),np.min(CA_S)])*1.1, np.max([np.max(CA_UK),np.max(CA_S)])*1.1))
ca_plot.plot(range(N), CA_UK, lw=3)
ca_plot.plot(range(N), CA_S, lw=3)
ca_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('trade balance')

income_plot = fig.add_subplot(344, xlim=(0, N), ylim=(0, np.max([np.max(Y_UK),np.max(Y_S)])*1.1))
income_plot.plot(range(N), Y_UK, lw=3)
income_plot.plot(range(N), Y_S, lw=3)
income_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('income')

tax_plot = fig.add_subplot(345, xlim=(0, N), ylim=(0, np.max([np.max(T_UK),np.max(T_S)])*1.1))
tax_plot.plot(range(N), T_UK, lw=3)
tax_plot.plot(range(N), T_S, lw=3)
tax_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('tax revenue')

tax_gdp_plot = fig.add_subplot(346, xlim=(0, N), ylim=(0, 1))
tax_gdp_plot.plot(range(N), T_UK/Y_UK, lw=3)
tax_gdp_plot.plot(range(N), T_S/Y_S, lw=3)
tax_gdp_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('tax revenue (% GDP)')

deficit_plot = fig.add_subplot(347, xlim=(0, N), ylim=(np.min([np.min(T_UK-G_UK),np.min(T_S-G_S)])*1.1, np.max([np.max(T_UK-G_UK),np.max(T_S-G_S)])*1.1))
deficit_plot.plot(range(N), T_UK-np.repeat(G_UK,N), lw=3)
deficit_plot.plot(range(N), T_S-G_S, lw=3)
deficit_plot.plot(range(N), ((T_UK-np.repeat(G_UK,N))+(T_S-G_S)), lw=3)
deficit_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('government budget')

debt_plot = fig.add_subplot(348, xlim=(0, N), ylim=(np.min(H_g_UK)*1.1,0))
debt_plot.plot(range(N), H_g_UK, lw=3)
debt_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('government debt')

gov_balance_gdp_plot = fig.add_subplot(3,4,9, xlim=(0, N), ylim=(-0.2, 0.2))
gov_balance_gdp_plot.plot(range(N-1), (T_UK[1:]-np.repeat(G_UK,N-1))/Y_UK[1:], lw=3)
gov_balance_gdp_plot.plot(range(N-1), (T_S[1:]-G_S[1:])/Y_S[1:], lw=3)
gov_balance_gdp_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('government balance (% GDP)')

private_balance_gdp_plot = fig.add_subplot(3,4,10, xlim=(0, N), ylim=(-0.2,0.2))
private_balance_gdp_plot.plot(range(N-1), np.diff(H_h_UK)/Y_UK[1:], lw=3)
private_balance_gdp_plot.plot(range(N-1), np.diff(H_h_S)/Y_S[1:], lw=3)
private_balance_gdp_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('private balance (% GDP)')

ca_gdp_plot = fig.add_subplot(3,4,11, xlim=(0, N), ylim=(-0.2,0.2))
ca_gdp_plot.plot(range(N-1), np.divide(CA_UK[1:],Y_UK[1:]), lw=3)
ca_gdp_plot.plot(range(N-1), np.divide(CA_S[1:], Y_S[1:]), lw=3)
ca_gdp_plot.grid()
# label axes
plt.xlabel('time')
plt.ylabel('trade balance (% GDP)')

# space subplots neatly
plt.tight_layout()









    



-c:103: RuntimeWarning: invalid value encountered in divide
-c:104: RuntimeWarning: invalid value encountered in divide



In [ ]: