Monetary Economics: Chapter 11

Preliminaries


In [1]:
# This line configures matplotlib to show figures embedded in the notebook, 
# instead of opening a new window for each figure. More about that later. 
# If you are using an old version of IPython, try using '%pylab inline' instead.
%matplotlib inline

from pysolve.model import Model
from pysolve.utils import is_close,round_solution

import matplotlib.pyplot as plt

Model GROWTHB


In [2]:
def create_growthb_model():
    model = Model()
        
    model.set_var_default(0)
    model.var('ADDl', desc='Spread between interest rate on loans and rate on deposits')
    model.var('Bbd', desc='Government bills demanded by commercial banks')
    model.var('Bbs', desc='Government bills supplied to commercial banks')
    model.var('Bcbd', desc='Government bills demanded by Central bank')
    model.var('Bcbs', desc='Government bills supplied by Central bank')
    model.var('Bhd', desc='Demand for government bills from households')
    model.var('Bhs', desc='Government bills supplied to households')
    model.var('Bs', desc='Supply of government bills')
    model.var('BLd', desc='Demand for government bonds')
    model.var('BLs', desc='Supply of government bonds')
    model.var('BLR', desc='Gross bank liquidity ratio')
    model.var('BUR', desc='Burden of personal debt')
    model.var('Ck', desc='Real consumption')
    model.var('CAR', desc='Capital adequacy ratio of banks')
    model.var('CG', desc='Capital gains on government bonds')
    model.var('CONS', desc='Consumption at current prices')
    model.var('Ekd', desc='Number of equities demanded')
    model.var('Eks', desc='Number of equities supplied by firms')
    model.var('ER', desc='Employment rate')
    model.var('Fb', desc='Realized banks profits')
    model.var('Fbt', desc='Target profits of banks')
    model.var('Fcb', desc='Central bank "profits"')
    model.var('Ff', desc='Realized entrepreneurial profits')
    model.var('Fft', desc='Planned entrepreneurial profits')
    model.var('FDb', desc='Dividends of banks')
    model.var('FDf', desc='Dividends of firms')
    model.var('FUb', desc='Retained earnings of banks')
    model.var('FUbt', desc='Targt retained earnings of banks')
    model.var('FUf', desc='Retained earnings of firms')
    model.var('FUft', desc='Planned retained earnings of firms')
    model.var('G', desc='Government expenditures')
    model.var('Gk', desc='Real government expenditures')
    model.var('GD', desc='Government debt')
    model.var('GL', desc='Gross amount of new personal loans')
    model.var('GRk', desc='growth_mod of real capital stock')
    model.var('Hbd', desc='Cash required by banks')
    model.var('Hbs', desc='Cash supplied to banks')
    model.var('Hhd', desc='Households demand for cash')
    model.var('Hhs', desc='Cash supplied to households')
    model.var('Hs', desc='Total supply of cash')
    model.var('HCe', desc='Expected historical costs')
    model.var('INV', desc='Gross investment')
    model.var('Ik', desc='Gross investment in real terms')
    model.var('IN', desc='Stock of inventories at current costs')
    model.var('INk', desc='Real inventories')
    model.var('INke', desc='Expected real inventories')
    model.var('INkt', desc='Target level of real inventories')
    model.var('K', desc='Capital stock')
    model.var('Kk', desc='Real capital stock')
    model.var('Lfd', desc='Demand for loans by firms')
    model.var('Lfs', desc='Supply of loans to firms')
    model.var('Lhd', desc='Demand for loans by households')
    model.var('Lhs', desc='Loans supplied to households')
    model.var('Md', desc='Deposits demanded by households')
    model.var('Ms', desc='Deposits supplied by banks')
    model.var('N', desc='Employment level')
    model.var('Nt', desc='Desired employment level')
    model.var('NHUC', desc='Normal historic unit cost')
    model.var('NL', desc='Net flow of new loans to the household sector')
    model.var('NLk', desc='Real flow of new loans to the household sector')
    model.var('NPL', desc='Non-Performing loans')
    model.var('NPLke', desc='Expected proportion of Non-Performing Loans')
    model.var('NUC', desc='Normal unit cost')
    model.var('OFb', desc='Own funds of banks')
    model.var('OFbe', desc='Short-run target for banks own funds')
    model.var('OFbt', desc='Long-run target for banks own funds')
    model.var('omegat', desc='Target real wage for workers')
    model.var('P', desc='Price level')
    model.var('Pbl', desc='Price of government bonds')
    model.var('Pe', desc='Price of equities')
    model.var('PE', desc='Price earnings ratio')
    model.var('PI', desc='Price inflation')
    model.var('PR', desc='Lobor productivity')
    model.var('PSBR', desc='Government deficit')
    model.var('Q', desc="Tobin's Q")
    model.var('Rb', desc='Interest rate on government bills')
    model.var('Rbl', desc='Interest rate on bonds')
    model.var('Rk', desc='Dividend yield of firms')
    model.var('Rl', desc='Interest rate on loans')
    model.var('Rm', desc='Interest rate on deposits')
    model.var('REP', desc='Personal loans repayments')
    model.var('RRl', desc='Real interest rate on loans')
    model.var('S', desc='Sales at current prices')
    model.var('Sk', desc='Real sales')
    model.var('Ske', desc='Expected real sales')
    model.var('T', desc='Taxes')
    model.var('U', desc='Capital utilization proxy')
    model.var('UC', desc='Unit costs')
    model.var('V', desc='Wealth of households')
    model.var('Vk', desc='Real wealth of households')
    model.var('Vfma', desc='Investible wealth of households')
    model.var('W', desc='Wage rate')
    model.var('WB', desc='The wage bill')
    model.var('Y', desc='Output at current prices (nominal GDP)')
    model.var('Yk', desc='Real output')
    model.var('YDhs', desc='Haig-Simons measure of disposable income')
    model.var('YDr', desc='Regular disposable income')
    model.var('YDkr', desc='Regular real disposable income')
    model.var('YDkre', desc='Expected regular real disposable income')
    model.var('YP', desc='Personal income')

    model.var('RRb', desc='Real interest rate on bills')
    model.var('RRbt', desc='Target real interest rate on bills')

    model.var('eta', desc='Ratio of new loans to personal income')
    model.var('phi', desc='Mark-up on unit costs')
    model.var('phit', desc='Ideal mark-up on unit costs')
    model.var('z1a', desc='Is one if bank liquidity ratio is below bottom range')
    model.var('z1b', desc='Is one if bank liquidity ratio is below bottom range')
    model.var('z2a', desc='Is one if bank liquidity ratio is above top range')
    model.var('z2b', desc='Is one if bank liquidity ratio is above top range')
    model.var('z3', desc='Parameter in wage aspiration equation')
    model.var('z4', desc='Parameter in wage aspiration equation')
    model.var('z5', desc='Parameter in wage aspiration equation')
    model.var('sigmase', desc='Opening inventories to expected sales ratio')
    
    model.param('alpha1', desc='Propensity to consume out of income')
    model.param('alpha2', desc='Propensity to consume out of wealth')
    model.param('beta', desc='Parameter in expectation formations on real sales')
    model.param('betab', desc='Spped of adjustment of banks own funds')
    model.param('bot', desc='Bottom value for bank net liquidity ratio')
    model.param('delta', desc='Rate of depreciation of fixed capital')
    model.param('deltarep', desc='Ratio of personal loans repayments to stock of loans')
    model.param('eps', desc='Parameter in expectation formations on real disposable income')
    model.param('eps2', desc='Speed of adjustment of mark-up')
    model.param('epsb', desc='Speed of adjustment in expected proportion of non-performing loans')
    model.param('epsrb', desc='Speed of adjustment in the real interest rate on bills')
    model.param('eta0', desc='Ratio of new loans to personal income - exogenous component')
    model.param('etan', desc='Speed of adjustment of actual employment to desired employment')
    model.param('etar', desc='Relation between the ratio of new loans to personal income and the interest rate on loans')
    model.param('gamma', desc='Speed of adjustment of inventories to the target level')
    model.param('gamma0', desc='Exogenous growth_mod in the real stock of capital')
    model.param('gammar', desc='Relation between the real interest rate and growth_mod in the stock of capital')
    model.param('gammau', desc='Relation between the utilization rate and growth_mod in the stock of capital')
    model.param('lambda20', desc='Parameter in households demand for bills')
    model.param('lambda21', desc='Parameter in households demand for bills')
    model.param('lambda22', desc='Parameter in households demand for bills')
    model.param('lambda23', desc='Parameter in households demand for bills')
    model.param('lambda24', desc='Parameter in households demand for bills')
    model.param('lambda25', desc='Parameter in households demand for bills')
    model.param('lambda30', desc='Parameter in households demand for bonds')
    model.param('lambda31', desc='Parameter in households demand for bonds')
    model.param('lambda32', desc='Parameter in households demand for bonds')
    model.param('lambda33', desc='Parameter in households demand for bonds')
    model.param('lambda34', desc='Parameter in households demand for bonds')
    model.param('lambda35', desc='Parameter in households demand for bonds')
    model.param('lambda40', desc='Parameter in households demand for equities')
    model.param('lambda41', desc='Parameter in households demand for equities')
    model.param('lambda42', desc='Parameter in households demand for equities')
    model.param('lambda43', desc='Parameter in households demand for equities')
    model.param('lambda44', desc='Parameter in households demand for equities')
    model.param('lambda45', desc='Parameter in households demand for equities')
    model.param('lambdab', desc='Parameter determining dividends of banks')
    model.param('lambdac', desc='Parameter in households demand for cash')
    model.param('psid', desc='Ratio of dividends to gross profits')
    model.param('psiu', desc='Ratio of retained earnings to investments')
    model.param('ro', desc='Reserve requirement parameter')
    model.param('sigman', desc='Parameter of influencing normal historic unit costs')
    model.param('theta', desc='Income tax rate')
    model.param('top', desc='Top value for bank net liquidity ratio')
    model.param('xim1', desc='Parameter in the equation for setting interest rate on deposits')
    model.param('xim2', desc='Parameter in the equation for setting interest rate on deposits')
    model.param('omega0', desc='Parameter influencing the target real wage for workers')
    model.param('omega1', desc='Parameter influencing the target real wage for workers')
    model.param('omega2', desc='Parameter influencing the target real wage for workers')
    model.param('omega3', desc='Speed of adjustment of wages to target value')


    model.param('ADDbl', desc='Spread between long-term interest rate and rate on bills')
    model.param('BANDb', desc='Lower range of the flat Phillips curve')
    model.param('BANDt', desc='Upper range of the flat Phillips curve')
    model.param('GRg', desc='growth_mod of real government expenditures')
    model.param('GRpr', desc='growth_mod rate of productivity')
    model.param('NCAR', desc='Normal capital adequacy ratio of banks')
    model.param('Nfe', desc='Full employment level')
    model.param('NPLk', desc='Proportion of Non-Performing loans')
    model.param('RA', desc='Random shock to expectations on real sales')
    model.param('Rbbar', desc='Interest rate on bills, set exogenously')
    model.param('Rln', desc='Normal interest rate on loans')
    model.param('sigmas', desc='Realized inventories to sales ratio')
    model.param('sigmat', desc='Target inventories to sales ratio')


    # Box 11.1 : Firms' equations
    # ---------------------------
    model.add('Yk = Ske + INke - INk(-1)')          # 11.1 : Real output
    model.add('Ske = beta*Sk + (1-beta)*Sk(-1)*(1 + (GRpr + RA))') # 11.2 : Expected real sales
    model.add('INke = INk(-1) + gamma*(INkt - INk(-1))')  # 11.3 : Long-run inventory target
    model.add('INkt = sigmat*Ske')                  # 11.4 : Short-run inventory target
    model.add('INk - INk(-1) = Yk - Sk - NPL/UC')   # 11.5 : Actual real inventories
    model.add('Kk = Kk(-1)*(1 + GRk)')              # 11.6 : Real capital stock
    model.add('GRk = gamma0 + gammau*U(-1) - gammar*RRl')  # 11.7 : Growth of real capital stock
    model.add('U = Yk/Kk(-1)')                      # 11.8 : Capital utilization proxy
    model.add('RRl = ((1 + Rl)/(1 + PI)) - 1')      # 11.9 : Real interest rate on loans
    model.add('PI = d(P)/P(-1)')                    # 11.10 : Rate of price inflation
    model.add('Ik = d(Kk) + delta*Kk(-1)')          # 11.11 : Real gross investment

    # Box 11.2 : Firms' equations
    # ---------------------------
    model.add('Sk = Ck + Gk + Ik')                  # 11.12 : Actual real sales
    model.add('S = Sk*P')                           # 11.13 : Value of realized sales
    model.add('IN = INk*UC')                        # 11.14 : Inventories valued at current cost
    model.add('INV = Ik*P')                         # 11.15 : Nominal gross investment
    model.add('K = Kk*P')                           # 11.16 : Nomincal value of fixed capital
    model.add('Y = Sk*P + d(INk)*UC')               # 11.17 : Nomincal GDP

    # Box 11.3 : Firms' equations
    # ---------------------------
    # 11.18 : Real wage aspirations
    model.add('omegat = exp(omega0 + omega1*log(PR) + omega2*log(ER + z3*(1 - ER) - z4*BANDt + z5*BANDb))')
    model.add('ER = N(-1)/Nfe(-1)')                 # 11.19 : Employment rate
    # 11.20 : Switch variables
    model.add('z3 = if_true(ER > (1-BANDb)) * if_true(ER <= (1+BANDt))')
    model.add('z4 = if_true(ER > (1+BANDt))')
    model.add('z5 = if_true(ER < (1-BANDb))')
    model.add('W - W(-1) = omega3*(omegat*P(-1) - W(-1))')  # 11.21 : Nominal wage
    model.add('PR = PR(-1)*(1 + GRpr)')             # 11.22 : Labor productivity
    model.add('Nt = Yk/PR')                         # 11.23 : Desired employment
    model.add('N - N(-1) = etan*(Nt - N(-1))')      # 11.24 : Actual employment
    model.add('WB = N*W')                           # 11.25 : Nominal wage bill
    model.add('UC = WB/Yk')                         # 11.26 : Actual unit cost
    model.add('NUC = W/PR')                         # 11.27 : Normal unit cost
    model.add('NHUC = (1 - sigman)*NUC + sigman*(1 + Rln(-1))*NUC(-1)')  # 11.28 : Normal historic unit cost

    # Box 11.4 : Firms' equations
    # ---------------------------
    model.add('P = (1 + phi)*NHUC')                 # 11.29 : Normal-cost pricing
    model.add('phi - phi(-1) = eps2*(phit(-1) - phi(-1))')  # 11.30 : Actual mark-up
    # 11.31 : Ideal mark-up
    model.add('phit = (FDf + FUft + Rl(-1)*(Lfd(-1) - IN(-1)))/((1 - sigmase)*Ske*UC + (1 + Rl(-1))*sigmase*Ske*UC(-1))')
    model.add('HCe = (1 - sigmase)*Ske*UC + (1 + Rl(-1))*sigmase*Ske*UC(-1)')  # 11.32 : Expected historical costs
    model.add('sigmase = INk(-1)/Ske')              # 11.33 : Opening inventories to expected sales ratio
    model.add('Fft = FUft + FDf + Rl(-1)*(Lfd(-1) - IN(-1))')  # 11.34 : Planned entrepeneurial profits of firmss
    model.add('FUft = psiu*INV(-1)')                # 11.35 : Planned retained earnings of firms
    model.add('FDf = psid*Ff(-1)')                  # 11.36 : Dividends of firms

    # Box 11.5 : Firms' equations
    # ---------------------------
    model.add('Ff = S - WB + d(IN) - Rl(-1)*IN(-1)')  # 11.37 : Realized entrepeneurial profits
    model.add('FUf = Ff - FDf - Rl(-1)*(Lfd(-1) - IN(-1)) + Rl(-1)*NPL')  # 11.38 : Retained earnings of firms
    # 11.39 : Demand for loans by firms
    model.add('Lfd - Lfd(-1) = INV + d(IN) - FUf - d(Eks)*Pe - NPL')
    model.add('NPL = NPLk*Lfs(-1)')                 # 11.40 : Defaulted loans
    model.add('Eks - Eks(-1) = ((1 - psiu)*INV(-1))/Pe')  # 11.41 : Supply of equities issued by firms
    model.add('Rk = FDf/(Pe(-1)*Ekd(-1))')          # 11.42 : Dividend yield of firms
    model.add('PE = Pe/(Ff/Eks(-1))')               # 11.43 : Price earnings ratio
    model.add('Q = (Eks*Pe)/(K + IN + Lfd)')        # 11.44 : Tobin's Q ratio

    # Box 11.6 : Households' equations
    # --------------------------------
    model.add('YP = WB + FDf + FDb + Rm(-1)*Md(-1) + Rb(-1)*Bhd(-1) + BLs(-1)')  # 11.45 : Personal income
    model.add('T = theta*YP')                       # 11.46 : Income taxes
    model.add('YDr = YP - T - Rl(-1)*Lhd(-1)')      # 11.47 : Regular disposable income
    model.add('YDhs = YDr + CG')                    # 11.48 : Haig-Simons disposable income
    # !1.49 : Capital gains
    model.add('CG = d(Pbl)*BLd(-1) + d(Pe)*Ekd(-1) + d(OFb)')
    # 11.50 : Wealth
    model.add('V - V(-1) = YDr - CONS + d(Pe)*Ekd(-1) + d(Pbl)*BLs(-1) + d(OFb)')
    model.add('Vk = V/P')                           # 11.51 : Real staock of wealth
    model.add('CONS = Ck*P')                        # 11.52 : Consumption
    model.add('Ck = alpha1*(YDkre + NLk) + alpha2*Vk(-1)')  # 11.53 : Real consumption
    model.add('YDkre = eps*YDkr + (1 - eps)*YDkr(-1)*(1 + GRpr)')  # 11.54 : Expected real regular disposable income
    model.add('YDkr = YDr/P - d(P)*Vk(-1)/P')  # 11.55 : Real regular disposable income

    # Box 11.7 : Households' equations
    # --------------------------------
    model.add('GL = eta*YDr')                       # 11.56 : Gross amount of new personal loans
    model.add('eta = eta0 - etar*RRl')              # 11.57 : New loans to personal income ratio
    model.add('NL = GL - REP')                      # 11.58 : Net amount of new personal loans
    model.add('REP = deltarep*Lhd(-1)')             # 11.59 : Personal loans repayments
    model.add('Lhd - Lhd(-1) = GL - REP')           # 11.60 : Demand for personal loans
    model.add('NLk = NL/P')                         # 11.61 : Real amount of new personal loans
    model.add('BUR = (REP + Rl(-1)*Lhd(-1))/YDr(-1)')  # 11.62 : Burden of personal debt

    # Box 11.8 : Households equations - portfolio decisions
    # -----------------------------------------------------

    # 11.64 : Demand for bills
    model.add('Bhd = Vfma(-1)*(lambda20 + lambda22*Rb(-1) - lambda21*Rm(-1) - lambda24*Rk(-1) - lambda23*Rbl(-1) - lambda25*YDr/V)')
    # 11.65 : Demand for bonds
    model.add('BLd = Vfma(-1)*(lambda30 - lambda32*Rb(-1) - lambda31*Rm(-1) - lambda34*Rk(-1) + lambda33*Rbl(-1) - lambda35*YDr/V)/Pbl')
    # 11.66 : Demand for equities - normalized to get the price of equitities
    model.add('Pe = Vfma(-1)*(lambda40 - lambda42*Rb(-1) - lambda41*Rm(-1) + lambda44*Rk(-1) - lambda43*Rbl(-1) - lambda45*YDr/V)/Ekd')
    model.add('Md = Vfma - Bhd - Pe*Ekd - Pbl*BLd + Lhd')  # 11.67 : Money deposits - as a residual
    model.add('Vfma = V - Hhd - OFb')               # 11.68 : Investible wealth
    model.add('Hhd = lambdac*CONS')                 # 11.69 : Households' demand for cash
    model.add('Ekd = Eks')                          # 11.70 : Stock market equilibrium

    # Box 11.9 : Government's equations
    # ---------------------------------
    model.add('G = Gk*P')                           # 11.71 : Pure government expenditures
    model.add('Gk = Gk(-1)*(1 + GRg)')              # 11.72 : Real government expenditures
    model.add('PSBR = G + BLs(-1) + Rb(-1)*(Bbs(-1) + Bhs(-1)) - T')  # 11.73 : Government deficit
    # 11.74 : New issues of bills
    model.add('Bs - Bs(-1) = G - T - d(BLs)*Pbl + Rb(-1)*(Bhs(-1) + Bbs(-1)) + BLs(-1)')
    model.add('GD = Bbs + Bhs + BLs*Pbl + Hs')      # 11.75 : Government debt

    # Box 11.10 : The Central bank's equations
    # ----------------------------------------
    model.add('Fcb = Rb(-1)*Bcbd(-1)')              # 11.76 : Central bank profits
    model.add('BLs = BLd')                          # 11.77 : Bonds are supplied on demand
    model.add('Bhs = Bhd')                          # 11.78 : Household bills supplied on demand
    model.add('Hhs = Hhd')                          # 11.79 : Cash supplied on demand
    model.add('Hbs = Hbd')                          # 11.80 : Reserves supplied on demand
    model.add('Hs = Hbs + Hhs')                     # 11.81 : Total supply of cash
    model.add('Bcbd = Hs')                          # 11.82 : Central bankd 
    model.add('Bcbs = Bcbd')                        # 11.83 : Supply of bills to Central bank
    # model.add('Rb = Rbbar')                         # 11.84 : Interest rate on bills set exogenously
    model.add('Rbl = Rb + ADDbl')                   # 11.85 : Long term interest rate
    model.add('Pbl = 1/Rbl')                        # 11.86 : Price of long-term bonds

    # Box 11.11 : Commercial Bank's equations
    # ---------------------------------------
    model.add('Ms = Md')                            # 11.87 : Bank deposits supplied on demand
    model.add('Lfs = Lfd')                          # 11.88 : Loans to firms supplied on demand
    model.add('Lhs = Lhd')                          # 11.89 : Personal loans supplied on demand
    model.add('Hbd = ro*Ms')                        # 11.90 : Reserve requirements of banks
    # 11.91 : Bills supplied to banks
    model.add('Bbs - Bbs(-1) = d(Bs) - d(Bhs) - d(Bcbs)')
    # 11.92 : Balance sheet constraint of banks
    model.add('Bbd = Ms + OFb - Lfs - Lhs - Hbd') 
    model.add('BLR = Bbd/Ms')                       # 11.93 : Bank liquidity ratio
    # 11.94 : Deposit interest rate
    model.add('Rm - Rm(-1) = z1a*xim1 + z1b*xim2 - z2a*xim1 - z2b*xim2')
    # 11.95-97 : Mechanism for determining changes to the interest rate on deposits
    model.add('z2a = if_true(BLR(-1) > (top + .05))')
    model.add('z2b = if_true(BLR(-1) > top)')
    model.add('z1a = if_true(BLR(-1) <= bot)')
    model.add('z1b = if_true(BLR(-1) <= (bot -.05))')

    # Box 11.12 : Commercial bank's equations
    # ---------------------------------------
    model.add('Rl = Rm + ADDl')                     # 11.98 : Loan interest rate
    model.add('OFbt = NCAR*(Lfs(-1) + Lhs(-1))')    # 11.99 : Long-run own funds target
    model.add('OFbe = OFb(-1) + betab*(OFbt - OFb(-1))')  # 11.100 : Short-run own funds target
    model.add('FUbt = OFbe - OFb(-1) + NPLke*Lfs(-1)')  # 11.101 : Target retained earnings of banks
    model.add('NPLke = epsb*NPLke(-1) + (1 - epsb)*NPLk(-1)')  # 11.102 : Expected proportion of non-performaing loans
    model.add('FDb = Fb - FUb')                     # 11.103 : Dividends of banks
    model.add('Fbt = lambdab*Y(-1) + (OFbe - OFb(-1) + NPLke*Lfs(-1))')  # 11.104 : Target profits of banks
    # 11.105 : Actual profits of banks
    model.add('Fb = Rl(-1)*(Lfs(-1) + Lhs(-1) - NPL) + Rb(-1)*Bbd(-1) - Rm(-1)*Ms(-1)')
    # 11.106 : Lending mark-up over deposit rate
    model.add('ADDl = (Fbt - Rb(-1)*Bbd(-1) + Rm*(Ms(-1) - (1 - NPLke)*Lfs(-1) - Lhs(-1)))/((1 - NPLke)*Lfs(-1) + Lhs(-1))')
    model.add('FUb = Fb - lambdab*Y(-1)')           # 11.107 : Actual retained earnings
    model.add('OFb - OFb(-1) = FUb - NPL')          # 11.108 : Own funds of banks
    model.add('CAR = OFb/(Lfs + Lhs)')              # 11.109 : Actual capital capacity ratio

    model.add('Rb = (1 + RRb)*(1 + PI) - 1')        # 11.111 : Interest rate on bills
    model.add('RRbt = (1 + Rb)/(1 + PI) - 1')       # 11.112 : Target real interest rate on bills
    model.add('RRb - RRb(-1) = epsrb*(RRbt - RRb(-1))')  # 11.113 : Real interst rate on bills
    return model

growthb_parameters = {'alpha1': 0.75,
                      'alpha2': 0.064,
                      'beta': 0.5,
                      'betab': 0.4,
                      'gamma': 0.15,
                      'gamma0': 0.00122,
                      'gammar': 0.1,
                      'gammau': 0.05,
                      'delta': 0.10667,
                      'deltarep': 0.1,
                      'eps': 0.5,
                      'eps2': 0.8,
                      'epsb': 0.25,
                      'epsrb': 0.9,
                      'eta': 0.04918,
                      'eta0': 0.07416,
                      'etan': 0.6,
                      'etar': 0.4,
                      'theta': 0.22844,
                      'lambda20': 0.25,
                      'lambda21': 2.2,
                      'lambda22': 6.6,
                      'lambda23': 2.2,
                      'lambda24': 2.2,
                      'lambda25': 0.1,
                      'lambda30': -0.04341,
                      'lambda31': 2.2,
                      'lambda32': 2.2,
                      'lambda33': 6.6,
                      'lambda34': 2.2,
                      'lambda35': 0.1,
                      'lambda40': 0.67132,
                      'lambda41': 2.2,
                      'lambda42': 2.2,
                      'lambda43': 2.2,
                      'lambda44': 6.6,
                      'lambda45': 0.1,
                      'lambdab': 0.0153,
                      'lambdac': 0.05,
                      'xim1': 0.0008,
                      'xim2': 0.0007,
                      'ro': 0.05,
                      'sigman': 0.1666,
                      'sigmase': 0.16667,
                      'sigmat': 0.2,
                      'phi': 0.26417,
                      'phit': 0.26417,
                      'psid': 0.15255,
                      'psiu': 0.92,
                      'omega0': -0.20594,
                      'omega1': 1,
                      'omega2': 2,
                      'omega3': 0.45621
                      }

growthb_exogenous = [('ADDbl', 0.02),
                     ('BANDt', 0.01),
                     ('BANDb', 0.01),
                     ('bot', 0.05),
                     ('GRg', 0.03),
                     ('GRpr', 0.03),
                     ('Nfe', 87.181),
                     ('NCAR', 0.1),
                     ('NPLk', 0.02),
                     ('Rbbar', 0.035),
                     ('Rln', 0.07),
                     ('RA', 0),
                     ('top', 0.12),

                     ('ADDl', 0.04592),
                     ('BLR', 0.1091),
                     ('BUR', 0.06324),
                     ('Ck', 7334240),
                     ('CAR', 0.09245),
                     ('CONS', 52603100),
                     ('ER', 1),
                     ('Fb', 1744130),
                     ('Fbt', 1744140),
                     ('Ff', 18081100),
                     ('Fft', 18013600),
                     ('FDb', 1325090),
                     ('FDf', 2670970),
                     ('FUb', 419039),
                     ('FUf', 15153800),
                     ('FUft', 15066200),
                     ('G', 16755600),
                     ('Gk', 2336160),
                     ('GL', 2775900),
                     ('GRk', 0.03001),
                     ('INV', 16911600),
                     ('Ik', 2357910),
                     ('N', 'Nfe'),
                     ('Nt', 'Nfe'),
                     ('NHUC', 5.6735),
                     ('NL', 683593),
                     ('NLk', 95311),
                     ('NPL', 309158),
                     ('NPLke', 0.02),
                     ('NUC', 5.6106),
                     ('omegat', 112852),
                     ('P', 7.1723),
                     ('Pbl', 18.182),
                     ('Pe', 17937),
                     ('PE', 5.07185),
                     ('PI', 0.0026),
                     ('PR', 138659),
                     ('PSBR', 1894780),
                     ('Q', 0.77443),
                     ('Rb', 0.035),
                     ('Rbl', 0.055),
                     ('Rk', 0.03008),
                     ('Rl', 0.06522),
                     ('Rm', 0.0193),
                     ('REP', 2092310),
                     ('RRb', 0.03232),
                     ('RRl', 0.06246),
                     ('S', 86270300),
                     ('Sk', 12028300),
                     ('Ske', 'Sk'),
                     ('T', 17024100),
                     ('U', 0.70073),
                     ('UC', 5.6106),
                     ('W', 777968),
                     ('WB', 67824000),
                     ('Y', 86607700),
                     ('Yk', 12088400),
                     ('YDr', 56446400),
                     ('YDkr', 7813270),
                     ('YDkre', 7813290),
                     ('YP', 73158700),
                     ('z1a', 0),
                     ('z1b', 0),
                     ('z2a', 0),
                     ('z2b', 0),
                     ]

                          
growthb_variables = [('Bbd', 4388930),
                     ('Bbs', 4389790),
                     ('Bcbd', 4655690),
                     ('Bcbs', 4655690),
                     ('Bhd', 33396900),
                     ('Bhs', 'Bhd'),
                     ('Bs', 42484800),
                     ('BLd', 840742),
                     ('BLs', 'BLd'),
                     ('GD', 57728700),
                     ('Ekd', 5112.6001),
                     ('Eks', 'Ekd'),
                     ('Hbd', 2025540),
                     ('Hbs', 'Hbd'),
                     ('Hhd', 2630150),
                     ('Hhs', 'Hhd'),
                     ('Hs', 'Hbd + Hhd'),
                     ('IN', 11585400),
                     ('INk', 2064890),
                     ('INke', 2405660),
                     ('INkt', 'INk'),
                     ('K', 127444000),
                     ('Kk', 17768900),
                     ('Lfd', 15962900),
                     ('Lfs', 'Lfd'),
                     ('Lhd', 21606600),
                     ('Lhs', 'Lhd'),
                     ('Md', 40510800),
                     ('Ms', 'Md'),
                     ('OFb', 3473280),
                     ('OFbe', 3782430),
                     ('OFbt', 3638100),
                     ('V', 165395000),
                     ('Vfma', 159291000),
                     ('Vk', 22576100),
                     ]

Scenario: Model GROWTHB, baseline


In [3]:
baseline = create_growthb_model()
baseline.set_values(growthb_parameters)
baseline.set_values(growthb_exogenous)
baseline.set_values(growthb_variables)

# run to convergence
# Give the system more time to reach a steady state
for _ in xrange(100):
    baseline.solve(iterations=200, threshold=1e-6)

Scenario: Model GROWTHB, increase rate of growth in government expenditure


In [4]:
from pysolve.model import SolutionNotFoundError

grg = create_growthb_model()
grg.set_values(growthb_parameters)
grg.set_values(growthb_exogenous)
grg.set_values(growthb_variables)

# run to convergence
# Give the system more time to reach a steady state
for _ in xrange(10):
    grg.solve(iterations=200, threshold=1e-6)

grg.set_values({'GRg': 0.035})

for _ in xrange(90):
    try:
        grg.solve(iterations=200, threshold=1e-6)
    except SolutionNotFoundError:
        grg._update_solutions(grg.solutions[-1])
Figure 11.6A

In [5]:
caption = '''
    Figure 11.6A  Evolution of the growth rate of real output, with the growth rate
    of real pure government expenditures being forever higher than in the baseline
    solution, when the central bank attempts to keep the real interest rate on bills
    at a constant level, but with a partial adjustment function.'''

data = list()
for i in xrange(5, 80):
    s = grg.solutions[i]
    s_1 = grg.solutions[i-1]
    data.append((s['Yk']/s_1['Yk']) - 1)

fig = plt.figure()
axes = fig.add_axes([0.1, 0.1, 1.1, 1.1])
axes.tick_params(top='off', right='off')
axes.spines['top'].set_visible(False)
axes.spines['right'].set_visible(False)
#axes.set_ylim(0.975, 1.034)

axes.plot(data, linestyle='-', color='b')

# add labels
plt.text(50, 0.028, 'The growth rate')
plt.text(50, 0.0275, 'of real output')
fig.text(0.1, -.15, caption);


Figure 11.6B

In [6]:
caption = '''
    Figure 11.6B  Evolution of the nominal bill rate, with the growth rate or real pure
    government expenditures being forever higher than in the baseline solution, when
    the central bank attempts to keep the real interest rate on bills at a constant level,
    but with a partial adjustment solution.'''

data = [s['Rb'] for s in grg.solutions[5:80]]

fig = plt.figure()
axes = fig.add_axes([0.1, 0.1, 1.1, 1.1])
axes.tick_params(top='off', right='off')
axes.spines['top'].set_visible(False)
axes.spines['right'].set_visible(False)
#axes.set_ylim(0.975, 1.034)

axes.plot(data, linestyle='-', color='b')

# add labels
plt.text(27, 0.085, 'Nominal bill rate')
fig.text(0.1, -.15, caption);



In [ ]: