Model LP2#

Model LP2 extends Model LP from Chapter 5 of Godley and Lavoie [2006] by making bond prices endogenous through a target-proportion mechanism and replacing static bond-price expectations with adaptive expectations. In all other respects, LP2 inherits the equations of LP unchanged.

Module Contents#

As with all MacroStat models, LP2 is divided into Variables, Parameters (fixed constants), Scenarios, and the Behavior (model initialization and steps). The module-level documentation can be seen in:

The remainder of this page gives an introduction to the model, notes on how it is implemented in MacroStat and then shows some of the model dynamics by replicating the relevant graphs of Godley and Lavoie (2006).

Model Overview#

Behavioral Equations#

Model LP2 inherits equations 1–9 and 13–20 unchanged from Model LP. The static-expectations equation (\(p_{bl}^e = p_{bl}\), equation 10 in LP) is replaced by two new equations that make bond prices endogenous and expectations adaptive.

New equation 10a — Target Proportion:

(1)#\[TP(t) = \frac{BL_h(t-1) \cdot p_{bl}(t-1)} {BL_h(t-1) \cdot p_{bl}(t-1) + B_h(t-1)}\]

New equation 10b — Endogenous Bond Price:

(2)#\[z_1 = 1 \text{ iff } TP > \text{top}, \quad z_2 = 1 \text{ iff } TP < \text{bot}\]
(3)#\[p_{bl}(t) = (1 + z_1 \cdot \beta - z_2 \cdot \beta) \cdot p_{bl}(t-1)\]

New equation 10d — Adaptive Expectations:

(4)#\[p_{bl}^e(t) = p_{bl}^e(t-1) - \beta_e \cdot \left(p_{bl}^e(t-1) - p_{bl}(t)\right)\]

With adaptive expectations, the expected capital gain \(\chi \cdot (p_{bl}^e - p_{bl})/p_{bl}\) in the expected return on bonds \(ERr_{bl}\) is no longer zero at each period, generating richer portfolio dynamics than in LP.

Transaction Flow Matrix#

Accounting Transaction Matrix for Model LP2#

Household

Production

Government

CentralBank

CentralBank

Total

Current

Current

Current

Current

Capital

Consumption Household

\(-C(t)\)

\(+C(t)\)

\(0\)

Consumption Government

\(+G(t)\)

\(-G(t)\)

\(0\)

National Income

\(+Y(t)\)

\(-Y(t)\)

\(0\)

Taxes

\(-T(t)\)

\(+T(t)\)

\(0\)

Interest On Bills Household

\(+r_b(t-1) \cdot B_h(t-1)\)

\(-r_b(t-1) \cdot B_h(t-1)\)

\(0\)

Bond Coupon Income Household

\(+BL_h(t-1)\)

\(-BL_h(t-1)\)

\(0\)

Central Bank Profits

\(+r_b(t-1) \cdot B_{CB}(t-1)\)

\(-r_b(t-1) \cdot B_{CB}(t-1)\)

\(0\)

Change in Bill Stock

\(+\Delta B_h(t)\)

\(-\Delta B_s(t)\)

\(+\Delta B_{CB}(t)\)

\(0\)

Change in Bond Stock

\(+\Delta BL_h(t)\)

\(0\)

Change in Bond Supply

\(-\Delta BL_s(t)\)

\(0\)

Change in Cash Stock

\(+\Delta H_h(t)\)

\(0\)

Change in Money Stock

\(-\Delta H_s(t)\)

\(0\)

Total

\(0\)

\(0\)

\(0\)

\(0\)

\(0\)

\(0\)

Balance Sheet Matrix#

Balance Sheet for Model LP2#

Household

Production

Government

CentralBank

Total

Current

Current

Current

Capital

Wealth

\(-V(t)\)

\(+V(t)\)

0

Bill Stock

\(+B_h(t)\)

\(-B_s(t)\)

\(+B_{CB}(t)\)

0

Bond Stock

\(+BL_h(t)\)

0

Bond Supply

\(-BL_s(t)\)

0

Cash Stock

\(+H_h(t)\)

0

Money Stock

\(-H_s(t)\)

0

Implementation in MacroStat#

Transposing these equations to the MacroStat framework, we consider that there are:

  1. Sixteen parameters (fixed constants): \(\alpha_1\), \(\alpha_2\), \(\theta\), \(\chi\), the eight \(\lambda\) portfolio coefficients, \(\beta\) (bond price adjustment step), \(\beta_e\) (expectation adjustment speed), \(\text{top}\) (upper target proportion bound), and \(\text{bot}\) (lower target proportion bound) (see Parameters)

  2. Two scenario variables: \(G(t)\) and \(r_b(t)\) — bond prices are now endogenous (see Scenarios)

  3. The remaining tracked series are variables — including the new TargetProportion and ExpectedBondPrice (see Variables)

Model Dynamics#

Preparatory Steps#

%load_ext autoreload
%autoreload 2

import importlib
import logging
import sys

# Import the necessary libraries for plotting
from matplotlib import pyplot as plt
from matplotlib.ticker import PercentFormatter

# Import the MacroStat model components
from macrostat.models.GL06LP2 import GL06LP2, ParametersGL06LP2, ScenariosGL06LP2

# Custom matplotlib style for the documentation
plt.style.use("../../macrostat.mplstyle")
# We show the logging output in the notebook
importlib.reload(logging)
logging.basicConfig(stream=sys.stdout, level=logging.INFO)

Running the Simulation#

First, we run the model without any shocks to see the convergence to the steady state.

params = ParametersGL06LP2()
model = GL06LP2(parameters=params)
model.simulate()
output = model.variables.to_pandas()

We can check that the variables are healthy, meaning the redundant equation \(H_h = H_s\) holds and all stocks are positive.

Note

Due to floating point precision, the redundant equation will not hold exactly. We check that the absolute percentage error is below a tolerance.

model.variables.check_health(tolerance=1e-3)

True

An overview of the first 10 steps of the model:

df = model.variables.to_pandas()
df.head(10).T
time 0 1 2 3 4 5 6 7 8 9
ConsumptionHousehold Household 0.0 0.0 0.000000 16.124001 29.123171 40.025520 49.155052 56.800381 63.202782 68.564339
ConsumptionGovernment Government 0.0 0.0 20.000000 20.000000 20.000000 20.000000 20.000000 20.000000 20.000000 20.000000
NationalIncome Macroeconomy 0.0 0.0 20.000000 36.124001 49.123169 60.025520 69.155052 76.800385 83.202782 88.564339
Taxes Household 0.0 0.0 3.876000 7.000832 9.621614 11.816236 13.654072 15.193127 16.481977 17.561300
InterestOnBillsHousehold Household 0.0 0.0 0.000000 0.000000 0.189307 0.342600 0.471316 0.579104 0.669339 0.744877
BondCouponIncomeHousehold Household 0.0 0.0 0.000000 0.000000 0.334654 0.603168 0.828082 1.016420 1.174195 1.306373
CentralBankProfits CentralBank 0.0 0.0 0.000000 0.483720 0.491546 0.510596 0.526161 0.539205 0.550126 0.559269
CapitalGains Household 0.0 0.0 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
ExpectedCapitalGains Household 0.0 0.0 0.000000 0.009906 0.008927 0.006128 0.003761 0.002172 0.001208 0.000655
Wealth Household 0.0 0.0 16.124001 29.123169 40.025520 49.155052 56.800377 63.202778 68.564339 73.054283
HouseholdBillStock Household 0.0 0.0 0.000000 6.310246 11.420005 15.710543 19.303473 22.311302 24.829226 26.937166
GovernmentBillStock Government 0.0 0.0 16.124001 22.695127 28.439871 33.249241 37.276981 40.648834 43.471512 45.834629
CentralBankBillStock CentralBank 0.0 0.0 16.124001 16.384880 17.019867 17.538698 17.973509 18.337532 18.642286 18.897463
HouseholdBondStock Household 0.0 0.0 0.000000 0.334654 0.603168 0.828082 1.016420 1.174195 1.306373 1.417100
GovernmentBondSupply Government 0.0 0.0 0.000000 0.334654 0.603168 0.828082 1.016420 1.174195 1.306373 1.417100
HouseholdCashStock Household 0.0 0.0 16.124001 16.384882 17.019871 17.538708 17.973515 18.337540 18.642300 18.897470
CentralBankMoneyStock CentralBank 0.0 0.0 16.124001 16.384880 17.019867 17.538696 17.973509 18.337534 18.642288 18.897463
DisposableIncome Household 0.0 0.0 16.124001 29.123169 40.025520 49.155052 56.800381 63.202782 68.564339 73.054283
ExpectedDisposableIncome Household 0.0 0.0 0.000000 16.124001 29.123169 40.025520 49.155052 56.800381 63.202782 68.564339
ExpectedWealth Household 0.0 0.0 0.000000 16.124001 29.123167 40.025520 49.155052 56.800373 63.202782 68.564339
HouseholdBillDemand Household 0.0 0.0 0.000000 6.310246 11.420005 15.710543 19.303473 22.311302 24.829226 26.937166
HouseholdBondDemand Household 0.0 0.0 0.000000 0.334654 0.603168 0.828082 1.016420 1.174195 1.306373 1.417100
HouseholdCashDemand Household 0.0 0.0 0.000000 3.385713 6.117520 8.409175 10.328192 11.935135 13.280743 14.407526
InterestRateBills Macroeconomy 0.0 0.0 0.030000 0.030000 0.030000 0.030000 0.030000 0.030000 0.030000 0.030000
BondPrice Macroeconomy 20.0 20.0 19.600000 19.208000 19.208000 19.208000 19.208000 19.208000 19.208000 19.208000
BondYield Macroeconomy 0.0 0.0 0.051020 0.052062 0.052062 0.052062 0.052062 0.052062 0.052062 0.052062
ExpectedBondPrice Household 20.0 20.0 19.799999 19.504000 19.355999 19.282000 19.244999 19.226500 19.217251 19.212626
ExpectedReturnOnBonds Household 0.0 0.0 0.052041 0.053603 0.052832 0.052447 0.052254 0.052158 0.052110 0.052086
TargetProportion Government 0.0 0.0 0.000000 0.000000 0.504624 0.503600 0.503088 0.502832 0.502704 0.502640

Convergence to the Steady State#

In the steady state of Model LP2, the target proportion \(TP\) stabilises within the band \([\text{bot}, \text{top}]\) and bond prices stop adjusting. Adaptive expectations converge to the actual bond price, so \(p_{bl}^e \to p_{bl}\), and capital gains vanish in the long run.

The convergence is slower than in LP because of the expectation feedback, but the long-run steady state is qualitatively similar.

dfo = output.loc[:50]

fig, axs = plt.subplots(ncols=2, nrows=3, figsize=(8, 8))

# National Income
axs[0, 0].plot(dfo.index, dfo['NationalIncome'], color='k', label=r'National Income $Y$')
axs[0, 0].legend(loc='upper center', bbox_to_anchor=(0.5, -0.1), frameon=False)
axs[0, 0].set_title('National Income')

# Consumption and Disposable Income
axs[0, 1].plot(dfo.index, dfo['ConsumptionHousehold'], color='k', label=r'Consumption $C$')
axs[0, 1].plot(dfo.index, dfo['DisposableIncome'], color='g', linestyle='-.', label=r'Disposable Income $YD_r$')
axs[0, 1].legend(loc='upper center', bbox_to_anchor=(0.5, -0.1), frameon=False)
axs[0, 1].set_title('Consumption')

# Wealth
axs[1, 0].plot(dfo.index, dfo['Wealth'], color='k', label=r'Wealth $V$')
axs[1, 0].plot(dfo.index, dfo['ExpectedWealth'], color='g', linestyle='-.', label=r'Expected Wealth $V^e$')
axs[1, 0].legend(loc='upper center', bbox_to_anchor=(0.5, -0.1), frameon=False)
axs[1, 0].set_title('Wealth')

# Savings
axs[1, 1].plot(dfo.index, dfo['Wealth'].diff(), color='k', label=r'Savings, $\Delta V$')
axs[1, 1].axhline(y=0, color='r', linestyle='--', label=r'Steady State $\Delta V^\star=0$')
axs[1, 1].legend(loc='upper center', bbox_to_anchor=(0.5, -0.1), frameon=False)
axs[1, 1].set_title('Savings')

# Portfolio Shares
cash_share = output['HouseholdCashStock'] / output['Wealth']
bill_share = output['HouseholdBillStock'] / output['Wealth']
bond_share = (output['HouseholdBondStock'].mul(output[('BondPrice','Macroeconomy')], axis=0)) / output['Wealth']

axs[2, 0].plot(output.index, cash_share, color='b', label='Cash $H_h/V$')
axs[2, 0].plot(output.index, bill_share, color='k', label='Bills $B_h/V$')
axs[2, 0].plot(output.index, bond_share, color='r', label='Bonds $p_{bl} BL_h/V$')
axs[2, 0].legend(loc='center right', frameon=False)
axs[2, 0].set_xlim(0, 50)
axs[2, 0].set_title('Portfolio Shares')
axs[2, 0].yaxis.set_major_formatter(PercentFormatter(1))

# Bond Price and Target Proportion
ax_bp = axs[2, 1]
ax_tp = ax_bp.twinx()
ax_bp.plot(dfo.index, dfo[('BondPrice', 'Macroeconomy')], color='b', label=r'Bond Price $p_{bl}$')
ax_tp.plot(dfo.index, dfo['TargetProportion'], color='r', linestyle='--', label=r'Target Prop. $TP$')
ax_bp.set_title('Bond Price & Target Proportion')
ax_bp.legend(loc='upper left', frameon=False)
ax_tp.legend(loc='lower right', frameon=False)
ax_tp.yaxis.set_major_formatter(PercentFormatter(1))

fig.suptitle('Figure LP2.1: Model convergence to the steady state')
plt.tight_layout()
plt.show()
../../_images/c0485fea5c3edc6e33dd8e9928efe19596a62ff0a01d9ddec9d601a57c3608ec.png

Perturbation 1: Rise in the Bill Rate (Section 5.8, first experiment)#

Following Section 5.8 of Godley and Lavoie [2006], we increase the interest rate on Treasury bills from 3% to 4%, but without changing the bond price exogenously (unlike Scenario 1 of LP). In LP2, the bond price adjusts endogenously through the target-proportion mechanism.

When \(r_b\) rises, bills become more attractive relative to bonds. Households shift their portfolio toward bills, which raises the target proportion \(TP\) above the upper threshold. The Treasury responds by allowing bond prices to rise until \(TP\) falls back into the target band. This is Figure 5.5 of Godley and Lavoie (2006).

scenarios = ScenariosGL06LP2(parameters=params)
sc1 = scenarios.get_scenario_index("Scenario.1: Rise in bill rate")
model1 = GL06LP2(parameters=params, scenarios=scenarios)
model1.simulate(scenario=sc1)
output_sc1 = model1.variables.to_pandas()

trigger = params["scenario_trigger"]
dfo = output_sc1.loc[trigger - 2:]
dfo_base = output.loc[trigger - 2:]

fig, axes = plt.subplots(1, 2, figsize=(10, 4))

# Figure 5.5: Bond yield
axes[0].plot(dfo_base.index, dfo_base['BondYield'], 'k--', label='Baseline')
axes[0].plot(dfo.index, dfo['BondYield'], 'k-', label='Shock')
axes[0].axvline(x=trigger, color='grey', linestyle=':', alpha=0.5)
axes[0].set_title('Figure 5.5: Bond Yield $r_{bl}$')
axes[0].set_xlabel('Period')
axes[0].legend(frameon=False)

# Bond Price
axes[1].plot(dfo_base.index, dfo_base[('BondPrice','Macroeconomy')], 'b--', label='Bond Price (base)')
axes[1].plot(dfo.index, dfo[('BondPrice','Macroeconomy')], 'b-', label='Bond Price (shock)')
axes[1].axvline(x=trigger, color='grey', linestyle=':', alpha=0.5)
axes[1].set_title('Bond Price $p_{bl}$')
axes[1].set_xlabel('Period')
axes[1].legend(frameon=False)

fig.suptitle('Figure LP2.2: Rise in bill rate with endogenous bond price adjustment')
plt.tight_layout()
plt.show()
../../_images/ae6da741c0650c0cd1affa568c4ea900bbf332a31427a035cf9d8026b3d848e5.png

Perturbation 2: Anticipated Fall in Bond Prices (Section 5.8, second experiment)#

In this experiment, households form a one-period expectation of a fall in the bond price by 1 unit (via an ExpectedBondPriceShock of \(-1\)). The lower expected bond price reduces the expected return on bonds, causing households to demand fewer bonds and more bills and cash.

With adaptive expectations, this shock is only temporary — households gradually revise their expectations back toward the actual bond price. However, while the expectation shock persists, the portfolio reallocation depresses bond demand and, through the target-proportion mechanism, pushes bond prices lower. This is Figure 5.7 of Godley and Lavoie (2006).

sc2 = scenarios.get_scenario_index("Scenario.2: Expected bond price fall")
model2 = GL06LP2(parameters=params, scenarios=scenarios)
model2.simulate(scenario=sc2)
output_sc2 = model2.variables.to_pandas()

dfo2 = output_sc2.loc[trigger - 2:]

fig, axes = plt.subplots(1, 3, figsize=(14, 4))

# Figure 5.7: Bond yield
axes[0].plot(dfo_base.index, dfo_base['BondYield'], 'k--', label='Baseline')
axes[0].plot(dfo2.index, dfo2['BondYield'], 'k-', label='Shock')
axes[0].axvline(x=trigger, color='grey', linestyle=':', alpha=0.5)
axes[0].set_title('Figure 5.7: Bond Yield $r_{bl}$')
axes[0].set_xlabel('Period')
axes[0].legend(frameon=False)

# Bond Price vs Expected Bond Price
axes[1].plot(dfo2.index, dfo2[('BondPrice','Macroeconomy')], 'b-', label=r'$p_{bl}$ (shock)')
axes[1].plot(dfo2.index, dfo2['ExpectedBondPrice'], 'r--', label=r'$p_{bl}^e$ (shock)')
axes[1].plot(dfo_base.index, dfo_base[('BondPrice','Macroeconomy')], 'b:', label=r'$p_{bl}$ (base)')
axes[1].axvline(x=trigger, color='grey', linestyle=':', alpha=0.5)
axes[1].set_title(r'Bond Price vs Expected $p_{bl}^e$')
axes[1].set_xlabel('Period')
axes[1].legend(frameon=False)

# National Income
axes[2].plot(dfo_base.index, dfo_base['NationalIncome'], 'k--', label='Baseline')
axes[2].plot(dfo2.index, dfo2['NationalIncome'], 'k-', label='Shock')
axes[2].axvline(x=trigger, color='grey', linestyle=':', alpha=0.5)
axes[2].set_title('National Income $Y$')
axes[2].set_xlabel('Period')
axes[2].legend(frameon=False)

fig.suptitle('Figure LP2.3: Anticipated fall in bond prices with adaptive expectations dynamics')
plt.tight_layout()
plt.show()
../../_images/b188d0ed9609ac0a920dfa14b90ecbbc751323c6f47d47c285a29f8573ebaaa9.png