#!/usr/bin/env python # coding: utf-8 # # Monetary Economics: Chapter 9 # ### 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. get_ipython().run_line_magic('matplotlib', 'inline') from pysolve.model import Model from pysolve.utils import is_close,round_solution import matplotlib.pyplot as plt # ### Model DISINF2 # In[2]: def create_disinf2_model(): model = Model() model.set_var_default(0) model.var('C', desc='Consumption at current prices') model.var('Ck', desc='Real consumption') model.var('F', desc='Realized firm profits') model.var('Fb', desc='Realized bank profits') 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('Ld', desc='Demand for loans') model.var('Ls', desc='Supply of loans') model.var('Mh', desc='Deposits held by households') model.var('Mhk', desc='Real value of deposits held by households') model.var('Ms', desc='Supply of deposits') model.var('N', desc='Employment level') model.var('omegat', desc='Target real wage rate') model.var('P', desc='Price level') model.var('PIC', desc='Inflation rate of unit costs') model.var('Rl', desc='Interest rate on loans') model.var('Rm', desc='Interest rate on deposits') model.var('RRc', desc='Real interest rate on bank loans') model.var('S', desc='Sales at current prices') model.var('Sk', desc='Real sales') model.var('Ske', desc='Expected real sales') model.var('UC', desc='Unit costs') model.var('W', desc='Wage rate') model.var('WB', desc='The wage bill') model.var('YD', desc='Disposable income') model.var('YDk', desc='Real disposable income') model.var('YDkhs', desc='Haig-Simons measure of real disposable income') model.var('YDkhse', desc='Expected HS real disposable income') model.var('Yk', desc='Real output') model.set_param_default(0) model.param('alpha0', desc='Autonomous consumption') 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('eps', desc='Parameter in expectation formations on real disposable income') model.param('gamma', desc='Speed of adjustment of inventories to the target level') model.param('phi', desc='Mark-up on unit costs') model.param('sigmat', desc='Target inventories to sales ratio') model.param('omega0', desc='Exogenous component of the target real wage rate') model.param('omega1', desc='Relation between the target real wage rate and productivity') model.param('omega2', desc='Relation between the target real rate and the unemploment gap') model.param('omega3', desc='Speed of adjustment of the wage rate') model.param('ADD', desc='Spread of loans rate over the deposit rate') model.param('Nfe', desc='Full employment level') model.param('PR', desc='Labor productivity') model.param('Rlbar', desc='Rate of interest on bank loans, set exogenously') model.param('RRcbar', desc='Real interest rate on bank loans, set exogenously') # The production decision model.add('Yk = Ske + INke - INk(-1)') model.add('INkt = sigmat*Ske') model.add('INke = INk(-1) + gamma*(INkt - INk(-1))') model.add('INk - INk(-1) = Yk - Sk') model.add('Ske = beta*Sk(-1) + (1-beta)*Ske(-1)') model.add('Sk = Ck') model.add('N = Yk / PR') model.add('WB = N*W') model.add('UC = WB/Yk') model.add('IN = INk*UC') # The pricing decision model.add('S = P*Sk') model.add('F = S - WB + IN - IN(-1) - Rl(-1)*IN(-1)') model.add('P = (1 + phi)*(1+RRc*sigmat)*UC') # The banking system model.add('Ld = IN') model.add('Ls = Ld') model.add('Ms = Ls') model.add('Rm = Rl - ADD') model.add('Fb = Rl(-1)*Ld(-1) - Rm(-1)*Mh(-1)') model.add('PIC = (UC/UC(-1)) - 1') model.add('RRc = RRcbar') model.add('Rl = (1 + RRc)*(1 + PIC) - 1') # The consumption decision model.add('YD = WB + F + Fb + Rm(-1)*Mh(-1)') model.add('Mh - Mh(-1) = YD - C') model.add('YDkhs = Ck + (Mhk - Mhk(-1))') model.add('YDk = YD/P') model.add('C = Ck*P') model.add('Mhk = Mh/P') model.add('Ck = alpha0 + alpha1*YDk + alpha2*Mhk(-1)') model.add('YDkhse = eps*YDkhs(-1) + (1 - eps)*YDkhse(-1)') # The inflation process model.add('omegat = omega0 + omega1*PR + omega2*(N/Nfe)') model.add('W = W(-1)*(1 + omega3*(omegat(-1)-(W(-1)/P(-1))))') return model disinf2_parameters = {'alpha0': 15, 'alpha1': 0.8, 'alpha2': 0.1, 'beta': 0.9, 'eps': 0.8, 'gamma': 0.25, 'phi': 0.24, 'sigmat': 0.2, 'omega1': 1, 'omega2': 1.2, 'omega3': 0.3} disinf2_exogenous = {'ADD': 0.02, 'PR': 1, 'RRcbar': 0.04, 'W': 1} disinf2_variables = [('omega0', '0.8 - omega1*PR - omega2'), ('UC', 'W/PR'), ('P', '(1+phi)*(1+RRcbar*sigmat)*UC'), ('YDkhs', 'alpha0/(1-alpha1-alpha2*sigmat*UC/P)'), ('Ck', 'YDkhs'), ('Sk', 'Ck'), ('INk', 'sigmat*Sk'), ('IN', 'INk*UC'), ('Ld', 'IN'), ('Mh', 'Ld'), ('Mhk', 'Mh/P'), ('Ms', 'Mh'), ('Ls', 'Ld'), ('Ske', 'Sk'), ('YDkhse', 'YDkhs'), ('Rl', '(1 + RRcbar) - 1'), ('Rm', 'Rl -ADD'), ('omegat', 'W/P'), ('Nfe', 'Sk/PR'), ] # ### Scenario: Model DISINF2, increase in the target real wage # In[3]: omega0 = create_disinf2_model() omega0.set_values(disinf2_parameters) omega0.set_values(disinf2_exogenous) omega0.set_values(disinf2_variables) # run to convergence # Give the system more time to reach a steady state from pysolve.model import SolutionNotFoundError for _ in range(15): omega0.solve(iterations=100, threshold=1e-6) # shock the system omega0.set_values({'omega0': -1.35}) for _ in range(40): omega0.solve(iterations=100, threshold=1e-6) # ###### Figure 9.4 # In[4]: caption = ''' Figure 9.4 Evolution of (Haig-Simons) real disposable income and of real consumption following an increase in the rate of inflation, in a variant where households are blind to the capital losses inflicted by inflation.''' ydkhsdata = [s['YDkhs'] for s in omega0.solutions[5:]] ckdata = [s['Ck'] for s in omega0.solutions[5:]] 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(80.2, 84) axes.plot(ydkhsdata, linestyle='-', color='r') axes.plot(ckdata, linestyle='--', color='b') # add labels plt.text(6, 82.21, 'Real') plt.text(6, 82.1, 'consumption') plt.text(16, 82, 'Haig-Simons real disposable income') fig.text(0.1, -.1, caption); # ###### Figure 9.5 # In[5]: caption = ''' Figure 9.5 Evolution of real wealth, following an increase in the rate of inflation, in a variant where households are blind to capital gains and losses from inflation.''' data = [s['Mhk'] for s in omega0.solutions[5:]] 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(12.5, 13.3) axes.plot(data, linestyle='--', color='b') # add labels plt.text(30, 13.1, 'Real wealth') fig.text(0.1, -.1, caption); # ###### Figure 9.6 # In[6]: caption = ''' Figure 9.6 Evolution of the rate of price inflation, following a one-shot increase in the target real wage of workers.''' data = list() for i in range(5, len(omega0.solutions)): s = omega0.solutions[i] s_1 = omega0.solutions[i-1] data.append((s['P']/s_1['P'])-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(-.005, .025) axes.plot(data, linestyle='--', color='b') # add labels plt.text(20, .018, 'Inflation rate') fig.text(0.1, -.05, caption); # In[ ]: